Question

A system comprises of a solid non conducting sphere of charge $Q$ and radius $R$. A point charge $q$ is placed at a distance $r$ from the centre of sphere. Find the total energy of the system
$[k=\frac{1}{4\pi {\epsilon }_0}]$

• A. $\frac{3k{Q}^2}{5R}+\frac{kQq}{R}$
• B. $\frac{3k{Q}^2}{5R}$
• C. $\frac{3k{Q}^2}{5R}+\frac{kQq}{r}$
• D. $\frac{kQq}{r}$

Answer 1

The correct option is C $\frac{3k{Q}^2}{5R}+\frac{kQq}{r}$

Given that,

System comprises of a solid non conducting sphere of charge $Q$.

Radius of the sphere be $R$.

Let the point charge be placed at a distance $r$ as shown in the figure,


Self energy of the sphere $U_{sphere}=\frac{3k{Q}^2}{5R}$

Self energy of the point charge $U_{charge}=0$

Interaction energy between the point charge and sphere be $U_{int}=\frac{kQq}{r}$

The interaction energy can be positive or negative depending upon the interacting charges.

$\therefore$ Total energy of the system $U_{Total}=U_{sphere}+U_{charge}+U_{int}$

$\Rightarrow U_{Total}=\frac{3k{Q}^2}{5R}+\frac{kQq}{r}$

Hence, option (c) is the correct answer.