Electric Potential Due to a Point Charge

The potential due to a single, isolated point charge $q$ is the most fundamental potential formula in electrostatics, since the potential of any more complicated charge distribution is ultimately assembled by adding up contributions of exactly this same form (Section 2.6).

Consider a point charge $q$ fixed at the origin, and a point $P$ at distance $r$ from it. The potential at $P$, by definition, is the work done per unit charge bringing a positive test charge from infinity to $P$:

where $k = 1/(4\pi\epsilon_0) = 9\times 10^9\ \text{N}\,\text{m}^2/\text{C}^2$ and the field of the point charge, $E(r') = kq/r'^2$, has been substituted directly (the field points radially outward for $q>0$, along the same direction as the displacement, so the integral needs no extra sign adjustment beyond the one already written). Carrying out the integration,

giving the standard result

$V(r) = \frac{kq}{r} = \frac{q}{4\pi\epsilon_0 r}$ …