Torque and energy of a magnetic dipole

Immersed in a uniform field $\vec{B}$, a magnetic dipole $\vec{\mu}$ is not translated (the forces on the opposite sides of the loop balance out) but rotated: the field tends to align it with itself.

Law — Torque and energy of a magnetic dipole

For a dipole $\vec{\mu}$ in a uniform field $\vec{B}$: $\vec{M}_\text{torc} = \vec{\mu}\times\vec{B} \qquad E_\text{pot} = -\vec{\mu}\cdot\vec{B}$

The magnitude of the torque is $|\vec{M}| = \mu B\sin\beta$, with $\beta$ the angle between $\vec{\mu}$ and $\vec{B}$: maximum when the dipole is perpendicular to the field, zero when it is aligned. The potential energy, on the other hand, is minimum when $\vec{\mu}$ is parallel to $\vec{B}$ (stable equilibrium) and maximum when it is antiparallel (unstable equilibrium).

Key formula

$|\vec{\mu}| = i\,A$ $\vec{M}_\text{torc} = \vec{\mu}\times\vec{B}$ $E_\text{pot} = -\vec{\mu}\cdot\vec{B}$

This is the same behaviour as a compass needle, which is to all effects a magnetic dipole: it rotates until it aligns with the Earth’s field, orienting itself towards North. The tendency to align, combined with a mechanism that periodically reverses the current, is what makes the continuous rotation of an electric motor possible.

Collegamenti

Argomenti: Magnetismo Concetti: Dipolo magnetico · Momento di una forza

Esercizi collegati: Torque on a square loop · Which field gives maximum torque · Worked exercise — rectangular loop in a uniform field