Three concentric metallic shells A, B, C have radii a, b, c (with a < b < c) and surface charge densities +sigma, -sigma, +sigma respectively. The electric potential of shell B is:

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EEJ MAIN Physics QUESTION #6990

Question 1

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

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

Correct Answer Explanation

Potential at shell B is the sum of contributions from all three shells:

From shell A (radius $a$, charge $Q_A = 4\pi a^2 \sigma$): potential at B (outside A) $= \frac{Q_A}{4\pi\epsilon_0 b} = \frac{\sigma a^2}{\epsilon_0 b}$
From shell B (radius $b$, charge $Q_B = -4\pi b^2 \sigma$): potential at B $= \frac{Q_B}{4\pi\epsilon_0 b} = \frac{-\sigma b}{\epsilon_0}$
From shell C (radius $c$, charge $Q_C = 4\pi c^2 \sigma$): potential at B (inside C) $= \frac{Q_C}{4\pi\epsilon_0 c} = \frac{\sigma c}{\epsilon_0}$

$V_B = \frac{\sigma}{\epsilon_0}\left(\frac{a^2}{b} - b + c\right) = \frac{\sigma}{\epsilon_0}\left(\frac{a^2 - b^2}{b} + c\right)$

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