Torque on a Magnetic Dipole in a Uniform Field Guide

A magnetic dipole — a bar magnet or a current loop with magnetic moment m — placed in a uniform magnetic field feels no net force, but it does feel a torque that tries to rotate it. The torque turns the dipole until its moment lines up with the field, like a compass needle settling toward north.

This alignment effect runs compasses, galvanometers and electric motors, and it defines the stable and unstable equilibrium positions of every magnet. Knowing the torque also gives the potential energy, which explains why alignment is natural and anti-alignment needs work.

Key Concepts

Magnetic moment m: strength and axis of the dipole, unit ampere metre squared (A m²) or joule per tesla (J/T). Angle θ: between m and field B. Torque τ: rotational effect in newton metre (N m), maximum when m is perpendicular to B, zero when parallel or anti-parallel. Potential energy U: lowest when aligned (θ = 0°, stable), highest when opposite (θ = 180°, unstable). Uniform field: same B everywhere, so forces on the two poles cancel in translation but not in rotation.

Formulas

τ = m B sinθ, vector form τ = m × B; τ in N m, m in A m², B in tesla (T). Potential energy: U = −m B cosθ, in joule (J). Work to rotate from θ1 to θ2: W = mB(cosθ1 − cosθ2). Assumption: field uniform over the dipole's size. Limit: in a non-uniform field there is also a net force pulling the dipole toward stronger or weaker field regions, in addition to torque.

Worked Example

A bar magnet of moment m = 2.0 A m² sits in a uniform field B = 0.40 T at θ = 60° to the field.

1. τ = mB sinθ = 2.0 × 0.40 × sin60° = 0.80 × 0.866 = 0.69 N m.
2. U = −mB cosθ = −2.0 × 0.40 × 0.5 = −0.40 J.
3. At full alignment, Umin = −mB = −0.80 J, so rotating from 60° to 0° releases 0.40 J.

The field twists the magnet with 0.69 N m of torque toward alignment, where its energy is lowest and the position is stable.

Common Misconceptions

• A uniform field pulls the magnet sideways: Equal and opposite pole forces give zero net force; only torque acts.
• Torque is greatest when the magnet faces the field: It is zero at alignment and greatest at 90°.
• Anti-parallel is also stable: At 180° the torque is zero too, but energy is maximum — the slightest nudge flips the magnet, so it is unstable.

Real-World Applications

• Compass needles aligning with Earth's field for navigation.
• Moving-coil meters, where magnetic torque balances spring torque to indicate current.
• Electric motors and magnetic couplings that exploit continuous dipole torque.

Torque τ = mB sinθ rotates every magnetic dipole toward alignment with the field, and the energy U = −mB cosθ explains why that aligned resting state is the one nature prefers.