Question

A thick shell with inner radius $R$ and outer radius $3R$ has uniform charge density $\rho$ $c/m^3$. It has a spherical cavity of radius $R$ as shown in the figure. The electric field at the centre of the cavity is $\frac{\rho R}{x{ϵ}_0}$. Then x is equal to :

• A. $18$
• B. $6$
• C. $24$
• D. $12$

Answer 1

The correct option is D $12$

We can consider the electric field at center to be a sum of electric fields due to complete shell of charge density $\rho$ if it did not have any cavity, and due to a sphere of cavity of charge density $-\rho$.

Due to symmetry, the electric at center due to former is zero.
The later produces an electric field at center towards right given as

$E=\frac{1}{4\pi {ϵ}_0}\frac{q}{r^2}$

$=\frac{1}{4\pi {ϵ}_0}\frac{\rho \left(\frac{4}{3}\pi R^3\right)}{(2R{)}^2}$

$=\frac{\rho R}{12{ϵ}_0}$