Potential energy for electrostatic forces between two bodies

The electrostatic force exerted by a charge Q on another charge q separated by a distance r is given by Coulomb's Law

${\displaystyle \mathbf{F}=\frac{1}{4\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\epsilon <!--\; \epsilon \; -->}}_0}\frac{Qq}{r^2}\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{r}},}$

where ${\displaystyle \stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{r}}}$ is a vector of length 1 pointing from Q to q and ε₀ is the vacuum permittivity. This may also be written using Coulomb constant k_e = 1 ⁄ 4πε₀.

The work W required to move q from A to any point B in the electrostatic force field is given by the potential function

${\displaystyle U(r)=\frac{1}{4\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\epsilon <!--\; \epsilon \; -->}}_0}\frac{Qq}{r}.}$