Electrostatic potential energy for collection of point charges

The electric potential at a distance r from a point charge q1 is V = k q1/r. If a second charge q2 is now brought in from infinity to a point a distance r12 from q1, the work needed to do so is W = q2 V = k q1 q2/r12, and this work is defined to be the electrostatic potential energy of the resulting two-charge system: U = k q1 q2/r12. This potential energy is positive for two like charges (since work must be done against their mutual repulsion to bring them together) and negative for two unlike charges (since the electric force does positive work pulling them together, so an external agent must actually extract work, or equivalently negative work is required, to bring them to that separation). For a system of more than two charges, the same idea is applied by imagining the charges assembled one at a time, from infinity, in some order: each new charge is brought in against the combined field of every charge already in place, and the total potential energy of the final configuration is the sum of the pairwise potential-energy terms k qi qj/rij over every distinct pair (i, j) of charges in the system -- for three charges this gives U = k[q1q2/r12 + q1q3/r13 + q2q3/r23], and the pattern extends straightforwardly to any number of charges. This total potential energy represents exactly the amount of work an external agent must supply to assemble t …