Question

Three concentric metallic shells $A$, $B$ and $C$ of radii $a,b$ and $c(a<b<c)$ have surface charge densities, $\sigma ,-\sigma$ and $\sigma$ respectively. Find the potentials of three shells $A$, $B$ and $C$.

Answer 1

Potential at $A$,

$V_A=\frac{\sigma 4\pi a^2}{4\pi {ϵ}_{0}a}+\frac{(-\sigma )4\pi b^2}{4\pi {ϵ}_{0}b}+\frac{\sigma 4\pi c^2}{4\pi {ϵ}_{0}c}$
$=\frac{\sigma }{{ϵ}_0}(a-b+c)$

Similarly, Potential at $B$,
$V_B=\frac{\sigma 4\pi a^2}{4\pi {ϵ}_{0}b}+\frac{(-\sigma )4\pi b^2}{4\pi {ϵ}_{0}b}+\frac{\sigma 4\pi c^2}{4\pi {ϵ}_{0}c}$
$=\frac{\sigma }{{ϵ}_0}(\frac{a^2}{b}-b+c)$

and Potential at $C$,

$V_C=\frac{\sigma }{{ϵ}_0}(\frac{a^2}{c}-\frac{b^2}{c}+c)$