Result

Torque on a current loop

(E7.4.1) E7.4 · Torque on a Current Loop →

τ = μ \times B, μ = I A

Statement

The torque on a current loop of area A carrying current I, with magnetic moment μ = IA directed by the right-hand rule.

Symbols and units

$τ$	torque	N·m
$μ$	magnetic moment	A·m²
$I, A$	current, loop area	A, m²

When it applies

Any planar current loop in a uniform external field.

Don't use when

Assumes a uniform external field — in a non-uniform field the loop experiences a net force in addition to torque, which this formula does not capture.

Derivation

Forces on opposite sides of the loop are equal and opposite but offset, producing a net torque; collecting the geometry into a single magnetic moment vector gives:

τ = μ \times B

Watch out for

Exactly analogous to an electric dipole in a field: torque is maximum when the loop's plane is parallel to B, and zero when the moment is already aligned with B.

Connects to

Torque and energy