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)

$q_1=\underset{0}{\overset{R_1}{\int }}\rho \times 4\pi r^{2}dr=4\pi \underset{0}{\overset{R_1}{\int }}\frac{{\rho }_0}{r}\times r^{2}dr=2\pi {\rho }_0{R}_1^2$
and $q_2=-\sigma \times 4\pi R_2^2$
Given $q_1+q_2=0$
or $2\pi {\rho }_0{R}_1^2-\sigma \times 4\pi R_2^2=0$
$\therefore \frac{R_2}{R_1}=\sqrt{\frac{{\rho }_0}{2\sigma }}$