(a) Derive an expression for the potential energy of a system of two point charges. (b) Derive the potential energy of an electric dipole in a uniform electric field, and state its positions of stable and unstable equilibrium.

1. 1 · Define potential energy. The work done in bringing the charges slowly from infinity to their places is stored as potential energy.
2. 2 · Two charges. Bringing q₁ needs no work. Bringing q₂ needs q₂ × q₁/(4πε₀r₁₂). So U = q₁q₂/(4πε₀r₁₂).
3. 3 · Draw the dipole. [Uniform field lines; dipole at angle θ; forces qE on each charge; p and θ marked]
4. 4 · Work against the torque. W = ∫ pE sin θ dθ from θ₀ to θ₁ = pE(cos θ₀ − cos θ₁).
5. 5 · Choose the zero. Take U = 0 at θ₀ = 90°. Then U(θ) = −pE cos θ = −p·E.
6. 6 · State the equilibrium. Stable at θ = 0° (U = −pE, the least). Unstable at θ = 180° (U = +pE, the greatest).
7. 7 · Real life. Food and fuel store energy as the potential energy of charges in molecules.
8. 8 · Close in one line. A system of charges stores the work done to assemble it, and a dipole has the least energy when it lines up with the field.

Scoring tip: In part (b), say where you take U = 0 (at 90°). Leaving it out makes the result look unexplained.