The electrostatic force is conservative, so a configuration of charges can be assigned an electrostatic potential energy $U$.

For conservative forces,

$$\Delta U=-W_{\rm electric},$$

where $W_{\rm electric}$ is the work done by the electrostatic force as the configuration changes.

Two point charges

Choose

$$U=0$$

when two point charges are infinitely far apart. For charges $q_1$ and $q_2$ separated by distance $r$, the electrostatic potential energy is

$$\boxed{U(r)=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}}.$$

The sign contains physical information.

• For like charges, $q_1q_2>0$, so $U>0$. External work must be supplied to bring the charges together from infinity against their repulsion.
• For opposite charges, $q_1q_2<0$, so $U<0$. The attractive interaction lowers the energy as the charges approach.

Potential energy belongs to the configuration of the interacting charges, not to either charge alone.

Work and kinetic energy

If only the electrostatic force does work, mechanical energy is conserved:

$$K_i+U_i=K_f+U_f.$$

Therefore

$$\Delta K=-\Delta U.$$

A decrease in electrostatic potential energy becomes an increase in kinetic energy, while moving charges into a higher-energy configuration requires kinetic energy or external work.

Example

Two equal positive charges

$$q_1=q_2=1.0\times10^{-9},\mathrm C$$

are separated by

$$r=0.10,\mathrm m.$$

Their electrostatic potential energy is

$$U =\frac{(8.99\times10^9)(1.0\times10^{-9})^2}{0.10} \approx8.99\times10^{-8},\mathrm J.$$

The positive value reflects the work required to assemble two like charges at this separation from infinite distance.

Several point charges

For a collection of point charges, the total electrostatic potential energy is the sum over distinct pairs:

$$\boxed{U=\sum_{i<j}\frac{1}{4\pi\varepsilon_0}\frac{q_iq_j}{r_{ij}}}.$$

Each interacting pair is counted once.

For example, for three charges,

$$U=\frac{1}{4\pi\varepsilon_0}\left( \frac{q_1q_2}{r_{12}}+ \frac{q_1q_3}{r_{13}}+ \frac{q_2q_3}{r_{23}} \right).$$

This is the energy required to assemble the configuration quasistatically from infinitely separated charges, under the chosen zero of energy.

Potential energy and electric potential are different quantities

Electrostatic potential energy $U$ belongs to a particular charge configuration and is measured in joules. Electric potential removes the dependence on a chosen test charge by describing potential energy per unit charge.

Keeping these two quantities distinct prevents a common confusion: a location in space can have a well-defined electric potential before any test charge is placed there, whereas the potential energy depends on what charge is actually present.