Question

Derive the expression for potential energy of a system of two charges in the absence of the external electric field.

Answer 1

Coulombic force acting on a test charge $F=\frac{kQq}{r^2}$ where $k=\frac{1}{4\pi {ϵ}_o}$

Potential energy of the two charges $U=-\int F.dr$

$\therefore$ $U=-\int \frac{kQq}{r^2}dr$

Or $U=-kQq\int \frac{1}{r^2}dr$

Or $U=-kQq\times \frac{-1}{r}$

$⟹$ $U=\frac{kQq}{r}$