A magnetic dipole in a field

Any current loop or coil is a magnetic dipole. For N turns of area A carrying current I:

m = NIA, pointing along the coil's axis by the right-hand rule. A solenoid and a bar magnet are both magnetic dipoles.

Example: 100 turns of area 0.01 m² carrying 2 A: m = 100 × 2 × 0.01 = 2 J/T.

In a uniform field B:

• No net force: the pull on the N end and the push on the S end cancel.
• A torque τ = m × B, size mB sin θ, turning m towards B.

Like a compass needle, the dipole swings until it lines up with the field. In 0.5 T, the coil above feels at most 2 × 0.5 = 1 N m.

Potential energy: U = –mB cos θ.

Work to turn the dipole from θ₁ to θ₂: W = mB(cos θ₁ – cos θ₂). For the coil above, turning from 0° to 90° takes 2 × 0.5 × (1 – 0) = 1 J.

If the magnet is free to swing, it oscillates about the field direction, faster in a stronger field. This is one way to measure B.