Question

A spherical conducting shell of inner radius $r_{1}$ and outer radius $r_{2}$ has a charge $Q$. A charge $-q$ is placed at the centre of the shell. The surface charge density on the inner and outer surfaces of the shell will be

• A. $\frac{q}{4 \pi r_{1}^{2}}$ and $\frac{Q}{4 \pi r_{2}^{2}}$
• B. $\frac{-q}{4 \pi r_{1}^{2}}$ and $\frac{Q+q}{4 \pi r_{2}^{2}}$
• C. $\frac{q}{4 \pi r_{1}^{2}}$ and $\frac{Q-q}{4 \pi r_{2}^{2}}$
• D. 0 and $\frac{Q-q}{4 \pi r_{2}^{2}}$

Answer 1

Answer: (C)

Since the charge $-q$ is placed at the centre of the shell, therefore it will induce a charge $+q$ on the inner surface and charge $-q$ on the outer surface of the shell.
$\therefore \quad$ Surface charge density on the inner surface of the shell,
$\sigma_{1}=\frac{q}{4 \pi r_{1}^{2}}$
Total charge on the outer surface of the shell $ = Q -q$
$\therefore $ Surface charge density on the outer surface of the shell,
$\sigma_2 = \frac{Q -q}{4\pi r_2^2}$