Illustrated Physics
physics, drawn
Concept

Canonical momentum — where it appears

The momentum that a symmetry conserves, which for a charge in a magnetic field is its ordinary momentum plus its charge times the vector potential. It is conserved when the fields do not depend on the matching coordinate, and fixes motion that ordinary momentum alone cannot.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Pushed forward, and left at rest. The path of an electron, initially at rest, overtaken by a 6-cycle pulse of linearly polarised light travelling to the right, for peak field strengths a₀ = 0.5, 1, 2, where a₀ = 1 is the strength at which the electron's quiver becomes relativistic; lengths in wavelengths of the light, the vertical axis along the electric field. While the pulse passes, the electron quivers across the beam and is pushed along it. When the pulse has gone it is at rest again — not moving, with none of the light's energy — but displaced forward by 0.14, 0.56, 2.25 wavelengths respectively, a distance growing as a₀². The quiver's width grows only as a₀: at a₀ = 2 the electron is pushed forward several times further than it swings sideways.

The light that moves an electron and pays it nothing

A pulse of light strong enough to throw an electron forward at a large fraction of the speed of light passes over it, and the electron is left exactly as it was found — at rest, with none of the light's energy — except that it is somewhere else. The reason is two quantities the electron cannot change while a plane wave is passing, and the same two quantities fix the figure eight it traces inside the wave and the angle at which it leaves a real, focused beam carrying the energy it managed to keep.

relativity · Relativistic dynamics 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.

Vector potentialAngular momentumCausalityConservation lawCoulomb gaugeCyclotron motionE cross b driftFigure eight orbitGauge freedomLaser accelerationLawson woodward theoremThe Lorentz factor

All concepts