Question

A solid sphere of radius $R_1$ and volume charge'density $\rho=\frac{\rho_0}{r}$ is enclosed by a hollow sphere of radius $R_2$ with negative surface charge density $\sigma$, such that the total charge in the system is zero. $\rho_0$ is a positive constant and $r$ is the distance from the centre of the sphere. The ratio $R_2$ $/ R_1$ is

• A. $\frac{\sigma}{\rho_0}$
• B. $\sqrt{\frac{2 \sigma}{\rho_0}}$
• C. $11 \sqrt{\frac{\rho \sigma}{2 \sigma}}$
• D. $\frac{\rho_0}{\sigma \sigma}$

Answer 1

Answer: (D)

$q_1=\int\limits_0^{R_1} \rho \times 4 \pi r^2 d r=4 \pi \int\limits_0^{R_1} \frac{\rho_0}{r} \times r^2 d r=2 \pi \rho_0 R_1^2$
and $q_2=-\sigma \times 4 \pi R_2^2$
Given $q_1+q_2=0$
or $2 \pi \rho_0 R_1^2-\sigma \times 4 \pi R_2^2=0$
$\therefore \frac{R_2}{R_1}=\sqrt{\frac{\rho_0}{2 \sigma}}$