Question

A soild sphere of radius $R_1$ and volume charge density $\rho =\frac{{\rho }_{0}o}{r}$ 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 center of the sphere.Then the ratio$R_2/R_1$ is

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

Answer 1

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

It is given that a solid sphere of radius R₁ and have volume charge density $\rho =\frac{{\rho }_0}{r}$. So, charge contained in volume R₁ is $\begin{align}

$Q=\underset{0}{\overset{R_1}{\int }}4\pi r^2\frac{{\rho }_0}{r}dr$

$Q=2\pi R_1^2{\rho }_0.........\left(1\right)$

This sphere is enclosed by a hollow sphere of radius R₂

Total charge on outer sphere $Q=4\pi R_2^2\sigma ........\left(2\right)$

Taking ratios of equation (1) and (2)

$\frac{R_2}{R_1}=\sqrt{\frac{{\rho }_0}{2\sigma }}$