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Find the potential on the edge of a disc of charge density $\;\sigma$ and radius $R$

$(a)\;\sigma\;k\;R\qquad(b)\;2\;\sigma\;k\;R\qquad(c)\;4\;\sigma\;k\;R\qquad(d)\;16\;\sigma\;k\;R$

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Answer : (c) $\;4\;\sigma\;k\;R$
Explanation : $\;dV=\large\frac{k\;d\;q}{r}$
$dq=\sigma\;(2r\theta)\;dr$
$dV=\large\frac{1}{4\;\pi\;\in_{0}}\;\large\frac{\sigma\;2r\;dr\;\theta}{r}=\large\frac{\sigma}{2\;\pi\;\in_{0}}\;\theta\;dr$
$r=2\;R\;cos\;\theta$
$r=2\;R\;cos\;\theta$
$dr=-2\;R\;sin\;\theta\;d \theta$
$dr=-\large\frac{\sigma}{2\;\pi\;\in_{0}}\;2\;R\;\theta\;sin\;\theta\;d \theta$
$V=\int_{\large\frac{\pi}{2}}^{0}\;dV=\large\frac{\sigma\;R}{\pi\;\in_{0}}\;\int_{0}^{\large\frac{\pi}{2}}\;\theta\;sin\;\theta\;d \theta$
$V=\large\frac{\sigma\;R}{\pi\;\in_{0}}=4\;k\;\sigma\;R\;.$
answered Feb 4, 2014 by yamini.v
edited Feb 4, 2014 by yamini.v
 

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