Question

A solid sphere of radius $R_1$ and volume charge density $p=\frac{p_0}{r}$ 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. $p_0$ is a positive constant and r is the distance from the centre of the spheere. The ratio $R_2/R_1$ is

• A. $\sigma /p_0$
• B. $\sqrt{2\sigma /p_0}$
• C. $\sqrt{p_0/2\sigma }$
• D. $p_0/\sigma$

Answer 1

The correct option is C $\sqrt{p_0/2\sigma }$

The charge contain in the solid sphere is $q_1={\int }_0^{R_1}p4\pi r^{2}dr$

or $q_1={\int }_0^{R_1}\left(p_0/r\right)4\pi r^{2}dr=4\pi p_0{\int }_0^{R_1}rdr=4\pi p_0\left(\frac{R_1^2}{2}\right)=2\pi p_0{R}_1^2$

The charge in the hollow sphere is $q_2=-\sigma \times 4\pi R_2^2$

As total charge is zero, $q_1+q_2=0$

or $2\pi p_0{R}_1^2-\sigma 4\pi R_2^2=0$

or $\frac{R_2}{R_1}=\sqrt{\frac{p_0}{2\sigma }}$