Illustrated Physics
physics, drawn
Concept

Magnetic monopole — where it appears

An isolated magnetic pole, which has never been found and whose absence is the statement that magnetic field lines have no ends. Its non-existence is one of Maxwell's four equations, and it is why a magnet cut in half gives two magnets rather than a north and a south.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The field of a current loop, seen edge on. Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.

The field with no ends, and the force that does no work

Magnetic field lines never start and never stop. That single absence is a law, it has survived every attempt to break it, and it makes the magnetic field a different kind of object from the electric one.

electromagnetism · Magnetism A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

electromagnetism · Magnetism Two quantities that are never computed and never change. A two-dimensional field integrated for 110 steps using only the two equations that contain a time derivative — Faraday's and Ampère's. The two that do not, Gauss's law for the electric field and the statement that there are no magnetic charges, are never imposed and never checked during the run. Their residuals are plotted: the divergence of B stays below 2.4e-16 of the field's own size and the divergence of E below 2.4e-16, over the whole run, while the field itself moves and changes by a factor of 8.59. That is not a numerical coincidence. Taking the divergence of Faraday's law gives the divergence of a curl, which vanishes identically, so ∂(∇·B)/∂t is zero whatever the fields are doing; the same manoeuvre on Ampère's law gives ∂(∇·D)/∂t = −∇·J. So the two constraints are initial conditions, propagated for ever by the two that are laws of motion, and Maxwell's four equations are two dynamical ones and two statements about how the field was set up.

The two equations that are not laws of motion

Maxwell's equations are usually presented as four laws of equal standing. Two of them contain no time derivative at all, which means they cannot be evolution equations: they are conditions on the field at one instant. What makes them consistent with the other two is that the other two preserve them exactly — and one of the two preservations holds only because charge is conserved.

electromagnetism · Maxwell equations The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which.

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

electromagnetism · Maxwell equations

Named alongside it

The objects these essays reach for when they reach for this one.

The Lorentz forceMagnetic fieldMaxwell equationsSymmetryBiot–SavartCharge conservationConsistencyConstraintContinuity equationDipole radiationDivergenceEquilibrium

All concepts