tag:blogger.com,1999:blog-88700971630703036902024-02-10T11:00:06.479+05:30HIMADRI'S BLOGA JEE AND NEET ASPIRANT ( 2021-22 ). LIKE TO DO A LITTLE BLOGGING IN FREE TIME AND SHARE SOME QUALITY INFORMATION.Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.comBlogger320125tag:blogger.com,1999:blog-8870097163070303690.post-39825585071997526742020-12-18T09:15:00.001+05:302020-12-18T09:15:58.634+05:30BIOMOLECULES CHEMISTRY CLASS 12<p> <b><i><u><span style="font-size: x-large;">BIOMOLECULES ( CHEMISTRY )</span></u></i></b><span> </span> </p><p><br /></p><p><br /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Carbohydrates: </span>Polyhydroxy aldehydes or polyhydroxy ketones or compounds on hydrolysis give carbohydrates.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Classification of carbohydrates:</span><br />Monosaccharides<br />(a) Simplest carbohydrates<br />(b) It cannot be hydrolysed into simpler compounds<br />(c) Examples – Glucose, mannose<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Oligosaccharides</span><br />(a) Carbohydrates which gives 2 to 10 monosaccharide units on hydrolysis<br />(b) Examples – Sucrose, Lactose, Maltose<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Polysaccharides</span><br />(a) Carbohydrates which on hydrolysis give large number of monosaccharide units.<br />(b) Examples – Cellulose, starch</li><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Anomers: </span>Pair of optical isomers which differ in configuration only around C1 atom are called anomers. Examples – ALPHA-D-glucopyranose and BETA-D-glucopyranose.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Epimers: </span>Pair of optical isomers which differ in configuration around any other C atom other than C1 atom are called epimers. E.g. D-glucose and D- mannose are C2epimers.<br /><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3068.4031007751937" height="129" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_01.png" style="height: auto; max-width: 100%;" width="208" /></li></ul><h3 style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; font-weight: 400; line-height: 1.2em; margin: 0px 0px 5px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Preparation of glucose (also called dextrose, grape sugar):</span></h3><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="13367.414634146342" height="41" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_02.png" style="height: auto; max-width: 100%;" width="288" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">From starch</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="9711.862068965518" height="58" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_03.png" style="height: auto; max-width: 100%;" width="296" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Structure of glucose</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="13388.964285714286" height="28" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_04.png" style="height: auto; max-width: 100%;" width="197" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Structure elucidation of glucose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">a) D – glucose with HI</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="6793.267441860466" height="86" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_05.png" style="height: auto; max-width: 100%;" width="307" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">b) D – glucose with HCN</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="8272.763888888889" height="72" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_06.png" style="height: auto; max-width: 100%;" width="313" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">c) D – glucose with NH<span style="border: 0px; font-size: 12px; height: 0px; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.5ex; vertical-align: baseline;">2</span>OH</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="7844.073170731707" height="82" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_33.png" style="height: auto; max-width: 100%;" width="338" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">d) D- glucose with Fehling’s reagent</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="8486.351351351352" height="74" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_07.png" style="height: auto; max-width: 100%;" width="330" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">e) D – glucose with Tollen’s reagent</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="7496.666666666667" height="99" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_08.png" style="height: auto; max-width: 100%;" width="390" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">f) D – glucose with nitric acid</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="1569.7898832684823" height="257" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_11.png" style="height: auto; max-width: 100%;" width="212" /> <img data-large_image_height="1903" data-large_image_width="9909.962264150943" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_34.png" style="height: auto; max-width: 100%;" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">g) D – glucose with (CH<span style="border: 0px; font-size: 12px; height: 0px; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.5ex; vertical-align: baseline;">3</span>CO)<span style="border: 0px; font-size: 12px; height: 0px; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.5ex; vertical-align: baseline;">2</span>O and ZnCl<span style="border: 0px; font-size: 12px; height: 0px; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.5ex; vertical-align: baseline;">2</span></span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="6302.408602150538" height="93" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_12.png" style="height: auto; max-width: 100%;" width="308" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">h) D – glucose with bromine water</span></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="4004.2291666666665" height="96" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_13.png" style="height: auto; max-width: 100%;" width="202" /></p><h3 style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; font-weight: 400; line-height: 1.2em; margin: 0px 0px 5px; padding: 0px;">i) Glucose with phenylhydrazine to form osazone</h3><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3494.032786885246" height="183" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_14.png" style="height: auto; max-width: 100%;" width="336" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">Glucose and fructose gives the same osazone because the reaction takes place at C1 and C2 only.</p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Other Reactions of Glucose (Presence of ring structure)</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="5955.077586206897" height="116" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_15.png" style="height: auto; max-width: 100%;" width="363" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">Glucose does not give Schiff’s test and does not react with sodium bisulphite and NH3. Pentaacetyl glucose does not react with hydroxyl amine. This shows the absence of –CHO group and hence the presence of ring structure.</p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Cyclic structure of glucose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="4676.364341085271" height="129" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_16.png" style="height: auto; max-width: 100%;" width="317" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Haworth representation of glucose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="6296.095744680852" height="94" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_17.png" style="height: auto; max-width: 100%;" width="311" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Cyclic structure of fructose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="5542.070175438596" height="114" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_18.png" style="height: auto; max-width: 100%;" width="332" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Haworth representation of fructose</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="7005.734513274336" height="113" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_19.png" style="height: auto; max-width: 100%;" width="416" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Glycosidic linkage: </span>The oxide linkage formed by the loss of a water molecule when two monosaccharides are joined together through oxygen atom is called glycosidic linkage.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Sucrose (invert sugar):</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">a)</span> Sucrose is a non-reducing sugar because the two monosaccharide units are held together by a glycosidic linkage between C1 of ALPHA-glucose and C2 of BETA– fructose. Since the reducing groups of glucose and fructose are involved in glycosidic bond formation, sucrose is a non-reducing sugar.<br /><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="2184.508875739645" height="169" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_20.png" style="height: auto; max-width: 100%;" width="194" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">b)</span> Sucrose is dextrorotatory but on hydrolysis it gives dextrorotatory & laevorotatory and the mixture is laevorotatory.</p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="9219.706896551725" height="58" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_21.png" style="height: auto; max-width: 100%;" width="281" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Haworth Projection of Sucrose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3792.3093525179856" height="139" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_22.png" style="height: auto; max-width: 100%;" width="277" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Maltose:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Maltose is composed of two α-D-glucose units in which C1 of one glucose (I) is linked to C4 of another glucose unit (II).</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">The free aldehyde group can be produced at C1 of second glucose in solution and it shows reducing properties so it is a reducing sugar.<br /><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="1919.3347639484978" height="233" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_23.png" style="height: auto; max-width: 100%;" width="235" /></li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Haworth projection of maltose:</span><br /><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3854.7948717948716" height="156" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_24.png" style="height: auto; max-width: 100%;" width="316" /></li><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Lactose (Milk sugar):</span>It is composed of β-D-galactose and β-D-glucose. The linkage is between C1 of galactose and C4 of glucose. Hence it is also a reducing sugar.<br /><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="2419.9566929133857" height="254" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_25.png" style="height: auto; max-width: 100%;" width="323" /></li><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Haworth projection of lactose:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3399.871951219512" height="164" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_26.png" style="height: auto; max-width: 100%;" width="293" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Starch: </span>It is a polymer of -glucose and consists of two components — Amylose and Amylopectin.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Amylose:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is a water soluble component</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is a long unbranched chain polymer</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It contains 200 – 1000 ALPHA-D-(+)- glucose units held by 1,5– glycosidic linkages involving C1 – C4glycosidic linkage</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It constitutes about 15-20% of starch</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Amylopectin</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is a water insoluble component</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is branched chain polymer</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It forms chain by C1 – C4glycosidic linkage whereas branching occurs by C1 – C6glycosidic linkage</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It constitutes about 80-85% of starch</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Cellulose:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It occurs exclusively in plants.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is a straight chain polysaccharide composed only of ALPHA-D-glucose units which are joined by glycosidic linkage between C1 of one glucose unit and C4 of the next glucose unit.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Glycogen:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">The carbohydrates are stored in animal body as glycogen.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is also known as animal starch because its structure is similar to Amylopectin.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is present in liver, muscles and brain.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">When the body needs glucose, enzymes break the glycogen down to glucose.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Amino acids:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">Amino acids contain amino (–NH2) and carboxyl (–COOH) functional groups.</p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img data-large_image_height="1903" data-large_image_width="5223.127659574468" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_27.png" style="height: auto; max-width: 100%;" /></p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">Where R – Any side chain<br />Most naturally occurring amino acids have L – Config.</p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="6672.544303797468" height="79" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_28.png" style="height: auto; max-width: 100%;" width="277" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Types of amino acids:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">a). <span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Essential amino acids: </span>The amino acids which cannot be synthesised in the body and must be obtained through diet, are known as essential amino acids. Examples: Valine, Leucine</p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">b). <span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Non-essential amino acids: </span>The amino acids, which can be synthesised in the body, are known as non-essential amino acids. Examples: Glycine, Alanine</p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="font-weight: 700;">Zwitterion form of amino acids:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Amino acids behave like salts rather than simple amines or carboxylic acids. This behaviour is due to the presence of both acidic (carboxyl group) and basic (amino group) groups in the same molecule. In aqueous solution, the carboxyl group can lose a proton and amino group can accept a proton, giving rise to a dipolar ion known as zwitter ion. This is neutral but contains both positive and negative charges.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">In zwitterionic form, amino acids show amphoteric behaviour as they react both with acids and bases.</li></ol><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="6074.961538461539" height="78" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_29.png" style="height: auto; max-width: 100%;" width="249" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Isoelectronic point: </span>The pH at which the dipolar ion exists as neutral ion and does not migrate to either electrode cathode or anode is called isoelectronic point.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Proteins: </span>Proteins are the polymers of ALPHA-amino acids and they are connected to each other by peptide bond or peptide linkage. A polypeptide with more than hundred amino acid residues, having molecular mass higher than 10,000u is called a protein.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Peptide linkage: </span>Peptide linkage is an amide linkage formed by condensation reaction between –COOH group of one amino acid and –NH2 group of another amino acid.</li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="4507.926229508196" height="122" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_30.png" style="height: auto; max-width: 100%;" width="289" /><br />Peptide link age</p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Primary structure of proteins: </span>The sequence of amino acids is said to be the primary structure of a protein.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Secondary structure of proteins: </span>It refers to the shape in which long polypeptide chain can exist. Two different types of structures:</li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">ALPHA-Helix:</p><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It was given by Linus Pauling in 1951</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It exists when R- group is large.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Right handed screw with the NH group of each amino acid residue H – bonded to – C = O of adjacent turn of the helix.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Also known as 3.613 helix since each turn of the helix hasapproximately 3.6 amino acids and a 13 – membered ring is formed by H – bonding.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">C = O and N – H group of the peptide bonds are trans to each other.</li></ol><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;">BETA-pleated sheet:</p><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It exists when R group is small.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">In this conformation, all peptide chains are stretched out to nearly maximum extension and then laid side by side which are held together by hydrogen bonds.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Tertiary structure of proteins: </span>It represents the overall folding of the polypeptide chain i.e., further folding of the 2° structure.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Types of bonding which stabilize the 3° structure:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Disulphide bridge (-S – S-)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">H – bonding – (C = O … H – N)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Salt bridge (COO– … + Na+)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Hydrophobic interactions</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">van der Waals forces</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Two shapes of proteins:</span></li></ul><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Fibrous proteins</span><br />a) When the polypeptide chains run parallel and are held together by hydrogen and disulphide bonds, then fibre– like structure is formed.<br />b) These proteins are generally insoluble in water<br />c) Examples: keratin (present in hair, wool, silk) and myosin (present in muscles), etc</p><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Globular proteins</span><br />a) This structure results when the chains of polypeptides coil around to give a spherical shape.<br />b) These are usually soluble in water.<br />c) Examples: Insulin and albumins</p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Quaternary structure of proteins:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Some of the proteins are composedof two or more polypeptide chains referred to as sub-units.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">The spatial arrangement of these subunits with respect to each other is known as quaternary structure of proteins.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Denaturation of proteins:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">The loss of biological activity of proteins when a protein in its native form, is subjected to physical change like change in temperature or chemical change like change in pH. This is called denaturation of protein.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Example: coagulation of egg white on boiling, curdling of milk.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Nucleoside:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Base + sugar</li></ol><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3018.0390625" height="128" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_31.png" style="height: auto; max-width: 100%;" width="203" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Nucleotide:</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Base + sugar + phosphate group</li></ol><p style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1em; padding: 0px;"><img alt="Biomolecules Class 12 Notes Chemistry" class="alignnone" data-large_image_height="1903" data-large_image_width="3128.219178082192" height="146" src="http://media.mycbseguide.com/images/static/revise/12/chemistry/12_chemistry_ch14_32.png" style="height: auto; max-width: 100%;" width="240" /></p><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Nucleic acids (or polynucletides):</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Long chain polymers ofnucleotides.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Nucleotides are joined by phosphodiester linkage between 5’ and 3’ C atoms of a pentose sugar.</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Two types of nucleic acids:</span><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">DNA</span></li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It has a double stranded ALPHA-helix structure in which two strands are coiled spirally in opposite directions.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Sugar present is <span class="cke_widget_wrapper cke_widget_inline cke_widget_selected" style="border: 0px; margin: 0px; padding: 0px;"><img class="" data-large_image_height="1903" data-large_image_width="NaN" height="14" src="https://elpiscart.com/cgi-bin/mathtex.cgi?\beta" style="height: auto; max-width: 100%;" width="10" /></span>–D–2-deoxyribose</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Bases:<br />i) Purine bases: Adenine (A) and Guanine (G)<br />ii) Pyrimidine bases: Thymine (T) and cytosine (C)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It occurs mainly in the nucleus of the cell.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is responsible for transmission for heredity character.<span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">RNA</span></li></ol><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It has a single stranded ALPHA-helix structure.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Sugar present is ALPHA–D–ribose</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Bases:<br />i) Purine bases: Adenine (A) and Guanine (G)<br />ii) Pyrimidine bases: Uracil (U) and cytosine (C)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It occurs mainly in the cytoplasm of the cell.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">It helps in protein synthesis.</li></ol><h3 style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; font-weight: 400; line-height: 1.2em; margin: 0px 0px 5px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Double helix structure of DNA:</span></h3><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">It is composed of two right handed helical polynucleotide chains coiled spirally in opposite directions around the same central axis.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Two strands are anti-parallel i.e., their phosphodiester linkage runs in opposite directions.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Bases are stacked inside the helix in planes PERPENDICULAR to the helical axis.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Two strands are held together by H – bonds (A = T, G=- C).</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">The two strands are complementary to each other because the hydrogen bonds are formed between specific pairs of bases.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Adenine forms hydrogen bonds with thymine whereas cytosine forms hydrogen bonds with guanine.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Diameter of double helix is 2 nm.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Double helix repeats at intervals of 3.4 nm. (One complete turn)</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Total amount of purine (A + G) = Total amount of pyramidine (C + T)</li></ol><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Vitamins: </span>Vitamins are organic compounds required in the diet in small amounts to perform specific biological functions for normal maintenance of optimum growth and health of the organism.</li></ul><ul style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Classification of vitamins: </span>Vitamins are classified into two groups depending upon their solubility in water or fat.</li></ul><ol style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em 3em; padding: 0px;"><li style="border: 0px; margin: 10px 0px; padding: 0px;">Water soluble vitamins <span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">i) </span>These vitamins are soluble in water.<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">ii) </span>Water soluble vitamins must be supplied regularly in diet because they are readily excreted in urine and cannot be stored (except vitamin B12) in our body.<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">iii) </span>Example: Vitamin C, B group vitamins.</li><li style="border: 0px; margin: 10px 0px; padding: 0px;">Fat soluble vitamins<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">i) </span>These vitamins are soluble in fat and oils but insoluble in water.<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">ii) </span>They are stored in liver and adipose (fat storing) tissues.<br /><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">iii) </span>Example: Vitamin A, D, E and K</li></ol><h3 style="background-color: #fcfcfc; border: 0px; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; font-weight: 400; line-height: 1.2em; margin: 0px 0px 5px; padding: 0px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Important vitamins, their sources and their deficiency diseases:</span></h3><table border="1" cellpadding="1" cellspacing="0" class="mobile" style="background-color: #fcfcfc; border-spacing: 0px; border: 1px solid black; color: #545454; font-family: -apple-system, system-ui, BlinkMacSystemFont, "Segoe UI", Helvetica, Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px; margin: 0px 0px 1.5em; padding: 0px; width: 808px;"><tbody style="border: 0px; margin: 0px; padding: 0px;"><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Name of vitamins</span></td><td style="border: 1px solid black; margin: 0px; padding: 8px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Sources</span></td><td style="border: 1px solid black; margin: 0px; padding: 8px;"><span style="border: 0px; font-weight: 700; margin: 0px; padding: 0px;">Deficiency diseases</span></td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin A</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Fish liver oil, carrots, butter and milk</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">xerophthalmia<br />(hardening of cornea of eye)<br />Night blindness</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin B1<br />(Thiamine)</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Yeast, milk, green vegetables and cereals</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Beriberi<br />(loss of appetite, retarded growth)</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin B2<br />(Riboflavin)</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Milk, egg white, liver, kidney</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Cheilosis<br />(fissuring at corners of mouth and lips), digestive disorders and burning sensation of the skin.</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin B6 (Pyridoxine)</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Yeast, milk, egg yolk, cereals and grams</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Convulsions</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin B12</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Meat, fish, egg and curd</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Pernicious anaemia<br />(RBC deficient in haemoglobin)</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin C<br />(Ascorbic acid)</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Citrus fruits, amla and green leafy vegetables</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Scurvy<br />(bleeding gums)</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin D</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Exposure to sunlight, fish and egg yolk</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Rickets<br />(bone deformities in children) and<br />osteomalacia<br />(soft bones and joint pain in adults)</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin E</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vegetable oils like wheat germ oil, sunflower oil, etc.</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Increased fragility of RBCs and muscular weakness</td></tr><tr style="border: 0px; margin: 0px; padding: 0px;"><td style="border: 1px solid black; margin: 0px; padding: 8px;">Vitamin K</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Green leafy vegetables</td><td style="border: 1px solid black; margin: 0px; padding: 8px;">Increased blood clotting time</td></tr></tbody></table>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-39062186838196583672020-12-18T09:09:00.000+05:302020-12-18T09:15:36.485+05:30CAPACITORS CLASS 12<p> </p><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><p style="text-align: left;"><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif;"><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><b><i><u><span style="font-size: x-large;">CAPACITORS</span></u></i></b><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span></span></p></blockquote><p><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><br /></span></p><p><b><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;">A</span><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;"> </span><em style="background-color: white; box-sizing: border-box; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;">capacitor</em><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;"> </span><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;">is a device that stores electrical energy. It is an arrangement of two conductors carrying charges of equal magnitudes and opposite sign and separated by an insulating medium.</span></b></p><div id="ds-book-content" style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px;"><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> the following points about capacitors:</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">1) The net charge on the capacitor as a whole is zero. When we say that a capacitor has a charge <em style="box-sizing: border-box;">q</em>, we mean the positively charged conductor has a charge +<em style="box-sizing: border-box;">q</em> and negatively charged conductor has a charge -<em style="box-sizing: border-box;">q</em>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">2) The positively charged conductor is at a higher potential than the negatively charged conductor.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">3) The potential difference <em style="box-sizing: border-box;">V</em> between the conductors is proportional to the charge <em style="box-sizing: border-box;">q</em>. The ratio <em style="box-sizing: border-box;">q</em>/<em style="box-sizing: border-box;">V</em> is known as <em style="box-sizing: border-box;">capacitance</em> <em style="box-sizing: border-box;">C</em> of the capacitor. Thus,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-1-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 3.772em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.979em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1002.98em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-3" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-4" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-5" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-6" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-7" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-8" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-9" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">4) Capacitance depends on the size and shape of the plates and the material between them. It does not depend on <em style="box-sizing: border-box;">q</em> or <em style="box-sizing: border-box;">V</em> individually.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">5) The SI units of capacitance are <span style="box-sizing: border-box; font-weight: 700;">farad</span>(F), which is equivalent to coulomb/volt. Practical values of capacitances are usually measured in microfarad (<span class="MathJax" id="MathJax-Element-2-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-10" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 1.689em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.342em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1001.34em, 2.532em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.18em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-11" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-12" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">μ</span><span class="mi" id="MathJax-Span-13" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.185em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>).</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-3-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-14" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.542em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.955em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1005.96em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-15" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-16" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="mi" id="MathJax-Span-17" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">μ</span><span class="mi" id="MathJax-Span-18" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-19" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-20" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-21" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-22" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.987em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.95em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-23" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.995em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-24" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-25" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-26" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-27" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">6</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-28" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.503em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">6) It is a scalar, having dimensions</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-4-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-29" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 10.766em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 8.534em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.838em, 1008.29em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-30" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-31" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-32" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-33" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-34" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-35" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span></span></span></span><span class="mi" id="MathJax-Span-36" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-37" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-38" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-39" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-40" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-41" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span></span><span class="mo" id="MathJax-Span-42" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-43" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-44" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-45" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-46" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mrow" id="MathJax-Span-47" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-48" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-49" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-50" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-51" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-52" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-53" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.318em, -999.998em); left: 7.01562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-54" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">Q</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 7.01562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-55" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.7em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.697em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-56" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span><span class="texatom" id="MathJax-Span-57" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-58" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-59" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-60" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-61" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-62" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-63" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-64" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-65" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mrow" id="MathJax-Span-66" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-67" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-68" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-69" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-70" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-71" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-72" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.85em, 4.318em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-73" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-74" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-75" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-76" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-77" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-78" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">Q</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-79" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-80" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.995em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-81" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.441em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-5-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-82" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 6.401em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.062em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.887em, 1004.81em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-83" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-84" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-85" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-86" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-87" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-88" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-89" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-90" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-91" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-92" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-93" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">s</span></span></span></span></span><span class="mi" id="MathJax-Span-94" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-95" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-96" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-97" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-98" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 8.51562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-99" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.318em, -999.998em); left: 8.51562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-100" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">Q</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.85em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.846em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-101" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">or, <span class="MathJax" id="MathJax-Element-6-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-102" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 18.752em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 14.883em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.887em, 1014.78em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-103" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-104" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span><span class="mi" id="MathJax-Span-105" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-106" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span><span class="mo" id="MathJax-Span-107" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mrow" id="MathJax-Span-108" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-109" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-110" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-111" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-112" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-113" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-114" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.929em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1001.69em, 4.169em, -999.998em); left: 29.5156px; line-height: normal; margin: 0px 0px 0px -0.841em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-115" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-116" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-117" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-118" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-119" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-120" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-121" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-122" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-123" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-124" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-125" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">T<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-126" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1002.78em, 4.169em, -999.998em); left: 29.5156px; line-height: normal; margin: 0px 0px 0px -1.386em; padding: 0px; position: absolute; top: -3.47em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-127" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-128" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-129" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">M<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-130" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-131" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-132" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-133" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">L</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-134" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-135" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-136" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-137" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-138" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">T<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-139" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-140" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-141" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-142" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.93em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.929em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-143" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span><span class="mo" id="MathJax-Span-144" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mo" id="MathJax-Span-145" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span><span class="texatom" id="MathJax-Span-146" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-147" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-148" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.086em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1001.04em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-149" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">M<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 1.143em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-150" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-151" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-152" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-153" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-154" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-155" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-156" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.639em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.65em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-157" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">L</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-158" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-159" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-160" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-161" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-162" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-163" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-164" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.243em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-165" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">T<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.796em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-166" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-167" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-168" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-169" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-170" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-171" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-172" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Isolated Conducting Sphere as a Capacitor</span></h3><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">A conducting sphere of radius <em style="box-sizing: border-box;">R</em> carrying a charge <em style="box-sizing: border-box;">q</em> can be treated as a capacitor. The high-potential conductor is the sphere itself and the low potential conductor is a sphere of infinite radius. The potential difference between these two sphere is</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-7-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-173" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 12.254em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 9.725em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1009.73em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-174" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-175" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-176" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-177" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-178" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-179" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.036em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-180" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.94em, 4.268em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.94em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-181" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-182" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-183" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-184" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-185" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-186" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-187" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-188" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-189" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-190" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.036em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-191" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-192" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span class="mo" id="MathJax-Span-193" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-194" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-195" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-196" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.036em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-197" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.94em, 4.268em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.94em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-198" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-199" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-200" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-201" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-202" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-203" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-204" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-205" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-206" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-207" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.036em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.816em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Hence, its capacity is (the capacitance of an isolated conductor is normally called <em style="box-sizing: border-box;">capacity</em>)</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-8-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-208" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 8.931em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.096em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1007.1em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-209" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-210" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-211" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-212" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-213" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-214" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-215" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-216" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-217" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-218" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-219" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-220" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-221" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-222" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-223" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-224" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-225" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The <em style="box-sizing: border-box;">capacity of a spherical conductor is directly proportional to its radius</em>. As the potential of earth is assumed to be zero, the capacity of the earth or of any conductor connected to earth (irrespective of its shape or charge on it) will be</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-9-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-226" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 9.377em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.443em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1007.39em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-227" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-228" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-229" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-230" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-231" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-232" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-233" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-234" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-235" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-236" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-237" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-238" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-239" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.35em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-240" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-241" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-242" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">∞</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">However, if we assume the earth to be a conducting sphere of radius 6400 km, its capacitance will be,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-10-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-243" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 17.314em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 13.743em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.044em, 1013.74em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-244" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-245" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-246" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-247" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-248" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-249" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-250" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-251" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-252" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-253" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-254" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span class="mo" id="MathJax-Span-255" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-256" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-257" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-258" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.078em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1002.98em, 4.169em, -999.998em); left: 31.0156px; line-height: normal; margin: 0px 0px 0px -1.486em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-259" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-260" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-261" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">6400</span><span class="mo" id="MathJax-Span-262" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-263" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-264" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-265" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-266" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-267" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-268" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-269" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1001.94em, 4.169em, -999.998em); left: 31.0156px; line-height: normal; margin: 0px 0px 0px -0.94em; padding: 0px; position: absolute; top: -3.519em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-270" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-271" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-272" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">9</span><span class="mo" id="MathJax-Span-273" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-274" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-275" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-276" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-277" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-278" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-279" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-280" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">9</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1003.08em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 3.078em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-281" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-282" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">711</span><span class="texatom" id="MathJax-Span-283" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-284" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-285" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-286" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-287" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">μ</span><span class="mi" id="MathJax-Span-288" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span></span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Energy Required to Charge a Conductor</span></h3><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When a conductor is charged its potential changes from 0 to <em style="box-sizing: border-box;">V</em>. In this process, work is done against repulsion between charge stored on the conductor and charge coming from the charging body. This work is stored as <em style="box-sizing: border-box;">electrostatic potential energy</em> <em style="box-sizing: border-box;">U</em>. So, if <em style="box-sizing: border-box;">dq</em> charge is given to a conductor at potential <em style="box-sizing: border-box;">V</em>,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-11-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-289" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 9.625em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.641em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1007.64em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-290" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-291" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-292" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-293" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-294" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-295" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-296" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-297" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-298" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-299" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-300" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-301" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-302" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-303" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-304" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.753em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-12-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-305" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 10.965em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 8.683em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.895em, 1008.68em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-306" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-307" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-308" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-309" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-310" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-311" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-312" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-313" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-314" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.391em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-315" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.7em, 4.318em, -999.998em); left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-316" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-317" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-318" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-319" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-320" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-321" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-322" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-323" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-324" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.598em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-325" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-326" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-327" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-328" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-329" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-330" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-331" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-332" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mn" id="MathJax-Span-333" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-334" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-335" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-336" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-337" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-338" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-339" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-340" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-341" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-342" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.417em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.66em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-343" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-344" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-345" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-346" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.524em, 1000.3em, 4.169em, -999.998em); left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-347" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.524em, 1000.3em, 4.169em, -999.998em); left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.817em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-348" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-349" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-350" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-351" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-352" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.995em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span> (where <em style="box-sizing: border-box;">q</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">0</span> is the total charge given to the conductor)</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Since <span class="MathJax" id="MathJax-Element-13-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-353" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.268em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.375em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1003.38em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-354" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-355" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-356" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-357" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mi" id="MathJax-Span-358" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>, we can say that the energy stored in a charged (conductor) is</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-14-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-359" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 14.784em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 11.709em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.945em, 1011.71em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-360" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-361" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-362" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-363" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-364" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-365" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-366" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-367" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-368" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-369" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-370" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-371" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-372" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-373" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-374" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.524em, 1000.3em, 4.169em, -999.998em); left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-375" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.673em, 1000.3em, 4.169em, -999.998em); left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-376" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">o</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-377" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.8em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.796em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-378" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-379" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-380" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-381" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-382" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-383" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-384" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-385" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-386" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-387" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-388" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">o</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-389" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-390" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-391" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-392" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-393" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-394" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-395" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-396" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-397" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-398" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-399" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-400" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-401" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that if the charging source (say, battery) supplies charge at constant potential (say <em style="box-sizing: border-box;">V</em>), the work done by the charging source <span class="MathJax" id="MathJax-Element-15-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-402" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.665em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.673em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1003.67em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-403" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-404" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-405" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-406" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-407" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>, whereas the energy stored in the charged conductor is <span class="MathJax" id="MathJax-Element-16-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-408" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.211em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.12em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1004.12em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-409" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-410" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-411" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-412" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-413" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-414" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-415" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-416" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-417" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-418" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span>. Thus, <em style="box-sizing: border-box;">in charging a body 50 % of the energy is wasted as heat</em>.</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Application 1<br style="box-sizing: border-box;" /></em></span><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Small identical droplets of distilled water (radius 0.1 mm) are found to have a charge 2 pc each. If 64 of these coalesce to form a single drop, calculate (a) the charge on it, and (b) its potential.</em></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Solution:</em></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<em style="box-sizing: border-box;">a</em>) From conservation of charge, we have</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-17-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-419" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 24.506em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 19.447em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.094em, 1019.4em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-420" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-421" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">Q</span><span class="mo" id="MathJax-Span-422" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-423" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mi" id="MathJax-Span-424" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-425" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-426" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">64</span><span class="mo" id="MathJax-Span-427" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="mn" id="MathJax-Span-428" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mo" id="MathJax-Span-429" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="msubsup" id="MathJax-Span-430" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.433em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1001.64em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-431" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-432" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-433" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.639em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.95em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-434" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.995em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-435" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 1.639em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-436" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-437" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-438" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">12</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-439" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-440" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-441" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-442" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-443" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-444" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-445" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main-bold; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1<span style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">8</span></span></span></span></span></span><span class="mo" id="MathJax-Span-446" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-447" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-448" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-449" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-450" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-451" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main-bold; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span></span></span><span class="msubsup" id="MathJax-Span-452" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.086em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1001.19em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-453" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-454" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-455" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-456" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-457" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-458" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-459" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-460" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main-bold; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-461" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 1.193em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-462" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-463" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-464" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-465" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-466" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-467" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-468" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main-bold; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1<span style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-469" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-470" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-471" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-472" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-473" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main-bold; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C</span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.628em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<em style="box-sizing: border-box;">b</em>) From conservation of mass, we have</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-18-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-474" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 17.959em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 14.239em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1014.24em, 2.83em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.329em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-475" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-476" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-477" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="mrow" id="MathJax-Span-478" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-479" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span></span><span class="texatom" id="MathJax-Span-480" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-481" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-482" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="texatom" id="MathJax-Span-483" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-484" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-485" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-486" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span class="mi" id="MathJax-Span-487" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-488" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-489" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-490" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-491" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-492" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-493" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span class="mi" id="MathJax-Span-494" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ρ</span><span class="mo" id="MathJax-Span-495" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-496" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="mo" id="MathJax-Span-497" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="mrow" id="MathJax-Span-498" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-499" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span></span><span class="texatom" id="MathJax-Span-500" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-501" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-502" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="texatom" id="MathJax-Span-503" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-504" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-505" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-506" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span class="texatom" id="MathJax-Span-507" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-508" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-509" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-510" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-511" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-512" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-513" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-514" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-515" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-516" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-517" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span class="mi" id="MathJax-Span-518" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ρ</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.334em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.628em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.497em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">or, <span class="MathJax" id="MathJax-Element-19-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-519" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 27.383em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 21.729em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1021.73em, 2.631em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-520" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-521" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span class="mo" id="MathJax-Span-522" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-523" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-524" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-525" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-526" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="msubsup" id="MathJax-Span-527" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.54em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1000.3em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-528" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-529" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-530" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-531" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="texatom" id="MathJax-Span-532" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-533" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-534" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-535" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-536" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span class="mo" id="MathJax-Span-537" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-538" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-539" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-540" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mn" id="MathJax-Span-541" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">64</span><span class="msubsup" id="MathJax-Span-542" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.54em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1000.3em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-543" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-544" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-545" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-546" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="texatom" id="MathJax-Span-547" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-548" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-549" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-550" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-551" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="mn" id="MathJax-Span-552" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0.1</span><span class="mo" id="MathJax-Span-553" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-554" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-555" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-556" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.987em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.95em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-557" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.995em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-558" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-559" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-560" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-561" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-562" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-563" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0.4</span><span class="mo" id="MathJax-Span-564" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-565" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-566" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-567" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.987em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.95em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-568" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.995em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-569" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-570" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-571" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-572" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-573" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">m</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.566em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.372em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-20-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-574" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 19.199em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 15.231em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.044em, 1015.18em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-575" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-576" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-577" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-578" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-579" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-580" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.036em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.318em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-581" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">Q</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.94em, 4.268em, -999.998em); left: 20.5156px; line-height: normal; margin: 0px 0px 0px -0.94em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-582" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-583" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-584" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-585" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-586" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-587" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-588" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-589" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-590" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-591" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.036em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-592" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-593" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-594" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-595" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.856em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1005.71em, 4.169em, -999.998em); left: 59.0156px; line-height: normal; margin: 0px 0px 0px -2.875em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-596" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-597" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-598" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">128</span><span class="mo" id="MathJax-Span-599" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-600" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-601" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-602" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.639em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-603" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-604" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-605" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-606" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-607" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-608" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-609" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">12</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-610" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="mn" id="MathJax-Span-611" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">9</span><span class="mo" id="MathJax-Span-612" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-613" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-614" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-615" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-616" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-617" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-618" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-619" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">9</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1002.88em, 4.169em, -999.998em); left: 59.0156px; line-height: normal; margin: 0px 0px 0px -1.436em; padding: 0px; position: absolute; top: -3.519em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-620" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-621" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-622" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0.4</span><span class="mo" id="MathJax-Span-623" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">×</span><span class="texatom" id="MathJax-Span-624" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-625" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-626" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.391em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.7em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-627" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-628" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-629" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">10</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-630" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-631" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-632" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-633" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1005.86em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 5.856em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-634" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-635" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2880</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.128em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-21-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-636" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1000.8em, 2.334em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.18em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-637" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-638" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.185em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.003em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.059em; width: 0px;"></span></span></nobr></span></span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Sharing of Charge</span></h3><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Let us have two isolated spherical conductors of radii <em style="box-sizing: border-box;">R</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">1</span> and <em style="box-sizing: border-box;">R</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">2</span>, charged at potentials <em style="box-sizing: border-box;">V</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">1</span> and <em style="box-sizing: border-box;">V</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">2</span>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-22-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-639" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.558em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.417em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1004.42em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-640" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-641" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-642" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-643" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-644" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-645" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-646" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-647" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-648" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-649" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-650" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-651" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-652" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-653" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-654" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-655" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-656" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-23-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-657" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.558em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.417em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1004.42em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-658" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-659" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-660" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-661" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-662" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-663" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-664" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-665" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-666" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-667" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-668" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-669" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-670" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-671" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-672" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-673" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-674" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">where <span class="MathJax" id="MathJax-Element-24-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-675" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.195em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.707em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1005.71em, 2.532em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-676" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-677" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-678" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-679" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-680" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-681" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-682" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-683" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-684" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-685" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-686" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-687" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-688" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-689" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-690" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-691" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-692" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-693" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-694" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-25-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-695" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.195em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.707em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1005.71em, 2.532em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-696" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-697" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-698" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-699" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-700" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-701" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-702" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-703" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-704" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-705" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-706" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-707" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-708" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-709" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-710" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-711" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-712" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-713" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-714" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The combined charge is <em style="box-sizing: border-box;">q</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">1</span> + <em style="box-sizing: border-box;">q</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">2</span> and combined capacitance is <em style="box-sizing: border-box;">C</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">1</span> + <em style="box-sizing: border-box;">C</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">2</span>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Now if they are connected through a wire, charge will flow from conductor at higher potential to that at lower potential till both acquire the same potential,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-26-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-715" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 20.042em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 15.875em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.995em, 1015.88em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-716" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-717" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-718" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-719" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-720" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-721" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.83em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1002.28em, 4.368em, -999.998em); left: 28.5156px; line-height: normal; margin: 0px 0px 0px -1.188em; padding: 0px; position: absolute; top: -4.611em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-722" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-723" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-724" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-725" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-726" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-727" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.598em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-728" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-729" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-730" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-731" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-732" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-733" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.598em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-734" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-735" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-736" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1002.63em, 4.368em, -999.998em); left: 28.5156px; line-height: normal; margin: 0px 0px 0px -1.337em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-737" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-738" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-739" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-740" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-741" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-742" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-743" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-744" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-745" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-746" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-747" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-748" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-749" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-750" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-751" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.83em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.83em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-752" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-753" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-754" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-755" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.723em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1003.62em, 4.268em, -999.998em); left: 37.5156px; line-height: normal; margin: 0px 0px 0px -1.783em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-756" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-757" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-758" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-759" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-760" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-761" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-762" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-763" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-764" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-765" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-766" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-767" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-768" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-769" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-770" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-771" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-772" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-773" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-774" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-775" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-776" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-777" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-778" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1002.18em, 4.268em, -999.998em); left: 37.5156px; line-height: normal; margin: 0px 0px 0px -1.089em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-779" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-780" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-781" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-782" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-783" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-784" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-785" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-786" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-787" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-788" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-789" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-790" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-791" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1003.72em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 3.723em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-792" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-793" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-794" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-795" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.772em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1003.67em, 4.268em, -999.998em); left: 38.0156px; line-height: normal; margin: 0px 0px 0px -1.833em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-796" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-797" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-798" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-799" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-800" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-801" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-802" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-803" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-804" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-805" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-806" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-807" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-808" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-809" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-810" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-811" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-812" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-813" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-814" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-815" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-816" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-817" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-818" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1002.24em, 4.268em, -999.998em); left: 38.0156px; line-height: normal; margin: 0px 0px 0px -1.089em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-819" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-820" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-821" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-822" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-823" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-824" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-825" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-826" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-827" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-828" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-829" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-830" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-831" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1003.77em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 3.772em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.316em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.809em; width: 0px;"></span></span></nobr></span></span></p><p id="XVlxmZx" style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-128061 aligncenter jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 519px) 100vw, 519px" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581f1483bf200.png?w=1170&ssl=1" srcset="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581f1483bf200.png?w=519&ssl=1 519w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581f1483bf200.png?resize=500%2C177&ssl=1 500w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581f1483bf200.png?resize=300%2C106&ssl=1 300w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 5px auto; max-width: 100%; vertical-align: middle;" /></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">And hence, if <em style="box-sizing: border-box;">q'</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">1</span> and <em style="box-sizing: border-box;">q'</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">2</span> are the charges on the two conductors after sharing,</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-27-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-832" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.31em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.219em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1004.22em, 2.681em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-833" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-834" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-835" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-836" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-837" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.449em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.872em, 1000em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-838" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-839" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-840" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-841" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-842" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-843" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-844" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-845" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-846" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-847" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.434em; width: 0px;"></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-28-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-848" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.31em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.219em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1004.22em, 2.681em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-849" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-850" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-851" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-852" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-853" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.449em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.872em, 1000em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-854" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-855" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-856" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-857" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-858" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-859" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-860" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-861" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-862" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-863" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.434em; width: 0px;"></span></span></nobr></span> with <span class="MathJax" id="MathJax-Element-29-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-864" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 13.495em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 10.717em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1010.72em, 2.681em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-865" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-866" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-867" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-868" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-869" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-870" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.449em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.872em, 1000em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-871" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-872" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-873" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-874" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mi" id="MathJax-Span-875" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-876" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-877" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-878" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.449em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.872em, 1000em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-879" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-880" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-881" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-882" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="mo" id="MathJax-Span-883" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mrow" id="MathJax-Span-884" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-885" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-886" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-887" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-888" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-889" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-890" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-891" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-892" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-893" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-894" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-895" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-896" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-897" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-898" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-899" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span class="mo" id="MathJax-Span-900" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-901" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.441em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.434em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">So, <span class="MathJax" id="MathJax-Element-30-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-902" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.889em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.252em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.995em, 1006.25em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-903" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-904" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-905" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-906" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.89em, 4.318em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-907" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-908" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-909" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-910" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-911" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.55em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-912" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-913" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msup" id="MathJax-Span-914" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.548em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-915" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-916" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-917" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.318em, -999.998em); left: 10.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-918" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-919" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-920" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-921" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-922" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-923" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-924" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msup" id="MathJax-Span-925" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.548em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-926" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.214em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-927" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-928" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.995em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-929" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-930" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-931" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-932" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.8em, 4.268em, -999.998em); left: 9.51562px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-933" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-934" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-935" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-936" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-937" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-938" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-939" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.8em, 4.268em, -999.998em); left: 9.51562px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-940" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-941" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-942" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-943" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-944" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-945" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-946" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.95em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.945em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-947" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-948" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-949" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-950" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.85em, 4.268em, -999.998em); left: 9.51562px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-951" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-952" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-953" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-954" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-955" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-956" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-957" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.85em, 4.268em, -999.998em); left: 9.51562px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-958" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-959" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-960" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-961" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-962" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-963" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-964" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.95em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.945em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Thus, <em style="box-sizing: border-box;">the charge is shared in proportion to capacity</em>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Some energy is lost in sharing charges. This energy is lost mainly as heat when charge flows from one body to the other through the connecting wire and also as light and sound if sparking takes place. The loss in energy is</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-31-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-965" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 29.566em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 23.465em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.292em, 1023.47em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.329em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-966" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-967" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-968" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-969" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-970" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-971" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.094em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-972" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-973" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">I<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-974" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-975" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-976" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-977" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-978" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-979" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-980" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mrow" id="MathJax-Span-981" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-982" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span></span><span class="texatom" id="MathJax-Span-983" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-984" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-985" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-986" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-987" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-988" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-989" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-990" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-991" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-992" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-993" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-994" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-995" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-996" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-997" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-998" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-999" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1000" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1001" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1002" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1003" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1004" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1005" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1006" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1007" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1008" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1009" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1010" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1011" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1012" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1013" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-1014" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1015" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1016" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1017" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1018" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span class="mo" id="MathJax-Span-1019" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-1020" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1021" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1022" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1023" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1024" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mrow" id="MathJax-Span-1025" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1026" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1027" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1028" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1029" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1030" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1031" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1032" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1033" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1034" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1035" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1036" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1037" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1038" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1039" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1040" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span class="texatom" id="MathJax-Span-1041" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1042" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1043" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1044" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1045" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1046" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1047" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1048" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1049" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.334em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-32-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1050" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 11.411em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 9.03em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.094em, 1009.03em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1051" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1052" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1053" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1054" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1055" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.177em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.64em, 4.268em, -999.998em); left: 32.0156px; line-height: normal; margin: 0px 0px 0px -0.791em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1056" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1057" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1058" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1059" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1060" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1061" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1062" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1063" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1064" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1065" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1066" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1067" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1003.03em, 4.368em, -999.998em); left: 32.0156px; line-height: normal; margin: 0px 0px 0px -1.535em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1068" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1069" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1070" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mo" id="MathJax-Span-1071" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1072" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1073" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1074" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1075" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1076" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1077" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1078" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1079" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1080" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1081" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1082" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1083" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1003.18em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 3.177em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1084" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1085" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1086" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1087" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1088" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1089" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1090" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1091" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1092" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">∼</span><span class="texatom" id="MathJax-Span-1093" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1094" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1095" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1096" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1097" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-1098" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1000.3em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1099" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1100" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.809em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Application 2</em></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Two isolated metallic solid spheres of radius R and 2R are charged such that both of these have same charge density <span class="MathJax" id="MathJax-Element-33-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; font-style: normal; font-weight: normal; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1101" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.79em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.32px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.593em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.577em, 1000.59em, 2.315em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.163em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1102" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1103" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.168em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 0.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.059em; width: 0px;"></span></span></nobr></span></em></span><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">. The spheres are located far away from each other and connected by a thin conducting wire, find the new charge density on the bigger sphere.<br style="box-sizing: border-box;" /></em></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Solution:</em></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">As charge density on both spheres is same, the total charge,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-34-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1104" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 25.002em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 19.844em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.143em, 1019.84em, 2.631em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1105" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1106" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1107" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1108" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1109" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1110" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1111" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1112" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1113" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1114" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1115" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1116" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1117" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1118" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1119" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-1120" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1121" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1122" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1123" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1124" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.987em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1001.44em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1125" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1126" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1127" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span class="mo" id="MathJax-Span-1128" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 1.54em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1129" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1130" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1131" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mn" id="MathJax-Span-1132" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1133" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1134" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1135" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1136" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.483em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1001.94em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1137" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1138" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1139" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1140" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1141" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1142" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span class="mo" id="MathJax-Span-1143" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 2.036em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1144" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1145" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1146" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-1147" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">20</span><span class="mi" id="MathJax-Span-1148" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1149" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1150" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1151" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1152" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1153" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1154" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.628em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.372em; width: 0px;"></span></span></nobr></span> ....(i)</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Now in sharing, the charge is shared in proportion to capacity (i.e., radius), so the charge on the bigger sphere,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-35-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1155" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 15.727em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 12.453em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.094em, 1012.45em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1156" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1157" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1158" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1159" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1160" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1161" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1162" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1163" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1164" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1165" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1166" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.879em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.85em, 4.268em, -999.998em); left: 29.0156px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1167" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1168" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1169" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1170" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1171" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1172" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1173" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1002.73em, 4.368em, -999.998em); left: 29.0156px; line-height: normal; margin: 0px 0px 0px -1.386em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1174" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1175" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1176" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1177" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1178" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1179" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1180" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1181" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1182" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1183" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1184" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1185" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1186" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1187" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1188" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.88em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.879em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1189" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1190" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1191" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1192" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1193" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.086em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1194" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1195" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1196" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1197" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.99em, 4.219em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.99em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1198" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1199" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1200" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span class="mo" id="MathJax-Span-1201" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mn" id="MathJax-Span-1202" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1203" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.09em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.086em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1204" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1205" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1206" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1207" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1208" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1209" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.35em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1210" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1211" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.128em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.809em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-36-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1212" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 19.149em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 15.181em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.995em, 1015.18em, 3.127em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1213" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1214" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1215" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1216" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.6em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1217" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.475em, 1000.25em, 4.12em, -999.998em); left: 0.647em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1218" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1219" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1220" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1221" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1222" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1223" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.631em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.89em, 4.318em, -999.998em); left: 26.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1224" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1225" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1226" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1227" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1228" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.55em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1229" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1230" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msup" id="MathJax-Span-1231" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.548em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1232" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1233" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">′</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1234" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1002.48em, 4.368em, -999.998em); left: 26.5156px; line-height: normal; margin: 0px 0px 0px -1.238em; padding: 0px; position: absolute; top: -3.42em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1235" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1236" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1237" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1238" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1239" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1240" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1241" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.739em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1001.39em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1242" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1243" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1244" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mn" id="MathJax-Span-1245" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1246" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span class="mo" id="MathJax-Span-1247" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 1.441em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1248" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.63em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.631em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1249" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1250" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1251" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1252" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.086em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1001.94em, 4.368em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.99em; padding: 0px; position: absolute; top: -4.611em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1253" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1254" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1255" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mn" id="MathJax-Span-1256" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="texatom" id="MathJax-Span-1257" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1258" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1259" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-1260" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">3</span><span class="mo" id="MathJax-Span-1261" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="mi" id="MathJax-Span-1262" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1001.94em, 4.169em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.99em; padding: 0px; position: absolute; top: -3.47em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1263" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1264" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1265" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">16</span><span class="mi" id="MathJax-Span-1266" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1267" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1268" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1269" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1270" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1271" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.09em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.086em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1272" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1273" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1274" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1275" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.086em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1276" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1001.94em, 4.169em, -999.998em); left: 21.0156px; line-height: normal; margin: 0px 0px 0px -0.99em; padding: 0px; position: absolute; top: -3.47em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1277" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1278" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1279" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">24</span><span class="mi" id="MathJax-Span-1280" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1281" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1282" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1283" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1284" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">R</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1285" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.09em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.086em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1286" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1287" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1288" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1289" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1290" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">5</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.35em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1291" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">6</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1292" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.503em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.997em; width: 0px;"></span></span></nobr></span> [using Eqn. (i)]</span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Capacitance of some Capacitors</span></h3><p id="liMkDSO" style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-127991 alignright jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 172px) 100vw, 172px" src="https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?w=1170&ssl=1" srcset="https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?w=172&ssl=1 172w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?resize=75%2C75&ssl=1 75w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?resize=122%2C120&ssl=1 122w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?resize=69%2C69&ssl=1 69w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581de35a195bd.png?resize=50%2C50&ssl=1 50w" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px 0px 20px 20px; max-width: 100%; vertical-align: middle;" /></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<span style="box-sizing: border-box; font-weight: 700;">1</span>) <span style="box-sizing: border-box; font-weight: 700;">Parallel Palate Capacitor</span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-37-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1293" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 3.524em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.78em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1002.78em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1294" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1295" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1296" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1297" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1298" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1299" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1300" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1301" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-38-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1302" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.592em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.004em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.243em, 1006em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1303" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1304" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1305" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1306" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1307" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1308" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.747em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.4em, 4.169em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1309" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.65em, 4.268em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1310" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1311" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1312" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1313" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1314" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1315" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1316" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.75em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.747em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1317" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1318" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1319" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1320" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1321" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1322" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1323" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1324" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1325" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1326" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1327" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1328" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1329" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.878em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-39-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1330" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 8.038em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.352em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.044em, 1006.35em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1331" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1332" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-1333" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1334" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mi" id="MathJax-Span-1335" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1336" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1337" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1338" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1339" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.318em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1340" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1341" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1342" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1343" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1344" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1345" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1346" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1347" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1348" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1349" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1350" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1351" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-40-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1352" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.393em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.856em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.044em, 1005.86em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1353" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1354" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1355" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1356" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1357" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1358" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1359" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1360" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1361" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1362" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1363" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1364" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1365" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1366" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1367" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1368" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1369" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1370" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1371" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1372" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.35em, 4.169em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1373" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.941em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that the capacitance is independent of charge given, potential raised, nature of metal or thickness of plates.</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<span style="box-sizing: border-box; font-weight: 700;">2</span>)<span style="box-sizing: border-box; font-weight: 700;"> Spherical Capacitors<br style="box-sizing: border-box;" /></span>The induced charge <em style="box-sizing: border-box;">q'</em> on shell <em style="box-sizing: border-box;">B</em> is equal and opposite to charge <em style="box-sizing: border-box;">q</em> on inner sphere <em style="box-sizing: border-box;">A</em>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">As the charges on the two conductors are equal and opposite, the system is a capacitor.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The electric field at a point <em style="box-sizing: border-box;">P</em> between the shells,</span></p><p id="wGRzDjw" style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-127993 alignright jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581eb2a8f3f71.png?w=1170&ssl=1" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px 0px 20px 20px; max-width: 100%; vertical-align: middle;" /></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-41-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1374" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 12.701em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 10.072em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1010.07em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1375" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1376" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1377" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1378" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1379" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1380" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.342em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1381" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1382" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1383" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="texatom" id="MathJax-Span-1384" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1385" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1386" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.342em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1387" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1388" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">B</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1389" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1390" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1391" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1392" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1393" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1394" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1395" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1396" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1397" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1398" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1399" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1400" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1401" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1402" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1403" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1404" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1405" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.747em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1406" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.65em, 4.169em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1407" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1408" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1409" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1410" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1411" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1412" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.214em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1413" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.75em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.747em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.878em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span> [as <span class="MathJax" id="MathJax-Element-42-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1414" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.691em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.104em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1006.05em, 2.532em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1415" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1416" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1417" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1418" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.342em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1419" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1420" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">B</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1421" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1422" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1423" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1424" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1425" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.747em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1426" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1427" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1428" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span><span class="mi" id="MathJax-Span-1429" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1430" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-1431" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span>]</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">or <span class="MathJax" id="MathJax-Element-43-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1432" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.344em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.806em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.143em, 1005.81em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1433" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1434" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1435" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1436" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.59em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.49em, 4.219em, -999.998em); left: 16.0156px; line-height: normal; margin: 0px 0px 0px -0.742em; padding: 0px; position: absolute; top: -4.462em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1437" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1438" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1439" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mi" id="MathJax-Span-1440" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1441" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.169em, -999.998em); left: 16.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1442" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1443" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1444" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1445" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.59em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.59em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1446" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1447" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1448" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1449" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1450" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1451" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1452" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1453" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1454" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1455" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1456" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1457" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1458" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1459" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1460" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1461" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1462" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.747em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1463" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.65em, 4.169em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1464" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1465" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1466" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1467" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1468" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1469" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.214em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1470" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.75em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.747em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.941em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-44-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1471" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.443em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.905em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.292em, 1005.71em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.329em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1472" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1473" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1474" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-1475" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1476" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1477" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1478" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1479" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1480" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1481" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-1482" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">s</span></span></span></span></span><span class="mi" id="MathJax-Span-1483" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1484" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mo" id="MathJax-Span-1485" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-1486" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1487" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1488" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.044em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1489" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1490" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1491" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1492" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1493" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1494" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1495" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1496" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.044em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1497" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.334em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.753em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-45-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1498" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 22.473em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 17.81em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.143em, 1017.61em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.329em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1499" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1500" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-1501" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mo" id="MathJax-Span-1502" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="msubsup" id="MathJax-Span-1503" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1504" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1505" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1506" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1507" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1508" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1509" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1510" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mn" id="MathJax-Span-1511" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.6em, 4.169em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1512" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1513" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1514" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1515" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1516" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1517" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1518" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1519" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1520" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span><span class="mi" id="MathJax-Span-1521" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1522" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-1523" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1524" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1525" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1526" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1527" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1528" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1529" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1530" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1531" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1532" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1533" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1534" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1535" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1536" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-1537" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.094em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1538" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.4em, 4.169em, -999.998em); left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1539" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1540" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1541" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1542" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1543" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1544" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1545" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.45em, 4.169em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1546" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1547" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1548" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1549" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1550" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1551" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1552" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1553" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1554" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span><span class="texatom" id="MathJax-Span-1555" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1556" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1557" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1558" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1559" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1560" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1561" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1000.65em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1562" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1563" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1564" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1565" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1566" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1567" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.3em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.214em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1568" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.8em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.796em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1569" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1570" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1571" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1572" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1573" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1574" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1575" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1576" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1577" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1578" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1579" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1580" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1581" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1582" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1583" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1584" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1585" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1586" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mrow" id="MathJax-Span-1587" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1588" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-1589" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1590" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1591" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1592" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1593" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1594" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1595" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1596" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-1597" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1598" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1599" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1600" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1601" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1602" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.334em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-46-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1603" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 8.534em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.749em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.094em, 1006.75em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1604" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1605" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1606" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1607" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1608" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1609" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1610" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1611" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1612" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1613" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1614" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1615" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.185em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1002.09em, 4.268em, -999.998em); left: 22.0156px; line-height: normal; margin: 0px 0px 0px -1.039em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1616" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1617" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1618" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1619" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1620" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1621" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1622" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1623" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1624" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1625" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-1626" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1001.54em, 4.368em, -999.998em); left: 22.0156px; line-height: normal; margin: 0px 0px 0px -0.791em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1627" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1628" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1629" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span><span class="mi" id="MathJax-Span-1630" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span class="mo" id="MathJax-Span-1631" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mi" id="MathJax-Span-1632" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mo" id="MathJax-Span-1633" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.18em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.185em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.809em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(1) As <span class="MathJax" id="MathJax-Element-47-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1634" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 3.822em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.028em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1002.98em, 2.383em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1635" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1636" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span class="mo" id="MathJax-Span-1637" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">→</span><span class="mi" id="MathJax-Span-1638" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">∞</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.003em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.059em; width: 0px;"></span></span></nobr></span>, the capacitance reduces <span class="MathJax" id="MathJax-Element-48-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1639" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 3.127em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.483em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1002.48em, 2.532em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1640" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1641" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1642" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1643" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1644" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1645" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1646" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1647" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1648" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.247em; width: 0px;"></span></span></nobr></span>. This shows that a spherical conductor is a spherical capacitor with its other plate of infinite radius.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(2) As <em style="box-sizing: border-box;">a</em> and <em style="box-sizing: border-box;">b</em> both become very large, maintaining the difference <span class="MathJax" id="MathJax-Element-49-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1649" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.062em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.02em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1004.02em, 2.483em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1650" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1651" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mo" id="MathJax-Span-1652" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mi" id="MathJax-Span-1653" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.201em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span class="mo" id="MathJax-Span-1654" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1655" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.128em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.184em; width: 0px;"></span></span></nobr></span> (finite), the expression for <em style="box-sizing: border-box;">C</em> reduces to <span class="MathJax" id="MathJax-Element-50-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1656" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.566em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.624em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.044em, 1003.62em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1657" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1658" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1659" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1660" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1661" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1662" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1663" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1664" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1665" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1666" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1667" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1668" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1669" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1670" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.35em, 4.169em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1671" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.941em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span>. This shows that a spherical capacitor behaves as a parallel plate capacitor if its spherical surfaces have large radii and are close to each other.</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">(3)</span> <span style="box-sizing: border-box; font-weight: 700;">Cylindrical Capacitor :<br style="box-sizing: border-box;" /></span></span></p><p id="JgjWvoB" style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-127996 alignright jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" src="https://i2.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_581ebb72d4173.png?w=1170&ssl=1" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px 0px 20px 20px; max-width: 100%; vertical-align: middle;" /></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The field at a point <em style="box-sizing: border-box;">P</em> is</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-51-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1672" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 6.252em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.963em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1004.96em, 2.879em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1673" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1674" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1675" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1676" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1677" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1678" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1679" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1680" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1681" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1682" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1683" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1684" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1685" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1686" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1687" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1688" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1689" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1690" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1691" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 9.01562px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1692" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1693" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1694" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1695" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">λ</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 9.01562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1696" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.89em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.895em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.878em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">But <span class="MathJax" id="MathJax-Element-52-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1697" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.31em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.219em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1004.22em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1698" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1699" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1700" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mo" id="MathJax-Span-1701" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-1702" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1703" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1704" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.044em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1705" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1706" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1707" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1708" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1709" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1710" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1711" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1712" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.044em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.753em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-53-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1713" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 10.667em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 8.435em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.499em, 1008.43em, 3.375em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1714" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1715" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="munderover" id="MathJax-Span-1716" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.598em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1717" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.268em, -999.998em); left: 0.002em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.123em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1718" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">v</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.35em, 4.169em, -999.998em); left: 0.25em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -5.007em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1719" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1720" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1721" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1722" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1723" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span></span><span class="mo" id="MathJax-Span-1724" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1725" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1726" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1727" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1728" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1729" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1730" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1731" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">λ</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1732" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1733" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1734" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1735" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1736" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1737" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1738" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1739" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1740" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-1741" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.094em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1742" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.4em, 4.169em, -999.998em); left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1743" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1744" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1745" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1746" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1747" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1748" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1749" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.45em, 4.169em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1750" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1751" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1752" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1753" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1754" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1755" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1756" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1757" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1758" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1759" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1760" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1761" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.7em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1762" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1763" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1764" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1765" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1766" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.8em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.796em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 3.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -1.309em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-54-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1767" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 9.774em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.741em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.292em, 1007.59em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.329em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1768" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1769" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">⇒</span><span class="mi" id="MathJax-Span-1770" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-1771" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1772" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1773" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1774" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.491em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1775" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1776" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1777" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1778" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">λ</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.39em, 4.268em, -999.998em); left: 15.0156px; line-height: normal; margin: 0px 0px 0px -0.692em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1779" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1780" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1781" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1782" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1783" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1784" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1785" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1786" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1787" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.49em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.491em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1788" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1789" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1790" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1791" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1792" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">l</span><span class="mi" id="MathJax-Span-1793" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span></span></span></span></span><span class="mrow" id="MathJax-Span-1794" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1795" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span></span><span class="texatom" id="MathJax-Span-1796" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1797" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1798" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1799" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1800" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1801" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1802" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1803" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.334em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.878em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.684em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-55-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1804" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 17.066em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 13.544em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.094em, 1013.54em, 2.979em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1805" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1806" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1807" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1808" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1809" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1810" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1811" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1812" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1813" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1814" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1815" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1816" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.062em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 51.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1817" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1818" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1819" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">λ</span><span class="mi" id="MathJax-Span-1820" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">L</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1004.86em, 4.368em, -999.998em); left: 51.0156px; line-height: normal; margin: 0px 0px 0px -2.478em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1821" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1822" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1823" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1824" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">λ</span><span class="texatom" id="MathJax-Span-1825" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1826" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1827" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mn" id="MathJax-Span-1828" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">4</span><span class="mi" id="MathJax-Span-1829" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1830" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1831" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1832" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1833" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1834" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1835" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="texatom" id="MathJax-Span-1836" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1837" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1838" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1839" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1840" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">l</span><span class="mi" id="MathJax-Span-1841" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="texatom" id="MathJax-Span-1842" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1843" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1844" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1845" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1846" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1847" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span class="texatom" id="MathJax-Span-1848" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1849" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1850" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-1851" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mo" id="MathJax-Span-1852" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1005.06em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 5.062em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1853" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1854" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1855" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1856" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.334em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.84em, 4.268em, -999.998em); left: 23.5156px; line-height: normal; margin: 0px 0px 0px -0.94em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1857" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1858" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1859" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="mi" id="MathJax-Span-1860" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">π<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1861" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1862" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1863" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1864" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1865" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1866" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">L</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.326em, 1002.13em, 4.368em, -999.998em); left: 23.5156px; line-height: normal; margin: 0px 0px 0px -1.089em; padding: 0px; position: absolute; top: -3.569em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1867" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1868" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1869" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">l</span><span class="mi" id="MathJax-Span-1870" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="texatom" id="MathJax-Span-1871" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1872" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1873" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1874" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1875" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1876" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">b</span><span class="texatom" id="MathJax-Span-1877" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1878" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1879" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-1880" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mo" id="MathJax-Span-1881" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.33em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.334em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.809em; width: 0px;"></span></span></nobr></span></span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Energy Stored in a Capacitor</span></h3><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">If <em style="box-sizing: border-box;">dq</em> charge is given to a capacitor at potential <em style="box-sizing: border-box;">V</em>, the work done is</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-56-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1882" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.096em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.608em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1005.51em, 2.631em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1883" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1884" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1885" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-1886" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1887" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1888" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mrow" id="MathJax-Span-1889" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1890" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1891" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="mo" id="MathJax-Span-1892" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.372em; width: 0px;"></span></span></nobr></span> [as <span class="MathJax" id="MathJax-Element-57-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1893" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.268em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.375em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1003.38em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1894" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1895" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1896" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1897" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mi" id="MathJax-Span-1898" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>]</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">or, <span class="MathJax" id="MathJax-Element-58-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1899" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 24.754em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 19.645em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1019.64em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1900" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1901" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-1902" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="msubsup" id="MathJax-Span-1903" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.094em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.078em, 1000.6em, 4.467em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1904" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Size1; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;">∫<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.4em, 4.318em, -999.998em); left: 0.697em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1905" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1906" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1907" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1908" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1909" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1910" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1911" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1912" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1913" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1914" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1915" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1916" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1917" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mn" id="MathJax-Span-1918" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1919" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1920" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1921" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="mi" id="MathJax-Span-1922" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="texatom" id="MathJax-Span-1923" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1924" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1925" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-1926" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1927" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="mi" id="MathJax-Span-1928" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1929" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-1930" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1931" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1932" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1933" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1934" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1935" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1936" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1937" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1938" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1939" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1940" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-1941" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1942" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1943" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1944" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1945" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-1946" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1947" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-1948" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-1949" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-1950" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="mo" id="MathJax-Span-1951" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1952" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1953" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1954" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1955" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1956" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1957" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-1958" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1959" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1960" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1961" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1962" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1963" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1964" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1965" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1966" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1967" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1968" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-1969" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-1970" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span> [as <span class="MathJax" id="MathJax-Element-59-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1971" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.268em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.375em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.391em, 1003.38em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1972" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1973" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mo" id="MathJax-Span-1974" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1975" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mi" id="MathJax-Span-1976" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.253em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>]</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">This work is stored as electrical potential energy,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-60-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1977" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 16.471em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 13.048em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.945em, 1013.05em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1978" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1979" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-1980" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1981" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">W<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-1982" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-1983" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-1984" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1985" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-1986" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.8em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-1987" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.895em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1988" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1989" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-1990" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1991" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-1992" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1993" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1994" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1995" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1996" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-1997" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-1998" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-1999" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2000" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2001" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2002" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2003" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2004" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2005" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2006" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2007" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2008" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2009" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2010" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2011" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2012" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2013" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.8em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.796em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2014" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2015" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2016" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2017" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2018" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2019" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2020" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2021" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2022" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2023" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2024" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2025" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2026" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2027" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2028" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-2029" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><em style="box-sizing: border-box;">This energy is not localized on the charges or the plates but is distributed in the <span style="box-sizing: border-box; font-weight: 700;">field</span></em>.</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">In case of a parallel plate capacitor, the field is limited between the plates, in a volume <em style="box-sizing: border-box;">A</em> x <em style="box-sizing: border-box;">d</em>. We can determine the <em style="box-sizing: border-box;">energy density u</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">E</span> in this volume,</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-61-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2030" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 26.094em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 20.687em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.639em, 1020.44em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2031" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2032" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2033" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2034" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.143em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2035" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">u</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2036" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2037" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2038" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2039" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2040" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2041" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2042" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2043" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2044" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.483em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 25.0156px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2045" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1002.33em, 4.169em, -999.998em); left: 25.0156px; line-height: normal; margin: 0px 0px 0px -1.188em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2046" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2047" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2048" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2049" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2050" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2051" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2052" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V</span><span class="mi" id="MathJax-Span-2053" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">o</span><span class="mi" id="MathJax-Span-2054" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">l</span><span class="mi" id="MathJax-Span-2055" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">u</span><span class="mi" id="MathJax-Span-2056" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">m</span><span class="mi" id="MathJax-Span-2057" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">e</span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.48em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.483em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2058" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2059" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2060" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2061" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.185em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1002.09em, 4.467em, -999.998em); left: 22.0156px; line-height: normal; margin: 0px 0px 0px -1.039em; padding: 0px; position: absolute; top: -4.71em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2062" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2063" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2064" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2065" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2066" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.35em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.524em, 1000.2em, 4.169em, -999.998em); left: 3.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2067" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.524em, 1000.25em, 4.169em, -999.998em); left: 3.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2068" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.35em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.188em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.35em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2069" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="texatom" id="MathJax-Span-2070" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2071" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2072" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2073" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.647em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2074" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 22.0156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2075" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2076" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2077" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span class="mi" id="MathJax-Span-2078" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1002.18em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 2.185em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2079" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2080" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2081" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2082" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2083" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2084" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mrow" id="MathJax-Span-2085" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2086" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-2087" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2088" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2089" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2090" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2091" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2092" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2093" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2094" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2095" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2096" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2097" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2098" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2099" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.35em, 4.169em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2100" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2101" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span><span class="texatom" id="MathJax-Span-2102" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2103" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2104" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.044em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1000.95em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2105" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2106" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2107" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2108" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2109" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2110" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.647em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2111" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.89em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2112" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2113" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2114" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span class="mi" id="MathJax-Span-2115" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.044em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2116" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2117" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2118" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2119" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"></span></span></span></span><span class="mrow" id="MathJax-Span-2120" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2121" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-2122" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2123" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2124" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2125" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2126" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2127" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2128" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-2129" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">s</span></span></span></span></span><span class="mi" id="MathJax-Span-2130" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-2131" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2132" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2133" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2134" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2135" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2136" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2137" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2138" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2139" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2140" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2141" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2142" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.35em, 4.169em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2143" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2144" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-62-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2145" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 14.338em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 11.362em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.143em, 1011.36em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2146" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2147" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2148" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2149" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.193em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.55em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2150" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">u</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2151" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2152" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2153" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2154" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2155" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2156" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2157" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2158" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2159" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2160" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2161" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2162" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2163" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2164" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2165" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.979em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.127em, 1002.48em, 4.417em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2166" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2167" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="texatom" id="MathJax-Span-2168" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2169" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2170" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.201em;"></span></span><span class="texatom" id="MathJax-Span-2171" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2172" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2173" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-2174" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-2175" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 2.582em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2176" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2177" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2178" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2179" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2180" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2181" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2182" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2183" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2184" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2185" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2186" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2187" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2188" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2189" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2190" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.243em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2191" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.846em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2192" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.816em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span> [as <span class="MathJax" id="MathJax-Element-63-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2193" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 3.822em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.028em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.193em, 1003.03em, 2.78em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2194" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2195" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2196" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2197" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2198" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">V<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.151em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.35em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2199" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2200" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-2201" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.753em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.559em; width: 0px;"></span></span></nobr></span>]</span></p><h3 style="box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Force Between the Plates</span></h3><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The plates carry equal and opposite charges. There is a force of attraction between them. To calculate this force, we use the fact that electric field is conservative, for which <em style="box-sizing: border-box;">F </em>= -(<em style="box-sizing: border-box;">dU/dx</em>).</span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">In case of a parallel plate capacitor</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-64-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2202" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 9.377em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 7.443em;"><span style="border: 0px; box-sizing: border-box; clip: rect(0.945em, 1007.44em, 2.929em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2203" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2204" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">U<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-2205" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2206" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2207" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2208" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2209" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2210" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2211" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2212" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2213" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2214" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2215" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2216" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2217" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2218" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2219" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2220" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 8.01562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2221" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.8em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.796em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2222" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2223" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2224" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2225" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2226" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2227" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2228" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2229" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2230" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1001.04em, 4.318em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.543em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2231" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2232" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2233" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2234" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2235" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2236" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2237" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2238" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2239" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2240" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2241" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2242" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2243" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2244" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2245" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2246" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-65-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2247" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 6.897em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 5.459em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.887em, 1005.21em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2248" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2249" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2250" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-2251" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2252" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2253" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2254" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2255" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2256" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2257" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-2258" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">s</span></span></span></span></span><span class="mi" id="MathJax-Span-2259" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">C<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-2260" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2261" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2262" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2263" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -4.511em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2264" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2265" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2266" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2267" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2268" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2269" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2270" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2271" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.169em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2272" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2273" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span><br style="box-sizing: border-box;" /><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">⸫ <span class="MathJax" id="MathJax-Element-66-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2274" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 14.635em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 11.61em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.887em, 1011.61em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2275" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2276" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span class="mo" id="MathJax-Span-2277" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2278" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2279" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2280" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.044em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.95em, 4.219em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.444em; padding: 0px; position: absolute; top: -4.462em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2281" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2282" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2283" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mi" id="MathJax-Span-2284" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.75em, 4.169em, -999.998em); left: 10.5156px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2285" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2286" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2287" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mi" id="MathJax-Span-2288" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.04em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.044em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mrow" id="MathJax-Span-2289" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.151em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2290" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-2291" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2292" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2293" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2294" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2295" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2296" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2297" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2298" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2299" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2300" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2301" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2302" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2303" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2304" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2305" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2306" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2307" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2308" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2309" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2310" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2311" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2312" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2313" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2314" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2315" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2316" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span class="mo" id="MathJax-Span-2317" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span><span class="mo" id="MathJax-Span-2318" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mo" id="MathJax-Span-2319" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="texatom" id="MathJax-Span-2320" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2321" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2322" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2323" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2324" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2325" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2326" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2327" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.292em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2328" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2329" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2330" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2331" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2332" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2333" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2334" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1001.14em, 4.268em, -999.998em); left: 13.0156px; line-height: normal; margin: 0px 0px 0px -0.593em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2335" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2336" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2337" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2338" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2339" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2340" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2341" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mi" id="MathJax-Span-2342" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.29em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.292em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The negative sign implies that the force is attractive. The force per unit area is</span></p><p style="box-sizing: border-box; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-67-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2343" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 15.379em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 12.205em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.838em, 1012.21em, 3.921em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.123em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2344" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2345" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2346" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0.895em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.3em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Main; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.271em; transition: none 0s ease 0s; vertical-align: 0px;">∣<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Main; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.924em; transition: none 0s ease 0s; vertical-align: 0px;">∣<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-2347" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2348" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2349" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2350" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2351" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.55em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2352" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">F<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.102em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2353" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2354" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0.895em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.3em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Main; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.271em; transition: none 0s ease 0s; vertical-align: 0px;">∣<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Main; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.924em; transition: none 0s ease 0s; vertical-align: 0px;">∣<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span class="mo" id="MathJax-Span-2355" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2356" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2357" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2358" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2359" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2360" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2361" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2362" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2363" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.59em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.7em, 4.318em, -999.998em); left: 16.0156px; line-height: normal; margin: 0px 0px 0px -0.345em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2364" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2365" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2366" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2367" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2368" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.697em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2369" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.399em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2370" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.227em, 1001.49em, 4.268em, -999.998em); left: 16.0156px; line-height: normal; margin: 0px 0px 0px -0.742em; padding: 0px; position: absolute; top: -3.47em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2371" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2372" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2373" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2374" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2375" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2376" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2377" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2378" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2379" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2380" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2381" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.548em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.313em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2382" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.59em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.59em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2383" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2384" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2385" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2386" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.094em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.276em, 1000.75em, 4.169em, -999.998em); left: 11.0156px; line-height: normal; margin: 0px 0px 0px -0.394em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2387" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2388" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2389" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2390" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2391" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.747em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.4em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2392" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.263em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2393" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1001em, 4.268em, -999.998em); left: 11.0156px; line-height: normal; margin: 0px 0px 0px -0.494em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2394" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2395" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2396" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span class="texatom" id="MathJax-Span-2397" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2398" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2399" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2400" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2401" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1001.09em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 1.094em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2402" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2403" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2404" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2405" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.499em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2406" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.3em, 4.169em, -999.998em); left: 5.01562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2407" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.5em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.499em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2408" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2409" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2410" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.895em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2411" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.449em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2412" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="texatom" id="MathJax-Span-2413" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2414" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2415" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.243em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.177em, 1000.75em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2416" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.846em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-2417" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.127em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span> <span class="MathJax" id="MathJax-Element-68-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-2418" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 12.949em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 10.27em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.887em, 1010.02em, 3.971em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.172em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2419" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2420" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-2421" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">[</span></span><span class="texatom" id="MathJax-Span-2422" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2423" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2424" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2425" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2426" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2427" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2428" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-2429" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">s</span></span></span></span></span><span class="texatom" id="MathJax-Span-2430" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2431" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2432" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.35em, 4.318em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.146em; padding: 0px; position: absolute; top: -4.561em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2433" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">q<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.5em, 4.169em, -999.998em); left: 6.51562px; line-height: normal; margin: 0px 0px 0px -0.246em; padding: 0px; position: absolute; top: -3.619em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2434" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.65em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.647em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2435" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-2436" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span class="mspace" id="MathJax-Span-2437" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="mspace" id="MathJax-Span-2438" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="mspace" id="MathJax-Span-2439" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="texatom" id="MathJax-Span-2440" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2441" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2442" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2443" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2444" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">a</span><span class="mi" id="MathJax-Span-2445" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mi" id="MathJax-Span-2446" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">d</span></span></span></span></span><span class="mspace" id="MathJax-Span-2447" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="mspace" id="MathJax-Span-2448" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="mspace" id="MathJax-Span-2449" style="border: 0px; box-sizing: border-box; display: inline-block; height: 0em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.151em;"></span><span class="mi" id="MathJax-Span-2450" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">E<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.052em;"></span></span><span class="mo" id="MathJax-Span-2451" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-2452" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2453" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-2454" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px 0.102em; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.747em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.4em, 4.169em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.196em; padding: 0px; position: absolute; top: -4.412em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2455" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.65em, 4.268em, -999.998em); left: 7.51562px; line-height: normal; margin: 0px 0px 0px -0.295em; padding: 0px; position: absolute; top: -3.668em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2456" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2457" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-2458" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2459" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-2460" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.647em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.574em, 1000.3em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2461" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ε</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.35em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.916em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-2462" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 10.08px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">o</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(0.846em, 1000.75em, 1.193em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.238em; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em; width: 0.747em;"></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1.044em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span></span><span class="mo" id="MathJax-Span-2463" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0em;"><span style="border: 0px; box-sizing: border-box; font-family: MathJax_Size2; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">]</span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 3.177em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 2.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.872em; width: 0px;"></span></span></nobr></span></span></p></div>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-58971522118454644222020-12-18T09:06:00.007+05:302020-12-18T09:15:36.486+05:30NUCLEAR PHYSICS CLASS 12<p style="text-align: left;"> <span style="font-size: x-large;"> </span><b><i><u><span><span style="font-size: x-large;">NUCLEAR PHYSICS</span></span></u></i></b></p><p style="text-align: left;"></p><p><span style="background-color: white; color: #333333; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: 700;"><br /></span></p><p><span style="background-color: white; color: #333333; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: 700;">Nucleus</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The entire positive charge and nearly the entire mass of atom is concentrated in a very small space called the nucleus of an atom.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The nucleus consists of protons and neutrons. They are called nucleons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Terms Related to Nucleus</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Atomic Number</span> The number of protons in the nucleus of an atom of the element is called atomic number (Z) of the element.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mass Number</span> The total number of protons and neutrons present inside the nucleus of an atom of the element is called mass number (A) of the element.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Size</span> The radius of the nucleus R ∝ A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">1/3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">⇒ R = R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">1/3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = 1.1 * 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-15</sup> m is an empirical constant.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iv) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Density</span> Nuclear density is independent of mass number and therefore same for all nuclei.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ρ = mass of nucleus / volume of nucleus ⇒ ρ = 3m / 4π R<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, m = average mass of a nucleon.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(v) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Atomic Mass Unit</span> It is defined as 1 / 12th the mass of carbon nucleus.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is abbreviated as arnu and often denoted by u. Thus</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 amu = 1.992678 * 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-26</sup> / 12 kg</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">= 1.6 * 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-27</sup> kg = 931 Me V</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isotopes</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms of an element having same atomic number but different mass numbers. are called isotopes.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">e.g., <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">1</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> are isotopes of hydrogen.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isobars</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms of different elements having same mass numbers but different atomic numbers, are called isobars.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">e.g., <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>He<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> and <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">10</sub>Na<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">22</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">10</sub>Ne<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">22</sup> are isobars.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isotones</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms of different elements having different atomic numbers and different mass numbers but having same number of neutrons, are called isotones.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">e.g., <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>He<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">4</sup> and <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">6</sub>C<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">14</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">8</sub>O<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">16</sup> are isobars.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isomers</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Atoms having the same mass number and the same atomic number but different radioactive properties are called isomers,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Force</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The force acting inside the nucleus or acting between nucleons is called nuclear force.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Nuclear forces are the strongest forces in nature.</p><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">It is a very short range attractive force.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">It is non-central. non-conservative force.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">It is neither gravitational nor electrostatic force.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">It is independent of charge.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">It is 100 times that of electrostatic force and 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">38</sup> times that of gravitational force.</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">According to the Yukawa, the nuclear force acts between the nucleon due to continuous exchange of meson particles.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mass Defect</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The difference between the sum of masses of all nucleons (M) mass of the nucleus (m) is called mass defect.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Mass Defect (Δm) = M – m = [Zm<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">p</sub> + (A – Z)m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub> – m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub>]</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Binding Energy</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The minimum energy required to separate the nucleons up to an infinite distance from the nucleus, is called nuclear binding energy.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Nuclear binding energy per nucleon = Nuclear binding energy / Total number of nucleons</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Binding energy, E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">b</sub> = [Zm<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">p</sub> + (A – Z) m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub> – m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">N</sub>]c<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Packing Fraction</span> (P)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">p = (Exact nuclear mass) – (Mass number) / Mass number</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">= M – A / M</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The larger the value of packing friction. greater is the stability of the nucleus.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">[The nuclei containing even number of protons and even number of neutrons are <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">most stable.</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The nuclei containing odd number of protons and odd number of neutrons are <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">most instable.]</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Radioactivity</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The phenomena of disintegration of heavy elements into comparatively lighter elements by the emission of radiations is called radioactivity. This phenomena was discovered by Henry Becquerel in 1896.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Radiations Emitted by a Radioactive Element</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Three types of radiations emitted by radioactive elements</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) α-rays</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) β-rays</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) γ – rays</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">α-rays consists of α-particles, which are doubly ionised helium ion.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">β-rays are consist of fast moving electrons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">γ – rays are electromagnetic rays.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">[When an α – particle is emitted by a nucleus its atomic number decreases by 2 and mass number decreases by 4.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Nuclear Physics" class=" lazyloading" data-src="https://farm6.staticflickr.com/5565/14823231201_c7c255a316_o.jpg" src="https://farm6.staticflickr.com/5565/14823231201_c7c255a316_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; padding: 0px; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a β -particle is emitted by a nucleus its atomic number is Increases by one and mass number remains unchanged.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Nuclear Physics" class=" lazyloading" data-src="https://farm6.staticflickr.com/5565/14803311716_de6b4e71f2_o.jpg" src="https://farm6.staticflickr.com/5565/14803311716_de6b4e71f2_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; padding: 0px; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a γ – particle is emitted by a nucleus its atomic number and mass number remain unchanged</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Radioactive Decay law</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The rate of disintegration of radioactive atoms at any instant is directly proportional to the number of radioactive atoms present in the sample at that instant.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Rate of disintegration ( – dN / dt) ∝ N</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">– dN / dt = λ N</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where λ is the decay constant.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The number of atoms present undecayed in the sample at any instant N = N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> e<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-λt</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> is number of atoms at time t = 0 and N is number of atoms at time t.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Half-life of a Radioactive Element</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The time is which the half number of atoms present initially in any sample decays, is called half-life (T) of that radioactive element.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between half-life and disintegration constant is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">T = log<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">e</sub> / λ = 0.6931 / λ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Average Life or Mean Life(τ)</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Average life or mean life (τ) of a radioactive element is the ratio of total life time of all the atoms and total number of atoms present initially in the sample.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between average life and decay constant τ = 1 / λ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between half-life and average life τ = 1.44 T</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The number of atoms left undecayed after n half-lifes is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">N = N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> (1 / 2)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">n</sup> = N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> (1 / 2) <sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">t/T</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, n = t / T, here t = total time.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Activity of a Radioactive Element</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The activity of a radioactive element is equal to its rate of disintegration.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Activity R = ( – dN / dt)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Activity of the sample after time t,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">R = R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> e <sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-λt</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is Becquerel (Bq).</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its other units are Curie and Rutherford.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 Curie = 3.7 * 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">10</sup> decay/s</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 Rutherford = 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">6</sup> decay/s</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Fission</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The process of the splitting of a heavy nucleus into two or more lighter nuclei is called nuclear fission.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a slow moving neutron strikes with a uranium nucleus (<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">92</sub>U<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">235</sup>), it splits into <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">56</sub>Ba<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">141</sup> and <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">36</sub>Kr<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">92</sup> along with three neutrons and a lot of energy.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Nuclear Physics" class=" lazyloading" data-src="https://farm4.staticflickr.com/3854/14846160273_60341ca26a_o.jpg" src="https://farm4.staticflickr.com/3854/14846160273_60341ca26a_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; padding: 0px; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Chain Reaction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If the particle starting the nuclear fission reaction is produced as a product and further take part in the nuclear fission reaction, then a chain of fission reaction started, which is called nuclear chain reaction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Nuclear chain reaction are of two types</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Controlled chain reaction</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Uncontrolled chain reaction</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Reactor</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The main parts of a nuclear reactor are following</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Nuclear Physics" class=" lazyloading" data-src="https://farm3.staticflickr.com/2933/14826162195_edf192fd7d_o.jpg" src="https://farm3.staticflickr.com/2933/14826162195_edf192fd7d_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; padding: 0px; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Fuel</span> Fissionable materials like <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">92</sub>U<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">235</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">92</sub>U<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">238</sup>, <sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">94</sub>U<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">239</sup> are used as fuel.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Moderator</span> Heavy water, graphite and beryllium oxide are used to slower down fast moving neutrons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coolant</span> The cold water, liquid oxygen, etc. are used to remove heat generated in the fission process.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iv) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Control rods</span> Cadmium or boron rods are good absorber of neutrons and therefore used to control the fission reaction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Atom bomb working is based on uncontrolled chain reaction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Nuclear Fusion</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The process of combining of two lighter nuclei to form one heavy nucleus, is called nuclear fusion.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Three deuteron nuclei (<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>) fuse, 21.6 MeV is energy released and nucleus of helium (<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>He<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">4</sup>) is formed.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Nuclear Physics" class=" lazyloading" data-src="https://farm4.staticflickr.com/3847/14639617760_b03b2da1d7_o.jpg" src="https://farm4.staticflickr.com/3847/14639617760_b03b2da1d7_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; padding: 0px; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">In this process, a large amount of energy is released.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Nuclear fusion takes place at very high temperature approximately about 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">7</sup> K and at very high pressure 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">6</sup> atmosphere.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Hydrogen bomb is based on nuclear fusion.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The source of Sun’s energy is the nuclear fusion taking place at sun.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Thermonuclear Energy</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The energy released during nuclear fusion is know as thermonuclear energy. Protons are needed for fusion while neutrons are needed for fission process.</p>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-38793889483730196712020-12-18T09:04:00.003+05:302020-12-18T09:15:36.484+05:30SEMICONDUCTORS CLASS 12<blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><p style="text-align: left;"><b><u><i><span style="font-size: x-large;">SEMI - CONDUCTORS</span></i></u></b></p></blockquote></blockquote></blockquote></blockquote><p><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><br /></span></p><p><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;">On the basis of</span><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;"> </span><span style="background-color: white; box-sizing: border-box; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt; font-weight: 700;"><em style="box-sizing: border-box;">electrical conductivity</em></span><span style="background-color: white; color: #555555; font-family: verdana, geneva, sans-serif; font-size: 12pt;">, the materials can be divided into three categories :</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(1) <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Conductors</em></span> (e.g., Cu, Al, Fe, etc.),<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(2) <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Insulators</em></span> (e.g., wood, diamond, mica, etc.), and<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(3) <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">Semiconductors</em></span> (e.g., Ge, Si, GaAs, … etc.)</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">At room temperature (<span class="MathJax" id="MathJax-Element-1-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.59em, 1000.75em, 2.334em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-3" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">≈</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 0.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0.003em; width: 0px;"></span></span></nobr></span>300 K), the conductivity of conductors is in the range 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">6</span> - 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">8</span> S<em style="box-sizing: border-box;">/</em>m, and that of insulators, 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">–8</span> to 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">4</span> S<em style="box-sizing: border-box;">/</em>m. The conductivity of semiconductors lies in between that of conductors and insulators.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Electronic devices (such as diodes, transistors, ICs) are made from silicon (Si) and germanium (Ge). In the early days (around 1950), mostly germanium was used, because it was comparatively easier to purify germanium. However, it was found that the devices made from silicon are more stable and their functioning is less dependent on temperature variations. Now, we have much improved metallurgical processes to purify silicon. So, now-a-days mostly the devices are of Si. An integrated circuit (IC) is made from a <em style="box-sizing: border-box;">silicon chip</em> or <em style="box-sizing: border-box;">wafer</em>.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">Energy Bands in Solids</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The electrons in an isolated atom have discrete energy levels. However, in a crystal an atom is surrounded by a large number of other atoms. Due to interatomic interactions, the energy levels are modified. This modification is more prominent for the electrons in the outermost shell. (The electrons in the inner shells are shielded by the electrons in outer shells and are not much affected by the electric fields of the neighbouring atoms.) Due to this modification, each energy level splits into a very large number of levels (~ 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">23</span>) lying close to one another. We can regard a bunch of these energy levels as a continuous energy distribution, and call it <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">energy band</em>.<br style="box-sizing: border-box;" /></span></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The energy bands, which are completely filled at 0 K are called valence bands (VB). The bands with higher energies are called conduction bands (CB). We are generally concerned with the highest valance band and the lowest conduction band.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that a <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">conduction band</em></span> is either <em style="box-sizing: border-box;">completely empty</em> or <em style="box-sizing: border-box;">partially filled</em>.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The difference between the highest energy in a valence band and the lower energy in the next higher conduction band is called <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">forbidden energy gap </em>(<em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span>).<br style="box-sizing: border-box;" /></span></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">As an example, consider a specimen of sodium (Na) containing N atoms. Its atomic number is 11 (1s<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">2</span>, 2s<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">2</span>, 2p<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">6</span>, 3s<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">1</span>). The details of its energy bands are shown in the table.</span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="alignnone size-full wp-image-129475 aligncenter jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 632px) 100vw, 632px" src="https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea81d4e0dc.png?w=1170&ssl=1" srcset="https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea81d4e0dc.png?w=632&ssl=1 632w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea81d4e0dc.png?resize=500%2C180&ssl=1 500w, https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea81d4e0dc.png?resize=300%2C108&ssl=1 300w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 5px auto; max-width: 100%; vertical-align: middle;" /></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">As temperature is raised, the electrons may collide with each other and with ions to exchange energy. The order of energy exchanged is <em style="box-sizing: border-box;">kT</em>. At room temperature (300 K), <em style="box-sizing: border-box;">kT</em> is 0.026 eV. Ordinarily, the energy gaps are much larger than <em style="box-sizing: border-box;">kT</em>. An electron in a completely filled band (i.e., a valence band) does not find an empty state with a slightly higher or lower energy. Hence, it cannot accept or donate any energy of the order of <em style="box-sizing: border-box;">kT</em>. However, the outermost electrons, which are in the highest occupied energy band, may take up this energy <span class="MathJax" id="MathJax-Element-2-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-4" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.995em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.796em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.59em, 1000.75em, 2.334em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-5" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-6" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">≈</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 0.691em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0.003em; width: 0px;"></span></span></nobr></span><em style="box-sizing: border-box;">kT</em> provided some empty states are available in the same band.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Similar thing happens when a piece of sodium is connected to a battery. The electric field can supply only a small amount of energy to the electrons. Only the electrons in the highest occupied band can accept this energy and then move according to the field. This gives rise to an electric current. The electrons in the inner (valence) bands cannot accept this small amount of energy and hence cannot take part in electric conduction.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;"><span style="box-sizing: border-box; font-weight: 700;">Why Materials Have Different Conductivities<br style="box-sizing: border-box;" /></span></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">There can be four broad types of energy band structures, as shown.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<em style="box-sizing: border-box;">A</em>) The highest occupied energy band is only partially filled at 0 K. (Such is the case with sodium, copper, etc.) When electric field is applied, the electrons in the partially filled band can accept energy from the field and can drift accordingly. Hence, such materials are <em style="box-sizing: border-box;">good conductors of electricity</em>.</span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-129476 aligncenter jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 546px) 100vw, 546px" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?w=1170&ssl=1" srcset="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?w=546&ssl=1 546w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?resize=500%2C395&ssl=1 500w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?resize=300%2C237&ssl=1 300w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?resize=480%2C380&ssl=1 480w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea8b4d6160.png?resize=63%2C50&ssl=1 63w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 5px auto; max-width: 100%; vertical-align: middle;" /></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<em style="box-sizing: border-box;">B</em>) The highest occupied energy band (VB) is completely filled at 0 K and next higher band is completely empty (CB). But the two are overlapping. (Zinc has such energy band structure.) Therefore, there are empty energy stats close to the occupied states. Hence, such solids are also <em style="box-sizing: border-box;">good conductors</em>.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<em style="box-sizing: border-box;">C</em>) The highest occupied energy band (VB) is completely filled and the next higher band is completely empty (CB). There is a large gap (<em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> > 5 eV) between these two bands. (Diamond is of this type.) The electrons in VB refuse to accept any energy from electric field, because there is no empty state nearby. Only when the energy supplied (either by applied field or by raising the temperature) is more that <em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> (~ 5 eV), an electron from VB can jump to CB, after which it can take part in electrical conduction. Therefore, at ordinary temperatures, these materials behave as <em style="box-sizing: border-box;">insulators</em>.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(D) The VB and CB are separated by small gap at 0 K (<em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> ~ 1 eV). [For Ge, <em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> = 0.72 eV ; and for Si, <em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> = 1.12 eV.] At 0 K, an ordinary battery cannot supply even this much energy. Hence electrical conduction cannot take place. That is, <em style="box-sizing: border-box;">at </em>0 K<em style="box-sizing: border-box;">, these materials behave as perfect insulators</em>. However, at room temperature, thermal energy pushes some of these electrons in VB to CB. Thus, small conduction becomes possible. Such solids are therefore called <em style="box-sizing: border-box;">semiconductors</em>.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 14pt;">INTRINSIC SEMICONDUCTORS</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">To make a diode or a transistor, the first step is to obtain a sample of semiconductor in its purest form. This is called <em style="box-sizing: border-box;">intrinsic semiconductor</em>. The impurity content is less than one part impurity in 100 million parts of semiconductor.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Crystal Structure of Semiconductors</span></h3><p id="TDEpSSL" style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-129481 alignright jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" src="https://i0.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea964800d0.png?w=1170&ssl=1" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px 0px 20px 20px; max-width: 100%; vertical-align: middle;" /></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Each atom of an intrinsic semiconductor (Ge or Si) has four valence electrons. These four electrons of each atom form <em style="box-sizing: border-box;">covalent bonds</em> with the four neighbouring atoms. Thus, the semiconductor has <em style="box-sizing: border-box;">tetrahedral </em>lattice structure. A simplified two-dimensional representation of this crystalline structure is shown in figure. The core represents the nucleus and all the orbiting electrons except the four valence electrons. Therefore the core has +4 charge. A covalent bond consists of two electrons, one from each adjacent atom. At 0 K, all the valence electrons are tightly bound to the parent atoms. No free electrons are available for electrical conduction. Hence, <em style="box-sizing: border-box;">the semiconductor behaves as a perfect insulator at </em>0 K.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Charge Carriers in Intrinsic Semiconductors</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">At room temperature, thermal energy is sufficient to make a valence electron jump to the conduction band. It starts orbiting the nucleus at a larger radius, and frequently it jumps from one nucleus to the other. It has become <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">free electron.</em></span></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="alignnone size-full wp-image-129482 aligncenter jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 344px) 100vw, 344px" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea9a61546c.png?w=1170&ssl=1" srcset="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea9a61546c.png?w=344&ssl=1 344w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582ea9a61546c.png?resize=300%2C229&ssl=1 300w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 5px auto; max-width: 100%; vertical-align: middle;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When an electron breaks a covalent bond and moves away, a vacancy is created in the bond. A <em style="box-sizing: border-box;">positive charge</em> is associated with this vacancy. This vacancy is called a <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">hole</em></span>. Free electrons and holes are always generated in pairs. At any time, the concentration of free electrons is same as the concentration of holes in an intrinsic semiconductor,</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-3-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-7" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.02em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.177em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.639em, 1003.18em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-8" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-9" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-10" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-11" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.945em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.6em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-12" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-13" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-14" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="texatom" id="MathJax-Span-15" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-16" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-17" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.846em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.5em, 4.368em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-18" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">p</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; left: 0.499em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.867em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-19" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 0.941em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Just as free electrons move <em style="box-sizing: border-box;">randomly</em> in the crystal, so do the holes. An electron from the neighbouring bond can make a jump to fill the vacancy, thereby shifting the vacancy (or the holes) to new location. Much energy is not needed to induce such a transfer, as all the electrons in the valence band have roughly the same energy. A free electron carries negative charge (–1.6 x 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">–19 </span>C) with it. A hole carries a positive charge (+1.6 x 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">–19</span> C) with it.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">A metal (such as Cu, Al, etc.) has only one type of charge carriers, namely, free electrons. But <em style="box-sizing: border-box;">a semiconductor has two types of charge carriers </em>¾ <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">free electrons</em></span> and <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">holes</em></span>.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When an electric field is applied, the free electrons (in the conduction band) drift opposite to the field and the holes drift along the field. Thus, both types of carriers contribute to electric conduction.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">In a semiconductor, not only the thermal generation of electron-hole pairs takes place, but also there is simultaneous <em style="box-sizing: border-box;">pair recombination</em>. When a free electron encounters a hole, during their random motion, the electron occupies the vacancy, re-establishing the covalent bond. The individual identity of both is lost. In this recombination process, same amount of energy is given out as was taken to generate electron-hole pair. <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">At equilibrium the rate of pair recombination is equal to the rate of pair generation</em></span>. If temperature increases, more bonds are broken, that is, the rate of generation increases. This increases the concentration of free electrons and holes, which in turn increases the rate of recombination. Equilibrium is again established.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that <em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> is more in silicon (<em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> = 1.12 eV) than in germanium (<em style="box-sizing: border-box;">E</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">g</span> = 0.72 eV). Therefore, at a given temperature, less number of electron-hole pairs will be generated in silicon than in germanium. Hence, <em style="box-sizing: border-box;">the conductivity of silicon is less than that of germanium</em><span style="box-sizing: border-box; font-weight: 700;">.</span></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;"> </span></span></p><p id="oWKymBB" style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="size-full wp-image-129488 alignright jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582eaa9285643.png?w=1170&ssl=1" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px 0px 20px 20px; max-width: 100%; vertical-align: middle;" /></span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Conduction in Intrinsic Semiconductor</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When a battery is connected across a semiconductor, the free electrons drift towards +ve terminal and holes drift towards –ve terminal. The total current <em style="box-sizing: border-box;">I</em> is summation of the current due to electron flow <em style="box-sizing: border-box;">I</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">n</span> and the current due to hole flow. The current in the connecting wire is due to electron flow.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Effect of Temperature on Conductivity of a Semiconductor</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When temperature is raised, more electron-hole pairs are generated. The higher the temperature, the higher is the concentration of charge carriers. Because of this, the conductivity increases with temperature. In other words, the resistivity (<span class="MathJax" id="MathJax-Element-4-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-20" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.417em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.475em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.342em, 1003.48em, 2.631em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-21" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-22" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">ρ</span><span class="mo" id="MathJax-Span-23" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mn" id="MathJax-Span-24" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="texatom" id="MathJax-Span-25" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-26" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mo" id="MathJax-Span-27" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">/</span></span></span><span class="mi" id="MathJax-Span-28" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">σ<span style="border: 0px; box-sizing: border-box; display: inline-block; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.002em;"></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.378em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.372em; width: 0px;"></span></span></nobr></span>) decreases with rise in temperature. That is, <em style="box-sizing: border-box;">the semiconductors have negative temperature coefficient of resistance</em>.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-size: 14pt;"><span style="box-sizing: border-box; font-size: 16pt;">EXTRINSIC SEMICONDUCTORS</span></span> </span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Intrinsic (pure) semiconductors are of little use. For making a semiconductor device, we deliberately add a tiny and controlled amount of desired impurity to the highly purified semiconductor. This process is called <span style="box-sizing: border-box; font-weight: 700;"><em style="box-sizing: border-box;">doping</em></span>. A doped semiconductor is called <em style="box-sizing: border-box;">extrinsic semiconductor</em>. The proportion of impurity added is about 1 in 10<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; top: -0.5em; vertical-align: baseline;">6</span>.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><em style="box-sizing: border-box;">N</em>-type, or Donor Type</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Pentavalent impurity (from group V, such as <em style="box-sizing: border-box;">phosphorous, arsenic</em>, or <em style="box-sizing: border-box;">antimony</em>) is added so that an impurity atom substitutes for a silicon atom in the crystalline structure. Four of its valence electrons make four covalent bonds with neighbouring atoms. The fifth electron remains unpaired, and is quite loosely bound to the nucleus. It needs very little energy (0.01 eV in Ge, 0.05 eV in Si) to free itself from the attractive force of the nucleus. At room temperature, the thermal energy is enough to do this job for all the impurity atoms added. Since each impurity atom <em style="box-sizing: border-box;">donates</em> one electron to the conduction band, this type of impurity is called <span style="box-sizing: border-box; font-weight: 700;">donor type</span>.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">After donating an electron, the impurity atom becomes +ve ion. However, this +ve charge is <span style="box-sizing: border-box; font-weight: 700;">immobile</span> as the ion is held in its place by covalent bonds. In addition to the free electrons donated by impurity atoms, there are some more due to breaking of covalent bonds. Thus, a few holes are also produced. If <em style="box-sizing: border-box;">N</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">D</span> is the concentration of donor (impurity) atoms, <em style="box-sizing: border-box;">n</em> that of free electrons and <em style="box-sizing: border-box;">p</em> that of hole, we have</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><em style="box-sizing: border-box;">N</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">D</span> + <em style="box-sizing: border-box;">p</em> = <em style="box-sizing: border-box;">n</em></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<span style="box-sizing: border-box; font-weight: 700;">+ve ions</span>) (<span style="box-sizing: border-box; font-weight: 700;">holes</span>) (<span style="box-sizing: border-box; font-weight: 700;">electrons</span>)</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">As a whole, the <em style="box-sizing: border-box;">N</em>-type semiconductor is neutral. It has electrons in majority and holes in minority (<em style="box-sizing: border-box;"><span class="MathJax" id="MathJax-Element-5-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; font-style: normal; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-29" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.12em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.276em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.54em, 1003.28em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-30" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-31" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-32" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">><span style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">></span></span><span class="mi" id="MathJax-Span-33" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">p</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span></em>).</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><em style="box-sizing: border-box;">P</em>- type, or Acceptor Type</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Here, trivalent impurity (from group III, such as <em style="box-sizing: border-box;">boron</em>, <em style="box-sizing: border-box;">aluminium</em>, <em style="box-sizing: border-box;">gallium</em> and <em style="box-sizing: border-box;">indium</em>) is added. The three valence electrons make only three covalent bonds with neighbours. The fourth bond remains incomplete. There exists a vacancy of an electron in this bond.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span style="box-sizing: border-box; font-weight: 700;">Note</span> that this vacancy is not a hole, as no charge is associated with it. However, the single electron in the incomplete bond has a great tendency to snatch an electron from neighbouring bonds. Only a little (about 0.01 eV) additional energy is needed by the electron in adjacent bond to jump and occupy the vacancy around the impurity atom. When this happens, a hole is now formed in the adjacent bond. This hole goes on moving around randomly in the crystal carrying +ve charge with it.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">When impurity atom accepts an electron to complete fourth bond, it becomes –ve ion (<span style="box-sizing: border-box; font-weight: 700;">immobile</span>). At room temperature, all impurity atoms (concentration <em style="box-sizing: border-box;">N</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">A</span>) convert into –ve ions. In addition to the holes created due acceptor impurity atoms, there will be some covalent bonds broken due to thermal energy.</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><em style="box-sizing: border-box;">N</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">A</span> + <em style="box-sizing: border-box;">n</em> = <em style="box-sizing: border-box;">p</em></span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">(<span style="box-sizing: border-box; font-weight: 700;">–ve ions</span>) (<span style="box-sizing: border-box; font-weight: 700;">electrons</span>) (<span style="box-sizing: border-box; font-weight: 700;">holes</span>)</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The holes are in majority (<span class="MathJax" id="MathJax-Element-6-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-34" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.12em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.276em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.54em, 1003.28em, 2.582em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-35" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-36" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">p</span><span class="mo" id="MathJax-Span-37" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">><span style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">></span></span><span class="mi" id="MathJax-Span-38" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.309em; width: 0px;"></span></span></nobr></span>).</span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Doping Level</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">If the doping level (i.e., the impurity concentration) is increased in an extrinsic semiconductor, the concentration of majority carriers increases. As a result, the chances of their recombining with minority carriers increase; and hence the concentration of minority carriers decreases. In fact, in an extrinsic semiconductor, we have</span></p><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><span class="MathJax" id="MathJax-Element-7-Frame" style="border: 0px; box-sizing: border-box; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr style="border: 0px; box-sizing: border-box; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-39" style="border: 0px; box-sizing: border-box; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 4.467em;"><span style="border: 0px; box-sizing: border-box; display: inline-block; font-size: 20.16px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 3.524em;"><span style="border: 0px; box-sizing: border-box; clip: rect(1.292em, 1003.52em, 2.681em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.23em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-40" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-41" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mi" id="MathJax-Span-42" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">p</span><span class="mo" id="MathJax-Span-43" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="msubsup" id="MathJax-Span-44" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.3em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.044em;"><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.6em, 4.169em, -999.998em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.015em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-45" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Math-italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.425em, 1000.45em, 4.169em, -999.998em); left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.363em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-46" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">2</span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: border-box; clip: rect(3.375em, 1000.25em, 4.169em, -999.998em); left: 0.598em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.718em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-47" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-48" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-49" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-50" style="border: 0px; box-sizing: border-box; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-51" style="border: 0px; box-sizing: border-box; display: inline; font-family: MathJax_Main; font-size: 14.2531px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">i</span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 4.02em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; box-sizing: border-box; display: inline-block; height: 2.235em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: border-box; display: inline-block; height: 1.566em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.434em; width: 0px;"></span></span></nobr></span><br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">where <em style="box-sizing: border-box;">n</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline;">i</span> is the intrinsic concentration of free electrons (or of holes).</span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;"><img alt="" class="alignnone size-full wp-image-129493 aligncenter jetpack-lazy-image jetpack-lazy-image--handled" data-lazy-loaded="1" data-recalc-dims="1" sizes="(max-width: 500px) 100vw, 500px" src="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582eabfac807c.png?w=1170&ssl=1" srcset="https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582eabfac807c.png?w=500&ssl=1 500w, https://i1.wp.com/www.dronstudy.com/wp-content/uploads//2016/11/img_582eabfac807c.png?resize=300%2C65&ssl=1 300w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 5px auto; max-width: 100%; vertical-align: middle;" /></span></p><h3 style="background-color: white; box-sizing: border-box; color: #333333; font-family: Montserrat; font-size: 20px; line-height: 1.1; margin-bottom: 10px; margin-top: 20px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Effect of Temperature on Extrinsic Semiconductor</span></h3><p style="background-color: white; box-sizing: border-box; color: #555555; font-family: "Open Sans"; font-size: 14px; margin: 0px 0px 30px;"><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">The number of charge carriers in an extrinsic semiconductor is much larger than that in intrinsic semiconductor. Hence, its conductivity is also many times that of an intrinsic semiconductor.<br style="box-sizing: border-box;" /></span><span style="box-sizing: border-box; font-family: verdana, geneva, sans-serif; font-size: 12pt;">Consider an <em style="box-sizing: border-box;">N</em>-type semiconductor. All the donors have already donated the electrons to the crystal at room temperature. If the temperature is raised further, more covalent bonds are broken. As a result, the concentration of minority carriers increases. Eventually a temperature is reached when the concentration of minority carriers becomes almost same as that of majority carriers. It will then behave like an intrinsic semiconductors (with higher conductivity). Any device made of <em style="box-sizing: border-box;">P-</em> and <em style="box-sizing: border-box;">N</em>-type semiconductor will fail at such a temperature. This <em style="box-sizing: border-box;">critical temperature</em> is about 85 °C for Ge and 200 °C for Si.</span></p>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-5405459406717144092020-10-03T11:25:00.000+05:302020-12-18T09:15:36.483+05:30ELECTROMAGNETIC INDUCTION FULL CHAPTER NOTES<p><b><i><u> <span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;">Whenever the magnetic flux linked with an electric circuit changes, an emf is induced in the circuit. This phenomenon is called</span><span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;"> </span><span style="border: 0px; box-sizing: border-box; color: #333333; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">electromagnetic induction</span><span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;">.</span></u></i></b></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Faraday’s Laws of Electromagnetic Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Whenever the magnetic flux linked with a circuit changes, an induced emf is produced in it.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) The induced emf lasts so long as the change in magnetic flux continues.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) The magnitude of induced emf is directly proportional to the rate of change in magnetic flux, i.e.,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E ∝ dφ / dt ⇒ E = – dφ / dt</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where constant of proportionality is one and negative sign indicates Lenz’s law.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Here, flux = NBA cos θ, SI unit of φ = weber,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">CGS unit of φ = maxwell, 1 weber = 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">8</sup> maxwell,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Dimensional formula of magnetic flux</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">[φ] = [ML<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>]</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Lenz’s law</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The direction of induced emf or induced current is always in such a way that it opposes the cause due to which it is produced.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Lenz’s law is in accordance with the conservation of energy.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Note To apply Lenz’s law, you can remember RIN or ® In (when the loop lies on the plane of paper)</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3851/14616673354_4143c1771c_o.jpg" src="https://farm4.staticflickr.com/3851/14616673354_4143c1771c_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) RIN In RIN, R stands for right, I stands for increasing and N for north pole (anticlockwise). It means, if a loop is placed on the right side of a straight current carrying conductor and the current in the conductor is increasing, then induced current in the loop is anticlockwise<br style="box-sizing: border-box;" /></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3851/14616673354_4143c1771c_o.jpg" src="https://farm4.staticflickr.com/3851/14616673354_4143c1771c_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) ®<span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">IN</span> In ® IN suppose the magnetic field in the loop is perpendicular to paper inwards ® and this field is increasing, then induced current in the loop is anticlockwise</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5513/14618230752_f05c3c05de_o.jpg" src="https://farm6.staticflickr.com/5513/14618230752_f05c3c05de_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Motional Emf</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If a rod of length 1 moves perpendicular to a magnetic field B, with a velocity v, then induced emf produced in it given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = B * v * I = bvl</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If a metallic rod of length 1 rotates about one of its ends in a plane perpendicular to the magnetic field, then the induced emf produced across its ends is given by</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5037/14432141038_218e40e97e_o.jpg" src="https://farm6.staticflickr.com/5037/14432141038_218e40e97e_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = 1 / 2 bωr<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> = BAf</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, ω = angular velocity of rotation, f = frequency of rotation and A = πr<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> = area of disc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The direction of induced current in any conductor can be obtained from Fleming’s right hand rule.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A rectangular coil moves linearly in a field when coil moves with constant velocity in a uniform magnetic field, flux and induced emf will be zero.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A rod moves at an angle θ with the direction of magnetic field, velocity E = – Blv sin θ.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">An emf is induced</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) When a magnet is moved with respect to a coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) When a conductor falls freely in East-West direction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) When an aeroplane flies horizontally.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iv) When strength of current flowing in a coil is increased or decreased, induced current is developed in the coil in same or opposite direction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(v) When a train moves horizontally in any direction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Fleming’s Right Hand Rule</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If we stretch the thumb, the forefinger and the central finger of right hand in such a way that all three are perpendicular to each other, th. if thumb represent the direction of motion, the forefinger represent tile direction of magnetic field, then centra} finger will represent the<br style="box-sizing: border-box;" />direction of induced current.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If R is the electrical resistance of the circuit, then induced current in the circuit is given by I = E / R</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If induced current is produced in a coil rotated in uniform magnetic field, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">I = NBA ω sin ωt / R = I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> sin ωt</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = NBA ω = peak value of induced current,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">N = number of turns in the coil ,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B= magnetic induction,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ω = angular velocity of rotation and</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A = area of cross-section of the coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Eddy Currents</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If a piece of metal is placed in a varying magnetic field or rotated high speed in a uniform magnetic field, then induced current set up<br style="box-sizing: border-box;" />the piece are like whire pool of air, called eddy currents.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The magnitude 0f eddy currents is given by i = – e / R = dφ / dt / R, where R is the resistance.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Eddy currents are also known as Facault’s current.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Self-Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The phenomena of production of induced emf in a circuit due to change in current flowing in its own, is called self induction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coefficient of Self-Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The magnetic flux linked with a coil</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">φ = LI</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, L = = coefficient of self induction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The induced emf in the coil</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = – L dl / dt</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">it unit of self induction is henry (H) and its dimensional formula is [ML<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>T <sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Self – inductance of a long solenoid is given by normal text</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> N<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> A / l = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> n<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> Al</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where. N = total number of turns in the solenoid,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 = length of the coil, n = number of turns in the coil and</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A = area of cross-section of the coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If core of the solenoid is of any other magnetic material, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">r</sub> N<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> A / l</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Self – inductance of a toroid L = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> N<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> A / 2πr</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Where, r = radius of the toroid</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Energy stored in an inductor E = 1 / 2 LI<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mutual Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The phenomena of production of induced emf in a circuit due to the change in magnetic flux in its neighbouring circuit, is called mutual induction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coefficient of Mutual Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If two coils are coupled with each, other then magnetic flux linked with a Coil (secondary coil)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">φ = MI</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where M is coefficient of mutual induction and I is current flow in through primary coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The induced emf in the secondary coil</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = – M dl / dt</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where dl / dt is the rate of change of current through primary coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The unit of coefficient of mutual induction is henry (H) and its dimension is [ML<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The coefficient of mutual induction depends on geometry of two coils, distance between them and orientation of the two coils.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Coefficient of Coupling</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Two coils are said to be coupled if full a part of the fuse produced by one links with the other.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">K = √M / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> L2, where L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> and L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> are coefficients of self-induction of the two coils and M is coefficient of mutual induction of the two coils.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Coefficient of coupling is maximum (K = 1) in case (a), when coils are coaxial and minimum in case (b), when coils are placed a right angles.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5561/14432140798_1d879dae84_o.jpg" src="https://farm6.staticflickr.com/5561/14432140798_1d879dae84_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Mutual inductance of two long coaxial solenoids is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M = μ N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> A / l</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">= μ n<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> n<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> Al</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> and N<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> are total number of turns in both coils, n<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> n<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> are number of turns per unit length in coils, A is area of cross-section of coils and 1 is length of the coils.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Grouping of Coils</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(a) When three coils of inductances L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>, L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> and L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub> are connected in series and the coefficient of coupling K = 0, as in series, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3923/14618230432_0452c1d8bc_o.jpg" src="https://farm4.staticflickr.com/3923/14618230432_0452c1d8bc_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(b) When three coils of inductances L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>, L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> and L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub> are connected in parallel and the coefficient of coupling K= 0 as in parallel, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = 1 / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + 1 / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + 1 / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Electromagnetic Induction" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2910/14595744166_fca9f9f7cd_o.jpg" src="https://farm3.staticflickr.com/2910/14595744166_fca9f9f7cd_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If coefficient of coupling K = 1, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">In series</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(a) If current in two coils are in the same direction, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + 2M</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(b) If current in two coils are in opposite directions, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> – 2M</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">In parallel</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(a) If current in two coils are in same direction, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> – M<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + 2M</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(b) If current in two coils are in opposite directions, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">L = L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> – M<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> / L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + L<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> – 2M</p>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-67677010736407752332020-10-03T11:23:00.003+05:302020-12-18T09:15:36.487+05:30MAGNETISM FULL CHAPTER NOTES<p><b><i><u> <span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;">The property of any object by virtue of which it can attract a piece of iron or steel is called</span><span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;"> </span><span style="border: 0px; box-sizing: border-box; color: #333333; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">magnetism</span><span style="background-color: white; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px;">.</span></u></i></b></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Natural Magnet</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A natural magnet is an ore of iron (Fe<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub>O<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">4</sub>), which attracts small pieces of iron, cobalt and nickel towards it.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetite or lode stone is a natural magnet.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Artificial Magnet</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A magnet which is prepared artificially is called an artificial magnet, e.g., a bar magnet, an electromagnet, a magnetic needle, a horse-shoe magnet etc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">According to molecular theory, every molecular of magnetic substance (whether magnetised or not) is a complete magnet itself.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The poles of a magnet are the two points near but within the ends of the magnet, at which the entire magnetism can be assumed to be concentrated.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The poles always occur in pairs and they are of equal strength. Like poles repel and unlike poles attract.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Properties of Magnet</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) A freely suspended magnet always aligns itself into north-south direction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Like magnetic poles repel and unlike magnetic poles attract each other.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) Magnetic poles exist in pair.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coulomb’s Law</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The force of interaction acting between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">F = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π . m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>, m<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = pole strengths, r = distance between poles and μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = permeability of free space.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Dipole</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic dipole is an arrangement of two unlike magnetic poles of equal pole strength separated by a very small distance, e.g., a small bar magnet, a magnetic needle, a current carrying loop etc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Dipole Moment</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The product of the distance (2 l) between the two poles and the pole strength of either pole is called magnetic dipole moment.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic dipole moment</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M = m (2 l)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is ‘joule/tesla’ or ‘ampere-metre<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>‘.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its direction is from south pole towards north pole.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Field Due to a Magnetic Dipole</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(1) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">On Axial Line</span></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3865/14610997962_d44bc4c091_o.jpg" src="https://farm4.staticflickr.com/3865/14610997962_d44bc4c091_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If r > > l, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4 π 2M / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">On Equatorial Line</span></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5033/14611529895_48b96b3692_o.jpg" src="https://farm6.staticflickr.com/5033/14611529895_48b96b3692_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4 π M / (r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> + l<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3 / 2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If r > > l, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4 π 2M / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Torque Acting on a Magnetic Dipole</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a Magnetic Dipole (M) is placed in a uniform magnetic field (B), then a Torque acts on it, Which is given by</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3864/14588524436_b9dccc90d6_o.jpg" src="https://farm4.staticflickr.com/3864/14588524436_b9dccc90d6_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">τ = M * B</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">or τ = MB sin θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Where θ is angle between the dipole axis and magnetic field.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Potential Energy of a Magnetic Dipole in a Uniform Magnetic Field</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The work done in rotating the dipole against the action of the torque is stored as potential energy of the dipole.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Potential Energy, U = W = – MB cos θ = – M . B</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Current Carrying Loop</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A current carrying loop behaves as a magnetic dipole. If we look the upper face of the loop and current is flowing anti-clockwise,then it has a north polarity and if current is flowing clockwise.then it has a south polarity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic dipole moment of a current carrying loop is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M = IA</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">For N such turns M = NIA</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Where I = current and A = area of cross-section of the coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Gauss’s Law in Magnetism</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Surface integral of magnetic field over any closed or open surface is always m.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3919/14611529875_9f8875b6ed_o.jpg" src="https://farm4.staticflickr.com/3919/14611529875_9f8875b6ed_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This law tells that the net magnetic flux through any surface is always zero.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">[When in an atom any electron revolve in an orbit it is equivalent to a current loop. Therefore, atom behaves as a magnetic dipole].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Moment of an Atom</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic moment of an atom M = 1 / 2 eωr<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where e = charge on an electron, ω = angular velocity of electron and r = radius of orbit.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">or M = n eh / 4πm</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where h = Planck’s constant and m ~ mass of an electron and eh / 4πm = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">B</sub>, called Bohr magneton and its value is 9.27 * 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-24</sup> A-m<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Earth’s Magnetism</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Earth is a huge magnet. There are three components of earth’s magnetism</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3847/14424925408_f0eae12ba6_o.jpg" src="https://farm4.staticflickr.com/3847/14424925408_f0eae12ba6_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Magnetic Declination (θ) The smaller angle subtended between the magnetic meridian and geographic meridian is called magnetic declination.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Magnetic Inclination or Magnetic Dip (δ) The smaller angle sub tended between the magnetic axis and horizontal is called magnetic inclination on magnetic dip.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) Horizontal Component of Earth’s Magnetic Field (H) If B is the intensity of earth’s magnetic field then horizontal component of earth’s magnetic field H = B cos δ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It acts from south to north direction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Vertical component of earth’s magnetic field</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">V = B sin δ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">∴ B = √H<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>+ V<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">and tan δ = V / H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Angle of dip is zero at magnetic equator and 90° at poles.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Meridian</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A vertical plane passing through the magnetic axis is called magnetic mendian.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Geographic Meridian</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">A vertical plane passing through the geographic axis is called geographic meridian.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Map</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic map is obtained by drawing lines on the surface of earth. which passes through different places having same magnetic elements.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The main lines drawn on earth’s surface are g1 en below</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isogonic Line</span> A line joining places of equal declination is called on isogonic line.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Agonic Line</span> A line joining places of zero declination is called an agonic line</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isoclinic Line</span> A line joining places of equal inclination or dip is called an aclinic line,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iv) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Aclinic Line</span> A hne joining places of zero inclination or dip is called an aclinic line.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(v) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Isodynamic Line</span> A line joining places of equal horizontal component of earth’s magnetic field (H) is called an isodynamic line.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Latitude</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) If at any place, the angle of dip is δ and magnetic latitude is λ then tan δ = 2 tan λ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) The total Intensity of earth’s magnetic field</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">I = I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">0</sub> √1 + 3 sin<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> λ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = M / R<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is assumed that a bar magnet of earth has magnetic moment M and radius of earth is R.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(Magnetic maps are maps obtained by drawing lines passing through different places on the surface of earth, having the same value of a magnetic element.)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Neutral Points</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Neutral point of a bar magnet is a point at which the resultant magnetic field of a bar magnet and horizontal component of earth’s magnetic field are zero.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When north pole of a bar magnet is placed towards south pole of the earth. then neutral point is obtained on axial line.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π 2Mr / (r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> – l<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> = H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If r > > l, then B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π 2M / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> = H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When north pole of a bar magnet is placed towards north pole of the earth, then neutral point is obtained on equatorial line</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π 2Mr / (r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> + l<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> = H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If r > > l, then B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π 2M / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> = H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tangent Law</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a bar magnet is freely suspended under the combined effect of two uniform magnetic fields of intensities B and H acting at 90° to each other, then it bar magnet comes to rest making an angle 0 with the direction of H, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = H tan θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Deflection Magnetometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is a device used to determine M and H. Its working is based on tangent law.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Deflection magnetometer can be used into two settings</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tangent A setting</span> In this setting the arms of the magnetometer are along east-west and magnet is parallel to the arms.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5536/14424925508_5b99a50572_o.jpg" src="https://farm6.staticflickr.com/5536/14424925508_5b99a50572_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">In equilibrium</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">B = H tan θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π 2M / d<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> = H tan θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tangent B setting</span> In this setting the arms of the magnetometer are along north-south and magnet is perpendicular to these arm in equilibrium</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5590/14608304811_5931cec3a8_o.jpg" src="https://farm6.staticflickr.com/5590/14608304811_5931cec3a8_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> / 4π M / d<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup> = H tan θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">In above setting the experiment can be performed in two ways.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(a) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Deflection method</span> In this method one magnet is used at a time and deflection in galvanometer is observed. Ratio of magnetic dipole moments of the magnets</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = tanθ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / tanθ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where θ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> and θ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> are mean values of deflection for two magnets.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(b) Null method In this method both magnets are used at a time and no deflection condition is obtained. If Magnets are at distance d<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> and d<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = (d<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / d<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">3</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tangent Galvanometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is a device used for detection and measurement of low electric currents. Its working is based on tangent law. If θ is the deflection produced in galvanometer when I current flows through it, then</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2924/14424959899_aa880e9228_o.jpg" src="https://farm3.staticflickr.com/2924/14424959899_aa880e9228_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">I = 2R / Nμ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> H tan θ = H / G tan θ = K tan θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Where G = Nμ / 2R is called galvanometer constant and K = H / G, is called reduction factor of tangent galvanometer.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Here N is number of turns in the coil and R is radius of the coil.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Tangent galvanometer is also called moving magnet type galvanometer.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Vibration Magnetometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is based on simple harmonic oscillations of a magnet suspended in uniform magnetic field.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Time period of vibrations is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">T = 2π √I / MH</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3886/14608306571_7a816fc93d_o.jpg" src="https://farm4.staticflickr.com/3886/14608306571_7a816fc93d_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, I = moment of inertia of the magnet,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M = magnetic dipole moment of the magnet and</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">H = horizontal component of earth’s magnetic field.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When two magnets of unequal size are placed one above the other and north poles of both magnets are towards geographic north then time period of oscillations is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">T<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> = 2π√I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> / (M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>) H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If north pole of first magnet and south pole of second magnet is towards geographic north, then time period of oscillations is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">T<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = 2π√(I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + I<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>) / (M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> – M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>) H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Then, M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> – T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup><sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Flux</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The number of magnetic lines of force passing through any surface is called magnetic flux linked with that surface.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic flux (phi;) = B . A = BA cos θ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where B is magnetic field intensity or magnetic induction, A is area of the surface.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its unit is ‘weber’.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Induction</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The magnetic flux passing through per unit normal area, is called magnetic induction.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic induction (B) = φ / A</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its unit is ‘waber/metre<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>‘ or ‘tesla’.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic of Material</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">To describe the magnetic properties of materials, following terms are required</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Permeability</span> It is the ability of a material to permit the passage of magnetic lines of force through it.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic permeability (μ) = B / H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where B is magnetic induction and H is magnetising force or magnetic intensity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetising Force or Magnetic Intensity</span> The degree up to which a magnetic field can magnetise a material is defined in terms of magnetic intensity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic intensity (H) = B / μ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Intensity of Magnetisation</span> The magnetic dipole moment developed per unit volume of the material is called intensity of magnetisation.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Intensity of magnetisation (I) = M / V = m / A</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where V = volume and A = area of cross-section of the specimen.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic induction B = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> (H + I)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iv) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Magnetic Susceptibility(χm)</span> The ratio of the intensity of magnetisation (1) induced in the material to the magnetising force (H) applied, is called magnetic susceptibility.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Magnetic Susceptibility(χm) = I / H</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">[Relation between Magnetic Permeability and Susceptibility is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">μ = μ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> (1 + χm) ]</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Classification of Magnetic Materials</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">On the basis of their magnetic properties magnetic materials are divided into three categories</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Diamagnetic substances</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Paramagnetic substances</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">iii) Ferromagnetic substances</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2907/14425141777_51b1184ed6_o.jpg" src="https://farm3.staticflickr.com/2907/14425141777_51b1184ed6_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2931/14425141277_f12901408c_o.jpg" src="https://farm3.staticflickr.com/2931/14425141277_f12901408c_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms of a paramagnetic substance contain even number of electrons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms or molecules of a paramagnetic substance do not possess any net magnetic moment.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The atoms of a paramagnetic material contain odd number of electrons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Every atom or molecule of a paramagnetic substance has its own magnet moment, i.e., its each atom or molecule is a tiny magnet.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">In a ferromagnetic substance, there are several tiny regions called domains. Each domain contain approximately 1010 atoms.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Each domain is a strong magnet as all atoms or molecules in a domain have same direction of magnetic moment.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Curie Law in Magnetism</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The magnetic susceptibility of a paramagnetic substance is inversely proportional to its absolute temperature.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">χm ∝ 1 / T ⇒ χm T = constant</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where χm = magnetic susceptibility of a para magnetic substance and T = absolute temperature.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Hysteresis</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The lagging of intensity of magnetisation (I) or magnetic induction (B) behind magnetising field (H), when a specimen of a magnetic substance is taken through a complete cycle of magnetisation is called hysteresis.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5536/14588524706_87d6b9d955_o.jpg" src="https://farm6.staticflickr.com/5536/14588524706_87d6b9d955_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Retentivity or Residual Magnetism</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The value of the intensity of magnetisation of a material, when the magnetising field is reduced to zero is called retentivity or residual magnetism of the material.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Coercivity</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The value of the reverse magnetising field that should be applied to a given sample in order to reduce its intensity of magnetisation or magnetic induction to zero is called coercivity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Permanent Magnets</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Commonly steel is used to make a permanent magnet because steel has high residual magnetism and high coercivity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Electromagnets</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electromagnets are made of soft iron because area of hysteresis loop for soft iron is small. Therefore, energy loss is small for a cycle of magnetisation and demagnetisation.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(Permanent magnets are made by the materials such as steel, for which residual magnetism as well as coercivity should be high. Electromagnets are made by the materials such as soft iron for which residual magnetism is high, coercivity is low and hysteresis loss is low).</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Important Points</span></p><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Magnetic length = 5 / 6 * geometric length of magnet.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">About 90% of magnetic moment is due to spin motion of electrons and remaining 10% of magnetic moment is due to the orbital motion of electrons.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">When a magnet having magnetic moment M is cut into two equal parts</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Parallel to its length</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = m / 2 * l = M / 2</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5556/14608304961_a239bd8524_o.jpg" src="https://farm6.staticflickr.com/5556/14608304961_a239bd8524_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Perpendicular to its length</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = m * l / 2 = M / 2</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3915/14424957909_69592aeaf7_o.jpg" src="https://farm4.staticflickr.com/3915/14424957909_69592aeaf7_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">When a magnet of length I, pole strength m and of magnetic moment M is turned into a semicircular arc then it new magnetic moment</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = m * 2R = m * 2 * 1 / π (πR = I)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">= 2M / π (M = m * l)</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3875/14425140107_e5c347c20d_o.jpg" src="https://farm4.staticflickr.com/3875/14425140107_e5c347c20d_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">A thin magnet of moment M is turned into an arc of 90°. Then new magnetic moment</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = 2√2M / π</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5487/14608304821_7ff2cb6fe9_o.jpg" src="https://farm6.staticflickr.com/5487/14608304821_7ff2cb6fe9_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">A thin magnet of moment M is turned at mid point 90°.Then new magnet moment</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = M / √2</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm6.staticflickr.com/5565/14610995832_131154dc5d_o.jpg" src="https://farm6.staticflickr.com/5565/14610995832_131154dc5d_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">A thin magnet of moment M is turned into an arc of 60°. Then new magnetic moment</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M = 3M / π</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3908/14609474484_ca2a444765_o.jpg" src="https://farm4.staticflickr.com/3908/14609474484_ca2a444765_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">A thin magnet of moment M is bent at mid point at angle 60°. Then new magnetic moment.</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">M’ = M / 2</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Magnetism" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2902/14424957409_db77c38004_o.jpg" src="https://farm3.staticflickr.com/2902/14424957409_db77c38004_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Original magnet MOS is bent at O, the mid point at 60°.All sidesare equal</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">The mutual interaction force between two small magnets of moments M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> and M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> is given by</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">F = K 6M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>M<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> / d<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">4</sup> in end- on position.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Here,d denotes the separation between magnets.</p><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Magnetic length = 5 / 6 * geometric length of magnet.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Cause of diamagnetism is orbital motion and cause of paramagnetism is spin motion of electrons. Cause of ferromagnetism lies in formation of domains.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">The perpendicular bisector of magnetic axis is known as neutral axis of magnet. Magnetism at neutral axis is zero and at poles is maximum.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">For steel coercivity is large. However, retentivity is comparatively smaller in case of steel. So, steel is used to make permanent magnets.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">For soft iron, coercivity is very small and area of hysteresis loop is small. So, soft iron is an ideal material for making electromagnets.</li></ul>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com1tag:blogger.com,1999:blog-8870097163070303690.post-47678452134295997272020-10-03T11:22:00.002+05:302020-12-18T09:15:36.485+05:30ELECTRIC CURRENT FULL CHAPTER NOTES<p> <span style="background-color: white; color: #333333; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: 700;">Electric Current (I)</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The rate of flow of charge through any cross-section of a wire is called electric current flowing through it.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electric current (I) = q / t. Its SI unit is ampere (A).</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The conventional direction of electric current is the direction of motion of positive charge.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The current is the same for all cross-sections of a conductor of non-uniform cross-section. Similar to the water flow, charge flows faster where the conductor is smaller in cross-section and slower where the conductor is larger in cross-section, so that charge rate remains unchanged.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If a charge q revolves in a circle with frequency f, the equivalent current,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">i = qf</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(In a metallic conductor current flows due to motion of free electrons while in electrolytes and ionized gases current flows due to electrons and positive ions.)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Types of Electric Current</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">According to its magnitude and direction electric current is of two types</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Direct Current (DC)</span> Its magnitude and direction do not change with time. A ceil, battery or DC dynamo are the sources of direct current.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Alternating Current (AC) An electric current whose magnitude changes continuously and changes its direction periodically is called alternating current. AC dynamo is source of alternating current.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Current Density</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The electric current flowing per unit area of cross-section of conductor is called current density.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Current density (J) = I / A</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its S1 unit is ampere metre<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup> and dimensional formula is [AT<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is a vector quantity and its direction is in the direction of motion positive charge or in the direction of flow of current.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Thermal Velocity of Free Electrons</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Free electrons in a metal move randomly with a very high speed of the order of 10<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">5</sup> ms-1. This speed is called thermal velocity of free electron.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Average thermal velocity of free electrons in any direction remains zero.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Drift Velocity of Free Electrons</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When a potential difference is applied across the ends of a conductor, the free electrons in it move with an average velocity opposite to direction of electric field. which is called drift velocity of free electrons.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Drift velocity v<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">d</sub> = eEτ / m = eVτ / ml</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, τ = relaxation time, e = charge on electron,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = electric field intensity, 1 = length of the conductor,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">V = potential difference across the ends of the conductor</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">m = mass of electron.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between electric current and drift velocity is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">v<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">d</sub> = I / An e</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mobility</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The drift velocity of electron per unit electric field applied is mobility of electron.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Mobility of electron (μ) = v<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">d</sub> / E</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is m<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>s<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup>V<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup> and its dimensional formula is [M<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup>T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>A].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Ohm’s Law</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If physical conditions of a conductor such as temperature remains unchanged, then the electric current (I) flowing through the conductor is directly proportional to the potential difference (V) applied across its ends.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">I ∝ V</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">or V = IR</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where R is the electrical resistance of the conductor and R = Ane<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> τ / ml.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Electrical Resistance</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The obstruction offered by any conductor in the path of flow of current is called its electrical resistance.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electrical resistance, R = V / I</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is ohm (Ω) and its dimensional formula is [ML<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup>T<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-3</sup>A<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-2</sup>].</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electrical resistance of a conductor R = ρl / A</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, l = length of the conductor, A = cross-section area and</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ρ = resistivity of the material of the conductor.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Resistivity</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Resistivity of a material of a conductor is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ρ = m / n<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> τ</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, n = number of free electrons per unit volume.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Resistivity of a material depend on temperature and nature of the material.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">It is independent of dimensions of the conductor, i.e., length, area of cross-section etc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Resistivity of metals increases with increase in temperature as</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ρ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">t</sub> = ρ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> (1 + αt)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where ρ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> and ρ<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">t</sub> are resistivity of metals at O°C and t°C and α temperature coefficient of resistivity of the material.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">For metals α is positive, for some alloys like nichrome, manganin and constantan, α is positive but very low.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">For semiconductors and insulators. α is negative.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Resistivity is low for metals, more for semiconductors and very high alloys like nichrome, constantan etc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(In magnetic field the resistivity of metals increases. But resistivity of ferromagnetic materials such as iron, nickel, cobalt etc decreases in magnetic field.)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Electrical Conductivity</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The reciprocal of resistivity is called electrical conductivity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electrical conductivity (σ) = 1 / ρ = 1 / RA = ne<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> τ / m</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI units is ohm<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup> m<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup> or mho m<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup> or siemen m<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">-1</sup>.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between current density (J) and electrical conductivity (σ) is given by</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">J = σ E</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, E = electric field intensity.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Ohmic Conductors</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Those conductors which obey Ohm’s law, are called ohmic conductors e.g., all metallic conductors are ohmic conductor.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">For ohmic conductors V – I graph is a straight line.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3888/14369395899_eebe1cd77f_o.jpg" src="https://farm4.staticflickr.com/3888/14369395899_eebe1cd77f_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Non-ohmic Conductors</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Those conductors which do not obey Ohm’s law, are called non-ohmic conductors. e.g., diode valve, triode valve, transistor , vacuum tubes etc.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">For non-ohmic conductors V – I graph is not a straight line.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3908/14369559567_e6d2697788_o.jpg" src="https://farm4.staticflickr.com/3908/14369559567_e6d2697788_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Superconductors</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">When few metals are cooled, then below a certain critical temperature their electrical resistance suddenly becomes zero. In this state, these substances are called <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">superconductors</span> and this phenomena is called <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">superconductivity</span>.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Mercury become superconductor at 4.2 K, lead at 7.25 K and niobium at 9.2 K</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Colour Coding of Carbon Resistors</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The resistance of a carbon resistor can be calculated by the code given on it in the form of coloured strips.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2937/14576094693_634aa15e75_o.jpg" src="https://farm3.staticflickr.com/2937/14576094693_634aa15e75_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><figure class="wp-block-table" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; overflow-x: auto; padding: 0px; vertical-align: baseline;"><table class="" style="border-collapse: collapse; border-spacing: 0px; border: 0px; color: #7b7b7b; display: block; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; margin: 1em 0px 30px; max-width: 100%; overflow: auto; padding: 0px; vertical-align: baseline; width: 750px; word-break: normal;"><tbody style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Colour coding</span></td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Colour</span></td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Figure</span></td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Black</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">0</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Brown</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">1</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Red</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">2</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Orange</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">3</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Yellow</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">4</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Green</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">5</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Blue</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">6</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Violet</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">7</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Grey</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">8</td></tr><tr style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">White</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">9</td></tr></tbody></table></figure><figure class="wp-block-table" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; overflow-x: auto; padding: 0px; vertical-align: baseline;"><table class="" style="border-collapse: collapse; border-spacing: 0px; border: 0px; color: #7b7b7b; display: block; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; margin: 1em 0px 30px; max-width: 100%; overflow: auto; padding: 0px; vertical-align: baseline; width: 750px; word-break: normal;"><tbody style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tolerance power</span></td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Colour</span></td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tolerance</span></td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Gold</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">5%</td></tr><tr style="border-bottom-color: rgb(238, 238, 238); border-bottom-style: solid; border-image: initial; border-left-color: initial; border-left-style: initial; border-right-color: initial; border-right-style: initial; border-top-color: initial; border-top-style: initial; border-width: 0px 0px 1px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">Silver</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">10%</td></tr><tr style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">No colour</td><td style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 0.85em; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 8px 20px; vertical-align: baseline;">20%</td></tr></tbody></table></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This colour coding can be easily learned in the sequence “B B ROY Great Bratain Very Good Wife”.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Combination of Resistors</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">1.In Series</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Equivalent resistance, R = R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Current through each resistor is same.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) Sum of potential differences across individual resistors is equal to the potential difference, applied by the source.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2921/14369398198_f68615a640_o.jpg" src="https://farm3.staticflickr.com/2921/14369398198_f68615a640_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">2. In Parallel</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Equivalent resistance</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 / R = 1 /R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + 1 / R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + 1 / R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">3</sub></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2920/14369398268_9908a66eb2_o.jpg" src="https://farm3.staticflickr.com/2920/14369398268_9908a66eb2_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Potential difference across each resistor is same.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Sum of electric currents flowing through individual resistors is equal to the be electric current drawn from the source.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Electric Cell</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">An electric cell is a device which converts chemical energy into electrical energy.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Electric cells are of two types</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Primary Cells</span> Primary ceUs cannot be charged again. Voltic, Daniel and Leclanche cells are primary cells.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Secondary Cells</span> Secondary cells can be charged again and again. Acid and alkali accumulators are secondary cells.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Electro – motive – Force</span> (emf) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">of a Cell</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The energy given by a cell in flowing unit positive charge throughout the circuit completely one time, is equal to the emf of a cell.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Emf of a cell (E) = W / q.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is volt.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Terminal Potential Difference of a Cell</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The energy given by a cell in flowing unit positive charge through till outer circuit one time from one terminal of the cell to the other terminal of the cell.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Terminal potential difference (V) = W / q.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Its SI unit is volt.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Internal Resistance of a Cell</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The obstruction offered by the electrolyte of a cell in the path of electric current is called internal resistance (r) of the cell. Internal resistance of a cell</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) Increases with increase in concentration of the electrolyte.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) Increases with increase in distance between the electrodes.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) Decreases with increase in area of electrodes dipped in electrolyte.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Relation between E. V and r</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = V + Ir</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">r = (E / V – 1) R</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If cell is in charging state, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = V – Ir</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Grouping of Cells</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">In Series</span> If n cells, each of emf E and internal resistance r are connected in series to a resistance R. then equivalent emf</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3893/14369319520_012792b29e_o.jpg" src="https://farm4.staticflickr.com/3893/14369319520_012792b29e_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> = E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + …. + E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub> = nE</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Equivalent internal resistance r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> = r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> + …. + r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub> = nr</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Current In the circuit I = E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> / (R + r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub>) = nE / (R + nr)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">In Parallel</span> If n cells. each of emf E and internal resistance r are connected to in parallel. then equivalent emf. E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> = E</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Equivalent internal resistance</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3840/14552599971_465a9e9b60_o.jpg" src="https://farm4.staticflickr.com/3840/14552599971_465a9e9b60_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">1 / r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> = 1 / r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + 1 / r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> + … + 1 / r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">n</sub> = n / r or r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub> = r / n</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Current In the circuit I = E / (R + r / n)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(iii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mixed Grouping</span> of Cells If n cells, each of emf E and internal resistance r are connected in series and such m rows are connected in parallel, then</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3926/14552600251_3bd7499699_o.jpg" src="https://farm4.staticflickr.com/3926/14552600251_3bd7499699_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Equivalent emf, E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Equivalent Internal resistance r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">eq</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Current in the circuit, I = nE / (R + nr / m)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">or I = mnE / mR + nr</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Note</span> Current in this circuit will be maximum when external resistance is equal to the equivalent internal resistance, i.e.,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">R = nr / m ⇒ mR = nr</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Kirchhoff’s Laws</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">There are two Kirchhoff’s laws for solving complicated electrical circuits</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(i) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Junction Rule</span> The algebraic sum of all currents meeting at a junction in a closed circuit is zero, i.e., Σ I = O.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This law follows law of conservation of charge.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">(ii) <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Loop Rule</span> The algebraic sum of all the potential differences in any closed circuit is zero, i.e.,</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">ΣV = 0 ⇒ ΣE = ΣIR</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This law follows law of conservation of energy.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Balanced Wheatstone Bridge</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Wheatstone bridge is also known as a <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">metre bridge</span> or <span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">slide wire bridge</span>.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This is an arrangement of four resistance in which one resistance is unknown and rest known. The Wheatstone bridge as shown in figure. The bridge is said to be balanced when deflection in galvanometer is zero, i.e., i<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">g</sub> = O.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3906/14576094253_db5273a62d_o.jpg" src="https://farm4.staticflickr.com/3906/14576094253_db5273a62d_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Principle of Wheatstone Bridge</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">P / Q = R / S</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The value of unknown resistance S can found. as we know the value of P,Q and R. It may be remembered that the bridge is most sensitive, when all the four resistances are of the same order.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Meter Bridge</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">This is the simplest form of Wheatstone bridge and is specially useful for comparing resistance more accurately.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2909/14369559217_7f07f06244_o.jpg" src="https://farm3.staticflickr.com/2909/14369559217_7f07f06244_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">R / S = l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / (100 – l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>)</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> is the length of wire from one end where null point is obtained.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Potentiometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Potentiometer is an ideal device to measure the potential difference between two points. It consists of a long resistance wire AB of uniform cross section in which a steady direct current is set up by means of a battery.</p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm3.staticflickr.com/2937/14369395319_9057ec6690_o.jpg" src="https://farm3.staticflickr.com/2937/14369395319_9057ec6690_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If R be the total resistance of potentiometer wire L its total length, then potential gradient, i.e., fall in potential per unit length along the potentiometer will be</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">K = V / L = IR / L</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">= E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> R / (R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> + R)L</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where, E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = emf of battery and R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">o</sub> = resistance inserted by means of rheostat Rh.</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Determination of emf of a Cell using Potentiometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">If with a cell of emf E on sliding the contact point we obtain zero deflection in galvanometer G when contact point is at J at a length I from the end where positive terminal of cell have been joined. then fall in potential along length i is just balancing the emf of cell. Thus, we have</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = Kl</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">or E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / E<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Determination of Internal Resistance of a Cell using Potentiometer</span></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">The arrangement is shown in figure. If the cell E is in open circuit and balancing length l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>, then</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">E = Kl<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub></p><figure class="wp-block-image" style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1em; padding: 0px; position: relative; vertical-align: baseline;"><img alt="CBSE Class 11 Physics Notes Current Electricity" class=" lazyloaded" data-pin-no-hover="true" data-src="https://farm4.staticflickr.com/3887/14555129192_461021a4b1_o.jpg" src="https://farm4.staticflickr.com/3887/14555129192_461021a4b1_o.jpg" style="border: 0px; box-sizing: border-box; font: inherit; height: auto; margin: 0px; max-width: 100%; opacity: 1 !important; padding: 0px; transition: opacity 0.4s ease 0s; vertical-align: middle;" /></figure><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">But if by inserting key K<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> circuit of cell is closed, then ooten difference V is balanced by a length l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> of potential where</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">V = Kl<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">Internal resistance of cell</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">r = E – V / V , R = l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> – l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> / l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> * R</p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;"><span style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: 700; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Important Points</span></p><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Potentiometer is an ideal voltmeter.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">Sensitivity of potentiometer is increased by increasing length of potentiometer wire.</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">If n identical resistances are first connected in series and then in parallel. the ratio of the equivalent resistance.</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">s</sub> / R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">p</sub>= n<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> / 1</p><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">If a skeleton cube is made with 12 equal resistance,each having a resistance R, then the net resistance across</li></ul><ol style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style-image: initial; list-style-position: initial; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">The diagonal of cube = 5 / 6 R</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">The diagonal of a face = 3 / 4 R</li><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">along a side = 7 / 12 R</li></ol><ul style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: Poppins, Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; list-style: square; margin: 0px 0px 1.5em; padding: 0px 0px 0px 2.14286em; vertical-align: baseline;"><li style="border: 0px; box-sizing: border-box; font: inherit; margin: 0px 0px 0.5em; padding: 0px; vertical-align: baseline;">If a resistance wire is stretched to a greater length, keeping volume constant, then</li></ul><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">R ∝ l<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup> ⇒ R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = (l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / l<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">2</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">and R ∝ 1 / r<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">4</sup> ⇒ R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub> / R<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> = (r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">2</sub> / r<sub style="border: 0px; bottom: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative;">1</sub>)<sup style="border: 0px; box-sizing: border-box; font-family: inherit; font-size: 10px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 0; margin: 0px; padding: 0px; position: relative; top: 0.25em;">4</sup></p><p style="background-color: white; border: 0px; box-sizing: border-box; color: #333333; font-family: "Open Sans", Helvetica, Arial, sans-serif; font-size: 16px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px 0px 1.25em; padding: 0px; text-rendering: optimizelegibility; vertical-align: baseline;">where l is the length of wire and r is the radius of cross-section area of wire.</p>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-32291360676091808262020-10-03T11:21:00.002+05:302020-12-18T09:15:36.483+05:30ELECTROSTATICS FULL CHAPTER NOTES<p> ELECTROSTATICS :- </p><pre style="background-color: white; border: 0px; box-sizing: border-box; color: #4c4c4c; font-family: "Open Sans", sans-serif !important; font-size: 14px; line-height: 20px; margin-bottom: 7px; margin-top: 7px; overflow-wrap: break-word; padding: 0px; text-align: justify; white-space: normal;"><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electrostatic:-</span></span><span style="border: 0px; box-sizing: border-box; font-size: 13px; font-weight: 600; line-height: 1.6; margin: 0px; padding: 0px;"> </span><span style="border: 0px; box-sizing: border-box; font-size: 13px; line-height: 1.6; margin: 0px; padding: 0px;">It is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges with no acceleration</span><span style="border: 0px; box-sizing: border-box; font-size: 13px; font-weight: 600; line-height: 1.6; margin: 0px; padding: 0px;">.</span></p></li><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Coulomb’s Law:- </span></span>It states that the electro-static force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and varies inversely as the square of the distance between the two bodies.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> F = K<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Here, K = 1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> = 9×10<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">9</sup> <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Nm</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-2</sup> (in free space)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Relative Permittivity (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">r</span>):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> The relative permittivity (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">r</span>) of a medium is defined as the ratio between its permittivity of the medium (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em>) and the permittivity (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) of the free space.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">r</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Coulomb force in vector form:-</span></span> The force on charge q<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> due to q<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> is,</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="\vec{F_{12}} = [q_{1}q_{2}/r^{2}]\hat{r_{21}}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vecf_12q_1q_2r2_hatr_21.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> If <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>>0, R.H.S is positive.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">If <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><0, a negative sign from <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> will change <img alt="\hat{r_{21}}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_hatr_21.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /> and <img alt="\hat{r_{12}}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_hatr_12.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" />. The relation will again be true, since, in that case have same directions.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Unit of Charge:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> C.G.S, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em> = ±1 stat-coulomb</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> S.I, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em> = ±1 Coulomb</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Relation between coulomb and stat-coulomb:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> 1 coulomb = 3×10<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">9</sup> stat-coulomb</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> 1 coulomb =(1/10) ab-coulomb (e.m.u of charge)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Dielectric constant:</span>-</span> The dielectric constant (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">r</span>) of a medium can be defined as the ratio of the force between two charges separated by some distance apart in free space to the force between the same two charges separated by the same distance apart in that medium.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> So, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">r</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">F</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">F</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">Here, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">F</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> and <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">F</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> are the magnitudes of the force between them in free space and in a medium respectively.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Charges:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">? </span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">Line charge, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">λ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">L</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Surface charge, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Volume charge, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ρ</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field (<img alt="\vec{E}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vece.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" />)</span> </span><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">:</span>-</span> The strength of an electric field is measured by the force experienced by a unit positive charge placed at that point. The direction of field is given by the direction of motion of a unit positive charge if it were free to move.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="\vec{E} = \vec{F}/q = Kq/r^{2}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vece_vecfqkqr2.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Unit of Electric field:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> E </em>= [Newton/Coulomb] or [Joule/(Coulomb) (meter)]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric lines of force:-</span> </span>An electric line of force is defined as the path<span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">, </span>straight or curved<span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">, </span>along which a unit positive charge is urged to move when free to do so in an electric field. The direction of motion of unit positive charge gives the direction of line of force.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-143818624-2284-capture.png" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Lines of Force" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; line-height: 1.6; margin: 0px; padding: 0px;"> Properties:-</span></span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) The lines of force are directed away from a positively charged conductor and are directed towards a negatively charged conductor.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) A line of force starts from a positive charge and ends on a negative charge. This signifies line of force starts from higher potential and ends on lower potential.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric field intensity due to a point charge:</span>-</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> E</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>)</li><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field Intensity due to a linear distribution of charge:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-144254169-9040-capture-1.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to a Linear Distribution of Charge." />?</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) At point on its axis.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> E = (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λ</em>/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) [1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em> – 1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">L</em>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Here, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">λ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the linear charge density.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) At a point on the line perpendicular to one end.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> ?<img src="https://files.askiitians.com/cdn1/images/2014925-14433870-2888-capture-2.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Here </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">λ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the line charge.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to ring of uniform charge distribution:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-144313793-3072-capture-3.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to Ring of Uniform Charge Distribution." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> At a point on its axis, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) [</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">qx</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/(</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">a</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">x</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">)</span><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">3/2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">]</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to uniformly charged disc:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-1451557-5923-capture-4.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to Uniformly Charged Disc." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Here </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">σ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the surface charge.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to thin spherical shell:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a)</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">out</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">)</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">in</span> = 0</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 0.75in; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-14527389-3099-capture-5.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to a Thin Spherical Shell" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field of a non-conducting solid sphere having uniform volume distribution of charge:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-145559322-2965-capture.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title=" Electric Field Due to a Non-conducting Solid Sphere Having Uniform Volume Charge Distribution" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) Outside Point:- </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">out</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">)</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Inside Point:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">in</span> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Qr</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">3</sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c) On the Surface:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">surface</span> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Here, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the total charge</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field of a cylindrical conductor of infinite length having line charge <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λ</em>:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-145643797-9082-capture-1.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title=" Electric Field Due to a Cylindrical Conductor of Infinite Length and Having Line Charge ?." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> (a) Outside the cylinder:- E = λ/2πε0r</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> (b) Inside the cylinder:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> = 0</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field of a non-conducting cylinder having uniform volume density of charge:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> (a) Outside the cylinder:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λ</em>/2π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> (b) Inside a point:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ρr</em>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric field of an infinite plane sheet of charge surface charge (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em>) :</span>-</span> <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-15053602-8114-capture.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to an Infinite Plane Sheet Having Surface Charge s." /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to two oppositely infinite charged sheets:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) Electric field at points outside the charged sheets:-</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> E<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">P </span></span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">= </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">R </span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">= 0</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: lower-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Electric field at point in between the charged sheets:-</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> E<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">Q</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric Dipole:</span>-</span> An electric dipole consists of two equal and opposite charges situated very close to each other.</li></ul><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Dipole Moment:</span>-</span> Dipole moment (<img alt="\vec{P}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vecp.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" />) of an electric dipole is defined as the product of the magnitude of one of the charges and the vector distance from negative to positive charge.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="\vec{P} = q\vec{a}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vecpq_veca.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Unit of Dipole Moment:-</span></span> coulomb meter (S.I), stat coulomb cm (non S.I)</li></ul><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to an electric dipole:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) At any point on the axial line:-</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-1596458-7644-capture-1.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to an Electric Dipole on the Axial line." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 1in; padding: 0px;">Alt Tag: Electric field due to an electric dipole on the axial line.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-15918555-6424-capture-2.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) At a point on the equatorial line (perpendicular bisector):-</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-15930570-2695-capture-3.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Electric Field Due to an Electric Dipole on the Equatorial Line." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span><img src="https://files.askiitians.com/cdn1/images/2014925-15232674-6428-capture-3.png" style="border: 0px; box-sizing: border-box; height: auto; line-height: 1.6; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (c) At any point:-</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-152252195-2274-capture-2.png" style="border: 0px; box-sizing: border-box; height: auto; line-height: 1.6; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title=" Electric Field Due to an Electric Dipole at any Point." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span><img src="https://files.askiitians.com/cdn1/images/2014925-152243342-793-capture-1.png" style="border: 0px; box-sizing: border-box; height: auto; line-height: 1.6; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Torque (</span><img alt="\tau" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_tau.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">) acting on a electric dipole in a uniform electric field (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em>):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="\tau" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_tau.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" />= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">pE</em> sin<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">θ</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> </em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">Here, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">p</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the dipole moment and </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">θ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the angle between direction of dipole moment and electric field </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric Flux:</span>-</span> Electric flux <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">?</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">E</span> for a surface placed in an electric field is the sum of dot product of <img alt="\vec{E}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_vece.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /> and <img alt="d\vec{a}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latexd_veca.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /> for all the elementary areas constituting the surface.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="\Phi _{E} = \int _{s} \vec{E}.d\vec{a}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_phi_e_int_s_vece.d_veca.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; line-height: 1.5; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Gauss Theorem:-</span> </span><span style="border: 0px; box-sizing: border-box; line-height: 1.5; margin: 0px; padding: 0px;">It states that, for any distribution of charges, the total electric flux linked with a closed surface is 1/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.5; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.5; margin: 0px; padding: 0px;"> times the total charge with in the surface.</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-1548552-2919-capture-2.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em>) of an infinite rod at a distance (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em>) from the line having linear charge density (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λ</em>):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> E = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">λ</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/2π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> The direction of electric field <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> is radially outward for a line of positive charge.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field of a spherically symmetric distribution of charge of Radius <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-15494205-8893-capture-3.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="A Spherically Symmetric Distribution of Charge of Radius R." /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">a) Point at outside (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> > </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">R</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">):- </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">), Here </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the total charge.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 80px; padding: 0px;"> (b) Point at inside (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em> < <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>):- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">qr</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">3</sup>), Here <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em> is the total charge.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; line-height: 1.6; margin: 0px; padding: 0px;">Electric field due to an infinite non-conducting flat sheet having charge σ:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> E = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> <img src="https://files.askiitians.com/cdn1/images/2014925-155542825-2039-capture.png" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" title="Infinite Flat Conductor Sheet Carrying Surface Charge s" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">This signifies, the electric field near a charged sheet is independent of the distance of the point from the sheet and depends only upon its charge density and is directed normally to the sheet.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"> <span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric field due to an infinite flat conductor carrying charge:- </span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"> ?</span>E<span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric pressure (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">P</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">elec</span>) on a charged conductor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> P</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">elec</span> = (½<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><div class="tablecont" style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"><table border="1" cellpadding="1" cellspacing="1" style="background: rgb(252, 252, 252); border-color: rgb(219, 219, 219); border-image: initial; border-style: solid; border-width: 1px 0px 0px 1px; font-size: 14px; line-height: 32px; margin: 10px 0px; max-width: 100%; padding: 0px; width: 1162px;"><tbody style="border: 0px solid rgb(219, 219, 219); box-sizing: border-box; font-size: 12px; line-height: 16px; margin: 0px; padding: 0px;"><tr style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px solid rgb(219, 219, 219); box-sizing: border-box; line-height: 16px; margin: 0px; padding: 0px;"><td style="border-color: rgb(219, 219, 219); border-image: initial; border-style: solid; border-width: 0px 1px 1px 0px; box-shadow: rgb(255, 255, 255) 0px 1px 1px inset; box-sizing: border-box; color: #4c4c4c; height: 32px; line-height: 16px; margin: 0px; padding: 0px 5px; text-align: unset; vertical-align: top; width: 376px;"><h2 style="border: 0px; box-sizing: border-box; font-size: 20px; font-weight: normal; line-height: 24px; margin: 15px 0px 5px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; line-height: 28px; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electro-Static Potential and Capacitance:-</span></span></h2></td></tr></tbody></table></div><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric Potential:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (a) Electric potential, at any point, is defined as the negative line integral of electric field from infinity to that point along any path.</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> <img alt="V (\vec{r}) = -\int_{\infty }^{r}\vec{E}.d\vec{r}" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latexv_vecr-_int__inftyr_vece.d_vecr.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (b) V</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">(</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">kq</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (c)</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Potential difference, between any two points, in an electric field is defined as the work done in taking a unit positive charge from one point to the other against the electric field.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">AB</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em> [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">A</span>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">B</span>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">So, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em> = [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">A</span>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">B</span>] = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">Units:- volt (S.I), stat-volt (C.G.S)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">Dimension:- [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em>] = [ML<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>T<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-3</sup>A<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-1</sup>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;">Relation between volt and stat-volt:- 1 volt = (1/300) stat-volt</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Relation between electric field (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em>) and electric potential (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em>):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> E</em> = -<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dV</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dx</em> = --<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dV</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dr</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential due to a point charge:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential at point due to several charges:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> + <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> +<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+ </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+….</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential due to charged spherical shell:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) Outside, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">out</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">)</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Inside, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">in</span> = - (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/R)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c) On the surface, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">surface</span> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential due to a uniformly charged non-conducting sphere:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (a) Outside, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">out</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = (1/4π</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">r</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">)</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Inside, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">in</span> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>(3<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>)/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">3</sup>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c) On the surface, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">surface</span> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (d) In center, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">center</span> = (3/2) [(1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>)] = 3/2 [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">surface</span>]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Common potential (two spheres joined by thin wire):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) Common potential, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) [(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>)/(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>)]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>)/(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>) = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em>/ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> ; <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em>/ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> </p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> or <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> =<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential at any point due to an electric dipole:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em>,<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">θ</em>) = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">qa</em> cos<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">θ</em>/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">p</em> cos<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">θ</em>/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) Point lying on the axial line:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">p</em>/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Point situated on equatorial lines:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em> = 0</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">If <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">n</em> drops coalesce to form one drop, then,</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">n</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">1/3</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> Q</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">nq</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">n</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2/3</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">small</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (d)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> σ</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">n</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">1/3</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> σ</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">small</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (e)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> E</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">n</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">1/3</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> E</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">small</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric potential energy <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> or work done of the system <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em> having charge <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> and <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span></span>:-</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> W</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> = (1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">12</span>) = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>V<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Electric potential energy <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> or work done of the system <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em> of a three particle system having charge <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> and <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></span>:-</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> W = U = </em>(1/4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">12</span> + <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">13</span> + <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">23</span>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Electric potential energy of an electric dipole in an electric field:</span>-</span> Potential energy of an electric dipole, in an electrostatic field, is defined as the work done in rotating the dipole from zero energy position to the desired position in the electric field.</li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> <img alt="W = - \vec{p}.\vec{E} = - pE cos\theta" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latexw-_vecp._vece-pecos_theta.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) If<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> θ </em>= 90º, then <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em> = 0</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) If<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> θ </em>= 0º, then<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> W</em> = -<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">pE</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (c) If<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> θ </em>= 180º, then<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> W</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">pE</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Kinetic energy of a charged particle moving through a potential difference:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> K. E = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">mv</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2 </sup>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">eV</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Conductors:</span>-</span> Conductors are those substance through which electric charge easily.</p></li><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Insulators:</span>- </span>Insulators (also called dielectrics) are those substances through which electric charge cannot pass easily.</p></li><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Capacity:</span>- </span>The capacity of a conductor is defined as the ratio between the charge of the conductor to its potential</p></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> C = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: square; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Units:-</span></span></span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> S.I – farad (coulomb/volt)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> C.G.S – stat farad (stat-coulomb/stat-volt)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"> Dimension o</span>f <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em>:-</span> [M<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-1</sup>L<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-2</sup>T<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">4</sup>A<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Capacity</span> of an isolated spherical conductor:-</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">r</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Capacitor:-</span></span> A capacitor or a condenser is an arrangement which provides a larger capacity in a smaller space.</p></li><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;">Capacity</span> of a parallel plate capacitor:-<img src="https://files.askiitians.com/cdn1/images/201539-114515304-2682-cap.png" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px; max-width: 93%; padding: 0px; vertical-align: middle;" /></span></p></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> C</span><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">air </span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">= </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">A</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">d</em></p><p align="center" style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> </p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">med</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Kε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">d</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Here, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em> is the common area of the two plates and <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">d</em> is the distance between the plates.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Effect of dielectric on the capacitance of a capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em>/[<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">d</em>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>+(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/K)]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> Here </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">d</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the separation between the plates, </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">t</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the thickness of the dielectric slab </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">A</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the area and </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">K</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> is the dielectric constant of the material of the slab.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">If the space is completely filled with dielectric medium (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">t</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">=</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">d</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">), then,</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">C</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">KA</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> d</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Capacitance of a sphere:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">air</span> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">med</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">K</em> (4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Capacity of a spherical condenser:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) When outer sphere is earthed:<span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">-</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">air</span> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ab</em>/(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>)]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">med</span> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> [K<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ab</em>/(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>)]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: lower-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> (b) When the inner sphere is earthed:-</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>= 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ab</em>/(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>)]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">?</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Net Capacity, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C </em>'=4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>[<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Increase in capacity, Δ<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em> = 4π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> It signifies, by connecting the inner sphere to earth and charging the outer one we get an additional capacity equal to the capacity of outer sphere.</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Capacity of a cylindrical condenser:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">air</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λl</em> / [(<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">λ</em>/2π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>) (log<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">e </span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>)] = [2π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">l</em> /(log<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">e </span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>) ]</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> C<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">med</span> = [2π<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Kε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">l</em> /(log<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">e </span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">b</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">a</em>) ]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Potential energy of a charged capacitor (Energy stored in a capacitor):-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> W</em> = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">QV</em> = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em> = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">CV</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Energy density of a capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> U</em> = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> = ½ (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">σ</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">This signifies the energy density of a capacitor is independent of the area of plates of distance between them so long the value of </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">E</em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> does not change.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Grouping of Capacitors:-</span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;"> ?(a)</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (i) Capacitors in parallel:- <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>+…..+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">n</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;"> </span><img src="https://files.askiitians.com/cdn1/images/2014925-16844289-5567-capture.png" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px; max-width: 93%; padding: 0px; vertical-align: middle;" title="Three Capacitors C1, C2 and C3 are in Parallel Connection." /><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">The resultant capacity of a number of capacitors, connected in parallel, is equal to the sum of their individual capacities.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (ii)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (iii)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span> =<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em>, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span> =<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em>, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (iv) Energy Stored, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (b) </p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (i) Capacitors in Series:- 1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em> = 1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1 </span>+ 1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2 </span>+……+ 1/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">n</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><img src="https://files.askiitians.com/cdn1/images/2014925-16116942-9893-capture.png" style="border: 0px; box-sizing: border-box; float: right; height: auto; margin: 5px; max-width: 93%; padding: 0px; vertical-align: middle;" title="Three Capacitors C1, C2 and C3 are in Series Connection." /><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px; vertical-align: sub;">?</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">The reciprocal of the resultant capacity of a number of capacitors, connected in series, is equal to the sum of the reciprocals of their individual capacities.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> (ii)</em><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> =</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> =</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> (iii) V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span>= <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">q</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (iv) Energy Stored, <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Energy stored in a group of capacitors:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> (a) Energy stored in a series combination of capacitors:-</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> W </em><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">= ½ (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) + ½ (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) + ½ (</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">/</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">) = </span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">+</span><em style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"> Thus, net energ<span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;">y stored in the combination is equal to the sum of the energies stored in the component capacitors.</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> (b) Energy stored in a parallel combination of capacitors:-</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"> W = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V </em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> +½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V </em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> + ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V </em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">W</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">3</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; list-style-type: upper-alpha; margin: 5px 0px 8px 40px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; line-height: 1.6; margin: 0px; padding: 0px;"> The net energy stored in the combination is equal to sum of energies stored in the component capacitors.</span></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Force of attraction between plates of a charged capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> (a) F</em> = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">E</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> (b) F</em> = σ<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px 40px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> (c) F=Q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">ε</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">A</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Force on a dielectric in a capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> F</em> = (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>/2<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup>) (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dC</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dx</em>) = ½ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup> (<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dC</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">dx</em>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Common potential when two capacitors are connected:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> V = [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+ <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>] / [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>] = [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>]/ [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Charge transfer</span> <span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">when two capacitors are connected:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> Δ<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em> = [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>] [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>]</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Energy loss</span> <span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">when two capacitors are connected:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> Δ<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">U</em> = ½ [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span><em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>+<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">C</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>] [<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">1</span>-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">2</span>] <sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">2</sup></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Charging of a capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (a)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> Q</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">Q</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(1-e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">RC</em></sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (b)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(1-e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">RC</em></sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (c)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> I</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">I</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(1-e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">RC</em></sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (d) <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">I</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">R</em></p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Discharging of a capacitor:-</span></span></li></ul><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (a)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> Q = Q<span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-t/RC</sup>)</em></p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (b)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> V</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">V</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">RC</em></sup>)</p><p style="border: 0px; box-sizing: border-box; line-height: 22px; margin: 5px 0px 8px; padding: 0px;"> (c)<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;"> I</em> = <em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">I</em><span style="border: 0px; bottom: -0.25em; box-sizing: border-box; font-size: 10.5px; line-height: 0; margin: 0px; padding: 0px; position: relative; vertical-align: baseline;">0</span>(e<sup style="border: 0px; box-sizing: border-box; line-height: 0; margin: 0px; padding: 0px; position: relative; top: -0.5em;">-<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">t</em>/<em style="border: 0px; box-sizing: border-box; margin: 0px; padding: 0px;">RC</em></sup>)</p><ul style="border: 0px; box-sizing: border-box; margin: 13px 0px; padding: 0px;"><li style="border: 0px; box-sizing: border-box; line-height: 17px; margin: 6px 0px 6px 20px; padding: 0px; text-align: justify;"><span style="border: 0px; box-sizing: border-box; color: black; margin: 0px; padding: 0px;"><span style="border: 0px; box-sizing: border-box; font-weight: 600; margin: 0px; padding: 0px;">Time constant:-</span></span> <img alt="\tau = RC" src="https://files.askiitians.com/cdn1/cms-content/common/latex.codecogs.comgif.latex_taurc.jpg" style="border: 0px; box-sizing: border-box; height: auto; margin: 5px 10px 5px 5px !important; max-width: 93%; padding: 0px; vertical-align: middle;" /></li></ul></pre>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-12577372689505907922020-08-23T22:06:00.004+05:302020-08-23T22:07:15.445+05:30WORK, POWER, ENERGY<p> <u style="background-color: white; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem;"><span style="color: #04ff00;">Work and Energy</span></u></p><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;">The concepts of work and energy are closely tied to the concept of force because an applied force can do work on an object and cause a change in energy. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Energy</span> is defined as the ability to do work.<p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0021" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0021" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Work</a></h2><a name="huetinck3831c01-sec2-0021" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The concept of work in physics is much more narrowly defined than the common use of the word. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Work</span> is done on an object when an applied force moves it through a distance. In our everyday language, work is related to expenditure of muscular effort, but this is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">not</em> the case in the language of physics. A person that holds a heavy object does no physical work because the force is not moving the object through a distance. Work, according to the physics definition, is being accomplished while the heavy object is being lifted but not while the object is stationary. Another example of the absence of work is a mass on the end of a string rotating in a horizontal circle on a frictionless surface. The centripetal force is directed toward the center of the circle and, therefore, is not moving the object through a distance; that is, the force is not in the direction of motion of the object. (However, work was done to set the mass in motion.) Mathematically, work is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = F · x, where <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">F</span> is the applied force and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">x</span> is the distance moved, that is, displacement. Work is a scalar. The SI unit for work is the joule (J), which is newton‐meter or kg m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If work is done by a varying force, the above equation cannot be used. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0025">shows the force‐versus‐displacement graph for an object that has three different successive forces acting on it. The force is increasing in segment I, is constant in segment II, and is decreasing in segment III. The work performed on the object by each force is the area between the curve and the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> axis. The total work done is the total area between the curve and the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> axis. For example, in this case, the work done by the three successive forces is shown in Figure 1<xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0025">.<figureanchor figures="huetinck3831c01-fig-0025" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0025"> <img alt="Force needed to change momentum, from fixed position - Physics ..." src="data:image/png;base64,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" style="color: #222222;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;"><span> </span><span> </span><span> </span><span> </span><span> </span>Acting force changing with position.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In this example, the total work accomplished is (1/2)(15)(3) + (15)(2) + (1/2)(15)(2) = 22.5 + 30 + 15; work = 67.5 J. For a gradually changing force, the work is expressed in integral form, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = ∫ F · <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em>x.</p><sect1 id="huetinck3831c01-sec2-0022" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0022" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0022" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Kinetic energy</a></span></h2><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0022" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Kinetic energy</span> is the energy of an object in motion. The expression for kinetic energy can be derived from the definition for work and from kinematic relationships. Consider a force applied parallel to the surface that moves an object with constant acceleration.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">From the definition of work, from Newton's second law of motion, and from kinematics, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Fx</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">max</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> + 2 <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">ax</em>, or <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>)/2 <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em>. Substitute the last expression for acceleration into the expression for work to obtain <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>) or <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mv</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> − (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mv</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>. The right side of the last equation yields the definition for kinetic energy: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">K</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. = (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mv</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> Kinetic energy is a scalar quantity with the same units as work, joules (J). For example, a 2 kg mass moving with a speed of 3 m/s has a kinetic energy of 9 J.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The above derivation shows that the net work is equal to the change in kinetic energy. This relationship is called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">work‐energy theorem:</span> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">net</sub> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">K</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">K</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">K</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>is the final kinetic energy and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">K</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>is the original kinetic energy.</p><sect1 id="huetinck3831c01-sec2-0023" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0023" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0023" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Potential energy</a></span></h2><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0023" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Potential energy,</span> also referred to as stored energy, is the ability of a system to do work due to its position or internal structure. Examples are energy stored in a pile driver at the top of its path or energy stored in a coiled spring. Potential energy is measured in units of joules.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Gravitational potential energy</span> is energy of position. First, consider gravitational potential energy near the surface of the earth where the acceleration due to gravity <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(g)</em> is approximately constant. In this case, an object's gravitational potential energy with respect to some reference level is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P.E</em>. = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mgh</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">h</em> is the vertical distance above the reference level. To lift an object slowly, a force equal to its weight <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(mg)</em> is applied through a height <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(h)</em>. The work accomplished is equal to the change in potential energy: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mgh</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mgh</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>, where the subscripts <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(f</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o)</em> refer to the final and original heights of the body.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Launching a rocket into space requires work to separate the mass of the earth and the rocket to overcome the gravitational force. For large distances from the center of the earth, the above equation is inadequate because <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> is not constant. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">general form of gravitational potential energy</span> is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P.E</em>. = − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">GMm/r</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">M</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> refer to the masses of the two bodies being separated and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">r</em> is the distance between the centers of the masses. The negative sign is a result of selecting the zero reference at <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">r</em> equal to infinity, that is, at very large <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">separation.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Elastic potential energy</span> is energy stored in a spring. The magnitude of the force required to stretch a spring is given by <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> = − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">kx</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> is the distance of stretch (or compression) of a spring from the unstressed position, and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">spring constant.</span> The spring constant is a measure of the stiffness of the spring, with stiffer springs having larger <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> values. The potential energy stored in a spring is given by <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. = (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">kx</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Change in potential energy is equal to work. The gravitational force and the force to stretch a spring are varying forces; therefore, the potential energy equations given above for these two cases can also be derived from the integral form of work, Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em>. <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">E</em>. = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = ∫ F · <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em>x.</p><sect1 id="huetinck3831c01-sec2-0024" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0024" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0024" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Power</a></span></h2><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0024" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Power</span> is the rate of doing work, average <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W/t</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> is the time interval during which work <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(W)</em> is accomplished. Another form of power is found from <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">W</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and substitution of average velocity of the object during time <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> for Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>: average <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">P</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em>/Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em>(average <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em>).</p><sect1 id="huetinck3831c01-sec2-0025" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0025" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0025" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">The conservation of energy</a></h2><a name="huetinck3831c01-sec2-0025" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The principle of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">conservation of energy</span> is one of the most far‐reaching general laws of physics. It states that energy is neither created nor destroyed but can only be transformed from one form to another in an isolated system.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Because the total energy of the system always remains constant, the law of conservation of energy is a useful tool for analyzing a physical situation where energy is changing form. Imagine a swinging pendulum with negligible frictional forces. At the top of its rise, all the energy is gravitational potential energy due to height above the stationary position. At the bottom of the swing, all the energy has been transformed into kinetic energy of motion. The total energy is the sum of the kinetic and potential energies. It maintains the same value throughout the back and forth motion of a swing<xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0026">.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0026"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.prd.techspa.com/assets/10054.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <img alt="Pendulum - Wikipedia" src="data:image/png;base64,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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span>A pendulum obeys the law of conservation of energy.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></div><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay id="huetinck3831c01-mdis-0017" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9985.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">At point C, the potential energy is dependent upon the height, and the rest of the total energy is kinetic energy.</p><sect1 id="huetinck3831c01-sec2-0026" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Although total energy is conserved, kinetic energy need not be conserved. A collision between two objects with conservation of kinetic energy is called an <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">elastic collision.</span> Colliding objects interacting with losses of kinetic energy due to frictional losses or deformation of an object are called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">inelastic collisions.</span> In the macroscopic world, most collisions are inelastic; however, losses of kinetic energy are negligible in the nearly elastic collisions between atomic particles and subatomic particles. For these cases, the law of conservation of momentum and the conservation of kinetic energy yield useful equations.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Consider a simple head‐on elastic collision where one mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) with a given velocity ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) hits a second mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>) that is initially at rest. Apply the laws of conservation of momentum and conservation of kinetic energy to get <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> + <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>′ and (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> = (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′ <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> + (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em>′ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>, where the primes refer to velocities after the collision. Solving the equations gives the velocities of the two masses after the interaction: <mathdisplay id="huetinck3831c01-mdis-0018" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9988.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Three special cases are instructive:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For equal masses where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>, note that <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′ becomes zero and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>′ equals <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>; thus, for equal masses, the objects simply exchange velocities, as is sometimes observed with pool balls. (Pool balls have rotational energy and somewhat inelastic collisions, so their behavior only approximates the example.)</p><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> is massive, the numerator and denominator are nearly the same in the equation for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′. Then, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′ is approximately equal to <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> but in the opposite direction. The denominator of the expression for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>′ will be so large so that the velocity of the second mass after the collision will be small. In other words, the incoming mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) will bounce back off the second mass with nearly the initial speed, and the hit mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>) will move slowly after the collision.</p><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> is massive, then <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>′ is approximately equal to <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>, and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>′ is nearly twice <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>; or the incoming massive particle continues at nearly the same velocity, and the hit mass moves ahead at nearly twice the initial velocity of the first mass after the collision.</p><sect1 id="huetinck3831c01-sec2-0027" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0027" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0027" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Center of mass</a></h2><a name="huetinck3831c01-sec2-0027" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The concept of the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">center of mass</span> (CM) is useful to analyze the motion of a system of particles. The system of particles acts as if all of its mass is concentrated at the CM. In the absence of an external force, if the CM of the system is at rest, then it will remain at rest, and if it is initially in motion, it will maintain that motion. In other words, the CM moves in accordance with Newton's second law. The <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> coordinates of the center of mass are <mathdisplay id="huetinck3831c01-mdis-0019" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9991.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Consider the previous example of a head‐on collision of two equal masses that stick together after the collision. The CM is initially in motion at a constant velocity and maintains the same velocity after the collision. As the first mass rolls in toward the second mass, the CM is always halfway between the two masses. Before the collision, the CM covers one‐half the distance of the incoming object in the same time, and therefore, the velocity of the CM is one‐half the initial velocity of the incoming mass. For the instant that the two masses interact, the CM is right between the two objects. After the collision, the masses stick together and have one‐half the initial velocity because the effective mass has doubled. The CM continues midway between the masses. It maintains the same velocity of (1/2) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>after the collision. In Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0027">, the moving white ball impacts the stationary black ball. The numbered and circled positions of the CM correspond to the numbered positions of the balls.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10055.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;"><span> </span><span> </span><span> </span>Inelastic collision of two balls.</span></p></sect1></item1></item1></sect1></sect1></sect1></sect1></sect1></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-62903312990290138032020-08-23T22:04:00.000+05:302020-08-23T22:07:15.445+05:30DYNAMICS ( COMPLETE )<p> <u style="background-color: white; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem;"><span style="color: #ff00fe;">Dynamics</span></u></p><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;">The study of dynamics goes beyond the relationships between the variables of motion as illuminated in kinematics to the cause of motion, which is force.<p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0013" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0013" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Newton's laws of motion</a></span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Newton's first law of motion,</span> also called the law on inertia, states that an object continues in its state of rest or of uniform motion unless compelled to change that state by an external force. The law appears to contain two separate statements. The first statement—that a state of rest will continue unless a force is applied—seems intuitively correct. The second statement—that an object will continue with a constant velocity unless compelled to change by an impressed force—seems contrary to common experience. It is important to realize that objects observed to slow down are being compelled to change by a frictional force. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Friction</span> is a retarding force that is ever present in our everyday world. For the ideal—the absence of outside forces acting on the object, as described by the law—friction must be eliminated. The value of the law is the introduction of the concept of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">force</span> as a push or pull that causes a body to change its state of motion.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Newton's second law of motion</span> states that if a net force acts on an object, it will cause an acceleration of that object. The law addresses the cause and effect relationship between force and motion commonly stated as <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">a</span>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> is the proportionality constant (mass). Force is measured in SI units of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">newtons,</span> abbreviated N.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Newton's third law of motion</span> states that for every action there is an equal and opposite reaction. Therefore, if one object exerts a force on a second object, the second exerts an equal and oppositely directed force on the first one.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Mass and weight</a></h2><sect1 id="huetinck3831c01-sec2-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Mass</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">weight</span> are distinctly different physical quantities, a fact that cannot be emphasized too strongly. Mass is the property that lends an object a reluctance to change its state of motion. Mass is the measure of the amount of matter in an object. Masses are compared on an equal‐arm balance. If a loaded two‐pan balance is level on earth, it will be level in a different gravitational field, as for example, on the moon. Thus, mass is an invariant quantity; it is measured in units of kilograms. A mass of 1 kilogram will experience an acceleration of 1 m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> under the action of a force of 1 newton.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The force that the earth exerts on an object of specific mass is called the object's weight on earth. Weight is a force measured in units of newtons and is a vector quantity. The expression for weight is <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">W</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">g</span>, where <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">g</span> is the acceleration due to gravity. A spring scale translates the force of attraction between an object and the earth into a reading of weight. In contrast to a measurement of mass, weight is not an invariant. An object on a spring scale on earth would not weigh the same on the moon because the pull of gravity on the object differs in the two locations.</p><sect1 id="huetinck3831c01-sec2-0015" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0015" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0015" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Force diagrams</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To better understand the relationship between force and acceleration in a particular case, it is helpful to use a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">force diagram,</span> also called a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">free‐body diagram.</span> An object that is not moving is said to be in a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">static equilibrium.</span> An example is a weight hanging by two ropes from the ceiling (see Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0014">). To analyze this problem, consider the forces acting on the knot joining the ropes. Then,</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="1"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p></sect1></sect1><blockquote style="-webkit-tap-highlight-color: transparent; border: none; box-sizing: inherit; color: #6f6f6f; line-height: 1.6; margin: 0px 0px 0px 40px; padding: 0px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">1. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Make the force diagram.</span></p></sect1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">2. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Find the components of forces not directed along the coordinate axes and write the force equation for each axis.</span></p></item1></sect1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">3. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Solve the simultaneous equations for the tensions.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10030.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><img alt="Weight Suspended Stock Illustrations – 139 Weight Suspended Stock ..." height="323" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcRb2rXU6uNn6pUkFhH5DmrNeSft6uWpD2eNMA&usqp=CAU" width="243" /><span style="color: #222222;"> </span></sect1></blockquote><blockquote style="-webkit-tap-highlight-color: transparent; border: none; box-sizing: inherit; color: #6f6f6f; line-height: 1.6; margin: 0px 0px 0px 40px; padding: 0px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><span style="font-size: 1rem; font-weight: bold;">Force diagram of a suspended weight.</span></sect1></blockquote><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The following examples illustrate these procedures. (All of the following vector diagrams are drawn to scale.)</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 1:</span> What is the tension in each rope in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0014">?</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="1"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">1. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Make the force diagram.</span></p><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">The tension in the lower rope attached to the mass must be <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> directed downward; therefore, T <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">3</sub> = − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em>.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">2.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Find the components of forces not directed along the coordinate axes and write the force equation for each axis.</span></p><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">Components of the forces in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction are <mathdisplay id="huetinck3831c01-mdis-0011" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9974.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">Components of the forces in the y direction are <mathdisplay id="huetinck3831c01-mdis-0012" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9976.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></li></ul><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">3. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Solve the simultaneous equations for the tensions.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <img alt="" src="https://www.cliffsnotes.com/assets/10070.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To minimize computation errors, show the components on a separate force diagram as shown in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0014">.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 2:</span> Now, see if you can make the free‐body diagram and set up the force equations for a pail on the end of a rope that is accelerating upward. Find the tension in the rope and the acceleration of the pail. See Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0015">(a).<figureanchor figures="huetinck3831c01-fig-0015" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10033.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><img alt="Why is lifting a bucket of water using a pulley easier than ..." height="271" 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" width="289" /></div><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="color: black; font-size: 1rem;"><span> </span><span> </span><span> </span><span> </span>Force diagram of a bucket being lifted.</span></div><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0015"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The two forces acting on the pail are the tension of the rope ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">T</span>) and weight ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">W</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em>). By Newton's second law:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">net</sub> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">ma</em></bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 3:</span> Next, try to set up the equations for a two‐body system of unequal masses attached by a rope over a frictionless pulley (see Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0016">). A diagram must be made for each of the two objects of mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) and ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>). Find the acceleration of the system and the tension in the rope. </xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0016"><mathdisplay id="huetinck3831c01-mdis-0013" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9978.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><figureanchor figures="huetinck3831c01-fig-0016" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></mathdisplay></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10035.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><img alt="Problem: Two masses on a pulley | Phyley" height="281" src="data:image/png;base64,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" width="281" /></div><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="color: black; font-size: 1rem;"> Force diagram of two masses hung over a pulley.</span></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The second equation has a negative acceleration because <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> is descending. Because the objects are connected by a rope that does not expand, the tensions and accelerations are the same for each mass. By algebraic manipulation, the equations may be simultaneously solved for the following results:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <img alt="" src="https://www.cliffsnotes.com/assets/10071.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 4:</span> In Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0017">, one object sits on a frictionless surface, and the other object hangs off the edge of the table over a pulley. Make the free‐body diagram and write the force equations to find the acceleration and tension.<figureanchor figures="huetinck3831c01-fig-0017" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10036.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><img alt="Solved: Two blocks are connected by a massless rope as shown be ..." height="314" src="data:image/png;base64,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" width="391" /></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;"> Same as PREVIOUS with one mass on a frictionless table.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In the vertical direction, the forces on <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> are the weight downward and the normal force ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">N</span>) upward due to the surface, which is equal and opposite to the weight. Because there is no acceleration of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> in this direction, the net vertical force is zero. A horizontal force of the tension in the rope accelerates the mass to the right. Write the force equations separately for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> directions for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>. The forces on the second mass are the same as those in the last example. For <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em>; in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> = 0. For <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> = − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em>. Combining the two equations gives the relationships.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <img alt="" src="https://www.cliffsnotes.com/assets/10072.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Even more complicated problems can be separated into manageable parts to allow solution by using these problem‐solving methods.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 5:</span> Consider a mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) on an inclined plane attached to a mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>) over a pulley as in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0018">(a). Both the plane and pulley are frictionless. Set up the problem to find the acceleration.<figureanchor figures="huetinck3831c01-fig-0018" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10037.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Using a tilted Atwood's machine" height="273" src="data:image/png;base64,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" width="444" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span>Now the table is tilted at an angle.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In this case, the forces on <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> must be resolved in components along the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> axes. The coordinate system with the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> axis parallel to the surface of the plane is selected so that only one force, the weight of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>, needs to be converted into component form. The normal force is always perpendicular to the plane and in this case, therefore, is opposed only be the component of weight that is also perpendicular to the plane's surface. Note that the angle between the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> axis and the weight ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em>) is the angle of inclination of the plane, which can be proven by geometry. The coordinate system for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> has the same orientation as in the previous example. Assume <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> > <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>. Then, for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> in its <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> cos θ = 0; in its <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> sin θ = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em>. For <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>: <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> = − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em>. The acceleration is then</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <img alt="" src="https://www.cliffsnotes.com/assets/10073.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">and tension is <img alt="" src="https://www.cliffsnotes.com/assets/10074.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To analyze a physical situation by the use of free‐body diagrams, use the following steps:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="1"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p></xref></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1><blockquote style="-webkit-tap-highlight-color: transparent; border: none; box-sizing: inherit; color: #6f6f6f; line-height: 1.6; margin: 0px 0px 0px 40px; padding: 0px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">1. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Make a free‐body diagram for each object.</span> If one object is sitting on a surface, be sure to include the normal force.</p></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">2.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Resolve the forces that are not directed along the x and y axes into components along a preferred coordinate system.</span> For inclined planes, use a coordinate axis with the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> axis parallel to the surface of the plane. Put the components on a separate diagram; that is, do not put the force and its components on the same diagram because this combination might complicate the following steps.</p></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">3. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Write out the force equation for each mass along each axis, noting the correct sign for the acceleration of the body.</span></p></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">4. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Solve the equations simultaneously to find the desired value(s).</span></p></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></blockquote><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 id="huetinck3831c01-sec2-0016" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0016" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0016" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Friction</a></span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Friction</span> is the force opposing the motion of one body sliding or rolling over the surface of second object. Several aspects of friction are important at low velocities:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">The direction of the force of friction is opposite the direction of motion.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">The frictional force is proportional to the perpendicular (normal) force between the two surfaces in contact.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">The frictional force is nearly independent of the area of contact between the two objects.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">The magnitude of the frictional force depends on the materials composing the two objects in contact.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Static friction</span> is the force of friction when there is no relative motion between two objects in contact, such as a block sitting on an inclined plane. The magnitude of the frictional force is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> ≤ μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> is the magnitude of the normal force, and the coefficient of friction (μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub>) is the dimensionless proportionality constant. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">coefficient of friction</span> may be approximately .2 for normal lubricated surfaces and close to 1 for glass sliding on glass. This equation sets the upper limit for the static frictional force. If a greater external force is applied, the situation will no longer be static, and the object will begin to move.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Kinetic friction</span> is the force of friction when there is relative motion between two objects in contact. The magnitude of the friction force in this case is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> </sub>≤ μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">k</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> is the magnitude of the normal force and μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> </sub>is the coefficient of kinetic friction. Note that μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> </sub>is not strictly a constant, but this empirical rule is a good approximation for finding frictional forces. Values given for the coefficients of static and kinetic friction do vary with speed and surface conditions so that it is not necessarily true that static friction exceeds sliding friction.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The following problem highlights the differences between static and kinetic friction.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 6:</span> A block sits on an inclined plane. What is the maximum angle that which the block remains at rest? First, draw the free‐body diagrams and then write out the force equation for each direction of the coordinate system (see Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0019">).<figureanchor figures="huetinck3831c01-fig-0019" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" 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width="430" /></span></xref></item1></item1></item1></sect1></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></article><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0019"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: black; font-size: 1rem;"> A block on an inclined plane, with friction.</span></span></xref></item1></item1></item1></sect1></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></article><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0019"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><br /></span></xref><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Suppose the surface is tilted to θ, at which the block just begins to move. Then, the force down the plane must be equal to the maximum force of static friction; thus <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">friction</em> </sub>= μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>. Therefore, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">friction</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> sin θ = μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> = μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> cos θ) and solving for the coefficient of friction:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">s</sub> = tan θ</bi></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">At a greater angle of tilt, the object accelerates down the surface, and the force of friction is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> </sub>= μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">k</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Example 7:</span> If the surfaces in Figures <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0017">and <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0018">were <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">not</em> frictionless, the frictional force parallel to the surface and opposite the direction of motion must be included in the analysis. Pulling a block along a horizontal surface at a constant speed (zero acceleration) is an example of a problem involving friction, and such a block is analyzed in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0020">.<figureanchor figures="huetinck3831c01-fig-0020" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; margin: 0px; padding: 0px;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10043.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222;" target="huetinck3831c01-fig-0017"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0018"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0020"><img alt="What is a Free-Body Diagram and How to Draw it (with Examples ..." height="439" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQTikIsasU2JP_ji3bM0wfZ0Dq1r6oU_Rcazw&usqp=CAU" width="439" /></xref></xref></xref></item1></item1></item1></sect1></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></article><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222;" target="huetinck3831c01-fig-0017"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0018"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0020"><span style="color: black; font-size: 1rem;"><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span>Pulling a block on a plane with friction.</span></xref></xref></xref></item1></item1></item1></sect1></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></article><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><figureanchor figures="huetinck3831c01-fig-0014" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><list2 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="plain"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0017"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0018"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0020"><br /></xref></xref></xref><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> cos θ − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> = 0, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> is the friction force and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> is the tension in the rope. In the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em> + <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> sin θ − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> = 0; also, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> = μ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">k</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Solve the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction equation for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">N</em>, substitute the expression into the friction force equation, and then substitute friction into the first equation to obtain the following:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <mathdisplay id="huetinck3831c01-mdis-0014" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9981.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><bi style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">Solution: <img alt="" src="https://www.cliffsnotes.com/assets/10075.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /> solving for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em>.</bi></p><sect1 id="huetinck3831c01-sec2-0017" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0017" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0017" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Centripetal force</a></h2><a name="huetinck3831c01-sec2-0017" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">When an object rotates in a circle, a force must be directed to the center of the circle to maintain the motion; otherwise, the object will take off tangent to the path. This constraining force is called a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">centripetal force,</span> meaning center‐seeking. In the example of a mass rotating in a horizontal circle at the end of the string, the centripetal force is provided by the tension in the string. In the case of orbiting satellites, gravity provides the center‐seeking force. From the definition of force <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">F</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">a</span> and the expressions for circular accelerations, the following equations are obtained: </p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay id="huetinck3831c01-mdis-0015" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9983.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If an object moves in a circle, the net force is a centripetal force. One such example is the conic pendulum, a mass on the end of a string that rotates in a horizontal circle (see Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0021">).<figureanchor figures="huetinck3831c01-fig-0021" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="http://cliffsnotescms-v1.stage.techspa.com/assets/10045.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0021"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span> </span><span> </span><span> </span><span> </span><img alt="Conical Pendulum: Its time period, tension in string, numerical ..." height="269" src="data:image/png;base64,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" style="color: #222222;" width="456" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0021"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> <span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span></span>A conic pendulum<br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> cos θ − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mg</em> = 0. In the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction (or the radial direction), <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> sin θ = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mv</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">R</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">R</em> is the radius of circular path. Dividing the second equation by the first equation and solving for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> yields the following:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <mathdisplay id="huetinck3831c01-mdis-0016" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9984.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><sect1 id="huetinck3831c01-sec2-0018" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0018" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0018" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Universal gravitation</a></span></h2><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0018" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Newton's law of universal gravitation</span> states that every mass in the universe attracts every other mass with a gravitational force that is directly proportional to the product of their masses ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>) and inversely proportional to the square of distance <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(r)</em> between them. In mathematical form, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">F</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Gm</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">r</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>), where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">G</em> is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">universal gravitational constant.</span> In the metric system, the accepted value of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">G</em> is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">G</em> = 6.673 × 10 <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">−11</sup> ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Nm</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sup>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">kg</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Kepler found three empirical laws regarding the motion of satellites that Newton later showed from his law of universal gravitation. These are <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Kepler's laws of planetary motion:</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">The law of orbits:</span> All planets move in elliptical orbits with the sun at one focus.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">The law of areas:</span> A line joining a planet and the sun sweeps out equal areas in equal time.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">The law of periods:</span> The square of the period <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(T)</em> of any planet is proportional to the cube of the semi‐major axis ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">r</em>) of its orbit, or <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> = (4μ <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">GM</em>) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">r</em> <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">3</sup>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">M</em> is the mass of the planet.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><sect1 id="huetinck3831c01-sec2-0019" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0019" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline !important; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0019" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Momentum and impulse</a></h2><a name="huetinck3831c01-sec2-0019" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">According to Newton's second law, a mass experiencing a net average force ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">F</span>) for a time interval Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> will undergo an average acceleration ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">F</span>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">a</span>). The product of the average force acting on the body and time of contact is defined as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">impulse.</span> Because acceleration is change in velocity, the relationships between these variables are expressed as the Impulse = F (Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>) = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">i</em>, where v <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">i</sub> is the initial velocity and v <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> is the velocity after the force is no longer in contact with the body. Impulse is measured in units of newton‐seconds, or more simply, N‐s.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">When applying the impulse equation, be sure to calculate the vector change in velocity—for example, consider a mass of 10 kg acted on by a force that changes its velocity from −8 m/s to 3 m/s . This force imparted an impulse of (10 kg)[3 − (−8) m/s] = (10 kg)(11 m/s) = 110 N‐s.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The right side of the impulse equation is the change in the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">linear momentum</span> of the object. The definition of linear momentum is <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">p</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">v</span>. Linear momentum is measured in units of kilogram meters/second or, in abbreviated form, kg m/s. Newton originally stated his second law by saying that the rate of change of momentum with time is proportional to the impressed force and is in the same direction; thus, F = Δ ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v)/Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> or F = Δp/Δ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>.</p><sect1 id="huetinck3831c01-sec2-0020" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0020" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0020" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Conservation of momentum</a></h2><a name="huetinck3831c01-sec2-0020" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">An extremely important fundamental principle in physics is the law of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">conservation of momentum</span>. The law states that if there is no external force acting on a system, the total momentum remains a constant, which provides a powerful way to analyze interactions between systems of objects. For example, if a rolling ball on a frictionless surface collides with another ball, the total momentum before and after the collision is the same. An interaction, therefore, can be examined without knowing the forces involved and the length of interaction time, which might be difficult to measure.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">First, consider a head‐on collision (so it is not necessary to utilize two‐dimensional vectors, that is, consider only straight line motion). Imagine a mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) with velocity ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) hitting a mass ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>) that is initially at rest. The momentums are the following: before the collision, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>, and after the collision, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> + <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>, where the primes indicate velocities after the interaction. From the law of conservation of momentum, the two expressions may be set equal to each other. Consider the special case where the two masses are equal on frictionless surface and stick together after the collision (have the same primed velocity). Then, total momentum before the collision equals total momentum after the collision, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">mv</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v′ + <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v′; therefore, v′ = (1/2)(v <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>), or the final velocity is one‐half the original velocity because the effective mass has doubled.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Another way to state the law of conservation of momentum is that the change in momentum of the two objects must be equal and opposite. For example, two ice skaters are at rest in the center of frictionless ice (possible at least in the imagination). Let one have a relatively small mass <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(m)</em> and the other a larger mass <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(M)</em>. Because they begin at rest, the initial momentum is zero. They then push apart in opposite directions. The total momentum must remain zero.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">According to the law of conservation of momentum, Δp <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">m</sub> = −Δp <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">m</sub> or <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em>v′ − 0 = −( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">M</em>V′−0); therefore, if the large mass <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(M)</em> is three times the smaller mass <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(m)</em>, v′ = −3V′, where v′ is the velocity of the small mass after the collision and V′ is the velocity of the large mass after the collision. The negative sign indicates velocities in opposite directions.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">This same analysis holds for a person standing on frictionless ice who throws an object; it even holds for a rocket going to the moon. The ice skater throwing a glove attains equal momentum in the direction opposite to that of the thrown object. This basic principle is the same for a rocket accelerating in space. Spacecrafts utilize the law of conservation of momentum in getting an additional push from discharged rocket stages as well as from fuel. In particular, the Apollo space capsule returning from the moon was only a small percent of the total mass initially sent upward from the launch pad; therefore, acceleration of a rocket can be caused by either a change in velocity, by a change in its mass, or by changes in both velocity and mass (see Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0022">). Thus, the expression of Newton's second law of motion, stated in terms of the change of momentum, is broader than the expression given only in terms of mass and acceleration.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If two objects strike with a glancing blow, the motion will be two‐dimensional. For example, one ball ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub>) with an initial velocity hits a second ball ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>), which is initially at rest. Figures <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0023">and <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0024">depict this situation with the first ball initially moving up from the bottom of the page. For the sake of simplicity, allow the two masses to be equal.</xref></xref></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: bold; margin: 0px; padding: 0px;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The momentum vectors can be added to show the law of conservation of momentum. The vector addition in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0024">(a) shows that the total of the two momentum vectors, <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">p</span>1′ and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">p</span>2′, after the collision are equal to the total momentum before the collision. (Because only <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> was moving, there was only one initial momentum vector, <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">p</span>1.) Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0024">(b) also shows the alternate method of using the law of conservation of momentum, that the change (difference) in momentum of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">1</sub> is equal and opposite to that of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">m</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sub>.</xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">If the two masses are not equal, the velocity vectors must be adjusted so that the vectors represent momentum. For example, if one mass is three times the other, the velocity vectors of the larger mass must be lengthened by a factor of three before using the law of conservation of momentum.</p></sect1></sect1></item1></item1></sect1></sect1></item1></item1></item1></sect1></item1></item1></item1></item1></item2></item2></list2></item1></item2></list2></figureanchor></item1></item1></sect1></sect1></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-87168519138151964892020-08-23T21:51:00.003+05:302020-08-23T22:07:15.446+05:30KINEMATICS ( TWO DIMENSION )<p> <u style="background-color: white; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem;"><span style="color: red;">Kinematics in Two Dimensions</span></u></p><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;">Our world is three‐dimensional. For easier analysis, many motions can be simplified to two dimensions. For example, an object fired into the air moves in a vertical, two‐dimensional plane; also, horizontal motion over the earth's surface is two‐dimensional for short distances. Elementary vector algebra is required to examine the relationships between vector quantities in two dimensions.<p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0007" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0007" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Addition and subtraction of vectors: geometric method</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The vector <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> shown in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0004">(a) represents a velocity of 10 m/s northeast, and vector <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> represents a velocity of 20 m/s at 30 degrees north of east. (A vector is named with a letter in <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">boldface,</span> nonitalic type, and its magnitude is named with the same letter in regular, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">italic</em> type. You will often see vectors in the figures of the book that are represented by their magnitudes in the mathematical expressions.) Vectors may be moved over the plane if the represented length and direction are preserved.<figureanchor figures="huetinck3831c01-fig-0004" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0004"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10008.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;">Graphical addition of vectors, </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; white-space: nowrap;">A + B = C.</span></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0004">(b), the same vectors are positioned to be geometrically added. The tail of one vector, in this case <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span>, is moved to the head of the other vector ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>). The vector sum ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>) is the vector that extends from the tail of one vector to the head of the other. To find the magnitude of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C,</span> measure along its length and use the given scale to determine the velocity represented. To find the direction θ of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C,</span> measure the angle to the horizontal axis at the tail of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0005">(a) shows that <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A + B = B + A</span>. The sum of the vectors is called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">resultant</span> and is the diagonal of a parallelogram with sides <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B.</span> Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0005">(b) illustrates the construction for adding four vectors. The resultant vector is the vector that results in the one that completes the polygon.<figureanchor figures="huetinck3831c01-fig-0005" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0005"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10009.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;">(a) </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; white-space: nowrap;">A + B = B + A</span><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;">. (b) Graphical addition of several vectors.</span></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To subtract vectors, place the tails together. The difference of the two vectors ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">D</span>) is the vector that begins at the head of the subtracted vector ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>) and goes to the head of the other vector ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span>). An alternate method is to add the negative of a vector, which is a vector with the same length but pointing in the opposite direction. The second method is demonstrated in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0006">.<figureanchor figures="huetinck3831c01-fig-0006" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0006"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10010.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; white-space: nowrap;">Graphical subtraction of vectors,</span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; font-weight: 700; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; white-space: nowrap;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">A − B = D.</span></span></xref></p><sect1 id="huetinck3831c01-sec2-0008" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0008" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0008" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Addition and subtraction of vectors: Component method</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For precision in adding vectors, an analytical method using basic trigonometry is required because scale drawings do not give accurate values.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Consider vector <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> in the rectangular coordinate system of Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0007">. The vector <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> can be expressed as the sum of two vectors along the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> axes, A = A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">x</sub> + A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">y</sub>, where A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">x</sub> and A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">y</sub> are called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">components</span> of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span>. The direction of A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">x</sub> is parallel to the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> axis, and that of A <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">y</sub> is parallel to the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> axis. The magnitudes of the components are obtained from the definitions of the sine and cosine of an angle: cos θ = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> and sin θ = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em>, or</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0007"><mathdisplay id="huetinck3831c01-mdis-0008" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9969.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><figureanchor figures="huetinck3831c01-fig-0007" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></mathdisplay></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0007"><mathdisplay numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10014.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="font-size: 1rem;">Components of a vector.</span></mathdisplay></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To add vectors numerically, first find the components of all the vectors. The signs of the components are the same as the signs of the cosine and sine in the given quadrant. Then, sum the components in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction, and sum the components in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction. As shown in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0008">, the sum of the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> components and the sum of the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> components of the given vectors ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>) comprise the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> components of the resultant vector ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>).<figureanchor figures="huetinck3831c01-fig-0008" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0008"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10017.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-align: left; white-space: nowrap;">Component method of vector addition,</span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; font-weight: 700; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-align: left; white-space: nowrap;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; line-height: inherit;">A + B = C</span>.</span></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0008"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">These resultant components form the two sides of a right angle with a hypotenuse of the magnitude of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>; thus, the magnitude of the resultant is</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <mathdisplay id="huetinck3831c01-mdis-0009" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9970.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The direction of the resultant ( <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>) is calculated from the tangent because tan θ = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">C</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">C</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>. To solve for the angle θ, use θ = tan <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">−1</sup> ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">C</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">C</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub>).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The procedure can be summarized as follows:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="1"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p></sect1><blockquote style="-webkit-tap-highlight-color: transparent; border: none; box-sizing: inherit; color: #6f6f6f; line-height: 1.6; margin: 0px 0px 0px 40px; padding: 0px;"><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">1.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Sketch the vectors on a coordinate system.</span></p></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">2.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Find the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> components of all the vectors, with the appropriate signs.</span></p></item1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">3.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Sum the components in both the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> directions.</span></p></item1></item1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">4.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Find the magnitude of the resultant vector from the Pythagorean theorem.</span></p></item1></item1></item1></sect1><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">5.<span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"> Find the direction of the resultant vector using the tangent function.</span></p></item1></item1></item1></item1></sect1></blockquote><sect1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Follow the same procedure to subtract vectors by calculating the appropriate algebraic sum of the components in Step 3.</p><sect1 id="huetinck3831c01-sec2-0009" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0009" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0009" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Multiplication of vectors</a></span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">The dot product:</span> There are two different ways in which two vectors may be multiplied together. The first is the dot product, also called the scalar product, which is written <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>. This can be evaluated in two ways:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">B</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub>+ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">B</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em></sub></li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">AB</em> cos θ, where θ is the angle between the vectors when they are set tail to tail, and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">B</em> are the lengths of the vectors.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Note that the order of the vectors does not matter and that the result of the dot product is a scalar rather than a vector. Note that if two vectors are perpendicular, their dot product is zero according to the second rule above.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Cross product:</span> The second way to multiply vectors is called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cross product</span> or the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">vector product</span>. It is written <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>. It can be evaluated in two ways:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">B</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">A</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">B</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">z</span>, when the vectors <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> both are in <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x–y</em> plane. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">z</span> indicates that the result is a vector that points along the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">z</em> axis. In general, the vector resulting from a cross product is always perpendicular to both of the vectors being multiplied together.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">AB</em> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">z</span> sin θ, where θ is the angle between the vectors <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> when they are placed tail to tail. Again, the result is a vector perpendicular to <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> (and therefore points along the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">z</em> axis if <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> are in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x–y</em> plane).</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The result of a cross product does depend on the order of the vectors. Note from the first rule that <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = − <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> · <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span>. Also, if <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> are parallel, the second rule implies that their cross product is zero.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Finally, the cross product give rise to the “right hand rule,” which allows you to easily determine the direction of the resulting vector. For the general expression <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A</span> × <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span> = <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span>, point your thumb in the direction of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">A.</span> Now point your index finger in the direction of <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">B</span>; if necessary, flip over your hand. The vector <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">C</span> points outward from your palm.</p><sect1 id="huetinck3831c01-sec2-0010" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0010" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0010" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Velocity and acceleration vectors in two dimensions</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For motion in two dimensions, the earlier kinematics equations must be expressed in vector form. For example, the average velocity vector is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>)/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>are the initial and final displacement vectors and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> is the time elapsed. As noted earlier, the velocity and displacement vectors are shown in bold type, whereas the scalar <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(t)</em> is not. In similar fashion, the average acceleration vector is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>− <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>)/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>are the initial and final velocity vectors.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">An important point is that the acceleration can arise from a change in the magnitude of the velocity (speed) as well as from a change in the direction of the velocity. If an object travels around a circle at a constant speed, there is an acceleration due to the change in the direction of the velocity, even though the magnitude of the velocity does not change. A mass moves in a horizontal circle with a constant speed in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0009">. The velocity vectors at positions 1 and 2 are subtracted to find the average acceleration, which is directed toward the center of the circle. (Note that the average acceleration vector is placed at the midpoint of the path in the given time interval.)<figureanchor figures="huetinck3831c01-fig-0009" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0009"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10020.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; text-align: left; white-space: nowrap;">Velocity and acceleration vectors of an object moving in a circle</span></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0009"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; text-align: left;">The following discussion summarizes the four different cases for acceleration in a plane:</span><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 1:</span> Zero acceleration</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 2:</span> Acceleration due to changing direction but not speed</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 3:</span> Acceleration due to changing speed but not direction</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 4:</span> Acceleration due to changing both speed and direction.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Imagine a ball rolling on a horizontal surface that is illuminated by a stroboscopic light. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0010">(a) shows the position of the ball at even intervals of time along a dotted path. Case 1 is illustrated in positions 1 through 3; the magnitude and direction of the velocity do not change (the pictures are evenly spaced and in a straight line), and therefore, there is no acceleration. Case 2 is indicated for positions 3 through 5; the ball has constant speed but changing direction, and therefore, an acceleration exists. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0010">(b) illustrates the subtraction of v <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">3</sub> and v <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">4</sub> and the resulting acceleration toward the center of the arc. Case 3 occurs from positions 5 to 7; the direction of the velocity is constant, but the magnitude changes. The acceleration for this portion of the path is along the direction of motion. The ball curves from position 7 to 9, showing case 4; the velocity changes both direction and magnitude. In this case, the acceleration is directed nearly upward between 7 and 8 and has a component toward the center of the arc due to the change in direction of the velocity and a component along the path due to the change in the magnitude of the velocity.<figureanchor figures="huetinck3831c01-fig-0010" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0010"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0010"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10022.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; text-align: left; white-space: nowrap;">(a) Path of a ball on a table. (b) Acceleration between points 3 and 4.</span></xref></xref></p><sect1 id="huetinck3831c01-sec2-0011" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><a name="huetinck3831c01-sec2-0011" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;">Projectile motion</h2></a></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Anyone who has observed a tossed object—for example, a baseball in flight—has observed <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">projectile motion</span>. To analyze this common type of motion, three basic assumptions are made: (1) acceleration due to gravity is constant and directed downward, (2) the effect of air resistance is negligible, and (3) the surface of the earth is a stationary plane (that is, the curvature of the earth's surface and the rotation of the earth are negligible).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">To analyze the motion, separate the two‐dimensional motion into vertical and horizontal components. Vertically, the object undergoes constant acceleration due to gravity. Horizontally, the object experiences no acceleration and, therefore, maintains a constant velocity. This velocity is illustrated in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0011">where the velocity components change in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> direction; however, they are all of the same length in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> direction (constant). Note that the velocity vector changes with time due to the fact that the vertical component is changing.<figureanchor figures="huetinck3831c01-fig-0011" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10023.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;"> Projectile motion.</span></div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;">In this example, the particle leaves the origin with an initial velocity ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>), up at an angle of θ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">o</sub>. The original <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> components of the velocity are given by <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x0</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y0</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>sin θ <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>.</div><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">With the motions separated into components, the quantities in the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> directions can be analyzed with the one‐dimensional motion equations subscripted for each direction: for the horizontal direction, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x0</em> </sub>and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x0</em> </sub><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>; for vertical direction, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>= <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y0</em> </sub>− gt and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y0</em> </sub>− (1/2) gt <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>, where <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">x</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> represent distances in the horizontal and vertical directions, respectively, and the acceleration due to gravity ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em>) is 9.8 m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>. (The negative sign is already incorporated into the equations.) If the object is fired down at an angle, the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> component of the initial velocity is negative. The speed of the projectile at any instant can be calculated from the components at that time from the Pythagorean theorem, and the direction can be found from the inverse tangent on the ratios of the components:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <mathdisplay id="huetinck3831c01-mdis-0010" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9971.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Other information is useful in solving projectile problems. Consider the example shown in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0011">where the projectile is fired up at an angle from ground level and returns to the same level. The time for the projectile to reach the ground from its highest point is equal to the time of fall for a freely falling object that falls straight down from the same height. This equality of time is because the horizontal component of the initial velocity of the projectile affects how far the projectile travels horizontally but not the time of flight. Projectile paths are parabolic and, therefore, symmetric. Also for this case, the object reaches the top of its rise in half of the total time <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(T)</em> of flight. At the top of the rise, the vertical velocity is zero. (The acceleration is always <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em>, even at the top of the flight.) These facts can be used to derive the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">range</span> of the projectile, or the distance traveled horizontally. At maximum height, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">y</em> </sub>= 0 and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em>/2; therefore, the velocity equation in the vertical direction becomes 0 = <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>sin θ − <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em>/2 or solving for <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em>, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> = (2 <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">0</sub> sin θ)/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em>.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Substitution into the horizontal distance equation yields <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">R</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>cos θ) <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em>. Substitute <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">T</em> in the range equation and use the trigonometry identity sin 2θ = 2 sin θ cos θ to obtain an expression for the range in terms of the initial speed and angle of motion, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">R</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>/ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">g</em>) sin 2θ. As indicated by this expression, the maximum range occurs when θ = 45 degrees because, at this value of θ, sin 2θ has its maximum value of 1. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0012">sketches the trajectories of projectiles thrown with the same initial speed at differing angles of inclination.<figureanchor figures="huetinck3831c01-fig-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10024.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></div><sect1 id="huetinck3831c01-sec2-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; font-weight: 700; line-height: 21px; white-space: nowrap;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; white-space: nowrap;">Range of projectiles launched at different angles.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Uniform circular motion</a></h2><a name="huetinck3831c01-sec2-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">For uniform motion of an object in a horizontal circle of radius </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">(R)</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">, the constant speed is given by </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">v</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> = 2π </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">R</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">/ </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">T</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">, which is the distance of one revolution divided by the time for one revolution. The time for one revolution </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">(T)</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> is defined as </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; font-weight: 700; line-height: inherit;">period.</span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> During one rotation, the head of the velocity vector traces a circle of circumference 2π </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">v</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> in one period; thus, the magnitude of the acceleration is </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">a</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> = 2π </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">v</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">/ </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">T</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">. Combine these two equations to obtain two additional relationships in other variables: </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">a</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> = </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">v</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> </span><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">2</sup><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">/ </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">R</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> and </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">a</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> = (4π </span><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">2</sup><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">/ </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">T</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;"> </span><sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">2</sup><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">) </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif; line-height: inherit;">R</em><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Tahoma, Arial, sans-serif;">.</span></p></a><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0012" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The displacement vector is directed out from the center of the circle of motion. The velocity vector is tangent to the path. The acceleration vector directed to the center of the circle is called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">centripetal acceleration</span>. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0013">shows the displacement, velocity, and acceleration vectors at different positions as the mass travels in a circle on a frictionless horizontal plane.</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10028.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;"> Uniform circular motion.</span></p></sect1></sect1></item1></item1></item1></sect1></item1></item1></sect1></item1></item1></item1></item1></sect1></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-79494517954555440272020-08-23T21:49:00.000+05:302020-08-23T22:07:15.445+05:30KINEMATICS ( ONE DIMENSION )<div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><u><span style="color: #2b00fe;">Kinematics in One Dimension</span></u></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; color: #222222; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;">Kinematics analyzes the positions and motions of objects as a function of time, without regard to the causes of motion. It involves the relationships between the quantities <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">displacement</span> <i style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(d)</i>, <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">velocity</span> <i style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(v)</i>, <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">acceleration</span> <i style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(a)</i>, and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">time</span> <i style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">(t)</i>. The first three of these quantities are vectors.<p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0001" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0001" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Definition of a vector</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">A <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">vector</span> is a physical quantity with direction as well as magnitude, for example, velocity or force. In contrast, a quantity that has only magnitude and no direction, such as temperature or time, is called a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">scalar.</span> A vector is commonly denoted by an arrow drawn with a length proportional to the given magnitude of the physical quantity and with direction shown by the orientation of the head of the arrow.</p><sect1 id="huetinck3831c01-sec2-0002" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0002" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0002" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Displacement and velocity</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Imagine that a car begins traveling along a road after starting from a specific sign post. To know the exact position of the car after it has traveled a given distance, it is necessary to know not only the miles it traveled but also its heading. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">displacement</span>, defined as the change in position of the object, is a vector with the magnitude as a distance, such as 10 miles, and a direction, such as east. Velocity is a vector expression with a magnitude equal to the speed traveled and with an indicated direction of motion. For motion defined on a number line, the direction is specified by a positive or negative sign. Average velocity is mathematically defined as</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> <mathdisplay id="huetinck3831c01-mdis-0001" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9956.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Note that displacement (distance from starting position) is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">not</em> the same as distance traveled. If a car travels one mile east and then returns one mile west, to the same position, the total displacement is zero and so is the average velocity over this time period. Displacement is measured in units of length, such as meters or kilometers, and velocity is measured in units of length per time, such as meters/second (meters per second).</p><sect1 id="huetinck3831c01-sec2-0003" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0003" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1rem; font-weight: 700; line-height: inherit;"><a name="huetinck3831c01-sec2-0003" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: rgb(0, 0, 0) !important; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;">Average acceleration</a></span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Acceleration,</span> defined as the rate of change of velocity, is given by the following equation: </p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9957.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><mathdisplay numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Acceleration units are expressed as length per time divided by time such as meters/second/second or in abbreviated form as m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>.</p><sect1 id="huetinck3831c01-sec2-0004" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0004" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0004" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Graphical interpretations of displacement, velocity, and acceleration</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The distance versus time graph in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0001">shows the progress of a person (I) standing still, (II) walking with a constant velocity, and (III) walking with a slower constant velocity. The slope of the line yields the speed. For example, the speed in segment II is</xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0001"><mathdisplay id="huetinck3831c01-mdis-0003" numbered="no" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9958.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><figureanchor figures="huetinck3831c01-fig-0001" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></mathdisplay></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9999.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></span></div><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></div></span><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Figure 1</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span class="Apple-tab-span" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; white-space: pre;"> </span> </span>Motion of a walking person.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Each segment in the velocity versus time graph in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0002">depicts a different motion of a bicycle: (I) increasing velocity, (II) constant velocity, (III) decreasing velocity, and (IV) velocity in a direction opposite the initial direction (negative). The area between the curve and the time axis represents the distance traveled. For example, the distance traveled during segment I is equal to the area of the triangle with height 15 and base 10. Because the area of a triangle is (1/2)(base)(height), then (1/2)(15 m/s)(10 s) = 75 m. The magnitude of acceleration equals the calculated slope. The acceleration calculation for segment III is (−15 m/s)/(10 s) = −1.5 m/s/s or −1.5 m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>.<figureanchor figures="huetinck3831c01-fig-0002" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0002"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"></p><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; font-weight: 700; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; white-space: nowrap;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10003.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; font-family: latoregular, Helvetica, Arial, sans-serif; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /></span></div><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0002"><div style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 14px; line-height: 21px; margin: 0px; padding: 0px; white-space: nowrap;"><span style="color: #333333; font-family: Verdana, Arial, sans-serif;">Accelerating motion of a bicycle</span></div></xref><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0002"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The more realistic distance‐versus‐time curve in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003">(a) illustrates gradual changes in the motion of a moving car. The speed is nearly constant in the first 2 seconds, as can be seen by the nearly constant slope of the line; however, between 2 and 4 seconds, the speed is steadily decreasing and the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">instantaneous velocity</span> describes how fast the object is moving at a given instant.<figureanchor figures="huetinck3831c01-fig-0003" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></figureanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003"><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/10006.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /><span style="color: #333333; font-family: Verdana, Arial, sans-serif; font-size: 14px; white-space: nowrap;">Motion of a car: (a) distance, (b) velocity, and (c) acceleration change in time.</span></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;">Instantaneous velocity can be read on an odometer in the car. It is calculated from a graph as the slope of a tangent to the curve at the specified time. The slope of the line sketched at 4 seconds is 6 m/s. Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003">(b) is a sketch of the velocity‐versus‐time graph constructed from the slopes of the distance‐versus‐time curve. In like fashion, the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">instantaneous acceleration</span> is found from the slope of a tangent to the velocity‐versus‐time curve at a given time. The instantaneous acceleration‐versus‐time graph in Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003">(c) is the sketch of the slopes of the velocity‐versus‐time graph of Figure <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-fig-0003">(b). With the vertical arrangement shown, it is easy to compute the displacement, velocity, and acceleration of a moving object at the same time.</xref></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For example, at time <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> = 10 s, the displacement is 47 m, the velocity is −5 m/s, and the acceleration is −5 m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup>.</p><sect1 id="huetinck3831c01-sec2-0005" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0005" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0005" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Definitions of instantaneous velocity and instantaneous acceleration</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The instantaneous velocity, by definition, is the limit of the average velocity as the measured time interval is made smaller and smaller. In formal terms, <img alt="" src="https://www.cliffsnotes.com/assets/33081.gif" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />. The notation <img alt="" src="https://www.cliffsnotes.com/assets/33082.gif" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /> means the ratio <img alt="" src="https://www.cliffsnotes.com/assets/33083.gif" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" /> is evaluated as the time interval approaches zero. Similarly, instantaneous acceleration is defined as the limit of the average acceleration as the time interval becomes infinitesimally short. That is, <img alt="" src="https://www.cliffsnotes.com/assets/33084.gif" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />.</p><sect1 id="huetinck3831c01-sec2-0006" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0006" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px; text-decoration-line: underline;"></a></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 17px; font-weight: 400; line-height: 1.385; margin: 20px 0px 0px; outline: 0px; padding: 0px; text-rendering: optimizelegibility;"><a name="huetinck3831c01-sec2-0006" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #008cba; font-family: Verdana, Arial, sans-serif; font-size: 14px; line-height: 21px; margin: 0px; outline: 0px; padding: 0px;">Motion with constant acceleration</a></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">When an object moves with constant acceleration, the velocity increases or decreases at the same rate throughout the motion. The average acceleration equals the instantaneous acceleration when the acceleration is constant. A negative acceleration can indicate either of two conditions:</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><list1 compact="yes" format="hanging" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" type="custom"><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"></item1></list1></p><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 1:</span> The object has a decreasing velocity in the positive direction.</li></ul><ul style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: inherit; font-size: 1rem; line-height: 1.6; list-style-position: outside; margin: 0px 0px 1.25rem 1.1rem; padding: 0px;"><li style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Case 2:</span> The object has an increasing velocity in the negative direction.</li></ul><item1 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For example, a ball tossed up will be under the influence of a negative (downward) acceleration due to gravity. Its velocity will decrease while it travels upward (case 1); then, after reaching its highest point, the velocity will increase downward as the object returns to earth (case 2).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Using <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">o</sub> (velocity at the beginning of time elapsed), <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">f</sub> (velocity at the end of the time elapsed), and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> for time, the constant acceleration is </p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><mathdisplay id="huetinck3831c01-mdis-0004" linklabel="1" numbered="yes" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9961.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />(1)</mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Substituting the average velocity as the arithmetic average of the original and final velocities <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">avg</sub> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>+ <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>)/2 into the relationship between distance and average velocity <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em> = ( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">avg</sub>)( <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>) yields.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><mathdisplay id="huetinck3831c01-mdis-0005" linklabel="2" numbered="yes" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9963.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />(2)</mathdisplay></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Substitute <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>from Equation <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0004">1 into Equation <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0005">2 to obtain</xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0004"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0005"><mathdisplay id="huetinck3831c01-mdis-0006" linklabel="3" numbered="yes" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9965.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />(3)</mathdisplay></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Finally, substitute the value of <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em> from Equation <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0004">1 into Equation <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0005">2 for</xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0004"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-mdis-0005"><mathdisplay id="huetinck3831c01-mdis-0007" linklabel="4" numbered="yes" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><img alt="" color="unknown" copyright="Wiley Publishing, Inc." erights="yes" loc="fixed" prights="yes" src="https://www.cliffsnotes.com/assets/9968.jpg" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" xlink:embed="onLoad" xlink:show="embed" xlink:type="simple" />(4)</mathdisplay></xref></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">These four equations relate <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">o</em> </sub>, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">v</em> <sub style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">f</em> </sub>, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">t</em>, <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">a</em>, and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">d</em>. Note that each equation has a different set of four of these five quantities. Table <xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-tbl-0001">summarizes the equations for motion in a straight line under constant acceleration.<tableanchor style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" tables="huetinck3831c01-tbl-0001"></tableanchor></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-align: center; text-rendering: optimizelegibility;"><xref style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" target="huetinck3831c01-tbl-0001"><img alt="" height="361" src="https://www.cliffsnotes.com/~/media/387da48a5492469fb38fd6dfeb115d4e.ashx?la=en" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; display: inline-block; height: auto; margin: 0px 0.9375rem 0.5rem 0px; max-width: 100%; vertical-align: middle;" width="589" /><br style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;" /></xref></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">A special case of constant acceleration occurs for an object under the influence of gravity. If an object is thrown vertically upward or dropped, the acceleration due to gravity of −9.8 m/s <sup style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;">2</sup> is substituted in the above equations to find the relationships among velocity, distance, and time.</p></item1></sect1></sect1></sect1></sect1></sect1></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-51665087668242063492020-08-23T21:46:00.006+05:302020-08-23T21:47:09.940+05:30ANIMAL SKELETON<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><i><u><span style="color: #2b00fe;">Skeletons in Animals</span></u></i></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; color: black; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="Animal Skeleton 3D Models | CGTrader" 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" /></p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Movement is one of the essential features of living things. Cellular movement is observed in one-celled amoebas, ciliates, and flagellates. Flagella whip about to produce cellular motion, while cilia beat synchronously to propel a cell.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In animals, movement is essential for locating food, escaping predators, and seeking mates. In many animals, the movement process is centered in the muscle cell, which contracts and relaxes. The contraction yields great force, which is applied against a surface by means of a skeleton.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Skeletal systems provide structure and protection for a variety of organisms. A water-based skeleton provides the structure necessary for movement in worms. The hard external skeleton (exoskeleton) not only provides a protective mechanism for many organisms but also assists in movement of insects. The internal skeleton (endoskeleton) present in many animals provides the structural network for support, protection, and movement.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Hydrostatic skeleton</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Many animals have a water-based skeleton, or<span style="color: #6fa8dc;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">hydrostatic skeleton.</span></span> Hydrostatic skeletons do not contain hard structures, such as bone, for muscles to pull against. Rather, the muscles surround a fluid-filled body cavity. In a worm, for example, movement occurs when muscle cells contract, and the contractions squeeze internal fluid (the hydrostatic skeleton) against the skin, causing the worm to stiffen and the body to shorten and widen. The squirming motion of a worm also depends on a hydrostatic skeleton.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #800180;">Exoskeleton</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">A second type of skeleton, the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #e69138;">exoskeleton,</span></span> exists in arthropods and mollusks. In mollusks, the exoskeleton is a hard, protective outer covering. An example is a clamshell; when a clam’s muscles contract, they close the shell rapidly, creating a spurt of water that propels the clam. In arthropods, the exoskeleton also provides protection and movement. Usually, wings are attached by muscles in the hard body surface, which provides a foundation for the muscle contractions. Muscle contractions raise and lower the wings, allowing flight.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ffa400;">Endoskeleton</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Vertebrates have an internal skeleton called the<span style="color: #6aa84f;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #6aa84f;">endoskeleton</span>,</span> a framework of bones and cartilage that serves as a point of attachment for muscle. The endoskeleton thus transmits the force of muscle contractions. The endoskeleton also provides support for the body (for example, the legs) and protection (the skull).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Skeleton - Wikipedia" height="631" 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" width="475" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Bone contains concentric rings of tissue in which bone cells called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">osteoblasts</em> produce the inorganic materials (fibers and matrix) of bone. Much of this material is calcium phosphate, formed from calcium and phosphorus delivered by the blood. Living, mature bone cells called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">osteocytes</em> are also located in the bone. Bone-destroying cells called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">osteoclasts</em> break down bone, thus providing a turnover of bone material needed in other areas. The combination of bone cells and bone tissue comprises a unit called a <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Haversian system.</em> Blood vessels and nerves also exist within the Haversian system.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Bones come together to form a joint, which may be immovable, such as in the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">sutures</span> of the skull, or movable, such as in the joints of the elbow and shoulder. In a movable joint, a capsule of synovial fluid provides lubrication. Tough, fibrous tissues, known as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">ligaments,</span> link bones to one another. Connective tissues, called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">tendons,</span> attach muscles to bones.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human skeleton - Wikiwand" height="710" 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npZq7DnH0q881LNQA80003NSvQAjTaRNCgCbppNGmmgCe3DW+PHdG5/2j71zb5/4s/sD/c9SXo3hvipH/LHb+Jh/0mone1r4yTusPv8Aeudb2lX9snN0VSbZ0mGldkDTvGrrxjLRqwPdrn49vfSLYnZhhweDgaQu1y7X6KQ++zN7psOGhAHPn6+j2JXwmIiY5c8ts3YWRAh+I8+FemC2z/4fAmG2hIZCSVa4LFedhzZQLa8esLaWtKXZo3HhD969+2wqiNEVpmH5iVXv4AnXTlw7q4N29kzXO1do3aa14I5NMhtpIy+53C2mp42qQx8+z8GPXFw+aV7FC+ZnJtpl6T2AB2DQDyqA2EZ9q43psST6vCQ5jF8hYG8aW943FyTyHK4qkV4mX2Lns0TQxjpGWSZi0sptlHWvkRRxGVbDy765Y8W0kkkcZtIjGaAngbnrxN+k8x+q41ArrlxZ6WQG2mhOvZc8B2kjyqr7baZcRBPhrZkbMwLFQy6hl4cwSKmnuUxsdO1dnzROcds0kFjebDE9VjzKqdA3Hhx4jmD0bN3gOLjdEkEMyizLIhLRudBmjJFxfnz4aHSvLemHEwkY/BMxBAMsNsysDrnC8j228eRrp2DtxMWhliSPplWxDaMpINhmsdD2ity6mVyRWysRisGkzbTmQxMSqhus0hbiR2RkE6EX7ABxhfSCPx4je90bUdtwfoV+FTuzW9dSaDaOEKvDIpLH2bk3VUa/G2htcEG/Owhd+oHtFJIMrZ5QRx0Zzk/0op86cO5mXUq96WagVoWrqIDr0r0ylQA+9K9NoUgHXppNCmtQMn6bRprUgLx6MYuriJO1o1/hDE/7xVU3ga+KmP6/sK0LcDAmPAqx4ys0vkbKv+lQfOqBvPEVxcgPM5h4cPtXNW82svNYgiW9H86A4mOTVGRGK/pUsGcDtXMp8qsa4BMTIFxsYE2Hf8Gbis0dsyN+oa3seBBseNUHd3EiLGQyE2GYqb8OupTrd12F6uONaFViwMqyJC/Vw80ZYPBKp0hLDgQRoDpp2cFbHzCt+JE7Z3R2hicSXlaIJwDh7qiX/KRmJ+/OrNhxBs/BsU1jiBJOl5ZeFz33sLctByrzxGBnECwjFgng87Hr25lIwMpbkLmw4nNzgd6MICkGz4XWJWLyEyta4W5LOeJZmJPka2n8iawd+5ONM2GzSEM7PIzFiTqZCbDTQDSwqO312czooitmzXHWyaAa6mw+JqX3RwHQYcQvk6QEsxDXBLHQg24WtXHvpgTNCI7xhswylmIA5k3y9l6isa8/kr/aL0fbaeSJ8PKczwkWJ1JRjzPMg317CK9tobMwkcoxjXw7XsXVxGrk8mv1Tc+endVL2XiRgcYj9KkiWtI0ZzLkbQ69xAbyFaHtPGYYfgzugEqnqORlkXS9viNRV5L4JRfyRW05V2hFlw2IyCKaPUHQ5SCzG41sDcHhdahN98akscTo1xK7SJ3RoojHxJJ+NdmysJBgZVihJlaZ8zs3VEeHTMbk8LLzPvE+Aqu71Sq2LKILJEqRoo4Kqi9viTfvvWYLyCT8SLtQIp1CuoiMK00ivQ0CKAPOlTiKaRSAbQpGm0AT1ecx0NPBpr0gNnwrCOJIB7saqvflW32qob8bLVoxib5SFJ8eHVPebjzr13bxjvgVkLs7o5W9hob2RSeehXWojfLbWdFwY1ysGY9hANh8792lcNcZa8HXNrRkqU63UitCw+OCyyYu4OHaNZJ4zxRsitFOnZdLgke8hHhn5qX2dvAYIRkA6ZLxqWBKtAzZyjW/K17X5Oa6rIOXBzwlgmdjbKwck74+PFyTJGxlKPdFjcksC50uBbQZRw1vVdxST7SxUjxgNYaZiFAQGyjXmePmaG0tuvLH0CRRwR3zMsShA7drWrm2bsmTEFkhK5gAcrNlLj9PbbnfuoxhZYZy8F93J2TJBDkkBEmYllzKwHYAQeyx8zT98Nn+sQ9GbISRYsRYG4sTblTdztkyQQ5cQSJMxJUsGyrwChgSOAvp+anb1YETYd414sNPEHT52rlz5f7OjHiZ/tjZKYfKFxEcxa+YJbq2txsT293Cp/B46HE7P6OeLpmgFmUXL5RoroRqDl0JHYarm0tkPhiqSlMxXMVRsxXW1m00Nc+HneN+kiYqw5g28u8V2JZRzZwy2QY1MThlEQkWOKWOPK+rTixyRZ+Yz5Ljs1PKqttCbPPK4NwXax7QDYH4AV7HbeJLZzISbMo7BmFiQo0v31wKLURhh5CUsofQvSoVsyI0KJoUAA0KRoGgAGm2p1CgCYBoE0KBNAF/3Aj/ALDLlF26ZmA7Six2H9dtUB5CzF29piWPiTc1ofo6xAXCC/8A6l1/iRLfaqhvVgegxksfItnXwfX5G48q5635tFprxREsa6MBsqefWGIsPzGyrp3nj5V54WDpJY4vzyInkzAH61qODsrmNQAq2UAcAByFO23RsjNderkz7Ebq4xFzmIEcwrAkVH4HZmImZvV43LR2zWYIylrgcSDfQ8K2pFBFQeM3czOcRhZDDOrDrWzIyixKMlwGBv5GpL6iXDKOlejy3ZwUkUCpO8hlABbO2Yg/lvzA0HOm7yQs8EiK5DFTY3try4VNtHcm7fED6G9qi9oqRoD5AAX+AqKl7K49GZy7t4qNkjaMBpL5eup0FrsbHQaj49tWbA+jssAZp2B7EUC3mb3qyYLZZV1lkkaV+GZsoyqeCqqgAAG3K5qxKABVZXSeyMKqKMx2xuC8al8PIZLamNwAxH6WGhPdYVTga3ec86yLfXBCHHSKvBwsgH7d7/ME+dUotcnpZO2CSyiFoUqVdJAVCkaVAxGmGnE02gAUr0DSoYEtQNKgTQBY9gYsps/EsvGCeCfyOn/Cpf0n4cEQYled0J7QRmX6N8aru7RzR42D/MwxbzjYH/kanMXOcTsFJOLQ5Ae38N+juf3bHzrlaxZn8llvDBS4sQY3SUa9G6vbtysGt8q0/FOBMWX2XCup7QwuDWUvwrSsJiEkwuFZTdhAiNrwZAAQezW9H1C4Y6XyixYCa+ldrzpEpZmsBc1EbPuDc1Vd4N41OKOGBPt5DbgDwtroNbfGuZRbLtpEptLa8ha6KADqFbrE9+UaiuTC4+QsCwQa+z7A+JGvxr29SUdY9Qke1mzE+PIfOnw4INzVhrfOBlHhbjWvwBYMHiI3jJtlK8VPG/YO2u9xyrKsfvH6viSqLmEZFwGOUsDc6HsHzrUNm4xZ4UnTg6hh5ik4tbiUk9keTJras89KcQGIgce9EQf3SP8AqrS3TWsW3n2565PnAsiXRO0rf2j46VT6dPVkxc/HBF0DRFI13HKA0006gaAGmhRNNNAANAUqApMCWvTTSpGmBMbmN/bkQ8JUljP70bH6gVNblR58BjsGeKl9OwshH1jNVjYU/R4zDv2TRg+BYKfkTV13Wh6PamOg5MA3jqf/ANDXNbyy1ZnKnSrRuCP8QvZ0TAc73YE/T5VXJ48jun5WZf4WIq0bo7uklMTKxUNfJGLgspHtMewm1h4Gt3NaDNaeovmFi6t6x7eHClMVOrXuZHYeDsWBHx+VbVAQllJ0OgPPuv8Azqo+kPYZeH1qNCzRnrZeIT3jbmBp4W8a56ZYkWtjlHPuftj1nMjrlkjRetmuHGovkvob9naNalN49pDCQCW0byHRVLFCQbAlV1LWuLjTS+vblCbUfDMuJiJzIymwNs65hmQ9zDSrNvbtCPFxYfGRS5lIMZjPFGIzk249gPLhaqSqWtfBhWPR+SvwQvI4RetJI1h3sx4k/Mmts3fw3QYWKK98qhb2te3O3fxrNvRypbGECPMOjIaT/KvzHe3Dt48r1pe2cYsaKAQCWAXusLk/AVm95eB0rbJz7w7wQYVC0rdYg5UGrMe4ViSuNbaXJNvtWhby7HbGKBER0gbNdidRa2W/Ljfyrzxm7TLs4RSBRIozLbjmAuR3g3I+FOqUYL/IWRcmUOjXkj3FOvXWc46mmlehegBGmmjTTQA00hSNIUmBJ3pGm3pUwGyOVsw4ggjxGorSoTbbuYcJcIG8bkH/AI1mknCtS2DLh5ZMLL0g6aLBKrrzOZY9T4EN8TXPf8lajP8AeqLJi8Qv6y38QDfetJwD9JDGOF0QgjsKggCql6Q8MnSLiEPtFo28iSp+Rpu4G02zthmJItnjvysQGUd2t7dxrM/KtSXo1Hxm0yU3gmmheNUdmDvlva5U5WZT4XUDzr2m33kw2T1mLMjHKXTQg2v7B0Ol+fKu/eHHJAGkcC2U+ZNrAd97VWsDtHB43C9BinETAqxuwQgr70bHQ89O+pR33a2Kyf7Hbe2DsrE9eOUx9J1gIWGUk2OqsCF8BavHC7kYCJLyYmVlUDS6oLDlmA1qH2rtnA4PDMmDXpJSxJZi5jU3yhrtYtcAaDTvqB2dteYBXZs4IBKk6Hnp2VaMZNPklJxNAfeCCF4cFs1F/EcBtGGW5F2YnV2y349lQW9mMmChzKxYvlBvawAJNgP3abhdsYJZfWBG0cgBN8uY3K2IBHdcX0qC2njzPIZCLKBZV7B/M86IQ34CU9jTd2P8MkhFnIBaxvdiL3vz4/Om7c2qsK9LOxPJV4ljxsP65Uz0foTgEZiTdpLX/KGKgeGlUjfTHdNjGHux9UePvf13VOMNU2jcp4hkh1POnA02lXaco69KmmlegB16aaVCgBGgKVCkBJXpXptK9MBNWnbkogwcPRqvvFmIsxc5s+vYDp5dlZdIdK0R9perL6uFDerJgYWP5pJSc48Aov51z/UJtJItThPLFvBsQSxKC+VxmZScxGn5hfiRpeqZu9MY8bCTp+JkYft3Qg+Z+VWbe+dnw+Lykg4fFxDQ+4Y0t/8AZc/Cq7t3EJ0kEykGQqkkhXgSCpBtyOhrFedOl+xzxnKLZ6T0vh4z+sX+BH8qzZ4wa1T0hpmwGccmQ+WYD71lzVSjqZu7EXtlbRHxH1FdOCX8Nf2R9K8Ntf3XmPrXXhfYXwH0q3sl6HFBSIp96McRdljHF2VB4sQB9aYGr7GHQbMjZtMsIc+JGc/WskZyzM54sSx8zetU37n6LAMg0uAg8DYfSsrXhXPRvmRa3bCEKNChXQREaF6NNoAN6V6bSvQATSppNG9JgSF6F6F6VMDq2XkOIh6T2OkTN4Zhp9qte1YrwyS5lYzbSjawIJyiyqp7wSRVKV7MrccrA27bG9qnsdic+CmeM9UTpIBpmTOwupI4dcE2vzFQsT1JlYNYZY8Zs1nn2rB/mxQyp+1Z7H+JPlVA2tihK5ZVAUAKoH5RwPj5VbIttuZsNK2qYhGjfUjr3XnfXrC2vImqltFCJZVYWId7gCw9o8ByFKrnf/vQWcbGobRXptjkjUmAMPEKGrJidK1fcqUTbMVDyVoyP2SV+lqyiRCpKnipKnyNqKNsodu+GR22j+H512Yf2R4CuDbZ/D8/513ReyPCr+yXo9Kl90MP0mPgXkHzn9wFh8wKh6t/oyw2bEyS8o4svnIw+yH41i14gxwWZI7vSlitIou1i3ko/wC4qh1YvSFis+MC8kQfFjc/QVW70qViCNWvMg0CaF6F6qTDegTSoUAK9ClSoAVGm0TSA76VKlTABFSe7R/vk91lUMCAQbE/zNKlU7OpuvsdaoBs5W5pMCp7D09tPKube3/EnvRCe866mlSqcO37Ny6/otvosY+rSjkJWt/Ahqh7fUDFTAf5jfM0qVKr7kgn0RX9sDqef2Nd0fAUqVdC5Jeh1aH6MFHq87c+lAv3BFIHzPxpUqlf0N1dim7zG+Omv+YD/SKjrUaVbh1RmXZgtQtSpVsyK1AilSoAbSpUqQANONKlQB//2Q==" style="color: #222222;" width="419" /> </p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human skeleton showing the major bones of the body.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-73182484384197598442020-08-23T21:43:00.007+05:302020-08-23T21:47:09.940+05:30HUMAN MUSCLE<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #2b00fe;">Human Muscle</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: black;">Muscle is made up of thousands of muscle fibers, each composed of a single muscle cell. As shown in Figure 27-2, a muscle cell contains a series of ultramicroscopic filaments called</span><span style="color: #a64d79;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">myofibrils.</span></span> Each myofibril is a muscle cell that contains units called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #b45f06;">sarcomeres</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">.</span> Sarcomeres contain thick microfilaments composed of the protein<span style="color: #6fa8dc;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">myosin.</span></span> Sarcomeres also contain thin microfilaments composed of the protein <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #f6b26b;">actin.</span></span> The actin and myosin filaments are arranged parallel to one another, with the myosin filaments’ molecular “heads” protruding toward the actin filaments. In skeletal muscle, the overlapping actin and myosin filaments give the muscle fiber a banded, or striated, appearance. Hence, the muscle is <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">striated muscle.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><img alt="Muscle Structure And Function | Support Systems In Animals | Siyavula" height="254" 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" width="488" /></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Anatomical structure of the muscle.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #04ff00;">Muscle contraction</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">When a nerve impulse arrives at the muscle cells, it passes across the neuromuscular junction and enters the muscle cell membrane, which is known as the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #800180;">sarcolemma</span>.</span> The impulse spreads across the muscle cell and enters its cytoplasm, which is called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">sarcoplasm.</em></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Reactome | Muscle contraction" height="323" 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" width="585" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"> The nerve impulse causes the actin filaments to slide across the surface of the myosin filaments. The sliding filaments pull together the ends of the muscle cell, thereby causing it to contract. The sliding filaments require that calcium ions and energy in the form of ATP be available. Two proteins called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">tropomyosin</em> and <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">troponin</em> also function in the contraction. Cross-bridges hold the filaments together as the muscle contracts.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: black;">After the</span><span style="color: #01ffff;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">muscle contraction</span> </span>has taken place, the energy to sustain the contraction is used up, and the cross-bridges break. The filaments then slide back to their original position, and the muscle cell relaxes. There is no partial contraction of the muscle cell. Contraction is an all-or-none phenomenon.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Energy for contraction</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Adenosine triphosphate (ATP) supplies the energy for muscle contraction. The reactions of glycolysis, the Krebs cycle, and the electron transport system normally produce ATP during cellular respiration (see Chapter 6). During normal activity, ATP is regenerated as it is used up during muscle contraction. When a person engages in strenuous activity, however, ATP is quickly used up and creatin phosphate is used for energy. Creatin phosphate transfers its energy to new ATP molecules, which then function as additional energy sources.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Muscle Contractions | 3D Muscle Lab" height="319" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQwtyFosn_dllEsJuwG6Ol7Qwlrs-TotYAT_Q&usqp=CAU" width="605" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">When creatin phosphate is used up, muscle cells obtain their energy solely from the process of glycolysis. Because no oxygen is available, the metabolism is anaerobic. Under these conditions, two molecules of ATP are obtained per molecule of glucose metabolized. The pyruvic acid that forms is converted to lactic acid in the muscle. Lactic acid prevents overexertion of the muscle because as lactic acid accumulates, the person experiences fatigue. The fatigue induces the person to stop exerting the muscles and breathe deeply. This breathing provides a plentiful supply of oxygen to satisfy the oxygen debt. Lactic acid is converted back to pyruvic acid, which is then metabolized through the Krebs cycle and electron transport system to provide a new supply of ATP and creatin phosphate.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: red;">Types of muscle</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human body has three major types of muscle. The muscle type discussed earlier in this chapter is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">striated muscle</em> because the fibers’ overlapping actin and myosin filaments give it a banded appearance (see Figure 27-2). This muscle is found in the limbs and is also called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #bf9000;">skeletal muscle</span>.</span> It operates under voluntary control and so is additionally known as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #6aa84f;">voluntary muscle.</span></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Muscle Contractions 101 — CrossFit Central Houston" height="325" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQVU3A6wEnYmvTAcIsSCNuFbgCrSEEF6xQrnA&usqp=CAU" width="581" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: black;">The second muscle type is</span><span style="color: #f1c232;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #f1c232;">smooth muscle</span>,</span> which has few actin and myosin filaments; therefore, it has few striations. Smooth muscle is found in the linings of the blood vessels, along the gastrointestinal tract, in the respiratory tract, and in the urinary bladder. Because it operates without voluntary control, it is sometimes called<span style="color: #674ea7;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">involuntary muscle.</span></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The third muscle type is <span style="color: #45818e;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cardiac muscle,</span> </span><span style="color: black;">which is found in the heart. It has striations because it has multiple actin and myosin filaments, but it is an involuntary muscle. The actin and myosin filaments in cardiac muscle exist as intertwined branches that form a conducting network for nerve impulses.</span></p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-9932549312069100582020-08-15T09:34:00.006+05:302020-08-15T09:35:55.154+05:30Birth And Fetal Development<p><span style="color: #ffa400;"> <span style="background-color: white; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem;">Birth</span></span></p><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The birth process, which is complex, is regulated by various hormones. When diminished levels of progesterone remove the inhibition on uterine contractions, the uterine muscles contract. The posterior pituitary gland releases the hormone<span style="color: #800180;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #800180;">oxytocin</span>,</span> which stimulates further contractions.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="12 Baby Birth Videos That (Really) Prepare You For The Big Day ..." height="351" 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" width="548" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">During the first stage of labor, the cervix opens and the baby descends into the birth canal, or vagina. By this time, the amniotic sac has broken. In the second stage of labor, the baby passes through the birth canal, assisted by painful uterine contractions. The baby’s head normally appears first, but in a breech birth the buttocks may appear first. In the third stage, the placenta (afterbirth) is delivered.</p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #2b00fe;">Fetal Development</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">During the <b><span style="color: #93c47d;">third month</span></b>, the fetus definitely resembles a human, but the head is relatively large. During the ensuing months, the remainder of the body increases in size proportionally. Cartilage is replaced by bone, and the reproductive organs develop.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Baby Development Month By Month (VIDEO) | American Pregnancy Blog" height="325" 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" width="585" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">During the <b><span style="color: #3d85c6;">fourth month,</span></b> the length of the fetus increases to about 6 inches. The heartbeat can be heard through the mother’s abdominal wall, and the fetus moves about. Distinctive movements can be felt at the<b><span style="color: #a64d79;"> fifth month,</span></b> and by the<b><span style="color: #e69138;"> sixth month </span></b>the fetus weighs almost 2 pounds. By the end of six months, the fetus might be able to survive outside the mother’s body, but it would have little fat in its skin, so temperature control would be a problem. By the end of the ninth month, the fetus has an average length of about 20 inches and a typical weight of 6 to 8 pounds.</p></article></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-23433882391666880052020-08-15T09:30:00.007+05:302020-08-15T09:35:55.154+05:30Fertilization And Development Of The Embryo<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ffa400;">Fertilization and Development</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human fertilization insemination of human egg cel Vector Image" 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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">For fertilization to occur, sperm cells must be released in the vagina during the period that the egg cell is alive. The sperm cells move through the uterus into the fallopian tube, where one sperm cell may fertilize the egg cell. The fertilization brings together 23 chromosomes from the male and 23 chromosomes from the female, resulting in the formation of a<span style="color: #01ffff;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #01ffff;">fertilized egg </span>cell</span> with 46 chromosomes. The fertilized cell is a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #2b00fe;">zygote.</span></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; float: right; font-weight: 700; line-height: inherit; margin-bottom: 1em; margin-left: 1em;"><img alt="What Are the Steps of Fertilization in Humans?" 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" /></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The zygote undergoes mitosis to form two identical cells that remain attached. This takes place about 36 hours after fertilization. Mitosis then occurs more frequently. Soon a solid ball of cells, a<span style="color: #ff00fe;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">morula,</span> </span>results. Morula formation occurs about six days after fertilization. During that time, the cells are moving through the fallopian tube. Within the next two days, a hollow ball of cells called a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #04ff00;">blastocyst</span></span> forms. The blastocyst enters the uterus. At one end of the blastocyst, a group of cells called the<span style="color: #800180;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">inner cell mass</span></span> continues to develop.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">About eight days after fertilization, the blastocyst implants itself in the endometrium of the uterus. During implantation, the outer cells take root in the endometrium. This outer layer of cells, called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #a64d79;">trophoblast</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">,</span> gives rise to projections that form vessels. These vessels merge with the maternal blood vessels to form the<span style="color: #6aa84f;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">placenta.</span></span> The trophoblast also develops into three membranes: the amnion, the chorion, and the yolk sac membrane.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human Reproduction and Fertilization - Biology Online Tutorial" height="341" 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" width="555" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The inner cell mass undergoes changes to form three germ layers known as the ectoderm, the mesoderm, and the endoderm. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #f1c232;">ectoderm</span></span> becomes the skin and nervous system, the<span style="color: #e69138;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">mesoderm</span></span> becomes the muscles and other internal organs, and the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #674ea7;">endoderm</span></span> becomes the gastrointestinal tract. The<span style="color: #cc0000;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">embryo</span></span> is formed at about the fourth week when all the organs of the body have taken shape.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-86898525458032487622020-08-15T09:26:00.002+05:302020-08-15T09:35:55.154+05:30Human Reproductive System<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #800180;">Human Reproduction</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Reproduction is an essential process for the survival of a species. The functions of the reproductive systems are to produce reproductive cells, the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #01ffff;">gametes</span>,</span> and to prepare the gametes for fertilization. In addition, the male reproductive system delivers the gametes to the female reproductive tract. The female reproductive organs nourish the<b><span style="color: #ffa400;"> fertilized egg cell </span></b>and provide an environment for its development into an embryo, a fetus, and a baby.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Human reproduction takes place by the coordination of the male and female reproductive systems. In humans, both males and females have evolved specialized organs and tissues that produce<b><span style="color: #f1c232;"> haploid cells</span></b>, the sperm and the egg. These cells fuse to form a <b><span style="color: #ff00fe;">zygote</span></b> that eventually develops into a growing fetus. A hormonal network is secreted that controls both the male and female reproductive systems and assists in the growth and development of the fetus and the birthing process.</p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Female Reproductive System</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The organs of female reproduction include the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #2b00fe;">ovaries</span>,</span> two oval organs lying within the pelvic cavity, and adjacent to them, two <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #04ff00;">fallopian tubes.</span></span> Also known as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #cc0000;">oviducts</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">,</span> the fallopian tubes are the passageways that egg cells enter after release from the ovaries. The fallopian tubes lead to the<b><span style="color: red;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">uterus</span> (womb),</span></b> a muscular organ in the pelvic cavity. The inner lining of the uterus, called the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">endometrium,</em> thickens with blood and tissue in anticipation of a fertilized egg cell. If fertilization fails to occur, the endometrium degenerates and is shed in the process of menstruation.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human female reproductive system|Sureden:Your Education Partner" height="366" 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" width="521" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The opening at the lower end of the uterus is a constricted area called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #a64d79;">cervix</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">.</span> The tube leading from the cervix to the exterior is a muscular organ called the<span style="color: #674ea7;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">vagina.</span></span> During periods of sexual arousal, the vagina receives the penis and the semen. The sperm cells in the semen pass through the cervix and uterus into the fallopian tubes, where fertilization takes place.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In the human female, egg cell production begins before birth, when about 2 million primitive cells known as<span style="color: #f6b26b;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">oogonia</span></span> accumulate in the ovaries. These oogonia are formed in the early stages of meiosis. After the age of puberty, the oogonia develop into primary<span style="color: #3d85c6;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">oocytes</span> </span>and then into <span style="color: #6aa84f;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">egg cells</span> </span>at a rate of one per month. Egg cell production occurs by the process of meiosis.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Studies of the Human Female Reproductive System" height="325" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcSjChTeeARELKrbETsCuInWRCGO9o64xOSRLA&usqp=CAU" width="581" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The egg cells develop within the ovary in a cluster of cells called the<span style="color: #c27ba0;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Graafian follicle,</span></span> which secretes the female hormone<span style="color: #cc0000;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">estrogen</span> </span>that regulates the development of secondary female characteristics. Egg cell development within the follicle requires approximately 14 days. The development is controlled by two hormones: follicle-stimulating hormone (FSH) and luteinizing hormone (LH). Both hormones are secreted by the anterior lobe of the pituitary gland.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">During the 14 days of egg cell development, the endometrium increases its supply of blood and nutrients in anticipation of a fertilized egg cell. On about the 14th day, the release of the egg cell from the follicle takes place. This process is called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #8e7cc3;">ovulation.</span></span> The egg cell is swept into the fallopian tube and begins to move toward the uterus. Meanwhile, the follicle is changed into a mass of cells known as the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #38761d;">corpus luteum.</span></span> The LH stimulates the conversion. The corpus luteum then secretes the hormone<span style="color: #6aa84f;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">progesterone,</span> </span>which together with estrogen continues to regulate the buildup of tissue in the endometrium and inhibit contractions of the uterus.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The egg cell remains alive in the fallopian tubes for 24 to 72 hours. If fertilization by a sperm cell fails to occur, the egg cell moves toward the uterus, and the corpus luteum begins to degenerate. This degeneration causes the level of progesterone and estrogen to drop off. Within two weeks, the hormone levels decline to a point where they cannot inhibit contractions of the uterus. Uterine contractions then occur, and the endometrium is released in the process of<span style="color: #b45f06;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">menstruation.</span></span> Follicle development begins again, but in the opposite ovary.</p></article><section class="navigation-buttons" style="-webkit-tap-highlight-color: transparent; border-top: 2px solid rgb(255, 255, 255); box-sizing: inherit; color: #222222; padding: 0px 0px 1rem;"><div class="row" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px auto; max-width: none; padding: 0px; width: auto;"><div class="small-6 columns clear-padding-left" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; float: left; margin: 0px; padding: 0px 0.9375rem; position: relative; width: 301px;"><a class="nav-bttn-filled " href="https://www.cliffsnotes.com/study-guides/biology/biology/reproduction/human-reproduction" style="-webkit-tap-highlight-color: transparent; background-color: #cccccc; box-sizing: inherit; color: black; display: table; line-height: inherit; text-decoration-line: none; width: 286px;"></a></div></div></section></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-604385628317443532020-08-15T09:21:00.004+05:302020-08-15T09:35:30.341+05:30Human Sense Organs<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Human Senses</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The sense organs connect the nervous system to the external environment. They are the sources of stimuli that cause a response in the nervous system, and they are the sources of all information to the human body.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #800180;">Eye</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In the human eye, the nerve cells are located in a single layer called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #76a5af;">retina</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">,</span> located along the back wall of the eye. Light rays enter the eye through a curved, transparent structure called the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">cornea,</em> and then pass through the pupil, an opening in the eyeball. The iris regulates the size of the pupil. Next, the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #134f5c;">lens</span></span> focuses the light on the retina, which contains two types of light-sensitive cells, <span style="color: #3d85c6;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">rod cells</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cone cells,</span> </span>which detect light. Cone cells, which detect color, are concentrated in the central portion of the retina, while rod cells, which permit vision in dim light, are concentrated at the edge of the retina. A light-sensitive pigment called<span style="color: red;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">rhodopsin</span></span> functions in the detection of light.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Science still battles to understand the human eye | Financial Times" 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" /><img alt="The Human Eye and The Colorful World - Tutorialspoint" height="218" 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" width="360" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">From the eye, a series of impulses is generated for transmission to the brain. The <span style="color: #fcff01;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">optic nerve</span> </span>carries these impulses. The region of keenest vision, the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">fovea,</em> is located at the center of the retina. When vision is poor, light is not focusing on the fovea, and corrective lenses are prescribed.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #2b00fe;">Ear</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The ear is the organ of hearing in humans. The outer ear funnels vibrations to the<span style="color: #0b5394;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">eardrum,</span></span> or tympanic membrane, which transmits the vibrations to three inner ear bones: the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">malleus</em> (hammer), the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">incus</em> (anvil), and the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">stapes</em> (stirrup). These bones transmit the vibrations to the inner ear where the receptor of hearing, the cochlea, is located.<img alt="How The Ear Works | Your Summary Of Normal Hearing Function" height="245" 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" width="319" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The<span style="color: #01ffff;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cochlea</span></span> is a snail-like series of coiled tubes within the skull. As the ear bones vibrate, they push and pull a membrane at one end of the cochlea, causing fluid within the tubules to vibrate. The vibrations are detected by sensitive hair cells, and nerve impulses are generated. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #e69138;">auditory nerve</span></span> carries the impulses to the brain for interpretation.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #04ff00;">Taste and smell</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Specialized receptor cells called chemoreceptors transmit taste and smell. <span style="color: #f1c232;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Chemoreceptors</span> </span>of the human tongue distinguish four different tastes: sweet, sour, salty, and bitter. In the human nose, chemoreceptors detect a variety of scents, including minty, floral, musky, putrid, and pungent.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human tongue basic taste areas | Premium Vector" height="256" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcTzO_qqLRdnV6i9Hqk3ZP-rUIgODOQPsmvthQ&usqp=CAU" width="473" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In both taste and smell, chemoreceptors are stimulated by molecules and ions that reach the tongue and nose. Liquid materials affect the chemoreceptors in the taste buds of the tongue, while gaseous molecules affect the chemoreceptors in the upper reaches of the nose. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #6aa84f;">olfactory nerve</span></span> carries nerve impulses from the nose to the brain for interpretation.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Human Nose Stock Photos, Pictures & Royalty-Free Images - iStock" height="356" src="data:image/jpeg;base64,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" width="538" /></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ffa400;">Other senses</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The other senses of the body include receptors for touch, pain, temperature, and balance. Touch and pain receptors, called<span style="color: #3d85c6;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Pacinian corpuscles,</span></span> are located in the skin, muscles, and tendons. The sense of balance is centered in the semicircular canals of the inner ear. Visceral senses include stretch receptors in the muscles, as well as carbon dioxide receptors in the arteries.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Skin (epidermis) anatomy | HealthEngine Blog" height="221" 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" width="554" /></p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-68419164683707049652020-08-15T09:15:00.000+05:302020-08-15T09:35:30.342+05:30PNS OF Humans<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #f1c232;">Human Peripheral Nervous System</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="The Central and Peripheral Nervous Systems | Biology for Majors II" 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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #04ff00;">peripheral nervous system</span></span> is a collection of nerves that connect the brain and spinal cord to other parts of the body and the external environment. It is subdivided into the sensory somatic system and the autonomic nervous system.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: red;">sensory somatic system</span></span> carries impulses from the external environment and the senses. It consists of 12 pairs of cranial nerves and 31 pairs of spinal nerves. The sensory somatic system permits humans to be aware of the outside environment and react to it voluntarily.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The<span style="color: #2b00fe;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">autonomic nervous system</span> </span>works on an involuntary basis. It consists of two groups of motor neurons and a set of knotlike groups of cell bodies called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #e06666;">ganglia.</span></span> Motor neurons extend to and from the ganglia to the body organs. One subdivision of the autonomic nervous system is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #ff00fe;">sympathetic nervous system</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">.</span> Impulses propagated in this system prepare the body for an emergency. They cause the heart rate to increase, the arteries to constrict, the pupils to dilate, and other changes to take place. The other subdivision is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #800180;">parasympathetic nervous system</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">.</span> Impulses in this system return the body to normal after an emergency has occurred.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-2799831747655400132020-08-15T09:13:00.002+05:302020-08-15T09:35:30.341+05:30CNS Of Humans<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Human Central Nervous System</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; color: black; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="2: Schematic view of the Human CNS. Image courtesy of http ..." 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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human nervous system can be conveniently subdivided into two divisions: the central nervous system (the brain and spinal cord) and the peripheral nervous system (the nerves extending to and from the central nervous system).</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ffa400;">Spinal cord</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">spinal cord</span> of the central nervous system is a white cord of tissue passing through the bony tunnel made by the vertebrae. The spinal cord extends from the base of the brain to the bottom of the backbone. Three membranes called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">meninges</em> surround the spinal cord and protect it. The outer tissue of the spinal cord is white (white matter), while the inner tissue is gray (gray matter).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="spinal-cord-feat | The Daily" height="228" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcQcTQxU5y3n-mSz-9LlkXj-PqWPD_q6rbgBFQ&usqp=CAU" width="534" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Thirty-one pairs of projections called <span style="color: #ffd966;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">nerve roots</span> </span>extend out along each side of the spinal cord. The nerve roots are sites of axons belonging to sensory and motor neurons. A central canal in the spinal cord carries cerebrospinal fluid, which provides for the nutrition and gaseous needs of the cord tissue. The neurons of the spinal cord serve as a coordinating center for the reflex arc and a connecting system between the peripheral nervous system and the brain.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: red;">Brain</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The brain is the organizing and processing center of the central nervous system. It is the site of consciousness, sensation, memory, and intelligence. The brain receives impulses from the spinal cord and from 12 pairs of cranial nerves coming from and extending to the senses and to other organs. In addition, the brain initiates activities without environmental stimuli.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="ᐈ Human brain drawing stock illustrations, Royalty Free human ..." height="351" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcRingPpV9CMW0dvu4KS2gQA-1O8USBHOI67Bg&usqp=CAU" width="546" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Two major hemispheres, the left and the right hemispheres, make up the tissue of the brain. The outer portion of the brain consists of gray matter, while the inner portion is white matter. Three major portions of the brain are recognized: the hindbrain, the midbrain, and the forebrain.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #6aa84f;">hindbrain</span></span> consists of the medulla, cerebellum, and pons. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #3d85c6;">medulla</span></span> is the swelling at the tip of the brain that serves as the passageway for nerves extending to and from the brain. The<span style="color: #04ff00;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cerebellum</span></span> lies adjacent to the medulla and serves as a coordinating center for motor activity; that is, it coordinates muscle contractions. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #cc0000;">pons</span></span> is the swelling between the medulla and the midbrain. The pons acts as a bridge between various portions of the brain.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Coronavirus can infect the brain and replicate, new study claims" height="356" 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" width="538" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #674ea7;">midbrain</span></span> lies between the hindbrain and the forebrain. It consists of a collection of crossing nerve tracts and is the site of the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">reticular formation,</em> a group of fibers that arouse the forebrain when something unusual happens.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The<span style="color: #a64d79;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">forebrain</span></span> consists of the cerebrum, the thalamus, the hypothalamus, and the limbic system. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #e69138;">cerebrum</span></span> contains creases and furrows called convolutions that permit the cerebral hemisphere to accommodate more than 10 billion cells. Each hemisphere of the cerebrum has four lobes, and activities such as speech, vision, movement, hearing, and smell occur in these lobes. Higher mental activities such as learning, memory, logic, creativity, and emotion also occur in the cerebrum.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The<span style="color: #c27ba0;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">thalamus</span> </span>serves as an integration point for sensory impulses, while the<span style="color: #76a5af;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">hypothalamus</span></span> synthesizes hormones for storage in the pituitary gland. The hypothalamus also appears to be a control center for such visceral functions as hunger, thirst, body temperature, and blood pressure. The<span style="color: #ffd966;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">limbic system</span></span> is a collection of structures that ring the edge of the brain and apparently function as centers of emotion.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-67838037567927460942020-08-15T09:08:00.003+05:302020-08-15T09:35:30.342+05:30Nerve Cells Of Animals<p> </p><div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #2b00fe;">Animal Nerve Cells</span></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Nervous coordination enables an organism’s rapid response to an external or internal stimulus. Characteristic of animals only, nervous coordination is the function of the nervous system. The receptors for nervous coordination are generally located in the sense organs at the body surface, while the response in nervous coordination generally involves a gland or muscle. The function of coordination is accomplished by means of a set of signals conducted along a series of nerve cells.</p><div class="gts-placeholder-wrapper float left middle-for-small-only" data-google-query-id="COT68PWinOsCFUzccwEdbHkE4A" id="abstractAd" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; display: inline-block; float: left !important; margin: 0px 0.9375rem 0.9375rem 0px; padding: 0px; text-align: center; width: 367.219px;"><div id="google_ads_iframe_/57176166/Cliffsnotes_2015_LitNotes_Detail_2__container__" style="-webkit-tap-highlight-color: transparent; border: 0pt none; box-sizing: inherit; margin: 0px; padding: 0px;"><span style="color: black; font-size: 1rem; text-align: left;">Animal nerve cells are specialized cells called neurons. Depending upon function, these cells can be divided into sensory neurons, interneurons, and motor neurons. These three types of nerve cells coordinate with each other to receive external stimuli and to transmit the impulse to muscles or glands of the body for an appropriate response to the stimulus.</span></div></div><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br /></h2><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br /></h2><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: #222222; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><br /></h2><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #800180;">Neuron</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #3d85c6;">neuron</span></span> is the nerve cell. (Approximately 12 billion neurons exist in the human body, the great majority of them in the brain and spinal cord.) The main portion of the neuron is the cell body. Protruding from the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #6aa84f;">cell body</span></span> are one or more short extensions called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #76a5af;">dendrites</span></span> and one long extension called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">axon.</span> Axons are covered by a fatty layer of material called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #cc0000;">myelin sheath.</span></span> Bundles of axons bound together are referred to as a <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #ffd966;">nerve.</span></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><img alt="Understanding Neurons' Role in the Nervous System" height="364" 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" width="521" /></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">There are three types of neurons in animals: sensory neurons, interneurons, and motor neurons. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #674ea7;">Sensory neurons</span></span> receive stimuli from the external environment;<span style="color: #ffa400;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">interneurons</span></span> (or association neurons) connect sensory and motor neurons and carry stimuli in the brain and spinal cord; <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #741b47;">motor neurons</span></span> transmit impulses from the brain and spinal cord to the muscle or gland that will respond to the stimulus. The neurons are supported, protected, and nourished by cells of the nervous system known as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #bf9000;">glial cells.</span></span> Together with extracellular tissue, the glial cells make up the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #c27ba0;">neuroglia.</span></span></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #ff00fe;">Nerve impulse</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The nerve impulse is an electrochemical event that occurs within the neuron. In an inactive neuron, the cytoplasm is negatively charged with respect to the outside of the cell. This difference in electrical charge is maintained by the active transport of sodium ions out of the cytoplasm. A cell in this state is said to have a<span style="color: red;"> </span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: red;">resting potential</span>,</span> and it is polarized.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Nerve Impulse | CK-12 Foundation" height="380" 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kKmxjAwYXEIoUg5uYUXY9AN/M9B1Iw0LigE2l1HM66gaNKoG0L90AGB3JJEnqcEHu19w0mFbLbjrOV8uRo/kX+uPMhy8pgYZ+cnlaYBA3Y4wWvbfCsSLy2rmn1nSfCLAHaBb9eeAauQ2WkoTZwVzjfgddWmmRBO3nh+z7QhOHSJ2jZWAxTTNlPc0tdQ6QIPQkdewt6/XHXUhUxE5TlsxY4Vmb9ruONKiivu6LqGW177idpnc733xBtO1PYKuQIvG7NswzL5kRBks7cSPPEtOpnnKXp8IZnXDCFGG1CCLCLQEG5iyojjfExBlAIlYc+eMuZthPlzHKMexz2CEUanTDFaAwhP7JN+uG9nfOKx2ym/zOZyvEKYSoTRqAeF0Ngf40sHH16EXxfUpiuO62c5tMvs5uBcTBe0Hs/UytTTUAgyUdbo46qflI3E9xjlVKJQ2YWnVo1xUW6y01jl0SmLs13VuQ2AEXEfjq5YeHNJQo1OvlwgFRUYsdBpLszSowHdG0nmumet9p2ImeWGVFp2DMMvCAjVLlVOfjLspRoMDoFR1+0KjKiL0grJne1t8ZTpU2BCgkeNrTXq1F+q1/C94bQprY06dMIoIliysYlgAZsCTcnVE4eqDIoBbximqnRibxXxLJ5g1DVeorReaTSq+UfCO+JaiVS2Mnlu+I5KlPDhA575VknWtU8agEfbW0/zLsfMQfPHqZFVsx6/M9UvTXLl8Rv8AtIVtDIIPPcEdQeYxabA2IkuEkXBi/wBoaFNdJRd7+WI9pRVOUo2dmYZxZRzUeF2Mcjvp9DuMJFQgWJ/UowXNwJOvmIPjpgzsyEjV3vM/TAPUt9Q9RH0kv9J5x77O1tc09QqK32WEOp7HmLDrGkHlgO0BUqTcHjrK0pkMGAsfaep+1PCf2jheXIjVTXTeAbGMcTaEwYHXdcTs7KcVapTb+1j4aTxvi3CnpHxAjzGKKNa+UVtOylc4mqDliwG4nLdbQZ8FJ2EZZTNalFOqNSj4T9pP5T07bfSKke64X09xJXWxxLr94RXyBpgLujXLDmbwp6EC8dTzicG1IoLbjv665QBUDG++R4bQDVNAsWVgPOJ/I4ykt2wjeDNqNZbmD8QqlqpVbkkKB3gCPnbAVW75Ah01stzDaNClTnxM1WCCQAEUmxgzJIuNonyvllpk7z7COTE9tw95oPZb2f8A2hwoEyYHnjibbtDKQiamfSbBs9PAatXQTbcS9hqOSQ1q0GoP7qmNieZbsOn+2C2Zm2dgdoNydB8wKxp7YpXZ1sAMz+BPMeP8fr1Kh1NYGw5DtjrHaqjtcn4nCOyU0FgJGpSGZyo06VqUWI0/eVrkjkLkW7fOq3bUstQfv+5AT2VXPQ/iZ+gDqjEq62lDaRt+zNoDEhR1J/AbnFOA4byfEL2nBmqSj7VQ9SQq+gFx88BiRfGEVY+Ei+dUiRTT5Y01Rb6RMFM3zJg/7ZSNmpx3Vr/I2wHaIdV5RmBxoZblFTUPduP5XEH57E41Qt+6ecxr2zHKX1snmQTFNo7CfwwZp1xopiw9HewhubPu6hC/CYI/lNx8tvMHFr3R7RBW4zmv9nnXMUWoV4qJqBSd1MeFh6wPUjYnFIpisve3SGoxpPiSYni/D6hzNVX+NHKk7gwYB9d/XHKZDUcmdSnUVaYtpaGUsoKaMuZYKp2BksOchR5WJtikUwilapy94guXYGmM4JmM8yAKijRuGEMOk2sD5ycKesVFlGXONSkCbtrOftQ0CZM8+mF9oLZwsBvlLcpnl1DU23Pn5dxhq1Vvdj8xbU2tkIVRytNjNEgSbjYeh2H4d8UU6aNnTii7DKpG+Q4aCJfYfDeYPmOR6YrWnlnNrVqYHc1in2hyVRmOhS0DYXt5c/THN2qk5bIQtnqLbMxBXyFWkxStSZCNwQRiEgjIiXIytmpn1J1HhPiSbibjuDyOAxAZHMSlVJzGsecF4fDq0go0Q35HowxBtyFExA5de87P8awephOs999jMslfKtTqnVfnvA28/wDziTYz/wCikUqG8L+Vdtn2kPTymB9tuG6KjUacNeAGuPWdgN55DE+z46e0dkPedIutfY+1fXwnl2by9JqumkT0BCkg9SFJLged45DbHfQKxsvX5nzNUsou3X4lq8CRPHmKkL0p+JvWR4fUYsXZQudU8sz+pzX2gtlTHPrOM+G8IyuYkZZqi1VvorFQrADYFQbk99gbbkUJSpVP9d78DbOSVK1Wn/sAtxF8okTPPSd1PMkVFbmZuCOs4nFVlJHOPNNWAPKM+CU0NRqtI/CD4TusgwQefSOhNzirZ1Uv2i7t3CIrs2DA2+JMmStV6hBBUOQDuCZifriGnkxbheVvmoXjaRptfE7mwl9EXNp7B7BpTyOUGdzJGpyf2el1iR7xu0zHl5Y5b4cYci53D8nwE7N2ZOwU2X+x+yjxMV+0/tp+0gl3uLRsAO2Bp0KlSp2lU6Rj1qGzUjToDWeZZ2oCxIx1VE4dRoVwENLqfClRY1G1xMEczF9pvGLNlBuVOhnO2kjIjUQioyszFVU10s+pYmLFlXbUOc9JwRsSSPqGvzb784AuAP8A6nT4gVVnYwZY/M4AktCAAn37AR8cIDsWMY92dvqy85uO+kkq0admqFp5KsR6tY43/Gupv5fuYcbaC3n+p9Qq0QbUlbvUYkeqiBjLpuHPoTSG3nl0ZQ2fqLIUhR/AAv4XwPaONMvKFgU6+8tpZ2oQPCx7iTjRVaYaaxtUpmpc2Ybjp27KeXe3O/RPf85EWsb7jNX7EUouwIlhH+Uao+f4HFmziyGQbUbnKLP7ROIMubqimSqq0MRvqKrz3UW3BEmcc6tVdQLZCW7HTUoMWsw+bzJJviGo9zOgiWEroqd1JHkcCBvEIncY0p1tKXJJPIicODWEQVuYtOZaZsPJQPywnGb/API/CIbw7NNq8RJB6nD6NQ3ziaqC2Ua5njbwBSgRv3+UHFD7S2iydNnX+0u4fxxWIDLpKAsGnYgWIIAIvG+q/TG09oVsnGnXWsx9nIzU6yCZOsAAKhdTYENpcdomG8gTO4wJpVANbjjv/cMVEJ0sfaUVKbJZof8AmW48+a4Q6ldc+ucrpsG0yhXC86KbSLA7qbow78we/LElRFItx3bp0aFVlIP/AGb3hvHfchatNm0T126qeVxz7449T+Peg4qU726yne/9tDaUwVbYrdGG8Sr0Mw7s7agF0kgkatYMzF4KSQR947RfsbLsCl8bcOc4+1fyBp0uyp8eXRmI4v7S5Rhpy9BsuukK40rcg9ieguQTOLaddFXCBblOQ1CqTiZsUzYyQqsDSrEt3JnzJgN8lOMWkrm6NnBeqUHeXKdzNCvTZTTU6kIbWlyW6wLkDaSL3xr06qaDTeOP6ilem4z38Yb7VZcVkTNouksAtZfuOLXEWuCt+i9cHtKioorDfkfOBszFCaR3ZjygFFDSy5IYS5FucSdP0V/Rh1wK3p0st/XzCaz1M93XxJ0c0TSqF1BAgbeQjkR8YsCMEtQlCWF5hQBgFNpZwbJUcxVSkupCx3EmALsSGmwUE/Fywg06VTIZddb5UlepSGI2PXW6Fcd4jVrVyYOgALSUTC0lGlAOkAX7zPPEx2UiobD/AJLhtn+MAnz898Ttk2Bl9V+Sgk/PYYMbORreJbar6TmhhZKMH7zeJu29h8sNVGGSr+ZO1QHVpFeH5gsHYEGZBYjl2xoo1Sbn3izVp2sIwzGWCkVF+K0kuFAI59f9jih6YBxjXztEI5Iwn7SyvTLKGWpSS0uqt33ECY5npjSpIuGA42mAgGxBPnFVfK0tZ1ZgT10O3LrF+mJmRL5v7GUK72+n3E+ehRYD9/cf+m39ceK0yPq9jPBnB+n3lJpUV/xXP8tOP9TYXZBvPL9w7udw5/qT/aaY/u6U/wAVQ6j6KIUeurHu0UaDn8f9mimx1PLr4jXK5R3QMcxUUm8AmBe0RYeQxQiMy3xERT2U2CiaVfZ2kkGtmwGgHSinUZsVDH8xscdEUjcYpyjXJvhXKPuF1UD60tSpSQpYEwBLG22xPPn1xWLYc8t0le+m+YjipVsxVermFl3Zwq03ezMSFBI0kcoPTHIKBSQ7chf9TroTgGBfcCCVk4eDLrmTtZCiLPP4tTAepwqotDXPrnDU1zkMPufiFPkqJCvkUp1F0gstdz7xW5rBZUYeQ5bcya08r0gD65/cCB2jA2qkjy0/JlDO8/vssaY+8qlPUT4GH6nGMTezrb2/UIAf0a/v+4tznDNPi1AoZ0sLAx+B7csLahbM6Rq1r5DWcoVk2UbY8jLumMrb4UmVQprDQSbDvgsItcQcRvaXZuiqLUI/hSet5Y/+yfXDHAVSfTrlAVixHPrnBcvxLT4XMpEXkgdiBeJva4Nx3WtbD3W0jGpXzXWMaHvFXUrTT2hwXTe4DpMT20nvhoVrXU3HW8fqZiW9mGfv1zk8xkDp94opwZlS41KRH2ksVvuQCOfUg9AkYgB15bo6ntABwknl8yrLZ5qJl6baGsRIZG7TEfI4nA7M95TY8jKi2MZML+802R0vl2ai5CspUagCQyI4Rd7HxhRM/Djo0kVqRNPhac+tVYVQHmVzBpaYt737zpAPktxO24PaNhKVTf8AV4jWOxPfw8DA85UzBGnWW7I0W/lEE/LAP2trX5fEFezve3XnFNLM1EsrsvYEj6YnDsuhtHFVbUTXeyfGXrsctmIqJUGlQQBDEwIgbtNj1C4v2Wu1QmnUNwZBtVEIO0p5EQD2hoqtY0i3uym3NGkCCGFwsAAb7eeF7RbHh0tx0jaBJTFrfnL6nDP3CBmZZJMqDUB3j4AeWn5jphrUv8YBNvf7RYqf5DYX9vvC8pQp5bLVas3ce6p1CsMZP7xlBMqIBUc5knpgQgp0y17br7/Gw3cPvPFmqVAttMyN3h58YDRp169NqaJUWIKsSw1coZjYza2222PAVKqFVB8P+wy1Om+JiIKfZ8pes4F4ILQB5scL/wDIV+s+8P8A9Ib6BI1UpoPDVpr/ACAM3/VqP0jGEIujAeWfvNBZtVPrlBnoUWJLZjeN1eT6xgGWkTct7Qg1QCwX3hNGhRZSi1fmhH65YaiU2UqG9otmcENb3kqGQZoNJ1OkxvpM/h9ceSkTmhE81QD6hO1+CzcmGvK0/GD5RZT2PpjzbNfPf4Zzy1xp98orZ6SmNDk89R0x2gYmug3GUWY75MVV+zTX1lvxOPFhuAnlU3zMvo1DySnP8v8A5wBq23CVJRvvM01alWUwlIaYEW7A4Cr/ACPZuVFspRS/iWqIGN844/Y6Dhani5LJEGRyYG4aZ2kdOuPpkFNwGH7nyDM6kqZLjUU8uFQQahCxtCL4mG/iltMnYDVgNoOQUab5tEXe56vPP+IqWYFDKqAs89ySfKSccmtd2xLpOtSsq2bWWFlKw1zj2RFjMzBygYYg7RhO+M3QzL8UqUrhz5Ek/wC3phybQ6aGLair6iMaXHDUGlyHB3Vwp9VZh8Q5T8zhy7RiFjn529v3EnZ8JuMvK/vIklf8OjedJNICQIkEbhhIkHaehBI3tuHK3XjCOe884x4ZWBJX3SM6KagCoNxAWJ/iZcNWoBlhF4l0OuI20iji/FIIRFolVFyqeEtzK9RsJ5wTscT1No3AD9yilR3knnFKZwz8Kf8AThIqngOUeaY4mGZLijUWDqqT5R+F8NSuaZxACLakHFiTLG9oaxbUGC9FRQF725z3nAnaahbFf4hLQQC1vmHUM4tUF1XQR8SoL7E+FfhcR9mx6k4ctTHmB5/rj5awGBTInrx4ecacGzKhGC6dBYMGUW1LcFgfhfe1vIjFWzFLHDpJtoxXz169pnuO1DSqlIDJ8STfwm4jy29MRbQTTfARcbvKVUbVFxaHf5zvDlWojEoQiwC24BOwE7seSzym0TjaRV8rZQKt0N75x9/wrIkKK1YrUYCBVe4ufi0qDH8zeuKmo7OP9jZ9evOTCttGqLl4dfYRpw32XMgp7p6U+KqCqhRBmFCnVI5hjhnZKlsIBBiW2kn6rhuHXxLqfFMrmta0lTWpAQtOp05tqUBhJ1GBMTMXgFTqU62lsX3EFqVWjmxNt/gfeX0MoHrCn7qujLpEe6LKZ/8AUJKgQIkzJHXGtWGPCAeX5vBAsmK45/jWGce4plcuwSpTrVyoAC0l8KRIuwK3kDmdr911a+HQH0H5mUKLuLggefxMzxL2hyVTURRzlKppOkgmA0iJBqRp+KfD0xO20I+uIessTZ6qb1I8v1MrneGM7I9MWqmDNtLfa1dBuZPIE4mq0r2ZdDLEqgAht0W5umnvCtEsybKWFza5gdTMdoxMQL5RwJtdpcOHvALwi9XOn6bn0GG9m1s8vODjG7PyhWVq0kYQC7Xu1k2tCi59THbDabIrC2f25RTqxB3feSq+/qMQVcrfTCkKBPKLDGt2jsRY29p4YFUG4vKq2TrKQR4f86rf1OAanUU5ZeoH5hK6EZ5+hhdKszn/AJhqNQdS66x5MMNUlv8AZhPqLxZAX/XcehtLf+G0YZlrKAu5XVUA+QlR3JOPNRp2JDfc/jKElV7gFfxGHDcnQVlZnqVRMk0qa6Rfqz6vmgxHXpAKTcny6/E6ey1CXAIA8z+vzPVqCZV1VlWoQQCCGUch/Dj5+tRV3LWbPxHxPpFqV0GEFeR+Zk8uiqBTap4pWKjwFLkhFhV+ySTEk8zMDH6JlRS7m5n5cSajXUZfiYv2gOYY6CQQojw/F/Kw5fyrbrjlbQarZbvD8zqUBTXMa+P4iXL5Spuobz2H1tidKb6iUO66GG0cqSbsgPOGn8Nj2w5UucyAYlntoDLK6UlBBcz2Qz9TjXVF1PtMVnOg94D/AMpPibMHaYVOt9z0xP8A4eLch8x/+XcB7y0vkwI/5k//AIxz9eWPXogf29plqx4e8bcGzlFw1M06rLDMuuoCQUUvbSim6gjfn8qKTqwtY28+A8hEVUdbG45cfUwGvxKshBokUxyamI1DoxaWO91JjtthTu17r7de0aKakWbPz695Bc4laVzKhWO1ZFgg/wAaqIdfISO+Mxq+Tj1A+4Gv3m4GTOnyP4O77Rfnci9E6XG91YGVZeTKwswOFvTKZHryjFcOLjrzgrtOFmGJ2jT1HeALk9BjVF54m0serqI0+ELsPz88EWvpMAtrNL7NVPeK9Mj97U+EjdiizcDnBa/OcXbI17g6mQ7ULWI0H5j1PZRq1FVrOEKTt4mCkiAItza09OgiupsbVFVW66zki7YKbErneNavs2PdqlOsKKqJQ+61RO7A6hLmw1G4v2g32YqgSkbRa7Tdizi/rMzmvYLdlzik/wAdNlE92lo88Qt/G1NbiWr/ACA0KnrlA8pwzM5N5FekoP3X1K45jSPi+nKCN8ZT2etSN7gev4hvWo1RYgn0jNHo6hVIKVBJlvjJHNVBOoDqZI3vGKx2d8TCzff0kx7S2FTcdb4LxD2q4g2qnTrKRckqFDmYBMkAmRFxGI6tSsTZTy1lFPZ6A7xHPSI8vnM0SFXe/wDhoLAySSVsB1JgThS1q4yB9v1HtSo6n7mNU9q61HQgrCqQ0tCroUbaUlYaYuYiwC8ySXbKgOZvFnZKbD6bddfmG0+JU6yfu9KN/ig6iSvVQAbg7zNoAOKUq481PmPDw6PnEmkUPe9NNfHoRJn3alE5hipnSadMC3mSsHE1UlDmxt4D/kopgPoufif+xW2bpX/dlzyNRyfosd+eJi630v5n4tKArcbeQk8nxFwy6dKcvAoU/OJ+uDp1mxC2XlAekMJvn5yObqsWfUxaDaSTvfnjKhJc3M1AMItKgJF8DqIWhkWcbYy4mgGdpOVYMhII2IxlyDcQhY5GMMvV1NKeCoLwtg0dOjdtj2xpGLNcj941Xw5HMR9l/anOBRDSOpUEnAhapztf0lPboMr+877Q1RUB945XxBVIG0Juw6eVxbpjr7U6sAGNut8+e2amUNwLy3i/EV94XpAkOAWrkBodgC6oIhVnebm52jCcZXd668urzVp3Fjy059WgFfKtXEVTNSP3bz4X3MHvE/K1hBxkNXJ9dx3GGrClmum8cIiFUpKEQbgjoRyxKCVylJUNnK2zH3rj6j9dMYX4zQnCQOXm6mR9R5jljMF8xCx2yMrqJBwBFoQN4dwnP+7YG1iCJ2tyPY7YZTqYTFVKeIQrNhKUFAWoubAnb+Enk6zvz32ODJCZDQwVu2uo65QPMUY8QMqfhP5Hoe2BZbZ7oQN8odwniYC+5rr7yiSYHNCftL8yYBH4gnSq2GBhdft5QKtPPGuTffzkeI8AKL72gwq0jNxcrBvIgeewIFyBjHoWGJMxNSuCcL5Hrr8xTXOlQnq3nyHp+eEtkMMcuZvLOF5J69VKVMS7mB06knoAAST0BxiAsQBMdgiljpPY+E8JpZOmEQqW+3Vbd23IUD7I6dl5nHf2fZxTGevXX/J8/WrNWa504Q51RYcOuk3ubgzteZHMX59pNYZj3TAWg7bos4lxQ30iw52IPcRt5csD9IjTs7JYsNZmOIVZhoBnqLj+FvlbE1Rr5xyDdIZDJBqlMEWLAxvzv9MYtMEia7kKYD/aAFXNAJAK0k1RvqJd5PeGXHO2xv8AMbeEr2K5pZ8TFVCcyAmzrJ1bAARJMbLcWHO43Iwtf82W8b45v8Oe4ydTO6ZoqWF/isrOw2Lg2josx3JvjzVMPc6Pn8Ty08XeP/PL5gVRAQSyA3uyeEjzXYfIYWQLXI5fEZc3tfn8ylaJnVSe42nwN9bfXAhTe6H8GEWFrMPzHdDi8oUzFJW6kqRIgAExvEbjpita91IqL16SU0LMDTaU1+C0qhHuX0EzZzKn+VgNvOT54A7Mj/Qef4MMV3T6xy/IinMZCpSMsNouLjtPT1jEzUnpnMR61FcZGXZinqAcbEQex5fT8MMqC/fG+Aht3Tug6GRGFA3hkSK0S5CqpLEwAASSewFzjCLzb2kalJkMMpU9GBB+uPWI1m3B0jTLoFQVHEz8C9e57frpL1UBcR9Ioub4R6yRzYa76tR30lgPljcYbNr38JuYyGk0HGKVN9wAdbkAkrtpHIx1/IYfXZHNvODSpOntFlXMe4HhV1J+LSwIPYysEdmGFF+zGQPP9TDT7Q5kdes+yvE6KxrR6YPNCCJ5HQbCP4Y8jbB061NcyCPL4+Ip6TnQg+fz8xlxThNKuv7RQK1LANpY7xzWNSmLQRtHc4oeilYdomZ39WiKdZ6R7N8uEzGYCCQ1Mg/zfUdRiFsAyIPOWriOYPtKqDUdQ/vV8ipv/TAKaV945Qm7S24w6vQpMP7wBu40/Plihlptq2fKJVnXQZQCtwqqtwAw6qZ/84nbZqgzGflHLXQ65ecIyubekJEfxIwlWHRlO4ue9zEY8CVEwqGMJ4rRFF9EN7uoq1AjSHUNPhM7MpmDzEHY424F1OhmISwvvHLr7QAUSg1bg7Hrj2EqLwiQxtL+DZyotTVTYrzY7rAM+IfqDtBxtJjiuPXyg1UGGxEYZ3LU8yNaAI5IB5AEmF19Q2wfebNM6sNqKtXvDI9dX3b+MXTZqeRzHWnxy4R/7A8PGWpNmqoio4KoCPhQElmPmVgdl74o2DZ7XqNJNvrYyKa+snmuMGrWRbwSWvu1iflbbvi3tbuFGk3ZNnUuA0X5rPvJ8WPPVN5fUTPSFcNzRYFWuCDPoN/TBo9xYzeyxqw8J2nlmaQJa0zHIXBOAwEzkYwJoOA8LKuruPCg1H0BkHvYWPUYYVKrEPUDZCYXjORarXerUr0QajEwCzaeQQ+EDwiF9McZqLMSzMBedenVVVCqpylVfKgUxSoOCPjqOWVC7C6gDVOlRMSZJYmNoE0zbCp9x8wg/exMPLU/jfF+arX01lJYbGIYD8xgXbO1QZ+/7hou9DlPqlMkBtQ7OPSA3Tp+rCRlcc4Q1t7StpJCFDqmPCIPqNj15Yy98iJtt95bkV0mPeBhzQQfnJg/5ZwdPunW/h1+LwXzGnXXG0sqZ4UmKmlImd9MjyURjWcUzYr+JgQuLg/mG5HidJxpOtJsFZtS9oaNQ8r74dTroRY35/m3zFPRcG4t14X+JfXo0hJ0EIwIJUjTbcwBFrXG0SY2wZFMDTI9de8XdzlfMRLnaApGfdhlN1fUxBHXwxHl54lqJ2Z0y47pUjYxr6SAzzQQoVQRB0qASOYLfEQeYmDgC5OXXzNwjXr4kKDtIVWZQSBANrmNtjjFvewhG1rmHcTrK1TYCAAABECAQLW2gemHViC9uETSBC34wNq+FYozDPU/7SvZ8UCABGkbgQDJJJA25xy2xylruKnZvzn0hpU61DtkFs9J5m9d05yve4/qvpjopUZfKcarTVt2crd6dT4/Aeo2+g/L1w26PrlJMLLpnLsllq1JtdCoCexEEcwd1jsTgkSojYqZguyOMLiaHL5UZ5HV6eiqg1FluAJgsL3HUd98X2XaBZxhbjISx2cgqbrMnm+HPSfS48u+ObUotTazTopVFRbrDMtwPM5nxUaRZQDcwotvBYgNHOJjBClUqfSIDVqdLJjJZjhmYyr6Kymm20ggg2mJWQesY0JUpHPKD2lOqLrmJblOMtSqBoGpCGUkSJBkGOR7jDe3se8IBoArlLzlaOaYspZKzGYLa1qHc6XIsexjpjezpVTlkfO4Pr8zMdSkLGxHIj0+JGhwpnVvGq0g2gmpKePok7uOYHrjBSxZXFvHLPw8Zpq2OmeuWeXj4SL8CqqkIyMpgsVbfpaLx2nljW2RwtgQeM8NpQnMEQz2W4X+81u/7sSrjSQGJmUBNzESYFoGxIweyULvrlv+Ivaq1ksBnu+YdxLjJqOTEhQYk25qBpFovixqwvkMhJUo2GZi2hnbhio1AyIt+h2wpamd7ShboQymM6VGlWuG0nmGsRaTB2PnilcFTwlp2imRcm3h8GdJWn4VBYmxMGAOcCLz+ePEhcgIqttK4CtM675fw9ahJMlSx3kgken6vjaasfWcxyoyhXtbxUUqPuANbgBqokjwltWkkc5uRyAXrhG11wAVGfGFslEswc5cJjsuKdUeGmin7rQJ7Bov6454KMMlHXjOkcSnMnrwkPCDBDUyN+Y+u/0x7uaZiZ3tdYP7pwCABUWfh3v25qf5T88BgYCwzHXWUPEpNzkZKjoQ/E6HmrLqHluJ8iBHXGLhXfb3njc7r+0l71XBRJXqd2jpbdP4RfzsDpKsO7+/+eE9Yrm36/74xXmKDIYPmCLgjkQeYxMwI1jwQcxCsrniBofxL848pw5KpthbMRbU87rrLf2dWPhAYfwmCP8AKZwWAHTPyy9oOMjWFZPN6LFqi7XIB8geTDz9IwxHwHMmA6Y9LRlRy6OpFJhe8QdAaOlyoPTvYxAxSio6kIfTd/yTs7IwLj5ibP8ADzuEKteV3EjfSRY+X0HOSpRPCx60lSVRxuOtYFTYoQTuDt5HCASucaQGyjrN8LX3nvHbRSIEX8RgQOsSAG2Jvtix6IxY2Nh7yVKpw4FFz7SKcURRCooA2kCfXf8AE4EbQoyCiF2LHUmbjj/tctctTzE33LKVvyO9vMcsc7ZqAVrV9fKd/aq64LbOLDzmPzfCFJ/c1QSfhSpC6uekMTBPQGJjvi47MB9B9D86TkNtJ/uOXxEmZyTqboy9iPwP2vTCmpsuomCoGzBgniUyJU/I4XmIes0Hsrn8xSrJXDHTN9beFl2YX5d9gYxVs/aE3vl4yXaVplSts/CbLiPFuExqLM/P3GloB6agunSJOxO9sWNtKEYXz9Jz0obRfu5eN5V+xDPqXWuy0aSgKlGAqg/CgXlt9qOXLbWp0qtirG3h8TRUagbFRc8fmF5Hh6ktRZ2amZGjMQWgREMgk77XiJBWMUJQKrgOY4H7gxL1/wC4FjxH5HXjeCUfYaktWKoZ1YHTNQKBzjw3JHWdJnrsj/xU8WdzG/8AvYr3bCfZD2NpUVFWqVAiTpLkBtRA+1fbYib9sMo7JTU3g1dtd+6v4neLIQWZqSnSSoLAMBe8arCZvABPyw6oifWR5Qabt9IMBo5IKS60P3hLNYsQOtiYHlty8lLs6qbhcz5xhrlhYtlHVJAxpnMsqVFaKZWzMWHwldgAJt8o5vtmMRzk5NgcAy3yir7Ni6LpkwQwGkb/AAkkATvzwDKguv46EIVmyaSpexriCysfID6kTby37Y8tKnxnjtMGzXBipZRR0kyNQDEkbHdjEidhzOC7Ebh1zmitfUyFHg9QCWIpoN3c6VXzLYBu4Lk2hYwxsBcwfOe0VGgCmVirUP8AjMPCvdVPxHubfzC2Jam2bqfOPp7KWN6mnCZ6gW1+M6jULaibkzJMnE1MWax3ypjllukc3wl1PhEg49U2dhpPJXB1ljZVyqppDDmDy8jyxppthAmCooJMn/wYpqYE3uRzE3v1x7sCneEztw+UVVq7zDBWA6jCS7HXOPCruylfvgIOgA9QY/LGYhrhm2PGM6GcFRY92pIk6YBJ6shizdRsfkQ0OGH0i/WkWUKnUwPMVGVQ2lXpkxqAtP3WB+Fux9JF8LaoRnYEQ1UE2uQZSaCuNVMXG6/0/X9MCUVxiTlCDFTZucqGbYbGfO/13wIqMNDCwA6iE0M+qkHTpP3l/wBxI8ycMWqoN7Z8RFmmTlfnNBQo+/8A3lJjezyJRiBzBgk22UE2B54vVe176Hzvp6/qRs3Zd1x5W1685bXydIHwge80wKkSqnrYkqe/1G2CaigNwO9x3fPOCtViM9OG+L8xkmdWSo6MwuCGjnvDQYnts3YYmamzgoxF+vKPVwpDKDbrziN8jUBj3bW7HEZpONxlQqKd4jzS7JEn+EnxEH7hP3G5dDHU4owOVy68PIzDVUHPrx8xAFzzJukr9qNvUf1wC1SmomOgfQw+jmzVTTSqKefuat58pvP08sUK/aD/ABn/APJk7LgPfHqJKlUYiKdJUrC/uyPi70+vlPzx650VbNw+JhAGbG68fmfZbOOw/wCY0FbxqAUieh/M3741ajH/AGW9cuU8yKP9d/SA5qhliTpqxP2bkg+cG3qcIdaJOTRqtVAzWH8DmjqNGs6AiHIPhi5AOqnB5na18NoIFN0Yjryiqxx2DqDw6vLl9or2rk3kk6rnramB8xhg2sD+569IH/lv/QdesdZb2srsAmpKqjkwvzI8dMArF90O9zgqddg10a/n+viA2xJqRbrx+Zp+KZjSiHLItQk/vFaoAQCFPgNTw6rn+mKHeva4F5LToqScRtwy+8Q8bLVWplaZWq/xIX95oMvF9gYY7f7aiE94ixPtaFkl1vkN/KDcQZqTwtQiAAXDGFG0IdgZkkiT2tOMqMcWRsOtOEKmAy5jrx+Ijz/FmSUoFQ32z9ozuNW+ozczPcc5ale3co5cfGVJQv3qufDwlnD+NtTRi7M1MCCjElwSTCyY1rub3hT65T2jAMySOG8es9U2cOchY8d0vUsiVKlGoWLRpsBvcAkAfzeICTpHWWnFhLIb3i7KWCuLWih8znVma9dJN5rMo/1YganU33HmZWOyOgB9IDmldzNWtrI2L1NUepOANM7yOcYGC5Ae0uy+S8MllMdGGGLTyvcc4tqmdvxDsjTbWp0G23MfTD6QOIZRVS2HWH5nNBbVCV6MBMeY5jFVSooFmiwlx3IsqZtgJDTP2h+fQ9sRFzqDlGhBvEnmc64RWU36/T8sY7kAETyIL2MWZin7xS6ja7gcuUjt+GEsMQuJQvdNjFtVCLYSwMapE7TZkIIMEXGMDFTlNwhsoxr5g/3yACbVFiVJ3uDuD+PfDi5+tfUdcYAp/wBD6TlNV/vaKxpgvTJmBIuOZT6jrzHlI+tN2o63TGU/S+/Q9b5Xnct7wh6YJ1bqBsfIfr5jHnTH3kGs8jYe626FUshSpKGrtqPJFv8APr8wO52wwUkpi9Q58B+Yo1Gc2p8508ZaqfdsAqEBQqwLDly9IESFtjRtBc4DkOE92ATvDWRo5upS3OqNienIqdx5bdseSo9LImYyJUjThfEhUbS0t4TKlNUeE7Mo/EYqo7QHNj9r+8nrUCguPiOa/s2XOoUpkAyKRN4E309Zw99mQsTb2k6bWVFifeYPJ5oqe2x8scdHKzruuKFVGfUw1tIJvO/f1wy7X1i7LwkGz7A+IBu8AH5gfUg4ztSNc5vZgjKMcpxZtw4aPs1BBHdWX89u2KE2ltb+h/BEQ+zrw5fBl/EAjL79aS+L4xE+LmwYE2P578sFUCkdoq+f/fGZTLA9mW8v+RdkKNOq0LSM9iSB5kt+WJ6YRzYKecdULILlvaW8YzQ0e6o/ANyPtHz5j8YHIDBV3BXAmn3g0UIOJ9ftE9NYxETaXoLxpkqLl1NMsHGxUwR6jCqlcU+9e0so7Kavdteek8Gy1c0FNShrgtpZPCTIhtQ2gb6rQfp0ti/kDVXPOR7d/FmiSdIh43x4UCwoQzH43YSI+4scupnt1xVX20Lknrf7TnJsWLN/QfmCcLzq5n31R6BBKz7wtKSoKgAEbw58IPOTjNnftmLFfXd16xO0IaIChvTf16TN5uiquxL31GdNzMn5YjdQDrylasSLWjTKBGK03A+EVG1QeUgHlOnT5FvQ1Uyr2Q+fXp95O+Jbsvl16/aLeK5isKpbU4gwpBieu3M7x3xNXaoHJz8I+iqFABJ5OtUYgadbNyUQT/0wPUjBJUqHLWC6IBfSMM3Ry9ID3sNUO9KnB0GftuAL9hJnDm7FfrGfAfkxS9q306cT+BBM/nlZAvwLyRBpXzgTJ79zgKtVGW2g4DSHTpsGvr4mCcOpLrBVyCDPyudwOmFU0TECDG1GbCQRCq+Yep/iqw+6wDDy3MemGMzP/YHz6MWqqn9SJylQII8BUn7jage3u3g/I4xUYHTl8GaXU7+fzCK+SbQdI81uCO+kgEb9MG1M4e7y/UAVBiz5xNRrFGkeRB5g7g4kU4TeUkXElxCnpIKmUYSp/EeYxlXLTQw6Weushl8pVqWRGbyE4WqO2gjrqupjjhnC6pOh6bBXBE/PSe8H8cUUqNS+EjIxVV0tiBzE+4dwCujhnXQoMHVFwbERPTrGCo7LVDXIsJlWvTK2BvG9XJUsudKuKQuHLsruRygW5eYvt1swpQNgQON8yfSQktVFyCfLIRO7ZPU3jZ5vMsAfmhacTk7Nc5k9c4dq9hkBIaKZj3TBTyjSx+ThcbZT9BA68Zt2H1gnrwjHNayp0GHUAgaevxLDDrJG+8c5xRUxlbg5iT08AaxGRgnAOIVSGFiBGmw3J6bSY3jlhOy1XudOQjdppoAIS/tE6kroZ4J8RZfFc3+AwPXBHamBPdv15QRsyka2684jNCmoPiJ/yj/5YlKIN/t+5TiY7uuU7m6il/iYAhD8A5ov8WPORi14bvDznlBtpx3+PlK5Q/E0/wCUj8JwPd3mb3twlqhRdWX5x/qAwWQzBHXnMzOo69I14OgUM7P7tIhryDOwBUyp3POO1sVUAACSbDfJ6xJIAFzug/EKzoSlR1VDsFQ6WHLxIZY9bm84XVLKcLEW8jbnGUwrZqM/POfLkldNSQw2Oh/EO5DCI/zT8seFMMO7n5H8H5mFyp72XmPiRoez1QkmGAHXSfkFYk+gwg7IxO/2+ZWm0qOv1PRP7N/ZEVKvjcQI1DSZjpJsCf645e17CXYAtlO9sm3ihRZ1Ge6eke1VWgco9OgVgCCEgSv3Z5z9fW+Aq64aO7hFUFqrVL7Rvuc+M8PzC0FJ/aARaUQKKjdVLAsule0gnsL461MqFGIenzOTtGIscB/HLIwROI+8daNNWCE3ZyNTGLCFAVEAmFXrJJw+nVZmC7uvQSCpSCgsdeuZ8YAvDddUlPGjP4mW4QTLk9ALm/KMB2d3uMwTy4w8dkzyIHPhJcNrgnMZpyRAhFG5d2BVbmygLc3tG+PUmsWqnq/xMqLktMevkJSubqP7tVTWWOnTG5tEeh8hGM7VzYAX3Wm9mguSbRtxDO5ehro0tYY+GrUpRe0MiMxkIDIMBZvy3c1SmoKi442+3l94hEqOQzW8AfvlvidcnR1WqkdA4j5kWGE9lTv9XOPNSoRpylPEl06ZWx2IMg+uAq5WuIVPPQyAcLTcgbjT/wBRv9Afngb2QmFmWAi9UJwm1429oXQzRp/CxB7G2Gq5XQwGTFqIzyvHXVfEocTtHUWixXl0OKE2pgM84htmUnLKH+8o1Vl6UHaWBgTzn85UbYccFQXKnr3iwHQ2Bl9Pg1PQR8QB1KNUiY/A9ATj3/lGHS+8Z9fmGtc34ccpRU0A6Sx/lW8dCuqw+WIq1VE+o9fb2nS2ei7/AEjrrxjPLZBisoniOzMTI8r7+Xyxyan8siHujPiZ3aP8K7i7HLhKfaXIVlIqamC1FkAMYBHhYW7ifXFH/tNUhr5Ececkf+O7IFbaeHKIONpqqmQZK0zJ5/u1w+ocT+dvtIFTCvP7xVXTTbHtIphfOWIRAJw4EWkxveH8K4mdQpEnQxgSfgbYEdB1855YdR2gg4Tp9oqtRBGLfHVZDSGoAAn3lRl6NTpn+pb/ADYsZOzBI8TyElVsZAPgOZmek9cc/EZbYTuW4dWAvQrEf/Tb+mBUHhPM68RzkOJUiNBgjwAX3lSVgjkYA+YxtQaHwm0yMx4/eL3acKOcaMpOnTLkKoLHoBJxoBbITCQuZjHNVAipSUg6J1kGQzsbwRYgAKs9QcOLYQFG7XzPQEUFLEsegOrwnJ1BUT3RjWI0SYDDmh5G207fPDUIZcG/d4+HxFsMJxbt/wAyWWopSuQUqmbE6tPUDbxH9dD5FRDfQ87fbOecswtqOvPKWHNagQalQDmCu/n4owuu9wcz16yrZlAIyE3PAeP0cjw8rSDGrX+KpAUgGQFFzFgSDf4scsbOSCx0OU7L11uoFu7u3X4nS+4W8Ijz3thUCqKQCyCb+MkTAvAiQCLAHocU0qa0QCgsdYivtL1iRUPhwiCsq1iXRrm7gzvuSY59Tsd7GcVimKpxIc985r1SmTeknw+lUVySgIgwRD3jkRIX6G+HUkdWxEfmSVWRhYGcFfNMzGnUqmmYYGSBEzEdAbRgFSoTiW9prdkBYgXk83lENMk1EQFgxAIIDBSGELfn9Rhr01KXLAZ+efpAR2DWAJ+IyyOWGXpFlcGs4JQgHUixDMAQIYjbn9cNWkEp3vmRllu4xLOaj2IyGvnES5SkqAkv2BA/riUJTVbm/tKS7lrC3vK2alHwsT3t+DYG9O2hm2fjOUszTEoELKd1LGPO8wcYKiDK2XnCKMc7+0tz7oqALTBUkQZ3seo3Exg6hULkuV4NMMWzMCbOAC1NZ8gfywrtABkojMBJzJlQz7DZVHkI/A4HtjuAhdmOJlrZ1yCZg2NvOPzwZrMRAFNQZdk6rTrdjo2I+92t+vrHkdvqY5feaVGijOSr52qtXWGIIiINiNxtuDz9cKqVHV8V5VSpoUtNRwriQ95oKKyNDLyMHt8Pb4fXEu3bQQpFrgzp/wAbs6lgb2M9m9jMhlKtMkoNXQ2t2v8A0xydkp7PWuWGct/lK21UXAByg/HvY2nmFelRqUyQdSXEoY8SmJsbdLicPXZQAVpsDvHhxHrFp/IMLGuptaxy14H0nmvtn7CZukVb3eoBEEqZ+FQD35YoNVgwDDOw9hEqlOopNNr5nL1mF4nS0RKkHuI/HFmLKc5qeZgNWjAk7YdbK8jJztB6RE9sCus0zaVmlHV1IYUjUa3NqPiHqYtaMddjemb8L8xOWuTi3H7GZKob745ROc6QE9110xEIJP2qirJ/zOZOPogLnXkT+J81YwfiGVytRdGZC6dwSdXqtzH+XScBVpBxa1/P51h06lRDdJkc57HZSZoVdydPvl1DsJRwPmDiP/8An2Fxa/Xl+Zcu3ucmHKYzieUzNJ2pVBoAsYhEYbyIgMLg459RaqkqcvYTo03pOAy5+5ga00ghqgnooJ+thgAF3nlGEnUCF5WqEGoLB+ySfEe4+6O+GowXO3z+othfKXVc2tSDXVpAjWhEx/KbfXBM4fOoPUfEALhyQ+hlmVpUm8Iral2UlGkdB0/XLGoiMLYvaF2job4feafNez1R6SKtOow6StMRoQLq1S072Czvil9mBCqoJ9h6yddusxZiB7xbxs5Wm3ui7CFCk0wLQNiTJPORJ9MBXFFDgv4ZdXh0qlVxjt45zP8AFsq1AoysGRhqp1F2PXyYHcd++IqidmRY5bjKUqdoDcZ7xNTwOoHpnM1LqqgBCYDOCx0qOQASoxO8BeeOjSqEgOd4PMdfac+slj2Y46+HVrRFnfaSq5IqpTdfulSAP5RMD5H1xNU2t2ycAiUU9lRc1JBjzgXDaXujXqU2VWhqdCzFmBkOAYlYO0eIco3oo0lKY8ORztxtwk9aqwfADnx4eBiPP5itUrNUqsdci9wAB8OnoBy/rOJnLsxJ160lKBFUKukjUb3iz9ofJvTk344wnGPH7zwGE+EXofAThI+mOI71pTRbSpbmcAMheEczaE0MxNMI32yYP3SNIB/EeuGhrphO+AV71xugPumkiLjfCLHSNuJFhFtzj2k9rDKFAKPGd9l5nz6X/Qw5VAHei2Yk92RrZnUBNgNgNsA7loSLhMKp+On/ABJ/p/8AGN+tPEfb9RqnC3n94flq/wC6WDeix/6XM/6vxxPVpipStw+x/cu2esadW/H8fqafhPtdoWJIPYxjiPsRBus+hXbqbr/kF53Me0esypIbrP54fR2BTm2sRtH8lYYU0h+c9t6sU6dWozAot7EzcGQZDCRvvjsLRRSoYbhOEa7d4pYZmIuJVqw1FFSojX8Mgj0m3qMdDDVT6QCJzmqU3+okGD+GqB7ygF7eE+o8IP1w1cNQd5JC5KHutKKnA6HNSo6yR+bD6Yw7PR8R16zw2ir4Hr0jOrlUq1NQY6aqsp5jxKUbfSbGflio0xU0OotJg7U9RobydP2aycCbnmW95J7nTb5Yz/wURkfufmb/AO2sdPxGj8WqGzOWWJgmRbkQcVYhfQSbsxAg7FyRAn4l6yPsxvbkMCLloeQE4cxErFxzPOO23rjcVsp7DfOJfb5FqLQzAtY03HQr4l+YLf8ASccz+RW5FSXbASuJPWZjL0VA1tt9kfeP9OuIFUfUZexOgn1ZWaWm/bGsCc5gIGUnkKzMdBvPLG0nJyMGooGYms4JwZKf7yquqD4aQ59dZ5C8QLmeUX6dHZLZn9znVtpv3V5xvmeJLQDt45+JjaQ7ydMdjbffpio1+zQ5ZSdaJqMB1aI8xwWjmlLZeoPeBSxUjS5iJ1L9rf41nvPLn1KNOtmhs3A75ald6OTjLj19or4dlGWaVamWpP1MaWMQ6G4LbWvqFsISiw7jDLrMdZx71Qe8pzHVjNFWpZcUPdrmkWnRDpdXLB3+MmBpY3AsfstG+KbU1QhTkLi/iZKDUL4iuZsd2g03xLl8zkqF6atmqo2aqPd0lPUJux/m7G2JVNNM/qPjkPkyhlqvke6PDM89BBWzlWtWNSq5LFWA5AQCwA6XH4Y8Gd3ux3H5hYURMKjrSNqWZNRRrGsG1/iB3gNuAfUTipXLjPPrjJyoU5ZQOtkPAdPmORt+BGFNSGHKMWp3ouz4lNUXHxgW7ao788JqDu35/Mch71uUX1W8IxOdI4ayGZsFHRfqb/0xj7h4Ql3mG6DUQOLECHJsDGzfKx7jDQpdcXOLLBTblBmqoghLt94/l+vngMSr9OvGFhLfVB1Yk3N8BcmHa0nU7c740zBLMnmCjBvQjqOYxiEq14ZswtGeXqhHK/4biD3Rtj5i3yw0AK9tx+09iLLfePvINS0MVbcGP/PrvhJpYSQZQK+IXEuNZVEk4MACLLlsow4fWStpFtSTpm0gmSpPWZInqcV0cFS3ESSszJfgYTWqsCd1YehGLtJOcJWVVc/7yFJCv97YHseh74S1QMbaGJWnhF9ROZ8+6TxnU33Z288C47MXbMzU757uQjL2dzS1iFazLLDl4dJBHeDpN+pxRslTtGwkxG1UyguJoc5wjKM+p6ZLQsn3zrcKBsHEbdMUvsyOxY35n5kyV6qrYH2HxEdGgULA+Lwna4HPfnjyrhJBhs1xcSmpV/exeQFjlaJ2254At34YHdlmeMMY8QZZjn9P1bBPkYKZztPJJmqL0ohxDKQYll+ydRgEiVm0ThNVBVpEAaZw0qGlUB3TE57h+YLw1Gop2C6GEDoLfXHGanUY/SeU661KYH1DnL8rwbMAiU0jnqZR9Jk/LDqez1r6faKevStrHeU4f7oysAndzEgc45KOp/HF9PZxSz95G9Y1MvaOH4itNZRiBzPXy6eeKnqqEy0kq0iWziPM5lWyzlSYNQGOvw/mDiFnU0SRxlqoRVF+EzzagQykggyCDBB5ERscc9gTnLRbSMRxLMVQAxGvYPoUMw2gsBM99++HCrVYWJ684rs6aG4ElmaLpSVE5G9rbSd9xcYN1ZUCr11lBR1ZiTAVy2swq6anID4W7Cdj9PLCMF9Bn7R2LjpI8MLe9QmfiAPrbHqBOMTKoGAx9wMEFtQifxxbs41vJK5vpJZzXDkiOZ8xv8x9V74xyxv118zyBcojNXxAr9du4PbEpOdxKQMs5Otw8OJT1H3f/GCaiH+mYKuHWQzgpq5PxNMRyEW9dsZUCKx3n2hIXKjcJQmd8XiMqbEDoenfn6DCu0uc9IfZ20g9bLhTE+R5EciPPAFQDaGGJnFod8eCz2KTNOLcuvTG2tlMvLFyh+1ghT4we04QmiNS+7i4kp36r674IC4w8vibiscXOEU6XvVuf3icvvL/APIfUY0J2g8R9v1PGpgOWh+8AzCkixnCmUmGKgEoyuZamZGBUlDeecBxaanIcTFem9PQfeBZRp6G695mAP6Y6VOuaqld4E59Sj2bBr5XiPIpLmpUMIl/M8gOv+w54kp5sWbQSp8hhXUw2pWNbc+ODp7jfSfyOHFjV113fEUFFPy6zkeA1CK3vADNIE9LnwQfQn5YzZb9ri4f8ntoA7PDxmzzzvrOmmIt8SSSdIkz5zjsFiTe05SqLZmVZylUpVipuUNRGjcaZG3mMAjk2Ybx+IxlAuvj+Z9mitbQYC1ACNUQT0ny/PBMA5B3wVul+EpclqIGq6EHa8Gx8+WMOaa6TRk3nBs6GBUuxi/gFmPnyGBqAixJ9N8JCDpzglSswjT4FB2Bj1J3OEszbsowKu+V5b3Tltdm6k7nsf0PLArga99YTYxa0qqUWNQUiNKSDp69Cevbl+OEMCz4N0aCFTENZD2kcwEGPbYf6zdlH9oCtsuyA/cPnLE/1wjSlh8o/Wpi84NTr6ehOFhsMMreEZHNGo3iFlv/AE/XY4Ok5c5wKiBBlCOK1yEpsvMNP/VH1gYOs5sCPGBSXMgwXLVybm0XB5zhStfWMZbRrSrhXVrHXBJ7kwfqDilWAYHjJ2UlSOEHbiWipa99vXAGthaGKOJZoczU96qNsTyI37YsPfAYSNRgJESZzg4pamEum4A5A8jHIbT/ALYmfZxTBOolSV8ZA0MTUM++sRYTsNv/ADiRarYpS1MYZbnssru2m1Qkyh2N91P5YKqoZjh14fE9SJAF9OMD/YjiMm2UuWlvhVCmHXQfjX4O43K+fMeo6YYpxC2/dBamVN90CeieeBznismFjc4YJOwk/wBp07X6jlgw9tIGC8kyRDAmDcdQf0PpjSuVxpPX3SQqlzKmKgv01RzH8Xbnj2LFprPYba6SytFXxL4ao+JNtfVl/i6jn574e/mNeHX2nh3cjpF/vDMYVcxlhGOQqkERAP5bHFNIkEWiKgBGcIz/AO8H7sgqCSQObczbfnb1EjZlWzjuaD7wKd1+rWCE2kGCOeEnTKM85rfZ6mFU1iACYZpIu3woI7gs8dh1x1NkXLtDv695ztqa5wDrrSb/AID7E1MzQp12d1NQE6RsPEQN+wxlTbFRrHWCmzswuoyjH+072OqEvmsqpYvHvlWS3hFnUDcECGAvz5nEmybSAvZtz/Es2nZyWxjrxnniZB5VmhVI3PUHp646wpm4vOaagtYQnL0dbtTpA+IHxbktyjpfB3GYGnGCcgC0VcYrU8vYj3jj7INgTuXbl5C9uUjEW0Vlp5AXMqoUmqZ6DrSZ2vxqo4grTAGwCW+Z8X1xAdpqNrblLRs6Kcrz7LcSpm1WnpPJqf5qxNu4PocEm0D+45TGoH+h5zWUcuYRmEiLdh2x1UUNYmcx2sSBEHtMkFSDIYwMc/bRY5b5dshuPKLM5U/cMQBeoF7wFB/EnEtRu6bcbchKaY7wvwvzMVUaLM0Df8O5xOqsxsI9mAFzGFesqr7tLn7Tde36/rh7MqrgX1iQpY4m9JOo37qn5D6vXH/aMeJ7gHWpngO8euEDZDNzbCyDDBEuqVToQj7LOPlob8WOCLHCPCCFFzL8zlD79mWfdk61ZhEq1x58x6HbBNTJqEjTX0OcxXHZgHXTlNDwzNCoCoMgfo/r/fHS2ZlIsJKVCuC0bZarK6BBIkjy5j5fnig2tJav13mZ4nTKsWpKqQfEIvfYg/dO0cjHW3MrXBugtxltI3FnN5Tmc3VIBLQCNmiCdiCCLj+uFVXqEXJ5x9BUBsBymj9lamSzD+6zfhLWFRLAHkL/AJ2xzNrIexvY+H7nd2S6qQFB8/1G2d/s8am/gU1EuVdSSDB56YgjobjEDbU9K3dv4i9v15ToJs9Csub4TwOv7HjMt7S8LFNifdy0SRJH+YfmOW/l0aW0JWGIDPfObtOyNQNictxmYdkP2GB/nn6EYeGXh7/qc91Yb58y07fH/wC3DTg8faTjF4QrKNTIKQ5G4lh67DDaZQgrnF1Aws0FauEJHukt11T89WElguWEe/zG4Sd59oSuYoVCNQdWt4lg/M7nzicMD02+q4MAq66WMPanSJE3Y7OR4THWDv5jDrUydLnjuibva+gkM3nTSGhVEm7cvlpj9d8bUq9mMKjPfMSn2hxMYvXi7mxAg9z9JJxMu0NwEeaKxlk81UquqIhZjt4hbuSR4QOuKEqs5AC3MS1NUBYmwm59jOD1c3XFGipagpnMVmB06rGATEmLAC/MxOKm2oURhOfzJl2Zqve04eU9/wAnlVpU0poIVFCqOwEDHHdi7FjqZ1EUIoUaCWnAwp53/ank6YWmwpoGLEkhQCTa5MXOOhsVRywUk2kO1IoBawvMZwsxQzDCzBGgixFzseWOtV+jn+ZydXF557wszTM3tzxyaOYnWrZPFdQeFsIbSPXdDeBqCySAb88Ggyiqp1npmcUe4NuQx3l/1zhf3mFqiUqzeNJE8jrAkfM/PHPqZqb+E6dPURE//wDmB/8AVb/Scc8/6vUy8f7D5CafgtJRlVIUAkiSBBNufXHT2VQKNwJztoYmrYy7MZZDlahKKSqEqSASDqG3TG1kU0Cbbp6m7CsBeZFx+5p/yD/91fHL/oPL8tOh/c9bhOZj4BjH+kTy6yhv7n/7n/YMLP0+v4jB9fpGGfP/ACdDzcem8eU3xTVP/wAdPWJp/wC9/SN+BCAANv8AbFOz5ESXaN8Pye/r+eKhJ30lWfHjfyb+v5D5YBwMR8oSHuiZysZpGfv/AJY5dTNM+M6lHJ/Sc4V8Y88cvaPpnc2T6xP05/Z3fLAncqs97EX9AB6YVseh8hPfy3+7rwnjH9pZjMSLEMYjzOJthJFQ24zpbaL7Ol+EwnFlAquAIGrljtuLObT5vVRB6+2CaTDWV8O/vU8xjKX1jzhVPoMs4puv8v8A3MPwA+WD2j6oFH6ZVk/iwtNYx9IdkdnPf8jimlo3W6JqbpKgZyzTeDabxcbYFM6RvPNlVFoqTfEwjzPWf7K8pTakNSK2qqFaVBkaQYM7iTti+kStFiNZC4xVwDpafoLLZdKahKaKiKICqAqgdABYY597y+1pbj09P//Z" width="505" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">A nerve impulse is generated when the difference in electrical charge disappears. This occurs when a stimulus contacts the tip of a dendrite and increases the permeability of the cell membrane to sodium ions. The ions rush back into the cytoplasm, and the difference in electrical charges disappears. This creates a pulse of electrochemical activity called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #93c47d;">nerve impulse</span></span><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">.</span> A neuron displaying a nerve impulse is said to have an <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><span style="color: #2b00fe;">action potential.</span></span> The cell is depolarized.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Nerve Impulse | CK-12 Foundation" height="366" 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" width="521" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">More specifically, the influx of sodium ions into the neuron cytoplasm activates the adjacent portion of the cell membrane to admit sodium ions also. Successively, the adjacent areas of the neuron lose their differences of electrical charge, and a wave of depolarization is generated in the neuron. This wave of depolarization is the nerve impulse. After the wave of depolarization has passed, the neuron reestablishes the difference in charges by pumping potassium ions out of the cytoplasm and then pumping sodium ions in.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #01ffff;">Synapse</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The nerve impulse passes down the dendrite, through the cell body, and down the axon. At the end of the axon, the impulse encounters a fluid-filled space separating the end of the axon from the dendrite of the next neuron or from a muscle cell. This space is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">synapse.</span> A synapse located at the junction of a neuron and muscle fiber is a neuromuscular junction.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Nerve synapse, artwork Stock Photo - Alamy" 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" /><img alt="Nerve impulse - Simple English Wikipedia, the free encyclopedia" height="225" 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" width="350" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">As the impulse reaches the end of the axon, it induces changes in the cell membrane and the release of chemical substances called<span style="color: #ffa400;"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">neurotransmitters</span></span> (for example, acetylcholine). Molecules of neurotransmitters accumulate in the synapse and increase the membrane permeability of the next dendrite. This causes an influx of sodium ions, and a new nerve impulse is generated. After the nerve impulse has swept down the next dendrite, the neurotransmitters in the synapse are destroyed.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="color: #04ff00;">Reflex arc</span></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">reflex arc,</span> the simplest unit of nervous activity, involves the detection of a stimulus in the environment by sensory nerve endings, followed by impulses that travel via the sensory neurons to the spinal cord. Here the impulses synapse with interneurons, and the interneurons generate impulses to respond to the stimulus. The impulses travel along the motor neurons to muscles or glands that respond appropriately.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Reflex arc | definition of reflex arc by Medical dictionary" height="284" 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" width="431" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In some cases, a reflex arc involves an interpretation. For this activity, interneurons transmit impulses up the spinal cord to the conscious area of the brain, where an analysis occurs.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-35727082216938939352020-07-27T08:09:00.001+05:302020-07-27T08:10:30.439+05:30HUMAN ENDOCRINE SYSTEM<div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><font color="#800180">Human Endocrine System</font></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human endocrine system modulates several processes of the body by the function of hormones. The endocrine system secretes hormones that control how bodily functions work. Thus, the human endocrine system watches over and coordinates all the systems of the body with the use of hormones.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1.6875rem;"><font color="#2b00fe">Pituitary gland</font></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">pituitary gland</span> is located at the base of the human brain. The gland consists of two parts: the anterior lobe (adenohypophysis) and the posterior lobe (neurohypophysis).</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The anterior lobe secretes at least seven hormones. One hormone, the<b><u><font color="#93c47d"> human growth hormone (HGH)</font></u></b>, promotes body growth by accelerating protein synthesis. This hormone is also known as <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">somatotropin.</em> A deficiency of the hormone results in dwarfism; an oversecretion results in gigantism.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Pituitary Gland Hormones and Functions" height="475" 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" width="631" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Another hormone of the anterior pituitary is prolactin, also called<b><i><u><font color="#ff00fe"> lactogenic hormone (LH).</font></u></i></b> This hormone promotes breast development and milk secretion in females. A third hormone is <b><u><font color="#8e7cc3">thyroid-stimulating hormone (TSH)</font></u></b>. The function of TSH is to control secretions of hormones from the thyroid gland. A fourth hormone is adrenocorticotropic hormone (ACTH). This hormone controls the secretion of hormones from the adrenal glands.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">There are three more hormones produced in the anterior lobe of the pituitary gland. The first is <b><u><font color="#f1c232">follicle-stimulating hormone (FSH)</font></u></b>. In females, FSH stimulates the development of a follicle, which contains the egg cell; in males, the hormone stimulates sperm production. The next hormone is luteinizing hormone (LH). In females, LH completes the maturation of the follicle and stimulates the formation of the corpus luteum, which temporarily secretes female hormones. In males, LH is <b><u><font color="#a64d79">interstitial cell-stimulating hormone (ICSH)</font></u></b>, which stimulates the production of male hormones in the testes. The final hormone is melanocyte-stimulating hormone (MSH), which stimulates production of the pigment melanin.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The posterior pituitary gland stores and then releases two hormones that are produced in the hypothalamus of the brain. The first hormone is<b><u><font color="#e69138"> antidiuretic hormone (ADH).</font></u></b> This hormone stimulates water reabsorption in the kidneys. It is also called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">vasopressin.</em> The second hormone is oxytocin, which stimulates contractions in the muscles of the uterus during birth.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#04ff00">Thyroid gland</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">thyroid gland</span> lies against the pharynx at the base of the neck. It consists of two lateral lobes connected by an isthmus. The gland produces<b><i><u><font color="#f1c232"> thyroxine,</font></u></i></b> a hormone that regulates the rate of metabolism in the body. It also produces a second hormone, <b><i><u><font color="#3d85c6">calcitonin,</font></u></i></b> which regulates the level of calcium in the blood.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Goiter: Causes, Risk Factors, Symptoms, Diagnosis & Treatment" height="341" 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" width="559" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Thyroxine production depends on the availability of iodine. A deficiency of iodine causes thyroid gland enlargement, a condition called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">goiter.</em> An undersecretion of thyroxine during childhood development results in a condition known as <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">cretinism</em> (dwarfism with abnormal body proportions and possible mental retardation). In adults, an undersecretion can also result in <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">myxedema</em> (physical and mental sluggishness). Thyroxine oversecretion results in a high metabolic rate and Graves’ disease.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#ffa400">Parathyroid glands</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">parathyroid glands</span> are located on the posterior surfaces of the thyroid gland. They are tiny masses of glandular tissue that produce parathyroid hormone, also called<b><u><font color="#a64d79"> <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">parathormone.</em></font></u></b> Parathyroid hormone regulates calcium metabolism in the body by increasing calcium reabsorption in the kidneys and by increasing the uptake of calcium from the digestive system.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Parathyroid Gland Funcion, Definition and Facts" height="246" 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" width="494" /></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#ff0000">Adrenal glands</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">adrenal glands</span> are two pyramid-shaped glands lying atop the kidneys. The adrenal glands consist of an outer portion, the adrenal <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">cortex,</span> and an inner portion, the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">medulla.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The adrenal cortex secretes a family of steroids called <u><b><font color="#674ea7"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">corticosteroids.</span> </font></b></u>The two main types of steroid hormones are mineralocorticoids and glucocorticoids. Mineralocorticoids such as aldosterone control mineral metabolism in the body. They accelerate mineral reabsorption in the kidneys. Mineralocorticoid secretion is regulated by ACTH from the pituitary gland. Glucocorticoids such as cortisol and cortisone control glucose metabolism and protein synthesis in the body. Glucocorticoids are also anti-inflammatory agents.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Adrenal Glands: Facts, Function & Disease | Live Science" height="356" 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vpbRBte19DNcdyG2teTSUxb8JjHrc3TcHmJn4xbJMW536kcUJy5HIrgwmVDc+Qu0sbNfQq6mVcT4JlzeTaaIajqixsKSgbxJcZh5+PNHznkBjHBuoqkgqyYKccjjhFa17THIMzXaEFDshnldiM4uLygFiGGGnmyb2HVh5jl4hSuW5DUbN8MjNgeV7TE8XY8WI6FDvT+5Ck+JZQqVFCaaOspz8rSWnm0kEFTOa8jqnvnCQkVQ1Hgsuxco1YPgtpWK8EVkwxik3Y0Tz80lmDz1d7A+qLqp+nQ/zwBpjvuX45ERYRqkUIRQh9ChDoFRL21PFNxc0B34m913uL+a9BCXqVKRkY9OxxFuobqlJoaixt2RfZtSP+W397+a0NMZ+rWXH9nQcNb2FABxdcnxK5Jb7/0ZljcpCq+AazP590fmnexhS/yoF19WQC4nUrr4WQ0IZeBemqD6pWU2ORgghhkDYgJ591+4zi88P2VE9qywdj3+yH8mHEKpskrpJRme43IG4cgOgQ5KGcy5YaEZRjiPQx7H7LSVX2pYIacfOR3n/qxg7/xHQddyyPqf1qrRLbCKc/j4/f8A6HNNoZ3vl8HRIqdkLOzhYGN423u6udvcfFeF1et1Gsnvvln8eF+j1Gk0dVCxFEgfYjjz/glm3Hldobshui0zleK0LYqqWG3wSOaPC/d9iF9c0V8dRRC35SZ8/wBRB1zlH8lldE29x8gHXcjT+UL1yeOSyU5RfdcanyVU8HewHijI7C5HHif4pe5R4yM0uSGTAHiakETSD2XDjlJKJTJbUkKahONm5+QRU0lnaNv5LrjyFjPgBTR8xYpeS+RuLKspVcMsfWm+h3rq54ZzGD2wqyeDjRvo57HXcd6ZqnyAsjwbmOsUcC+g7hM+oVkKWx4Gqrh7ent87O838wl37LPwxWuWyRhwiWxHRFmsovYsmXbyQZHP4ujLT6ghL421NBtHzNL8nOKgfD4LPn4NmPk2YfHchGqXIObKtt6jvRwjcxtz+J/8gPVKfUZ5moLwF0Ufa5fIrrNHSKEIoQihBz2Olz00sJ3scHjwdofcD1Wv9OnmEoMztZHbOMvngxVsdirWLDOwYd2Xd9rIz7zB7OBTenfuYpqvsTOg41JdscQ42Rao8uRlV95FjGKkOflHwt08gmYoZrjxkX8SlzacAuWPKwNVLHJq2cwprs9RN/VRC9vvO4NS+MEutaxCPbAGJV0k8/aEX1s1o3AcAAlJOUp5G4QjCGB/2R2Ca4iqq/6satj3Z3cj05+izvrP1OGihiHNj6/H5DaGmWqnhfav9x4qJHP0Y2zRoA0WaANwHAL583OyTnLlvt/J7CmqupYMcsDx8p+v0XXCXwMxnD5K4hqhteC0hH2vhP8ApOQ8HNjf/wDW0fVpX0j+z1m7QwXxlf7nhvqkNl0l8gXDpC4vF9Dc/VbFb3GfYtqRrxhmSBrwTqPLRctW2OUVpe6bTFaSqc8a2JG64CTlNyQ8oqLL8Bxd0crS74L2c0aXbxXKbGnhnL6lKLx2dKrsCa6MT07i5rhextcJ2E+cPsyFa1xICVuAg6ObbysiOMZBI3tdAKv2alZqzUcuKDPTtfaNV6uL+4CPkLTZzNUs5Sj9yGklJcMsZXR8WLq1EfKOOqXhmmCWJ242R4W1sHKM0FYqcEAgg2+icik1wKSk0+TXSgtKt0CnyhywGo1CDdHKEbFhlVVB2czhw3jwKkJbo5CJ5iKW2dfe0fG1z4IGpniO1D+irx7hYk+XwSMvA+vIawKC7wmqV5F7nwJ+M1fazyScHONvDc32AWFdPfY5GnVDZBRMSEEIoQihCKED+xFTlq2tO6VrmHzF2/3gE5oZ7bkvkV1kN1WfjkKYzBZxWndHkVpllFdHWmGUSDWwGnMcQuRnslkk698do7RY2yXM8EgtbuPC6erlFrgy5UShhAGoqGakvGqI5xXbDxhLhYB4ljcQ0G5KE7Yt4QxskllmnaHG2wwtpmC5td/ieCX1GpVfC7OafTOcnZI0bDQAg1U1msBytuNBrq63TX0XNPKcoueOfBNY0pKtP9nX6mZn/tsADBwI+8T139br5ndOy++dlqe7PX/4ez0dEaaYxr6+QJX49C34p4m9M7foCjx0Oss5jXL/AEGP6nTQ7ms/sppcbhebMniceQe2/pe6rZotXV99cl/ASGp09nEZIKtbn3/FwPPoUu8SWPIRvZ10K23FJllfMdLUzPXNIPovcf2Zf/wG/wD7M8V9XlnVqK8nPMFqNQfL1WxRPLFr48BzEbPpLfd0TFqzFitTxbkT2x28ln7ccmlnJ5ntm5LksZOxzgc9jdo3Rt7J+sd/Tmm63vWDO1VGXuXZ02mcySMSZQ4F1hm390And4jevPfWvqNmklGut4bWWaP0P6dXqFKdqz4RccNhk0H2bvVvodQk9L/aS2HF6yvlDWt/s/DG6l4f+wv49sW147zdTucNxXqNPrqNVHMXn/k87ZG/SSxNf+jnGO7HSw3c0Zm9F2emT5gO0a2M+GK7mEGxSbi12PJp9GuhqywjVHpucWgdkFJB2LEHA77jh4J9WsSdSGbBMW1GiJnchK6kJba4m6KKKZrQc12knhxH5pXe68ldJUrJOLOa1lQZHF7jclKzk5PJswiorajxa7h0AVO2d8BsSdlSzS7iGEDxf3R9fZMWy9Olv8AEt9qQgrz5rEUIRQhFCEUIXUk5Y9jxvY4OH7JB/JWhLbJMrJbotHQNpIwTmG5wBHgRcL0VyysmPQ8cC/M29vAeyWfI0ngMYW7/AGeU83NHsnKPtYrd96A1Yy6DbEYgy7AbduCdwBPoCq08SJf9gv1cxe9x33J91lWzcptjsIqMUjqVFA1sdPAfhawvcOdml1vOwXoqo7a0jAsk3Oc/4FLaDEHyvOZxIvuubDwHBBuUU+EaFCaglkzYc8v7rt3M8OQXabG+Gdsio8o0VOGkeBAt1PH6FGdTyDjagvsxjFTC9rI5CW3tld3m+XLyISF/0nS6l5shz8rhjK+oXUR9j/h8hb9JeNOdD3sodIctm6AMaTbeT1Pmqx01Wi0/o1dfn8ilNk9Ve7Z+EIFDLly9TdcoljA3Yt2RvpojJSy5dSNVot5jwZze21ZFKV3c6k/T/NIyftNBLkombuPP/BQpIumaaCoLSRwsi0zcclLI5OxbHy3w2B3N0v8A5HD8gvDf2im5a9/pf8Hovo0EqsL/ALywoyZYyeDYcQjR4hbuu7zTvBTOn1M6Z763hiGq0ULouLRK2gY4Zm6t9x0PTqvbfTfq0dSviXlHhPqP0uzSy3LmP/AgbWbEtlBfEA2Tlwd/ArYk42LnsBptY63iXRyyrpXxuLXAgg6gpKdbg8M24TU1lG2hlu23Fv0TVM90f0CsjhhzCZLFOVidqGzaJva4XIeMeV3odUDULl/oU0r23pHNqdt2EpSCzHJsS7LKZt3ea5BZZyXCNu10uSlij4vcXHwYLD3d7Ln1CWK4wOaSObJS+OBMWOaJFCEUIRQhFCEUIdDjf2lFA/8AUynxYS38l6Gl76Isx5LbdJAmJtzboVSKCs27OjNHOz7tnfkj6R9xA6nhxZlkZvRJIsmfcJh0ncPljcfYoEY4jJ/gtbLmK/Is0kV3sHNzfqFjwg3NGhN4TOqY3KIQHD43MAvybbcOpXpo4xk87WnN4EOrN3XStvLNWvhF4jGUNGoIvpo6/PqiRisYKtvOTdBMS031uQOoGu/2Rk8IDKPIxYRhzYixx1fJ8AG5o+848+i634QnZY55XwKv6QK3PVZB8MYAHisnWyzJR+DT+nw21Z+TDT1BAaA0HQcOi7CXC4CzXL5HHZxpMUh3abrb07BcGbe0pIVKt7L2yDS/3v4JafeMD0M47Pj33a0Nj1HQ8+oXH1wjq75Z7ZK5rdWgeIVk9scs48N9nVNgoy/Dae3zGX/yO/gvn31ybnr5L8L/AIPUfS2oU8/95YdLGs4XPM7vIJGMVnEVlmlucuW8IxS7S07HZXSx35DvepaCAtOr6VrbI7o1vH54EbNbpYPDmshKgxaB4uyWPwzNsehBOin/APO1tM1ONbUkBs1Gmti4uSaZmbiDDI+O98pIB6ePHxXuaoTdUZyWG1yj5/qqo12yUHlZ4F/bTZRtQwyMAEgG/wC90KImprbItptS6nh9HKmQGKSzhaxsQVWEHVPk2nNTjlBal0cnI8MVlyh0ou/RVLOcTvYKl64Qh1dF/k51TCzCOl0lFYgbEvuNGFxXcF2lcnLXwZNu5rztj4RxtHm7vH6hJfUZZtx8INoY4rz8sW1njhFCEUIRQhFCEUIPezD81Db7kjh6gO/Mrc0DzRj8mVqli7PyjJaxd4Ike2dfRbs3UBlWGu0bK0tPid3uq0z2Xfs5qYOVWV4NGL05jJB4Ep6zhZA0y3GXBZe7UDi6MgeiXre5SS+Atyw4v8gemo7PaS5os4fVJQ07Uk2xqVqafA+bXR3LbEfA3f8AhC2oxbiY2mlhv9idPTOHI+aVnXI0ozRa+jc52mlrcyd3QK6qkznqJI1CQtcTY20BuN/M28dUwuALWUNmzU2a3EA6A7xofbQLs+jPujhiJtHSkzyO5uKyNTVmbkja09mIJFOcta3KbCw9ePuovalgt23kaNnp80UgPxAacPonaXuQhqFiSwLUtdLc2cUnOcs8D0YRweIq15IufouRnJs64LBbUykixN9DdXsfGCkFh5HLZDbMU2HugERdJESWnNZuWVxJvxuDf1WDqPoS1eodznhYS/PBo1/UPRSjtyB6jaeWqPZzEBpOgbcNHjz8Stf6doNPpltjHn5fLFtbqrrvc3x8eDJJDIzhca+gWq1KJnpxZdSVY8ESFiZSdYyU1WTHnv3o7G/NpNiD4XXZLDE3BZx8jTgeMB4AJSl1PlC04OLAP6QNngWmdg/Fb6qsZepHa+0N6S9xe19CfS6tafI+SPF5SY5LhsdMF0gmJ/4bv3Spb0hCf+IjnER08ln+DYfYZwCC7gj0RAXPCFLaSfPVTO/XI8m90fRYeplutk/yaWnjtqivwDUAMRQhFCEUIRQhFCDpsO+8E7eTmH1BH5LY+mv2yRm65e6LJUDU+CZfZRdAqtHEbwUrd8oYr+GM0Fc2sgyuIE7RYjdntxHVPUXK2GH2ITqdE8roB1sBpmPLtHv7rRxtxNkvf/cRbzy+hut+tJY6XYOjxBp+NgPUGxQK9Vn7kGlS/wDKzoOOsbLDDIHWzRsIv1aOK2apKUMmJXmFkov5FGopJG62uOY1CG4y7NGMoM+yOcXaX1APIbhqVfL4OJRwbopHusG3dYWva90eIGSUeWHoHOidECPiBcTbjci3kPqqz5eBXiabF/HWxsndfMA/XmOqz7YxhPnyP0OUoceDLFQNk+B7TyG5cjXGXTLuxx7Rswv7BxDzbTUItbVb5YK1eouDHXQs1kbq2/DmhTUX7kFrb6YMcAH/AOOKBhbg/g+TOF9VJuKfJIp+DZg9RGHlrjZsjcpPLcQ7yIRKJxTB3Qk48do11eGSwkOLbsPEatI6FMODg8oFC2M+PITpZ84MbnEEEZSd5G9NxeVgWnHa9yMuJUmQtda2beORG9UnHDyi9c9ywE8BmaLtdq1wIPgVdLKAXp5yjTE4wyFoO7ceYO4rhxpTjkd8MqWzRFjtbiyUtg4S3IRknFnPqrDzDI+M/K/TwKYjhrKNFWbkn+A5K/s8PnfzZlHi7RDveAEFuvSOew7kgjYYzbPttryBPoE3XxHIpdzwc2mfmcXcyT6lebk8ts2ksLB4VTpFCEUIRQhFCEUINuwbv94H6rT6OP8AFan018yQhrl9v7L6w94p2XYKK4BkhS7DIxuBGrfLoUBpx6CcPsZ8F2Oe8Coq3ERncwkl7uVyfhCYp0jm82f9/Ylfrow9lS5GeF1I0ZBTxZeVvzWktPHGEZ7lc3lyZrxTDo5aVoiGUNFgOAtw+ikVhuLBQtlGzMjmtS6WB5bctI4cPRLTc65cM24bLI5CUGIMlYRICS0XcRpbkcvHXRM1XKawwE6pQeYl1TSuDQITmAAvb4ue5Enl/aUi1n3oM4BMZY3RSElzRmYTvBG8eBC7l8MVuShLdHoH4rR9vFp8bSbdd2iFqafUjx2MUWenPnoU5Q5hAtlNted/FZnMPwzR4kWNridHjMOu9W9TP3clXXjo1imJjvGSNfhPgrut7cwKqa3YkVSsykZ2lpyjwVc4fuO9/ayiNrXH4h5qmYy8lnuXg+imH3wFNsf/ACJufwMuGOAhY17hkF7A8deF+C1aMKtGdcnvbj2aYqxjQ42BNxyBA6I7a8A/Tk2uTW50NTHYd17Tx43H8kPspidUgdFTuY6xV4rAWUlJG7Fmn7F/NpB/ZOn1VH2DpfaC+AVZBC5OO5ALohPaWiEjRKBroD5cUGh4zBg6rGngXtuqjs6SKHjI7MfBv80HUy7/ANBzQx3WOXwJkbbAJdLCNJjFhptHIeTH/ulMriD/AEKz+5fs5qvNG2fFCEUIRQhFCEUIRQg1bCfFP/0x+8Fp/Tfvl+hHXdR/ZfWHvJyb5BQ6MNUy2vAoU1hhIs04DE0y53C7YxmtzO5o9foiaeClLnwD1Emo4XkYcWrZHw5pS6MF2gFrkW5cE/JrbnoSqrjGeI8gKDKTaOZwdyeBY+Y3IEXn7Zf6jUsr7o/6DvslVFzHwyCz2963TcbexRXJ8NmZqoJPdHoFbX4WXw52i7mHeN5au3w3x4DaO7bPaxMparJpbjr111v5XsEnCzZxg1ZQ3ckNY7PnaSHbr87bjbwso7G57ok9NbcMcNnMS0dLMABG0uLx0HHzsnVY9mZGZfT7lGHnwVx78zHBwu7dv4cEXKfKZMeJIqqzE53fYCeKBOMX2gkHJLhmcNhb8LB9VVVxXSL7pvtnyeubltYj9n+as2kiRg8ng17HNyuaSALageyq2pLksoOL4ZhhwqF4zMeQfulKrTQlzFhnfOPEkZXbPOzl18zG3Lrb+jfPclv6CW/OeOwv9VHbjyeoHulzcC3UDkN1h4aJ+iTsyusAp4gFwO0h7lu0b8QsLkdLpptuOF2LfbPnox0ZeGOIvmzt3C3NDjux/IWe1v8Aga4WmUgFtiBq7gmEzPliC4CVVQRmKMOktYusR4BUy3J8AI2STbSKoaPJqxwcPdWz8lnZu4aD0VU0RntCA228myWsWHlANkm+DmW02KCqqrt/q2DK3wG8+aSnLfPC6Ruaer0q8PtmNpu5d7YTwHqAfZyD9R/7pTH+R/oWl9y/ZzZeaNs+KEIoQihCKEIoQihBs2FGlQ79Vg9ST+S1Ppq5kxDXP7UWVm8pqwHAzvILbFVeJRwWXDCGy0TrVJG8RC3jm4ImjypSA6tr2fspmlMlO8XuWuB8iLI05OdTXwSMVCxP5AjXFrkgm1IcxwPmA1QbGKp2+Brgf1gRoOuq08qUMsyLoe/015CNBVNqI/szZ+jnRE3cwneLfM3qiV3J9gLapVS56+RU2jwgtlu1pAcLnxQradzzEf02oTjyZKHA5HnRptxPAeJ4KsaPkLPURSPuPYkxsYpIXBwJBleNxI3MaeIB1J5gJbU3p/3cOi2npk5erP8AhfALqKpzXtykg67urih22uMltDxgpR5QUixS9u0AJ+9xTEL8/cLypx9polabZo7OHuPJHWXzEHxnEuDEa08WD3/iqObCqC+T1DU3+ULsZZ4OOOCMJafhAsq4aZHhhGlkc+I5dJM4H4tN3ujwy4vAvNKMueg7i0bIGBjA3ty0ZyRoDvyjzVk5YbQtXmyWZfaLFKyV5z/1b2nU7ri+vihVOc++GPz2R47QWxSs7GMAZS87zYA+P1Rpy2xFqob5Z8GSOpkcGxNJzO1d5/kopS4ReUIrMmF8ZlbAyBp7zg1x13am27yVZTSfIvTF2OTXRXhu1NnAODcp0Oir6sZcFrNJxlGDaqg7QGWKRxa3VzCdLcwgaqiUllP+A+kt2+2a5+RaidYJKLUUPPk2UjleDyDkhkwffbmCPXROx5WBSz5ObVEeVzmn5SR6Gy81JYbRtxeUmVqp0ihCKEIoQihCKEHXZSLJSPef7R+ngwW+pK2fp8cVOXyzN1bzYl8GeqdqrzZIg98lkDdgMlkO7HVP2skfF8Zt4s1t6XTeksW/AnrYPYn8M84fC3tTmd2YcbFrgSDc8LI9a2zZLJPZws4LZaeFlQ5l2WG4ubcou2CljBRSnKtSDMdU2aCWCNrQ8NOW1mhzuo5q0+ntF1BwsjOT8iNg8sjZw9pIc06nda2+6yqN2/JrXKLhg6cMX7ePPG4Oto7utLgeYuFq1qDXBgyqdUsSELallQH2fK98btWgmzfDKNFn6uuafLeDY0kq3HKSTAUEXeb4j2SsIe5DcpcHuc9+/K30VrPvOR+09h/VdTKlsVS9rg5psfb0RIzknlFXGLWGH6KJtSLWAkG/kU9CSsXPYnPNT46B07Oxvf4r7kKx+mgsX6hlixU3s8XF0vHVP/MGlR8Dbs4YXvZkfYtc1xBT9dkZJ7TO1CnFcouxyid/SHOLh8V0VLKQKma2YB2ITjOCbAOFrcNNSfZccknyHrjmOEZsReJY2Ob8pseg5Hw3KlmJpNF604NphbBqfKHScXENCNFY5Fr5ZaiCdqpHzVboogXdmGs0/VHe8NSVnXOU57YjeljGupOXkxUeFNzfay7tXBmtuhduUhp//KQSdzx7UMtAyGSKYxlxcGEAOtuA6JzKcXgSm5xnHd8iC16wt2WbGDfSvR4MFNDBhc1iE9WxSxCvthS9nVPtufZ4/a3+91i62vZc/wA8mhpJ7qkBUoMkUIRQhFCEUIRQh0WWHsaeKLi1gv8Aid3ne5K9FCPp1Rj+DHct9jl+QBUvS02MxQPlclpMMkTD6l0c0b2mzmubb1tr0suVzcZpok4qUGn8Bnai7auThrcdLclpXtqYnpsOpHqo+0LZW77DNz8bI2d+JI5H2JxZvwymkilD+F73R64NPkBbOMo4MO2jRFORGLNlHaX/ABbx63WfqpbHheRnRe+GX44MGzmJSQygs4mxHAqmmskpYCamqM48nQJ4qOsaGZ8r28N2q0Je7h8mPH1qHlLgB1mx743gtGZpvY+SpGmO7KGoa1SjyYv9WZXOPcOpVXp8vIX+ril2WDZFwNi4XXFpV8nP61eDy7Zcg/G3z0U/p18nVq8+DXTiOlHaOucp4cTbcr8VrJR7rXgWsXrO1fn56pS2alyO1V7FgzUVEZXZQNeaDGre8BJ2KCyxu2VwYRuc/MHWtu4LR02n9Nmbq9RvWMGv+lCrzFv9bHo5vFzRucOfIo8JxbaAut1JfDBmP0hLWBo1C5dDcuA2nsSzkrw/DJBvIF97SuV1SRay6I4YDhjmx98gBveb15G3ILspYwkZ91icsoXa98cbXRxStGY999u+4k6klUltisJ4HK1KWJSX6FetrGgdnH8PE8XJC21L2xNCFb+6QS2brbSNYNxNj1vomdNZngX1NfDYCrI8kr2/dc4e6yrFtm1+R2DzBMsgerwZySDNDLqE5XIWmj7tvT5oYphvaSw+B1H0PqgfUYZjGf8ABbRSxOUP5EtY5pEUIRQhFCEUIEMApe1qYmcC8X8BqfYFH08d9sY/kFfLbW3+Bzx2W7it25mXSuBanckJscijFIUCTCoocUPJdDZtI7tYYKof2jAHfib3Xe4WrN76ozRnULZZKt/wCI84a1zbgtNj4XuNPUIa3YTXgYe3LT8jHTWz5nOJDhcC/tZasPnJn2ZxhI87a0TpgyojsY2NDC0DvMPM8weaR1lMpNMLobYwzB99i7E8Qi+9/DogRapWfI207HjwUR1RLr3IN94VI2tyLutJDlh+1MsLDHfPl1u7VaG9eeWjNnpIzeeg07H5ZIW5crZCL9LIsYZjuQr6EYzw+gPNPO8m0wB42VGpvpjMVXHwCpMMmc7vSXHO5sg+lZnsYVtaXCCbIWGHsnytIJ5HQ+KPszHDF3JqW5I81OzORrXtyyCw3aofoItHVZeHwVRNyhw7LKSN4Csljwdbz5NNCRTxODj9pIbW+63qjQW1YYKxO2ax0hZpQ7tW5CQbixGh9QlIp7+B6WNnI8S4kYyTN2b3aWBaLjxItdOYx5wZfpKX28HluI3YXhrGkbiG3PlmJV8cZyc9PEtrMGA4259Q5r3EZwQC47zyS0bU5YaGL9Mo15iZtpMBkEl2NJDtTyHmqXVb3mJfTaiO3EhfqcKyHvyMaeXxH2SstOl3LA5G/PSDGzNFAJL9sC4C7QQQC7hcpnTwjF5TyLamybXQs17j20mYWdmNx1usm1v1JZNCtLYsHyJykWRhKkkTNbATQeq4+0opm8Q3MPFhB+l0xqI76JIXrlsuiznhXnjYPihCKEIoQihBk2CivVZvuRvd5kBv/wCk99Pjm7PwhPXPFWPlhbGHalalwrV0L05SExtGR5QGERSSqMuM+ytWHxS08rC6JoMmYHWM2tbXeHGwT+hsbzW+uxHVwxJWRfPX7MzJg2Nxb8WS19+jiExvSi3Em1uSTKcFlkLrNF2jUk7h5oemsscuOgl8YJcj9gNXC4lm/MLO5G61H7lwY10JxxITtpNm3wveWuztadfvAHUEjwWXfpZL3rk1dNqozST4AtGLHMdwQKuPcxmzngtFSSHuP+N6vG1tSZXYk0hhmmJhjcDa7QLrTUs1poRUcWNMpoKqzzY6kHf1Qoy8F5w4NUNfnaWuIHDiFaFmeykqknlGGpcGGxZ7mx6rkvaEj7vJazEjku0kDda5/iuqxtZOOpZ5KBibi6xc6yH6ryW9FYPLi4nNzRFuzk6sINbP0vZwds4XkdpGD9USmDUeexTUT3WbF0uzAzCZ3PL5L2vc671X05OWZBndWo4iMmz0Ehk7RzLRMabA/MUTMmJXuO3anyLGIUr3VbC6PIx8gGhuBc8+BSlym7U2uB+uUY0vD5wM21znvblY4/Z6Zb7wOKanBqGYiGlwpe5dnO6p5usm2TNqCwj1RSEOBC7RJqRyxZRv2ujAna8f2kbHHxIsfoq65Ytz8g9I814+GwUwpdDDN1K7VMQYKaG7ARmBYfma4eoIWhBbotfgQt4eTmbhY2PBeaawbiPi4QihCKEIoQcP0eM71Q7lG0erv5LT+mL3yf4M/XviK/Jdi51KduBVC/OUhIbiZXoLCIqKoyw0bIjNT1jR8WWM+QLr+5C0dBypIR1rxOD/AGYKOEvJjG9zbDxBB/Iq9cXL2l5yUVuNFe8Rt7Fm4fEfvHijy/u47IlIe972bNlqWV0oyNJ18kTTqS5fQHVShtwxxxR0RqSMwzBoa/kdNQma2nHkzob1DOBT2g2ZczWGz2E30O7okb9M2vYaOn1afE+GAH0MjWG7CLnkeiW9GUYPKHFZFyWGEsNaXwGMggt1abG3gmqMuvawFvtnuB0dQWvHRBjLEwzjmJuxJneNtA4Aj80S1cgq3weaCrLWkSd5g5jnyXKrHFe47ZBN+00snhIIyEHpuRVbF8YBuE1ymZpp4AfnQ52Vpl4xngLUUjHMvHGXFvNNQmpRzFC04tS9zCckWRjJpjYNZ3W9UTPGWATUpOMQfQ4o6RxcbloN8qrVZuC2UqKwe6jaWUP3aDc3go7MPo5HSxaPX+n4Lh4YWuuCW72m3RdV0WuTn9NZ14GLF6RruznDgzOAcrtDr0VoSzx8Clcmsx7EDGqJgkdaRm881n6iqO7s2KLHt5R9wfBnSyNa17NT978lynT85yS69Ri3gp2rqGunyt3RtDP+26W101K3C8F9JBqvL88gtiWQwzZTHVGgDkN+zDu+3xWlT0Z+o6Of4my00o5PePRxXnbeJv8AbNiv7F+jKqFyKEIoQihBy/R5/wCo/Cz6lan0z7pfwZ31DqP7PWLbym7gdXQBnSEhuJlegsIioqrLBLZ/Ff6NMHkXYQWyN5sdvt1GhHgi6e70rNwK+n1YbfPgaYadkVRHJH32SgvjO7Q6agrbr27lJeTNk5Sg4y4a7BuMubHIdO8dbb7X1tc71L5xhL8haE5xCOzWLEE23/l05K9Nu9YA6mhG3bGlMkLalvxg5XHdcEaE9Rb3VNRB4zEHo54k4PoW8N2gkh7rhmbyJSternDiXI9bpYWcrhh2PH6dzASHtv8AtJyOphKOWJvS2KXBZWVcL2BjXuBPJtkRzi0chCaeWYafZkl28a668kD0EnkNLVYQUqNnWk/ELAWGhV3BNgI6h4K37JBzLZgNb7jyVXVFrBZatpnmo2fjgaHSPJ03ALirikRaiU3hGajp6V7/AITbiSuqFbfRecrYx7CUOKU0XdYw/RHWFwLyrsny2UQubVktLnWO4EXy+CnEkWealnAWwnZkU7swfmJ6aeirBRj0At1TsXKLqqjpWHtJRd2osAbaq3L6RRTsaxEBw7JMlla9kgyl4JaQRpfchTqinuG/6yUYbWuTdt3XiItaBd+tuQC5K3ZD9gdDU5ttnNqmQk6rKsk2zcisIuoJS1wINiCiUTaZW2Ka5Nu1sAEzZB/bRtefxag/RV10FGzK8lNHJutxfh4BLUshhmunRoA5Dbs0O+3xWlT0IX9CJjJ/2ib/AKj/AN4rz1v3y/bNer7F+kY0MuRQh//Z" width="538" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The adrenal medulla produces two hormones:<b><u><font color="#b45f06"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">epinephrine</span> (adrenaline) and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">norepinephrine</span> (noradrenaline). </font></u></b>Epinephrine increases heart rate, blood pressure, and the blood supply to skeletal muscles. Epinephrine functions in stressful situations to promote the fight-or-flight response. Norepinephrine intensifies the effects of epinephrine. Both hormones prolong and intensify the effects of the sympathetic nervous system.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#a64d79">Pancreas</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The pancreas is located just behind the stomach. Its endocrine portion consists of cell clusters called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><u><font color="#f1c232">islets of Langerhans.</font></u></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><img alt="Exocrine pancreatic insufficiency - Wikipedia" height="491" 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" width="614" /></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The pancreas produces two hormones:<i><u><b><font color="#3d85c6"> insulin and glucagon</font></b></u></i>. Insulin is a protein that promotes the passage of glucose molecules into the body cells and regulates glucose metabolism. In the absence of insulin, glucose is removed from the blood and excreted in the kidney, a condition called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">diabetes mellitus.</em> Diabetes mellitus is characterized by glucose in the urine, heavy urination, excessive thirst, and a generally sluggish body metabolism.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The second pancreatic hormone, <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><u><font color="#8e7cc3">glucagon</font></u>,</span> stimulates the breakdown of glycogen to glucose in the liver. It also releases fat from the adipose tissue so the fat can be used for the production of carbohydrates.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#674ea7">Other endocrine glands</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Among the other endocrine glands are the ovaries and testes. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">ovaries</span> secrete <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">estrogens,</span> which encourage the development of secondary female characteristics. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">testes</span> secrete <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">androgens,</span> which promote secondary male characteristics. Testosterone is an important androgen.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Testes: Anatomy, definition and diagram | Kenhub" 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" /><img alt="Ovaries | You and Your Hormones from the Society for Endocrinology" 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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;"><u><b><font color="#a64d79">pineal gland</font></b></u></span> is a tiny gland in the midbrain. Its functions are largely unknown, but it seems to regulate mating behaviors and day-night cycles. The <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">thymus gland</span> is located in the neck tissues. It secretes<b><u><font color="#cc0000"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">thymosins,</span></font></u></b> which influence the development of the T-lymphocytes of the immune system.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Know your brain: Pineal gland — Neuroscientifically Challenged" height="385" 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" width="493" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">Prostaglandins</span> are hormones secreted by various tissue cells. These hormones produce their effects on smooth muscles, on various glands, and in reproductive physiology.<u><b><font color="#ff00fe"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Erythropoietin</span> </font></b></u>is a hormone produced by the kidney cells. Erythropoietin functions in the production of red blood cells. <b><u><font color="#6aa84f"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Gastrin</span> and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">secretin</span></font></u></b> are hormones produced by digestive glands to influence digestive processes.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-50377306399983196802020-07-27T08:00:00.002+05:302020-07-27T08:10:30.439+05:30ENDOCRINE SYSTEM<div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, helvetica, arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, helvetica, arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><font color="#ffa400">ENDOCRINE SYSTEM</font></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, helvetica, arial, sans-serif; font-size: 16px;"><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; color: black; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="The Endocrine System – An Overview | Thyroid UK" height="246" 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" width="494" /></p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The animal body has two levels of coordination: nervous coordination and chemical coordination. Chemical coordination is centered in a system of glands known as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#741b47">endocrine glands</font>.</span> These glands are situated throughout the animal body and include such organs as the pancreas, thyroid gland, and adrenal gland. The glands secrete <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#cc0000">hormones,</font></span> a series of chemical substances composed of protein or sterol lipids.</p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;">Hormones bring about changes that help coordinate body systems in a general way. For example, the pancreas secretes insulin, which facilitates the passage of glucose into all body cells for use in energy metabolism. Another example is thyroxine, a thyroid gland secretion that regulates overall body metabolism. In contrast to chemical coordination, the nervous system coordinates functions in the animal body on a more localized level as it delivers nerve impulses to contract body muscles or regulate gland activities.</span></p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="The Endocrine System: A Profile of Glands That Regulate the Body ..." height="350" src="https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcRHpRUkD4wpIShIQirpcp2BUzR9SZLyRSbEfA&usqp=CAU" width="542" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The endocrine glands secrete their hormones into the bloodstream, where the blood carries the hormones to the target organs. Because the endocrine glands have no ducts, they are often called<font color="#bf9000"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">ductless glands.</span> </font>Other glands of the body (such as the enzyme-secreting salivary glands) deliver their enzymes via ducts and are referred to as <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#0b5394">exocrine glands.</font></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The structure and physiology of hormones and endocrine glands are relatively similar in all animals; the emphasis in this chapter is on the human endocrine system.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-90880973876604376512020-07-27T07:57:00.001+05:302020-07-27T08:00:42.269+05:30HUMAN EXCRETORY SYSTEM<div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><font color="#2b00fe">Human Excretory System</font></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="Pituitary Gland | Endocrine Society" src="data:image/jpeg;base64,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" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human excretory system functions to remove waste from the human body. This system consists of specialized structures and capillary networks that assist in the excretory process. The human excretory system includes the kidneys and their functional unit, the nephron. The excretory activity of the kidneys is modulated by specialized hormones that regulate the amount of absorption within the nephron.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1.6875rem;"><font color="#800180">Kidneys</font></span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The human kidneys are the major organs of bodily excretion (see Figure 26-1). They are bean-shaped organs located on either side of the backbone at about the level of the stomach and liver. <font color="#ff0000">Blood enters the kidneys through <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">renal arteries</span> and leaves through <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">renal veins.</span> Tubes called <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">ureters</span> carry waste products from the kidneys to the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">urinary bladder</span> for storage or for release.</font></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Kidney (Renal) Trauma: Symptoms, Diagnosis & Treatment - Urology ..." height="438" 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" width="439" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#2b00fe">The product of the kidneys is <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">urine,</span> a watery solution of waste products, salts, organic compounds, and two important nitrogen compounds: uric acid and urea. <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">Uric acid</span> results from nucleic acid decomposition, and <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">urea</span> results from amino acid breakdown in the liver. Both of these nitrogen products can be poisonous to the body and must be removed in the urine.</font></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#01ffff"> </font></p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#04ff00">Nephron</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The functional and structural unit of the kidney is the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-weight: 700; line-height: inherit;">nephron.</span> The nephron produces urine and is the primary unit of homeostasis in the body. It is essentially a long tubule with a series of associated blood vessels. The upper end of the tubule is an enlarged cuplike structure called the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#3d85c6">Bowman’s capsule.</font></span> Below the Bowman’s capsule, the tubule coils to form the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">proximal tubule,</em> and then it follows a hairpin turn called the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">loop of Henle.</em> After the loop of Henle, the tubule coils once more as the <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">distal tubule.</em> It then enters a <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; line-height: inherit;">collecting duct,</em> which also receives urine from other distal tubules.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Nephrons | BioNinja" height="418" 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" width="459" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Within the Bowman’s capsule is a coiled ball of capillaries known as a <font color="#674ea7"><span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">glomerulus.</span> </font>Blood from the renal artery enters the glomerulus. The force of the blood pressure induces plasma to pass through the walls of the glomerulus, pass through the walls of the Bowman’s capsule, and flow into the proximal tubule. Red blood cells and large proteins remain in the blood.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">After plasma enters the proximal tubule, it passes through the coils, where usable materials and water are reclaimed. Salts, glucose, amino acids, and other useful compounds flow back through tubular cells into the blood by active transport. Osmosis and the activity of hormones assist the movement. The blood fluid then flows through the loop of Henle into the distal tubule. Once more, salts, water, and other useful materials flow back into the bloodstream. Homeostasis is achieved by this process: A selected amount of hydrogen, ammonium, sodium, chloride, and other ions maintain the delicate salt balance in the body.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The fluid moving from the distal tubules into the collecting duct contains urine. The urine flows through the ureters toward the urinary bladder. When the bladder is full, the urine flows through the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#a64d79">urethra</font></span> to the exterior.</p><h2 style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1.6875rem; font-weight: 400; line-height: 1.385; margin: 0.625rem 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><font color="#ffa400">Control of kidney function</font></h2><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The activity of the nephron in the kidney is controlled by a person’s choices and environment as well as hormones. For example, if a person consumes large amounts of protein, much urea will be in the blood from the digestion of the protein. Also, on a hot day, a body will retain water for sweating and cooling, so the amount of urine is reduced.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Pituitary Gland(Hypophysis) Anatomy, Functions and Conditions ..." height="325" src="data:image/jpeg;base64,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" width="585" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;">Some individuals suffer from a condition in which they secrete very low levels of ADH. The result is excessive urination and a disease called</span><span style="font-size: 1rem;"> </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: inherit;">diabetes insipidus.</em><span style="font-size: 1rem;"> </span><span style="font-size: 1rem;">Another unrelated form of diabetes,</span><span style="font-size: 1rem;"> </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: inherit;">diabetes mellitus,</em><span style="font-size: 1rem;"> </span><span style="font-size: 1rem;">is more widespread. People with this disease produce insufficient levels of insulin. Insulin normally transports glucose molecules into the cells. But when insulin is not available, the glucose remains in the bloodstream. The glucose is removed from the bloodstream in the nephron; to dilute the glucose, the nephron removes large amounts of water from the blood. Thus, the urine tends to be plentiful.</span></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Hormones from the <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;"><font color="#cc0000">cortex</font></span> of the adrenal glands also control the content of urine. These hormones promote reabsorption of sodium and chloride ions in the tubules. Thus, they affect the water balance in the body because water flows in the direction of high sodium and chloride content.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0tag:blogger.com,1999:blog-8870097163070303690.post-37692741226666329592020-07-27T07:52:00.000+05:302020-07-27T08:00:42.268+05:30EXCRETORY SYSTEM OF ORGANISMS<div class="study-guides-coursequiz-maincontent" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px; margin: 0px; padding: 0px;"><div class="title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; margin: 0px; padding: 0px;"><h2 class="section-title" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-family: latobold, Helvetica, Arial, sans-serif; font-size: 1.4rem; font-weight: 400; line-height: 1.385; margin: 1.875rem 0px; padding: 0px; text-rendering: optimizelegibility;"><font color="#2b00fe">Excretory Systems</font></h2></div></div><article class="copy" style="-webkit-tap-highlight-color: transparent; background-color: white; box-sizing: inherit; font-family: latoregular, Helvetica, Arial, sans-serif; font-size: 16px;"><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; clear: right; float: right; font-size: 1rem; line-height: 1.6rem; margin-bottom: 1em; margin-left: 1em; margin-top: 0px; padding: 0px; text-rendering: optimizelegibility;"><img alt="Animal organs. Excretory system. Atlas of plant and animal histology." 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" /></p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">The excretion process is concerned with the removal of waste products from the animal body. The relatively stable internal environment of the organism is dependent upon the process of excretion, a concept known as<font color="#ffa400"> <span style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-weight: 700; line-height: inherit;">homeostasis.</span></font> Homeostasis means “staying the same.” It refers to a steady-state equilibrium that exists internally in a healthy organism or cells, despite changes in the external environment. The excretory system plays a major role in homeostasis.</p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><span style="font-size: 1rem;">Because one-celled organisms are in constant contact with their environment, they do not need excretory organs. However, multicellular organisms need a mechanism to carry waste products from cells to the external environment. Flatworms, such as planaria, have a series of excretory cells called</span><span style="font-size: 1rem;"> </span><em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; font-size: 1rem; line-height: inherit;">flame cells.</em><span style="font-size: 1rem;"> </span><span style="font-size: 1rem;">Flame cells contain cilia that direct water and metabolic wastes to enter the cells and pass into excretory canals. The excretory canals join with other canals to form excretory tubules. Fluid from the excretory tubules leaves the body through pores. </span></p><p class="NormalwDropCap" style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Excretory System & Osmoregulation Lab#8 Bio110 - Biology 110 ..." height="256" 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" width="473" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">In earthworms, members of the phylum Annelida, the excretory system consists of structural units called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">nephridia</em> (the singular is <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">nephridium</em>)<em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">.</em> Each nephridium contains a ciliated tunnel that leads to a long, coiled tubule, which leads to a bladderlike sac (a primitive bladder). Fluid moves from the internal environment into the funnel. As fluid passes through the tubule, cells in the tubular lining absorb useful compounds such as glucose, amino acids, and salts. The remaining materials constitute metabolic waste, and they are passed into the bladderlike sac. The sac later opens through a pore in the earthworm’s skin, from which the waste products are discharged.</p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;"><img alt="Notes on Excretory system of Earthworm | Grade 11 > Biology ..." height="271" 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" width="454" /></p><p style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; color: black; font-size: 1rem; line-height: 1.6rem; margin: 0px 0px 1.25rem; padding: 0px; text-rendering: optimizelegibility;">Insects have a series of tubules for excretion called <em style="-webkit-tap-highlight-color: transparent; box-sizing: inherit; line-height: inherit;">Malpighian tubules.</em> Fluid enters at the upper end of the tubules and passes down their entire length. The cells in the tubular walls reabsorb precise amounts of water, salts, and other materials to maintain a delicate balance within the insect tissues. The tubules eventually lead to an insect’s intestine, where waste products are removed.</p></article>Himadri Mondalhttp://www.blogger.com/profile/02284648452264805309noreply@blogger.com0