Potential Energy for a System of N Point Charges

Extend the two-charge result of section 8.6.1 to an arbitrary system of N point charges by continuing to bring each new charge in from infinity, one at a time, and tracking the work done against ALL the charges already placed so far.

After $q_1,q_2$ are both in position (total work so far $=\dfrac{1}{4\pi\epsilon_0}\dfrac{q_1q_2}{r_{12}}$, exactly the two-charge case), bringing a third charge $q_3$ in from infinity requires work against the COMBINED potential of both $q_1$ AND $q_2$ at $q_3$'s new location: $W_3=\dfrac{1}{4\pi\epsilon_0}\left(\dfrac{q_1q_3}{r_{13}}+\dfrac{q_2q_3}{r_{23}}\right)$. Bringing a fourth charge $q_4$ in next requires work against $q_1,q_2$ AND $q_3$ together, and so the pattern continues, with each newly arriving charge contributing one new interaction term for every charge already present. …