Question

derive an expression for the potential energy of a point charge.

Answer 1

A_____________r_____________P
Let us take a point charge Q at point A and take a point P such that AP = r.
When the test charge is moved from infinity to the point P,
then the change in the electric potential energy of the system is given by,
$$\begin{gathered} {\mathrm{U}}_{\mathrm{P}}-{\mathrm{U}}_{\infty }=\frac{\mathrm{Qq}}{4\mathrm{\pi }{\in }_0\mathrm{r}} \\ \mathrm{Therefore}\mathrm{potential}\mathrm{at}\mathrm{point}\mathrm{P}\mathrm{due}\mathrm{to}\mathrm{a}\mathrm{point}\mathrm{charge}={\mathrm{V}}_{\mathrm{P}}=\frac{{\mathrm{U}}_{\mathrm{P}}-{\mathrm{U}}_{\infty }}{\mathrm{q}} \\ =\frac{\mathrm{Q}}{4\mathrm{\pi }{\in }_0\mathrm{r}} \\ \mathrm{And}\mathrm{potential}\mathrm{energy}=\frac{\mathrm{Qq}}{4\mathrm{\pi }{\in }_0\mathrm{r}} \end{gathered}$$