Potential Energy of a System of Charges

Energy belongs to the whole arrangement, not to any single charge. A lone charge sitting in empty space has no electric potential energy at all — energy only shows up when charges interact, and interaction takes two. So the natural unit of energy isn't a charge, it's a pair. Build the system by bringing charges in from far away one at a time: the first one costs nothing to place, the second costs energy because it feels the first, the third costs energy from its interactions with both charges already there, and so on. Add up every one of those costs and you get the system's total stored energy, a single number.

The bookkeeping trick is to count every pair exactly once. For N charges, that means C(N,2) = N(N-1)/2 pairs: 3 charges give 3 pairs, 4 charges give 6 pairs, and so on. For each pair you compute kq_iq_j/r_ij, with sign included — opposite charges give negative terms that lower the energy, same-sign charges give positive terms that raise it. Sum all the pair terms and you're done. There's no double counting because the pair (1,2) and the pair (2,1) are the same interaction, not two different ones.

This is the quieter cousin of Unit 9's main idea: potential energy of a system is a property of the whole configuration, while potential (coming up next) is a property of a single point in space, built charge by charge so you can later ask how much energy a new charge would need to be placed there. Keeping these straight — system energy in joules versus potential at a point in volts — stops a lot of sign and setup errors later in the unit.