Question

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:

• A. $\frac{\sigma}{\epsilon_0}[\frac{b^2-c^2}{b}+a]$
• B. $\frac{\sigma}{\epsilon_0}[\frac{b^2-c^2}{c}+a]$
• C. $\frac{\sigma}{\epsilon_0}[\frac{a^2-b^2}{a}+c]$
• D. $\frac{\sigma}{\epsilon_0}[\frac{a^2-b^2}{b}+c]$

Answer 1

Answer: (D)

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

Answer 2

The potential at radius $b$ is the sum of potentials due to each shell. Charge on shell A: $Q_A = \sigma \cdot 4\pi a^2$ Charge on shell B: $Q_B = -\sigma \cdot 4\pi b^2$ Charge on shell C: $Q_C = \sigma \cdot 4\pi c^2$

Potential at radius $b$ due to shell A (since $b > a$): $V_A = \frac{k Q_A}{b} = \frac{1}{4\pi\epsilon_0} \frac{\sigma \cdot 4\pi a^2}{b} = \frac{\sigma a^2}{\epsilon_0 b}$ Potential at radius $b$ due to shell B (since $b$ is on the surface): $V_B = \frac{k Q_B}{b} = \frac{1}{4\pi\epsilon_0} \frac{-\sigma \cdot 4\pi b^2}{b} = -\frac{\sigma b^2}{\epsilon_0 b} = -\frac{\sigma b}{\epsilon_0}$ Potential at radius $b$ due to shell C (since $b < c$): $V_C = \frac{k Q_C}{c} = \frac{1}{4\pi\epsilon_0} \frac{\sigma \cdot 4\pi c^2}{c} = \frac{\sigma c^2}{\epsilon_0 c} = \frac{\sigma c}{\epsilon_0}$

Total potential at $b$: $V_B = V_A + V_B + V_C = \frac{\sigma a^2}{\epsilon_0 b} - \frac{\sigma b}{\epsilon_0} + \frac{\sigma c}{\epsilon_0} = \frac{\sigma}{\epsilon_0} (\frac{a^2}{b} - b + c) = \frac{\sigma}{\epsilon_0} (\frac{a^2-b^2}{b} + c)$.