Potential Energy of a Single Charge in an External Electric Field

Up to this point, the "system" under discussion has always consisted entirely of the charges whose own mutual potential energy was being computed. This sub-section instead considers a single small charge $q$ placed inside an EXTERNAL electric field $\vec{E}$, produced by some OTHER source that is not one of the charges under study -- and is assumed to be completely unaffected by the presence of $q$ itself, an assumption that holds as long as $q$ is small enough not to disturb whatever external arrangement is producing $\vec{E}$.

The external source itself may be known in complete detail, only partly known, or left entirely unspecified -- none of that actually matters for this calculation, because everything needed is already captured in the resulting external potential $V(\vec{r})$ at each point in space (which may itself vary smoothly from point to point, exactly as any potential can). …