1.

A parallel-plate capacitor having plate area A and plate separation d is joined to a battery of emf E and internal resistance R at t=0. Consider a plane surface of area A/2, parallel to the plates and situated symmetrically between them. If the displacement current through this surface as a function of time is given as:id=EnRe−tdε0ARWhere, ε0 is the permittivity of free space.The value of n is :

Answer» A parallel-plate capacitor having plate area <span id="MathJax-Element-1-Frame" class="mjx-chtml"><span id="MJXc-Node-553024" class="mjx-math"><span id="MJXc-Node-553025" class="mjx-mrow"><span id="MJXc-Node-553026" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span> and plate separation <span id="MathJax-Element-2-Frame" class="mjx-chtml"><span id="MJXc-Node-553027" class="mjx-math"><span id="MJXc-Node-553028" class="mjx-mrow"><span id="MJXc-Node-553029" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span></span></span> is joined to a battery of emf <span id="MathJax-Element-3-Frame" class="mjx-chtml"><span id="MJXc-Node-553030" class="mjx-math"><span id="MJXc-Node-553031" class="mjx-mrow"><span id="MJXc-Node-553032" class="mjx-texatom"><span id="MJXc-Node-553033" class="mjx-mrow"><span id="MJXc-Node-553034" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.036em;">E</span></span></span></span></span></span></span> and internal resistance <span id="MathJax-Element-4-Frame" class="mjx-chtml"><span id="MJXc-Node-553035" class="mjx-math"><span id="MJXc-Node-553036" class="mjx-mrow"><span id="MJXc-Node-553037" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span> at <span id="MathJax-Element-5-Frame" class="mjx-chtml"><span id="MJXc-Node-553038" class="mjx-math"><span id="MJXc-Node-553039" class="mjx-mrow"><span id="MJXc-Node-553040" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span id="MJXc-Node-553041" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span id="MJXc-Node-553042" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span>. Consider a plane surface of area <span id="MathJax-Element-6-Frame" class="mjx-chtml"><span id="MJXc-Node-553043" class="mjx-math"><span id="MJXc-Node-553044" class="mjx-mrow"><span id="MJXc-Node-553045" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span><span id="MJXc-Node-553046" class="mjx-texatom"><span id="MJXc-Node-553047" class="mjx-mrow"><span id="MJXc-Node-553048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span id="MJXc-Node-553049" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span>, parallel to the plates and situated symmetrically between them. If the displacement current through this surface as a function of time is given as:<br><br><span id="MathJax-Element-7-Frame" class="mjx-chtml"><span id="MJXc-Node-553050" class="mjx-math"><span id="MJXc-Node-553051" class="mjx-mrow"><span id="MJXc-Node-553052" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-553053" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-553054" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span></span><span id="MJXc-Node-553055" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span id="MJXc-Node-553056" class="mjx-mstyle MJXc-space3"><span id="MJXc-Node-553057" class="mjx-mrow"><span id="MJXc-Node-553058" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.559em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.559em; top: -1.407em;"><span id="MJXc-Node-553059" class="mjx-texatom"><span id="MJXc-Node-553060" class="mjx-mrow"><span id="MJXc-Node-553061" class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.036em;">E</span></span></span></span></span><span class="mjx-denominator" style="width: 1.559em; bottom: -0.794em;"><span id="MJXc-Node-553062" class="mjx-mrow"><span id="MJXc-Node-553063" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span id="MJXc-Node-553064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.559em;" class="mjx-line"></span></span><span style="height: 2.201em; vertical-align: -0.794em;" class="mjx-vsize"></span></span></span></span><span id="MJXc-Node-553065" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-553066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.52em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-553067" class="mjx-texatom" style=""><span id="MJXc-Node-553068" class="mjx-mrow"><span id="MJXc-Node-553069" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span id="MJXc-Node-553070" class="mjx-mstyle" style="font-size: 141.4%;"><span id="MJXc-Node-553071" class="mjx-mrow"><span id="MJXc-Node-553072" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.564em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.564em; top: -1.396em;"><span id="MJXc-Node-553073" class="mjx-mrow"><span id="MJXc-Node-553074" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span id="MJXc-Node-553075" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span></span><span class="mjx-denominator" style="width: 2.564em; bottom: -0.964em;"><span id="MJXc-Node-553076" class="mjx-mrow"><span id="MJXc-Node-553077" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-553078" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">ε</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-553079" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><span id="MJXc-Node-553080" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span><span id="MJXc-Node-553081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.564em;" class="mjx-line"></span></span><span style="height: 2.36em; vertical-align: -0.964em;" class="mjx-vsize"></span></span></span></span></span></span></span></span></span></span></span><br><br>Where, <span id="MathJax-Element-8-Frame" class="mjx-chtml"><span id="MJXc-Node-553082" class="mjx-math"><span id="MJXc-Node-553083" class="mjx-mrow"><span id="MJXc-Node-553084" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-553085" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">ε</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-553086" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span> is the permittivity of free space.<br><br><br><br>The value of <span id="MathJax-Element-9-Frame" class="mjx-chtml"><span id="MJXc-Node-553087" class="mjx-math"><span id="MJXc-Node-553088" class="mjx-mrow"><span id="MJXc-Node-553089" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span></span> is :<tnl-filler id="2881" answer_ids="2263894,2263895,2303845" type="number" size="3" contenteditable="false"></tnl-filler><br><br><br><br><img alt="" height="218" src="https://df0b18phdhzpx.cloudfront.net/ckeditor_assets/pictures/1406709/original_Picture1.png" width="150"><br>


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