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 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. $\sqrt{{\rho }_0/2\sigma }$
• D. $\frac{{\rho }_0}{\sigma }$

Answer 1

Answer: (C)

$\sqrt{{\rho }_0/2\sigma }$

Charge inside the sphere of radius $R_1$,
$Q_1=\int \rho \cdot 4\pi r^{2}dr$
$={\int }_0^{R_1}\frac{{\rho }_0}{r}4\pi r^{2}dr$
$=4\pi {\rho }_0{\int }_0^{R_1}rdr$
$=\frac{4\pi {\rho }_0{R}_1^2}{2}$
$=2\pi {\rho }_0{R}_1^2$.
Charge on hollow sphere,
$Q_1=-\sigma \cdot 4\pi R_2^2$
$\because Q_1+Q_1=0$
$2\pi {\rho }_0{R}_1^2-\sigma 4\pi R_2^2=0$
${\rho }_0{R}_1^2-2\sigma R_2^2=0$
$\frac{R_2}{R_1}=\sqrt{\frac{{\rho }_0}{2\sigma }}$.