Torque on a current loop

A rectangular loop in a uniform field feels no net force, but it can feel a torque, just as an electric dipole does in a uniform electric field.

With B in the plane of the loop, the two sides across the field feel equal and opposite forces IbB, a distance a apart. The torque is τ = IabB = IAB, with A = ab.

When the normal to the loop makes an angle θ with B, the forces on the side arms cancel along the axis, and the couple on the other two arms has a lever arm a sin θ. So τ = IAB sin θ.

Define the magnetic moment m = IA, pointing along the area vector given by the right-hand thumb rule. Then τ = m × B, the twin of τ = p × E. For N turns, m = NIA.

The unit of m is A m² (also J/T), with dimensions [L²A].

The torque vanishes when m is parallel or antiparallel to B. Parallel is stable (a small turn is undone); antiparallel is unstable (a small turn grows). That is why a small magnet lines up with a field.

Example 4.10: a 100-turn coil, radius 10 cm, 3.2 A has B = 2 × 10⁻³ T at its centre and m = NIπr² = 10 A m². In a 2 T field, τ = 0 when m is along B and 20 N m after a quarter turn; with moment of inertia 0.1 kg m² it reaches ω = 20 s⁻¹ after turning 90°.

Example 4.11: a flat loop on a table cannot be spun about the vertical by any uniform field, since τ = IA × B always lies in the plane of the loop. A free loop settles with A along B, where the total flux through it is greatest; a flexible loop pulls into a circle, the shape enclosing the most area.