A current loop is a magnet
Bend a current into a loop and, seen from afar, its field is indistinguishable from a tiny bar magnet's. We capture its strength and orientation with a single vector, the magnetic dipole moment \vec{\mu}: its magnitude is current times area times number of turns, and its direction is set by curling your right fingers along the current so your thumb points out the loop's 'north' face.
The dipole moment of a flat coil of N turns, each of area A, carrying current I. Units: A·m².
Torque, motors, and energy
Place that loop in an external field and opposite sides carry current in opposite directions, so they feel opposite forces — a couple that twists the loop. This torque on a current loop tries to swing \vec{\mu} into alignment with \vec{B}, exactly as a compass needle swings to point north.
Torque on a magnetic dipole. It is maximal when the loop lies flat in the field (θ = 90°) and zero once aligned (θ = 0).
This is the beating heart of the electric motor: keep flipping the current direction every half-turn (with a commutator or with AC) so the torque always pushes the same way, and the loop spins continuously. The energy stored in a dipole's orientation is lowest when aligned with the field.
Potential energy of a dipole in a field: minimum (aligned) at θ = 0, maximum (anti-aligned) at θ = 180°.
- Quick problem. A coil of N = 50 turns, area A = 0.010\,\text{m}^2, carries I = 3.0 A in a field B = 0.40 T. Find the maximum torque.
- Moment. \mu = N I A = (50)(3.0)(0.010) = 1.5\,\text{A·m}^2.
- Torque. \tau_{\max} = \mu B = (1.5)(0.40) = 0.60\,\text{N·m} — enough to turn a small motor shaft.
Why iron is magnetic
Every atom is itself a bundle of tiny current loops — orbiting electrons and, dominantly, electron spin — so every atom is a small magnetic dipole. In most materials these dipoles point every which way and cancel. In a ferromagnetic material such as iron, a quantum interaction locks neighbouring spins parallel across whole regions called magnetic domains. Apply an external field and the favourably-aligned domains grow; the material magnetizes strongly and, in a hard magnet, stays magnetized after the field is removed.
Two milder responses round out the picture. In paramagnetism, dipoles align only weakly and only while the field is on; in diamagnetism, the field induces atomic currents that oppose it, giving a faint repulsion present in all matter. Heat a ferromagnet above its Curie temperature and thermal jostling scrambles the domains — the magnetism disappears.
The four laws and the birth of light
Step back and count what we now hold. Gauss's law tells us how electric charge sources the electric field; Gauss's law for magnetism says there are no magnetic charges; Ampère's law says currents make magnetic fields; and — in the very next track — Faraday's law will show that a changing magnetic field makes an electric field. James Clerk Maxwell noticed Ampère's law was incomplete, added a 'displacement current' term for changing electric fields, and closed the circle: the four Maxwell's equations.
The payoff is breathtaking. Together the equations allow a self-sustaining ripple in which a changing electric field creates a magnetic field, which in turn creates an electric field, leapfrogging through empty space at a speed built entirely from the two constants we have already met.
Plug in μ₀ and ε₀ and out falls the speed of light. Electromagnetic waves ARE light — magnetism and optics are one subject.
Where this leads
You now command the whole of magnetostatics: the field, the Lorentz force, motion in fields, the sources of fields, and dipoles in matter. Next comes electromagnetic induction — moving magnets and changing fields that generate electricity, powering every generator and transformer on the grid. Beyond that, optics reveals itself as electromagnetic waves in action, and special relativity delivers the final unification: electricity and magnetism are two faces of one field, seen from different frames of motion.