# Three concentric matallic shells A , B and C of radii R , 2R and 3R have surface charge densities $\;\sigma , -\sigma\;and\;\sigma\;$ respectively . Find the potential difference between A & C

$(a)\;\large\frac{\sigma\;R}{\in_{0}}\qquad(b)\;\large\frac{2\;\sigma\;R}{\in_{0}}\qquad(c)\;zero\qquad(d)\;\large\frac{4\;\sigma\;R}{\in_{0}}$

Explanation :
$V_{A}=\large\frac{k \sigma 4 \pi R^2}{R}-\large\frac{k \sigma 4 \pi(2R)^2}{2R}+\large\frac{k \sigma 4 \pi(3 R)^2}{3R}$
$V_{A}=\large\frac{\sigma R}{\in_{0}}-\large\frac{\sigma 2 R}{\in_{0}}+\large\frac{\sigma 3 R}{\in_{0}}=\large\frac{2 \sigma R}{\in_{0}}$
$V_{C}=\large\frac{k \sigma 4 \pi R^2}{3R}-\large\frac{k \sigma 4 \pi (2R)^2}{3R}+\large\frac{k \sigma 4 \pi (3R)^2}{3R}$
$V_{C}=\large\frac{ \sigma R}{ 3 \in_{0}}-\large\frac{4 \sigma R}{3 \in_{0}}+\large\frac{9 \sigma R}{3 \in_{0}}$
$V_{C}=\large\frac{2 \sigma R}{\in_{0}}$
$V_{A}-V_{C}=0\;.$