Question

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

• A. $\frac{\sigma }{{ϵ}_0}\left[\frac{b^2-c^2}{b}+a\right]$
• B. $\frac{\sigma }{{ϵ}_0}\left[\frac{b^2-c^2}{c}+a\right]$
• C. $\frac{\sigma }{{ϵ}_0}\left[\frac{a^2-b^2}{a}+c\right]$
• D. $\frac{\sigma }{{ϵ}_0}\left[\frac{a^2-b^2}{b}+c\right]$

Answer 1

The correct option is D $\frac{\sigma }{{ϵ}_0}\left[\frac{a^2-b^2}{b}+c\right]$

$V_{outer}=\frac{KQ}{r}$

where r is distance of point from the centre of shell

$V_{inside}=\frac{KQ}{R}$

where R is the radius of shell

$V_B=\frac{K{q}_A}{r_b}-\frac{K{q}_B}{r_b}+\frac{K{q}_C}{r_c}$

$V_B=\frac{1}{4\pi {ϵ}_o}[\frac{\sigma 4\pi a^2}{b}-\frac{\sigma 4\pi b^2}{b}+\frac{\sigma 4\pi c^2}{c}]$

$V_B=\frac{\sigma }{{ϵ}_o}[\frac{a^2-b^2}{b}+c]$