A bar magnet in a uniform field.

• A compass needle of moment m in a uniform field B feels no net force but a torque τ = m × B, of size mB sin θ, where θ is the angle between m and B. It is a restoring torque: it turns m towards B.
• The magnetic potential energy follows by integrating the torque over angle: U = −mB cos θ = −m·B. The zero is chosen at θ = 90°, where the needle is at right angles to the field.
• U is least, −mB, at θ = 0°: the most stable position. It is greatest, +mB, at θ = 180°: the most unstable position. Turning the magnet from 0° to 180° takes work 2mB.
• The torque is zero at both 0° and 180°, but only 0° is stable: a small nudge from 180° grows.
• Set oscillating, the needle swings about the field direction; the arrangement can be used to find B if m is known, or m if B is known.
• Example 5.1: in a uniform field a magnetised needle is only turned, never pulled. An iron nail near a bar magnet is in a non-uniform field; the magnet induces a moment in it, and because the nail's induced pole nearest the magnet is the unlike one, the nail is drawn in as well as turned.
• Example 5.1: a field source need not have a north and a south pole; that happens only if it has a net magnetic moment, which a toroid or an infinite straight wire does not.
• Example 5.1: of two identical-looking bars, a repulsion at some pairing shows both are magnets. If only one is, touch the end of one bar to the middle of the other: a magnet's field is strongest at its ends and weakest at its centre, so no pull at the middle of B means B is the magnet.

Teaching this idea? Put the simulation on your own page or LMS with this code.