Define electric dipole moment. Derive an expression for the electric field of a dipole at a point on its axis (end-on position).

1. 1 · Define. An electric dipole is ±q a distance 2l apart. Dipole moment p = q × 2l, in C m, from −q to +q.
2. 2 · Draw the figure. [−q at A, +q at B, centre O, AB = 2l; P on the axis beyond B with OP = r; E₁ away from B, E₂ towards A]
3. 3 · Field of each charge. E₁ = kq/(r − l)² along AP; E₂ = kq/(r + l)² along PA.
4. 4 · Subtract. E = kq[(r + l)² − (r − l)²]/(r² − l²)² = k(2lq)(2r)/(r² − l²)².
5. 5 · Use p. E = (1/4πε₀) 2pr/(r² − l²)², along p.
6. 6 · Short dipole. For r much larger than l: E = (1/4πε₀) 2p/r³.
7. 7 · Close in one line. The end-on field points along p and falls as 1/r³; it is twice the broad-on field at the same distance.

Scoring tip: Show the subtraction step, (r + l)² − (r − l)² = 4lr. This one line carries a mark in most marking schemes.