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.
$\therefore$ 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,
${\sigma }_2=\frac{Q-q}{4\pi r_2^2}$