Derive an expression for the torque on a rectangular current-carrying coil in a uniform magnetic field. Show that the coil behaves like a magnetic dipole.

1. 1 · Draw the figure. [Rectangular coil PQRS in uniform B; top view with forces on PQ and RS; normal m at angle θ to B; arm b sin θ]
2. 2 · Forces on the sides.
   • QR and SP: equal, opposite and along the same line, so they cancel.
   • PQ and RS: F = IaB each, equal and opposite, forming a couple.
3. 3 · Arm of the couple. Perpendicular distance between the two forces = b sin θ.
4. 4 · Torque. τ = IaB × b sin θ = IAB sin θ; for N turns, τ = NIAB sin θ.
5. 5 · Vector form and dipole moment. With m = NIA along the normal, τ = m × B, like τ = p × E for an electric dipole.
6. 6 · Equilibrium. τ = 0 at θ = 0° (stable) and θ = 180° (unstable); largest, NIAB, at θ = 90°.
7. 7 · Real life. This torque turns the coils of every electric motor, from fans to e-rickshaws.
8. 8 · Close in one line. So, a current loop is a magnetic dipole of moment m = NIA, turned by τ = m × B.

Scoring tip: Mark θ between the normal and B on your figure. Many students lose a mark by measuring it from the plane of the coil.