Illustrated Physics
physics, drawn
Concept

Vector potential — where it appears

The field A whose curl is the magnetic field, used because it turns two of Maxwell's equations into identities. It is not unique — three quite different arrangements describe one uniform field — so it is not measurable at a point, while its integral round a closed loop is the enclosed flux and is.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

electromagnetism · Magnetisation Three vector potentials for one magnetic field. Three different vector potentials, drawn as arrow fields over the same square, every one of which describes the same uniform magnetic field of 1 tesla out of the page. The symmetric gauge circulates about the origin; the two Landau gauges are unidirectional and point in perpendicular directions, and neither has any circulation about anything a reader can see. Underneath each is the field recovered from its own arrows, by adding up the potential round a small square and dividing by the area enclosed — which is what the curl is. The three numbers 1.000000, 1.000000, 1.000000 agree to the last digit computed. The picture is the argument: a vector potential has no physical direction, no physical magnitude and no physical circulation of its own, because a change of gauge alters all three and alters nothing that can be measured. What survives is the curl, and the curl is the field.

The potentials that are not unique

Nobody solves Maxwell's equations for the fields. They are solved for potentials instead, and the potentials are not unique — three completely different vector potentials describe the same uniform magnetic field, and one of them changes everywhere the instant a charge moves, at any distance, without anything having outrun light.

electromagnetism · Maxwell equations Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

electromagnetism · Induction 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 Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all.

The phase a magnet leaves on a path it never touched

An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

quantum · Matter waves Electrons that reach the anode, and electrons that turn back. Paths of electrons leaving a cathode 4 mm across at rest, in a cylindrical diode with an anode 8 mm across held at 4000 V, with no space charge, for axial magnetic fields of 0.80, 0.98, 1.02 and 1.30 times Hull's cut-off field of 142.2 millitesla; computed by integrating the Lorentz force. Below the cut-off the field only bends the path, and the electron still reaches the anode — at 0.98 of the cut-off it arrives almost tangentially. Just above it the electron turns back just short of the anode and returns to the cathode, then repeats the arc further round; at 1.30 times the cut-off it turns back well inside. Nothing reaches the anode, and the current stops completely within a few per cent of the field.

The field at which no electron reaches the anode

Put a hot wire inside a metal tube, make the tube positive, and electrons stream across from wire to tube. Add a magnetic field along the axis and they curve on the way, and still arrive — until, at one definite field, every one of them turns back a hair's breadth short of the tube and the current stops dead. The field at which it happens can be found without knowing anything about how the voltage is distributed between the wire and the tube, because one quantity the electron carries is fixed by the magnetic field alone. The microwave oven runs just inside that line.

electromagnetism · Magnetism

Named alongside it

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

Magnetic fluxAngular momentumCanonical momentumGauge freedomMaxwell equationsAntennaBiot–SavartBohr magnetonThe Boltzmann factorCausalityCoulomb gaugeCyclotron motion

All concepts