Electrostatic Potential Energy of a System of Two Point Charges

The electrostatic potential energy of a system of point charges is defined as the total work an external agent must do to assemble the charges, one at a time, bringing each one in from infinity to its final position, against the electrostatic forces exerted by the charges already placed -- moving each charge quasi-statically, so no kinetic energy is left over at the end.

Two point charges. Suppose charge $q_1$ is fixed first, at no cost (no other charge yet exists to exert a force on it). Bringing the second charge $q_2$ in from infinity to a final distance $r_{12}$ from $q_1$ then requires work equal to $q_2$ times the potential $q_1$ has already set up at that location (Section 2.4):

By definition, this work done IS the potential energy stored in the resulting two-charge system:

Sign and physical meaning. For two charges of the same sign (both positive or both negative), $U$ comes out positive -- physically sensible, since like charges repel, so positive work must be done by an external agent to push them together against that repulsion; if released, they would fly apart, converting this positive potential energy into kinetic energy. For two charges of opposite sign, $U$ comes out negative -- unlike charges attract, so the external agent must actually do NEGATIVE work (equivalently, the electrostatic force itself does positive work) bringing them together from infinity, exactly as for two oppositely-signed masses would never occur gravitationally, but is the everyday case for electrostatic attraction. In both cases, $U\to 0$ as $r_{12}\to\infty$, consistent with the same zero-potential-energy reference used for potential itself. …