Obtain an expression for the electric potential energy of a system of two point charges, and extend it to three charges.

1. 1 · Define. The potential energy of a system of charges is the work done in bringing them from infinity to their positions.
2. 2 · Bring the first charge. No other charge is present yet, so bringing q₁ needs no work.
3. 3 · Bring the second charge. q₁ sets up a potential V = q₁/(4πε₀r₁₂) at the place of q₂. Work = q₂V.
4. 4 · Write the result. U = (1/4πε₀) q₁q₂/r₁₂: positive for like charges, negative for unlike charges.
5. 5 · Draw the figure. [q₁ at P₁ and q₂ at P₂, a distance r₁₂ apart; for three charges, a triangle with sides r₁₂, r₁₃ and r₂₃]
6. 6 · Three charges. U = (1/4πε₀)(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃): one term for each pair.
7. 7 · Close in one line. So the energy of any group of charges is the sum of kqq/r over every pair, counted once.

Scoring tip: Say why bringing the first charge costs no work. It is a small step, but examiners look for it.