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

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Answer : (c) $\;4\;\sigma\;k\;R$

Explanation : $\;dV=\large\frac{k\;d\;q}{r}$

$dq=\sigma\;(2r\theta)\;dr$

$dV=\large\frac{1}{4\;\pi\;\in_{0}}\;\large\frac{\sigma\;2r\;dr\;\theta}{r}=\large\frac{\sigma}{2\;\pi\;\in_{0}}\;\theta\;dr$

$r=2\;R\;cos\;\theta$

$r=2\;R\;cos\;\theta$

$dr=-2\;R\;sin\;\theta\;d \theta$

$dr=-\large\frac{\sigma}{2\;\pi\;\in_{0}}\;2\;R\;\theta\;sin\;\theta\;d \theta$

$V=\int_{\large\frac{\pi}{2}}^{0}\;dV=\large\frac{\sigma\;R}{\pi\;\in_{0}}\;\int_{0}^{\large\frac{\pi}{2}}\;\theta\;sin\;\theta\;d \theta$

$V=\large\frac{\sigma\;R}{\pi\;\in_{0}}=4\;k\;\sigma\;R\;.$

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