Illustrated Physics
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Series

Field transformation — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · relativity
  2. 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.

    part 2 · relativity
  3. Two things that change and one that does not. How a boost treats a piece of charged matter: the charge density rises by the Lorentz factor because the same charges occupy a contracted length, and the length falls by the same factor. At β = 0.6 the density is 1.250 times what it was and the length is 0.800 times, and their product is one to fourteen decimals across the whole range drawn. So the total charge is the same number in every frame, and it is the only quantity in the transformation that is. Charge density is the time component of a four-vector and transforms like an energy; charge itself is a scalar, and nothing about the observer changes it.

    The one quantity a boost leaves alone

    Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.

    part 3 · relativity
  4. 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.

    part 4 · relativity
  5. 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.

    part 5 · relativity
  6. A charge going past, as the field it delivers. The electric field across the line of motion at a point a distance b from the path of a passing charge, against time in units of b/c, on a logarithmic field axis in units of q/4πε₀b², for Lorentz factors of 1.2, 3, 10, 30. The peak is γ times the Coulomb field at rest and the width at half height is 2.311 b/c at γ = 1.2, 0.542 b/c at γ = 3, 0.154 b/c at γ = 10, 0.0511 b/c at γ = 30: taller by γ and shorter by γ, so the area under each curve — the sideways impulse the passing charge delivers, 2q²/4πε₀bv — is the same for all of them to within the integration's precision. A fast charge's field arrives as a thin, intense flash.

    The charge that passes as a flash of light

    A charge at rest has a field that no change of frame can turn into light. A charge going past at nearly the speed of light delivers a field that is, for every practical purpose, a pulse of light anyway: thin, perpendicular to the motion, with its electric and magnetic parts equal and crossed. Enrico Fermi saw in 1924 that such a field could be counted as photons, and the Large Hadron Collider now collides lead nuclei's photons with each other.

    part 6 · relativity
  7. Two pushes that nearly cancel. The forces between two equal charges moving side by side at the same speed, measured in the laboratory, in units of the Coulomb force the same two charges exert at rest, against their speed. Electric repulsion grows as γ, because a moving charge's field is stronger across its motion: 2.29 at β = 0.9, 7.1 at 0.99. The magnetic attraction of two parallel currents grows as γβ² and eats it almost entirely: the net push is γ(1 − β²) = 1/γ — 0.436 at 0.9 and 0.141 at 0.99. Measured in the pair's own frame the push is exactly the Coulomb force at every speed; the difference is not a force that exists in one frame and not another but a difference in how fast the two frames' clocks run while the charges drift apart.

    The beam that stops pushing itself apart

    A beam of electrons is a crowd of like charges, and a crowd of like charges flies apart. A slow beam does, within centimetres. A fast one hardly does at all: at a gigaelectronvolt the same current travels kilometres before its width doubles. The electric repulsion has not weakened — it has grown, by the factor γ. What has grown faster is a magnetic attraction between parallel currents, which cancels all but 1/γ² of the push. Seen from the beam itself there is no magnetism at all, only Coulomb repulsion acting while the beam's clocks run slow.

    part 7 · relativity
  8. A muon's spin lapping its own momentum round a storage ring. Snapshots of a muon circling a storage ring at γ = 29.3, taken every four turns (not to scale in time: the muon is drawn at a different place on the ring for each snapshot so they do not overlap). The dashed arrow is the direction of motion, tangent to the ring; the solid arrow is the spin. On each turn the spin gains 12.3° on the momentum — 2πaγ — so after 29.3 turns it has gained a full lap. That lap takes 4.37 microseconds, and the muon, whose lifetime is stretched from 2.2 to 64.4 microseconds by its speed, lives for about fifteen of them.

    The spin that runs ahead of its own motion

    A charged particle circling in a magnetic field has two directions that turn: the direction it is moving, and the direction its spin points. If the particle's magnetism were exactly what Dirac's equation first gave — a g-factor of exactly two — the two would turn together at every speed, the spin locked to the motion as a compass needle is locked to a car. The muon's g is not quite two, and its spin creeps ahead by a fixed rate that is the same at a crawl and at 99.94 per cent of the speed of light. That constancy is a delicate cancellation between a magnetic field made stronger by motion and a rotation that exists only because the particle is being turned, and it is what made the most precise measurements of the muon possible — at one magic speed, where the electric fields that steer the particles stop affecting the spin at all.

    part 8 · relativity
  9. The pattern in which a spinning Earth turns gyroscopes. The direction in which the Earth's rotation turns a gyroscope near it, in a plane through the Earth's axis (spin upward): the lines are those of the precession rate Ω = (G/c²r³)[3(J·r̂)r̂ − J], which has exactly the shape of the magnetic field of a spinning charged ball. On the circle of a polar orbit 650 km up, arrows show the direction and relative size of the turning at twelve points. Over the poles a gyroscope is turned the same way the Earth spins, twice as fast as anywhere else at that distance; over the equator it is turned the opposite way. Averaged round a polar orbit, what is left is half the size of the over-the-equator rate and opposite to it — in the sense of the Earth's spin.

    The gyroscope a spinning Earth turns

    A moving charge makes a magnetic field, and a spinning charged ball makes a dipole field around it. Gravity has the same structure one level down: a spinning mass drags the local standards of rotation around with it, and a perfect gyroscope near the Earth turns, very slightly, in the pattern of a dipole field — with the Earth's spin over the poles and against it over the equator. Averaged round a polar orbit the turning is 41 milliarcseconds a year. It took forty-four years and a satellite carrying the roundest objects ever made to measure it.

    part 9 · relativity
  10. Loops that swell as the electric field approaches cB. A charge starting from rest in crossed uniform fields — the electric field up the page, the magnetic field out of it — for E/cB = 0.3, 0.6, 0.85, integrated exactly; lengths in units of mc/qB. Each traces a chain of loops drifting to the right at E/B. Newton's law gives loops of height 2mE/qB² — 0.6, 1.2, 1.7 — whatever the speed. The exact loops are 0.66, 1.88, 6.13 high, larger by γ² of the drift speed, 1.10, 1.56, 3.60: in the frame moving at the drift the electric field vanishes and the magnetic field is weakened to B/γ, and the charge circles at the drift speed on a larger orbit than a weaker field would explain alone. As E approaches cB the drift approaches c and the loop grows without limit.

    The orbit that opens when E reaches cB

    A charge in crossed electric and magnetic fields drifts sideways at E/B, looping as it goes, and the textbook leaves it there. The textbook result has a speed in it that can be made larger than light, and something has to give. What gives is the loop. As the electric field approaches c times the magnetic one, the loops swell without limit, and at E = cB the charge stops looping and is carried off for ever. Above that ratio no frame can make the electric field vanish; one can make the magnetic field vanish instead, and in that frame the charge simply runs away. The transition from a bound orbit to a run-away is decided by a single quantity every observer agrees on, E² − c²B².

    part 10 · relativity

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