Question

An isolated metal sphere of radius $r$ is given a charge $q$. The potential energy of the sphere is :

• A. $\frac{q^2}{4\pi {ϵ}_{0}r}$
• B. $\frac{q}{4\pi {ϵ}_{0}r}$
• C. $\frac{q}{8\pi {ϵ}_{0}r}$
• D. $\frac{q^2}{8\pi {ϵ}_{0}r}$

Answer 1

The correct option is D $\frac{q^2}{8\pi {ϵ}_{0}r}$

Let at any instant $q$ be the charge on the sphere
Now potential of the sphere $=\frac{kq}{r}$
Work done to add a $dq$ charge to this sphere
$dw=\frac{kq}{r}\cdot dq$ (As $V_{\infty }=0$)
$\therefore$ total work done to accumulate $q$ charge is
$\int dw={\int }_0^q\frac{kq}{r}\cdot dq$
$\Rightarrow w=\frac{k}{r}[\frac{q^2}{2}{]}_0^q$
$\Rightarrow w=\frac{k{q}^2}{2r}$
$\Rightarrow w=\frac{q^2}{4\pi {\epsilon }_{0}r\cdot 2}$
$\Rightarrow w=\frac{q^2}{8\pi {\epsilon }_{0}r}$