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Obtain equation of electric energy of a single charge.
Solution
The external electric field $\overrightarrow{\mathrm{E}}$ and the corresponding external potential $\mathrm{V}$ may vary from point to point.
According to definition of electric potential $V$ at a point $P$ is the work done in bringing a unit positive charge from infinity to the point $\mathrm{P}$. (We assume the potential at infinity to be zero.) Thus, work done in bringing a charge $q$ from infinity to the point $P$ in the external field is $\mathrm{W}=q \mathrm{~V}$.
This work is stored in the form of potential energy of $q$,
$\therefore \mathrm{U}=q \mathrm{~V}$
If the point $\mathrm{P}$ has position vector $\vec{r}$ relative to origin, then potential energy at point $\mathrm{P}$, $\mathrm{U}(\vec{r})=q \mathrm{~V}(\vec{r})$
Means potential energy in an external field = electric charge $\times$ electric potential in external field.
