Obtain the equation for the electric potential energy of a single charge in an external electric field.

(N/A) The external electric field $\overrightarrow{E}$ and the corresponding external potential $V$ may vary from point to point.
According to the definition of electric potential,$V$ at a point $P$ is the work done in bringing a unit positive charge from infinity to the point $P$. (We assume the potential at infinity to be zero.)
Thus,the work done in bringing a charge $q$ from infinity to the point $P$ in the external field is $W = qV$.
This work is stored in the form of potential energy $U$ of the charge $q$.
$\therefore U = qV$.
If the point $P$ has a position vector $\vec{r}$ relative to the origin,then the potential energy at point $P$ is $U(\vec{r}) = qV(\vec{r})$.
This means the potential energy in an external field is equal to the product of the electric charge and the electric potential at that point in the external field.