Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 28 - The Electric Potential - Exercises and Problems - Page 836: 62

Answer

$\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{R}$

Work Step by Step

We know that the electric field at the center of this shell is zero. And we know that the difference in potential between two points is the source of an electric field. $$E=\dfrac{\Delta V}{d} $$ where $d$ is the separation distance between the two points, the surface of the shell and its center. So, $d=R$ $$E_{\rm center}=\dfrac{\Delta V}{R}=0 $$ Hence, $$\Delta V=V_{\rm surface}-V_{\rm center}=0\;\rm V\tag 1$$ Recall that the electric potential on the surface of the shell (at $r=R$) is given by $$V_{\rm surface}=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{R}\tag 2$$ Plug into (1), $$\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{R}-V_{\rm center}=0$$ Therefore, $$\boxed{V_{\rm center}=\dfrac{1}{4\pi \epsilon_0}\dfrac{Q}{R}}$$
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