Torque on a Current Loop and the Magnetic Moment

Picture a rectangular loop carrying current, sitting in a uniform magnetic field. The two sides of the loop that run parallel to the rotation axis each feel a force F = IL x B, but those forces point in opposite directions because the current flows opposite ways around the loop. Equal and opposite forces that don't act along the same line form a couple: no net push on the loop as a whole, just a twist. That twist is the torque, and it's why a loop in a field spins toward alignment instead of flying off sideways.

The size of that twist depends on how far apart the forces act and how tilted the loop is. Working through the geometry gives tau = N I A B sin(theta), where N is the number of turns, A is the loop's area, and theta is the angle between the loop's normal vector and B. The torque is zero when the loop's normal already points along B, since sin(0) = 0, and maximum when the loop is edge-on to the field, at theta = 90 degrees. As the loop rotates toward alignment, it loses potential energy, exactly the way a compass needle settles into a magnetic field: the system's potential energy is U = -mu B cos(theta), lowest when mu points along B.

This is where the magnetic moment mu = NIA earns its name: it packages the loop's current, area, and turns into a single vector that behaves exactly like the moment of a bar magnet. Write the torque as tau = mu x B and the loop's behavior matches a compass needle or bar magnet feeling a field, right down to producing its own dipole field far away. This is the bridge from loops and wires toward treating magnetic dipoles as fundamental objects, same role electric dipoles play with E fields.