Question

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

• A. $\frac{\sigma }{{ϵ}_0}(a+b-c)$
• B. $\frac{\sigma }{{ϵ}_0}(a-b+c)$
• C. $\frac{\sigma }{{ϵ}_0}(b-a-c)$
• D. None

Answer 1

The correct option is C $\frac{\sigma }{{ϵ}_0}(b-a-c)$


Let, the charge on shells $\text{A, B}$ and $\text{C}$ are $q_A,q_B$ and $q_C$ respectively.

So, the net potential on $\text{A}$:

$V_A=\frac{1}{4\pi {ϵ}_0}(\frac{q_C}{c}+\frac{q_B}{b}+\frac{q_A}{a})$

According to problem
$q_A=-\sigma 4\pi a^2$
$q_B=+\sigma 4\pi b^2$
$q_C=-\sigma 4\pi c^2$

Substituting the values in the above equation we get,

$V_A=\frac{1}{4\pi {ϵ}_0}[-\frac{\sigma 4\pi c^2}{c}+\frac{\sigma 4\pi b^2}{b}-\frac{\sigma 4\pi a^2}{a}]$

$\therefore V_A=\frac{\sigma }{{ϵ}_0}(b-c-a)$

Hence, option (c) is the correct answer.