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 $\frac{R_2}{R_1}$ is

• A. $\frac{\sigma }{{\rho }_0}$
• B. $\sqrt{2\sigma /{\rho }_0}$
• C. $\sqrt{{\rho }_0/\left(2\sigma \right)}$
• D. $\frac{\sigma }{{\rho }_0}$

Answer 1

Answer: (C)

For solid sphere of radius $R_1$
$q_1=\underset{0}{\overset{R_1}{\int }}4\pi r^{2}dr\rho$
$=\underset{0}{\overset{R_1}{\int }}4\pi r^{2}dr\frac{{\rho }_0}{r}$
$q_1=4\pi \frac{R_1^2}{2}{\rho }_0$
$q_2=-4\pi R_2^2\sigma$
$q_1+q_2=0$
$4\pi \frac{R_1^2{\rho }_0}{2}-4\pi R_2^2\sigma =0$
${\left(\frac{R_1}{R_2}\right)}^2=\frac{2\sigma }{{\rho }_0}$
$\frac{R_2}{R_1}=\sqrt{\frac{{\rho }_0}{2\sigma }}$