A current loop placed in a magnetic field can experience a torque even when the net magnetic force on the loop is zero. This behavior is summarized by the loop's magnetic dipole moment.

Consider a planar loop carrying current $I$ and enclosing area $A$. Define an area vector

$$\mathbf A=A\hat{\mathbf n},$$

where $\hat{\mathbf n}$ is perpendicular to the loop according to the right-hand rule: curl the fingers in the current direction and the thumb gives $\hat{\mathbf n}$.

The magnetic dipole moment is

$$\boxed{\boldsymbol\mu=I\mathbf A}.$$

For a coil of $N$ identical tightly packed turns,

$$\boxed{\boldsymbol\mu=NI\mathbf A}.$$

Its SI unit is $\mathrm{A,m^2}$.

Torque on a rectangular loop

Place a rectangular current loop in a uniform magnetic field $\mathbf B$. Let $\theta$ be the angle between $\boldsymbol\mu$ and $\mathbf B$.

The magnetic forces on opposite sides of the loop are equal and opposite, so the net force is zero. However, one pair of forces can act along different lines and form a couple.

For a rectangle with side lengths $a$ and $b$, the relevant force magnitude on each side is

$$F=IbB$$

when that side is perpendicular to the field. The perpendicular lever-arm separation of the force lines is $a\sin\theta$, so the torque magnitude is

$$\tau=F(a\sin\theta) =IabB\sin\theta.$$

Since $A=ab$,

$$\boxed{\tau=IAB\sin\theta=\mu B\sin\theta}.$$

In vector form,

$$\boxed{\boldsymbol\tau=\boldsymbol\mu\times\mathbf B}.$$

The torque tends to rotate the loop so that its magnetic dipole moment aligns with the field.

Stable and unstable orientations

When

$$\boldsymbol\mu\parallel\mathbf B,$$

the torque is zero and a small angular displacement produces a restoring tendency. This is the stable orientation.

When

$$\boldsymbol\mu\text{ is antiparallel to }\mathbf B,$$

the torque is also zero, but a small displacement produces torque farther away from that orientation. It is unstable.

This is directly analogous to the torque on an electric dipole in an electric field.

Worked example

A coil has

$$N=100$$

turns, current

$$I=0.50,\mathrm A,$$

and area per turn

$$A=1.0\times10^{-2},\mathrm{m^2}.$$

Its magnetic dipole moment magnitude is

$$\mu=NIA=(100)(0.50)(1.0\times10^{-2}) =0.50,\mathrm{A,m^2}.$$

Place the coil in a uniform field

$$B=0.20,\mathrm T$$

with

$$\theta=30^\circ.$$

The torque magnitude is

$$\tau=\mu B\sin\theta =(0.50)(0.20)(0.50) =\boxed{5.0\times10^{-2},\mathrm{N,m}}.$$

Nonuniform magnetic fields

In a uniform field, the net force on an ideal closed current loop is zero even though torque can act. In a nonuniform field, the forces around the loop need not cancel, so a magnetic dipole can also experience a net translational force.

The current-loop dipole is the basic mechanical model behind galvanometers, electric motors, magnetic moments of matter, and many interactions between magnets and external fields.