Potential Energy of Charge Configurations

Bring one charge in from far away and it costs nothing — there's no field yet to push against. Bring in a second charge, and you do work against (or with) the first charge's field: that work is U = kq1q2/r. Bring in a third charge, and now it has to fight the fields of both charges already sitting there, one pairwise term for each. The total energy of the configuration is just the sum of these pairwise energies, U_total = kq1q2/r12 + kq1q3/r13 + kq2q3/r23. Order doesn't change the final total, since potential energy is a property of the finished arrangement, not the path taken to build it — but it can feel like different orders 'cost' different amounts along the way because each step depends on what's already in place.

For three equal charges q at the corners of an equilateral triangle with side r, all three pairwise distances and charges match, so each pair contributes the same energy kq²/r, and the total is simply U = 3kq²/r. Swap in different charges or an irregular triangle and you must compute each pairwise term separately and add them, since the three distances or charges are no longer identical.

This is the electrical cousin of gravitational potential energy for multiple masses: both come from a 1/r interaction, both sum pairwise contributions, and both represent the work needed to assemble a configuration from infinite separation. The sign differs in character, though — like charges make positive energy (it takes work to push them together), while gravity, always attractive, makes negative energy (masses fall together and release energy). This pairwise-sum idea also sets up potential itself, since the potential at a point from several charges is just the scalar sum of each charge's individual potential there.