Three concentric metal shells A, B and C of r | vedclass

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Question

(A) The potential at any point on the surface of a shell is the sum of potentials due to all three shells.For a point at distance $r$ from the center,

Vedclass · Accepted Answer

$\frac{\sigma}{\epsilon_0} \left[ \frac{a^2-b^2}{b}+c \right]$

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(A) The potential at any point on the surface of a shell is the sum of potentials due to all three shells.For a point at distance $r$ from the center,the potential due to a shell of radius $R$ and charge $Q$ is $V = \frac{KQ}{R}$ if $r \le R$ and $V = \frac{KQ}{r}$ if $r > R$,where $K = \frac{1}{4\pi\epsilon_0}$.The charges on the shells are $q_A = \sigma(4\pi a^2)$,$q_B = -\sigma(4\pi b^2)$,and $q_C = \sigma(4\pi c^2)$.For shell $B$ (radius $b$),the potential $V_B$ is the sum of potentials due to shell $A$ (at distance $b > a$),shell $B$ (at distance $b = b$),and shell $C$ (at distance $b $V_B = \frac{K q_A}{b} + \frac{K q_B}{b} + \frac{K q_C}{c}$$V_B = \frac{1}{4\pi\epsilon_0} \left[ \frac{\sigma(4\pi a^2)}{b} + \frac{-\sigma(4\pi b^2)}{b} + \frac{\sigma(4\pi c^2)}{c} \right]$$V_B = \frac{\sigma}{\epsilon_0} \left[ \frac{a^2}{b} - \frac{b^2}{b} + \frac{c^2}{c} \right]$$V_B = \frac{\sigma}{\epsilon_0} \left[ \frac{a^2 - b^2}{b} + c \right]$

Three concentric metal shells $A, B$ and $C$ of respective radii $a, b$ and $c$ $(a < b < c)$ have surface charge densities $+\sigma, -\sigma$ and $+\sigma$ respectively. The potential of shell $B$ is

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