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

Answer 1

Total charge=0
$\Rightarrow {\int }_0^{R_1}S.4\pi r^{2}dr=6.4\pi R_2^2=0\Rightarrow {\int }_0^{R_1}\frac{{\int }_0}{r},4\pi r^{2}dr-64\pi r_2^2=0\Rightarrow 4\pi {\int }_0{\left[\frac{r^2}{2}\right]}_0^{R_1}-6.4\pi R_2^2=0\Rightarrow \frac{{\int }_0{R}_1^2}{2}-6R_2^2=0\Rightarrow \frac{R_2^2}{R_1^2}=\frac{{\int }_0}{26}\therefore \frac{R_2}{R_1}=\sqrt{\frac{{\int }_0}{26}}$