Question

A conducting spherical shell carrying a charge $Q$ has inner radius $a$ and outer radius $b$ . If a small electric dipole is placed at its center as shown in the figure then which of the following statements is true

• A. Surface charge density on the inner surface of the shell is zero everywhere.
• B. Surface change density on the inner surface is uniform and equal to $\frac{\left(Q/2\right)}{4\pi a^2}$
• C. The electric field outside the shell is the same as that produced due to the point charge at the centre of the shell.
• D. Surface charge density on the outer surface depends on $|\overrightarrow{p}|$

Answer 1

The correct option is C The electric field outside the shell is the same as that produced due to the point charge at the centre of the shell.

As the dipole does not induce any charge on the inner surface, the Surface charge density solely depends on $Q$ (charge that is present on conducting spherical shell).

From gauss' law, we know

$\oint \overrightarrow{E}\cdot \overrightarrow{dA}=\frac{q_{enc}}{{\epsilon }_0}$

At a point outside (distance $r$ from the centre) the sphere

$E\times 4\pi r^2=\frac{Q}{{\epsilon }_0}$

$\Rightarrow E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2},r\ge R$

Therefore, the electric field outside the shell is the same as that of the point charge at the centre of the shell.

Hence, option $\left(B\right)$ is correct.