A metal ball of radius R is placed concentrically inside a hollow metal sphere of inner radius 2R and outer radius 3R. The ball is given a charge +2Q and the hollow sphere a total charge -Q. The electrostatic potential energy of this system is $\frac{n{Q}^2}{24\pi {\epsilon }_{0}R}.$ Find $n$

 

The electrostatic potential energy of this system=
By indication, inner wall of hollow sphere is induced with share −2Q and its outer surface with charge.
Two conducting concentric, hollow sphere A and B have radii a and b respectively.

Electrostatic potential energy $=$ Self energy of each shell + interaction energy of each pair of shells

$\mathrm{V}=\frac{1}{4\pi {ϵ}_0}\times \frac{\mathrm{Q}}{\mathrm{R}}$

when, it is given that,

$\begin{array}{l}{U}_{\text{self }}=U_1+U_2+U_3\\ =\frac{k(2Q{)}^2}{2\times R}+\frac{k(2Q{)}^2}{2\times 2R}+\frac{k{Q}^2}{2\times 3R}\\ =\frac{4k{Q}^2}{2R}+\frac{4k{Q}^2}{4R}+\frac{k{Q}^2}{6R}=\frac{19k{Q}^2}{6R}\end{array}$

$\begin{array}{l}{\text{ U}}_{\mathrm{int}}={\mathrm{U}}_{12}+{\mathrm{U}}_{23}+{\mathrm{U}}_{31}\\ =\frac{\mathrm{k}\left(2\mathrm{Q}\right)(-2\mathrm{Q})}{2\mathrm{R}}+\frac{\mathrm{k}\left(2\mathrm{Q}\right)\left(\mathrm{Q}\right)}{3\mathrm{R}}+\frac{\mathrm{k}(-2\mathrm{Q})\left(\mathrm{Q}\right)}{3\mathrm{R}}\\ =-\frac{4{\mathrm{kQ}}^2}{2\mathrm{R}}+\frac{2{\mathrm{kQ}}^2}{3\mathrm{R}}-\frac{2{\mathrm{kQ}}^2}{3\mathrm{R}}=\frac{-2{\mathrm{kQ}}^2}{\mathrm{R}}\end{array}$

Total energy = $U_{self}+U_{int}=\frac{19k{Q}^2}{6R}-\frac{2k{Q}^2}{R}=\frac{7k{Q}^2}{6R}$