Illustrated Physics
physics, drawn
Concept

Field transformation — where it appears

The rule by which electric and magnetic fields mix when the observer's frame changes, so that neither is a separate object. Two combinations survive it — E² − c²B² and E·B — and their values decide whether a frame exists in which one of the two fields vanishes.

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

The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Magnetism is electricity seen sideways

The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.

relativity · Field transformation What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance.

The field nobody can transform away

A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter.

relativity · Field transformation 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 Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Charge and current are one thing

Ask what a boost leaves alone and the answer is charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

relativity · Field transformation Six numbers, one object. The electromagnetic field tensor written out as the four-by-four array it is, for a field with E = (0.4, 1.2, 0) and cB = (0, 0, 0.7), and again after a boost of rapidity 0.9 along the first axis. The array is antisymmetric — checked entry by entry — so of its sixteen slots only six are independent, and those six are the three components of E and the three of cB. The boost does not act on E and B separately; it acts on the array, mixing the top row into the lower block, which is what 'the electric field in one frame is partly magnetic in another' means written down. Its two scalars, E·B = 0.000 and E² − c²B² = 1.110, are unchanged, and they are the only two an antisymmetric rank-two tensor has.

Six numbers, one object

Three components of E and three of B mix into each other under a boost and never into anything else. Six numbers that transform among themselves are the independent entries of a four-by-four antisymmetric array, and writing them that way is not notation — it turns Maxwell's four equations into two, makes the two invariants the only two there could be, and shows that "electric" is a choice of axes rather than a kind of field.

relativity · Field transformation Two factors of γ between a magnet and an X-ray. An electron of 6 GeV (γ = 11742) crossing an undulator, a row of magnets alternating every λᵤ = 3 cm. Top: the laboratory, where the electron wiggles once per period. Middle: the electron's own frame, where the row is contracted by γ to 2.55 μm and rushes past at nearly the speed of light; to the electron it is a wave of that wavelength, and the electron radiates at it, in the infrared. Bottom: that light, emitted forward and seen back in the laboratory, is Doppler-shortened by a further factor of about 2γ, to λᵤ/2γ² = 1.088 Å; the wiggle's own slowing of the electron adds a factor (1 + K²/2), giving 1.632 Å for a deflection parameter K = 1 — hard X-rays from centimetre magnets. The drawing is schematic: the three periods differ by factors of about twelve thousand and a hundred million, and cannot be drawn to one scale.

The magnet period that comes out as an X-ray

A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

relativity · Length contraction

Named alongside it

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

InvarianceLength contractionMaxwell equationsReference frameCharge densityElectric fieldFour-vectorThe Lorentz factorThe Lorentz transformationMagnetismRapidityRelativity

All concepts