Magnetism is electricity seen sideways

A wire carrying a steady current is electrically neutral. There are as many positive ions in a metre of it as there are conduction electrons, they cancel exactly, and a stationary charge placed beside it feels nothing at all. Set that charge moving parallel to the wire and it feels a force — which is called magnetic, and which every textbook computes from $q\mathbf{v}\times\mathbf{B}$.

Now ride alongside the charge. In that frame the charge is stationary, so there can be no magnetic force on it whatever. The force is still there, because whether a charge accelerates is not a matter of opinion. Something in the moving frame must be pushing it, and the only thing available is an electric field — from a wire that was supposed to be neutral.

That is the whole argument, and it has the peculiar property of being both entirely elementary and a genuine derivation. Nothing is invoked but the contraction of lengths, and what comes out is the existence of magnetism.

Getting the sign and the size right #

The details matter here more than usual, because the effect is a difference between two nearly equal quantities and the bookkeeping decides the sign.

In the laboratory the lattice is at rest with proper spacing $a$, and the electrons drift at $v$, so their proper spacing must be $\gamma a$ for their contracted spacing to be $a$ — which is what neutrality requires. Move to the electrons’ frame. Their spacing is now the proper one, $\gamma a$, larger than before. The lattice is now moving, so its spacing contracts to $a/\gamma$, smaller than before. Positive charges are closer together than negative ones, and the wire has a net positive charge.

A positive test charge, at rest in that frame, is therefore repelled — pushed away from the wire. And in the laboratory frame, a positive charge moving in the same direction as the conventional current beside a wire is also pushed away, by $q\mathbf{v}\times\mathbf{B}$. The two frames agree about which way the charge goes, which they had better.

The field the laboratory frame uses circles the wire and falls as $1/r$, and nothing about that picture is wrong. What the argument on this page shows is that it is not the only bookkeeping available: a description with no magnetic field in it at all is available for the same experiment, provided the observer is moving along with the current. Two descriptions, one experiment, and the same force on the test charge in both.

The numbers, both ways #

The argument is usually left qualitative, which is a pity, because the numbers are its most striking feature.

And the drift speed is the part that should give pause.

A relativistic correction in the twenty-sixth decimal place is not the sort of thing that usually produces a visible effect. What makes this one different is that it acts on a charge density of about $10^{10}$ coulombs per cubic metre — the conduction electrons of a metal, which is an enormous quantity of charge sitting exactly cancelled by an equally enormous quantity of the opposite sign. Multiplying $10^{10}$ by $10^{-26}$ leaves something measurable, and that something is the entire science of magnetism.

That is the reason relativity was discovered in electromagnetism rather than in mechanics. Every other relativistic effect is invisible below a good fraction of the speed of light, because there is no near-cancellation to amplify it. Electromagnetism has one, built into every neutral piece of conducting matter, and Maxwell’s equations were therefore relativistic thirty-five years before anyone knew what that meant.

The factor over the range where it is ordinarily met is 1.005 at a tenth of light speed, 1.155 at a half, 7.09 at 0.99 — and everything on that curve belongs to particle physics and astronomy. The wire on this page sits at the extreme left-hand end of it, at a drift velocity where the factor departs from 1 by about five parts in $10^{26}$. No plot could show it. That is the arresting part of the argument: an effect this far below any plausible threshold of relevance produces a force strong enough to start a motor, because it is multiplied by the enormous charge density it acts on.

The other force, which is not amplified #

The comparison that makes the amplification vivid is with gravity, where the same near-cancellation does not exist.

Gravitational charge — mass — comes in one sign only, so there is no such thing as a gravitationally neutral body in which two enormous opposite densities cancel. The gravitational analogue of magnetism does exist, and is called gravitomagnetism: a rotating mass drags inertial frames around with it, and a test body moving past a moving mass feels a velocity-dependent correction of exactly the shape derived above. Its size, for the Earth, is $v^2/c^2$ times the ordinary gravitational effect with no amplifying factor at all — which is why measuring it took a satellite with four fused-quartz gyroscopes cooled to 1.8 K, and why the answer came out to a precision of about 19 per cent after forty years of development.

Electromagnetism gets a factor of $10^{26}$ for free because matter is made of cancelled charge. That single structural fact is why a fridge magnet is an everyday object and frame dragging is a heroic measurement, and it is the reason the two forces feel so different despite having the same shape.

Why the fields cannot be separated #

The wire is a special case in one respect: a frame exists in which the magnetic field vanishes. That is not generally true, and understanding why is what stops the argument from being over-read.

Two combinations of the fields are the same in every frame: $E^2-c^2B^2$ and $\mathbf{E}\cdot\mathbf{B}$. A field with the first of those negative is magnetically dominated in every frame, and no boost will make it electric — the field inside a solenoid is such a field, and so is the field of a permanent magnet. A field with the first positive can be made purely electric, and the wire above is that case. A field with both invariants zero is a light wave, which is neither electric nor magnetic in any frame and cannot be transformed into a static field of either kind.

So the honest statement is not that magnetism is really electricity. It is that the split between them is frame-dependent, and that the object which is not frame-dependent is the pair taken together — a single antisymmetric tensor with six components, of which three are called $\mathbf{E}$ and three $\mathbf{B}$ by an observer who has chosen a frame.

The paper that did not need an experiment #

The argument on this page is not how Einstein arrived at relativity, but the situation it describes is, and the 1905 paper says so in its first paragraph — which is unusual, because that paragraph is about a piece of bookkeeping rather than about an experiment.

The observation is this. Move a magnet toward a stationary coil and a current flows; the textbook explanation is that the changing magnetic flux produces an electric field which drives the charges. Hold the magnet still and move the coil toward it, and the same current flows; the textbook explanation is completely different — the charges in the coil are now moving in a magnetic field, so $q\mathbf{v}\times\mathbf{B}$ pushes them. Same relative motion, same current, two unrelated mechanisms.

Einstein’s opening sentence calls that asymmetry one which does not appear to be inherent in the phenomena, and the whole of special relativity follows from taking it seriously. The theory’s first motivation was therefore not the Michelson–Morley experiment — which the paper does not cite — but the observation that electromagnetism was giving two accounts of one event and that the accounts differed by a choice of frame.

The wire on this page is the same complaint in its cleanest form. There is one experiment: a charge accelerates. There were two accounts of it, one magnetic and one electric, and they turn out to be the same account written in two coordinate systems. Once that is seen, the asymmetry is gone, and so is the question of which account is the real one.

What this buys, and what it does not #

It explains a coincidence that would otherwise be inexplicable. The magnetic force law contains $c$, in the sense that $\mu_0\varepsilon_0 = 1/c^2$, and there is no reason in classical electromagnetism why the constant governing the force between currents should be related to the speed of light. In the relativistic reading it is not a coincidence at all: the magnetic force is the electric one with a factor of $v u/c^2$, and the $c^2$ is there because it is a first-order relativistic correction.

It makes the force law’s structure inevitable. That the magnetic force is perpendicular to the velocity, that it does no work, that it depends on the velocities of both the source charges and the test charge — all of that follows from the transformation rather than being separate empirical facts. The prohibition on doing work becomes a statement about what a boost can and cannot change.

It does not explain permanent magnets. The magnetism of a lodestone comes from electron spin, which is an intrinsic angular momentum with no motion associated with it, and no amount of length contraction produces it. A ferromagnet’s field is a quantum-mechanical alignment effect, and the argument on this page has nothing whatever to say about why iron is magnetic and copper is not. The spin’s magnetic moment does transform correctly once it exists — relativity is consistent with it — but the moment itself comes out of the Dirac equation rather than out of any picture of circulating charge, and the classical attempt to model it as a spinning ball requires a surface moving faster than light.

It does not remove the field. A frame was found in which this particular field is purely electric; there is no frame in which fields are unnecessary. Charges still act on one another through something occupying the space between, and the derivation above uses the field concept twice over — once to say the laboratory has a magnetic field, and once to say the moving frame has an electric one. What has been removed is not the field but the impression that there are two of them.

The disc that has no answer #

Einstein’s opening complaint was about a magnet and a coil, and there is a sharper version of the same puzzle that has produced arguments for nearly two centuries. It is worth working through, because the resolution is exactly the lesson of this essay.

Faraday built the first electrical generator in 1831: a copper disc spinning between the poles of a magnet, with one brush at the axis and one at the rim, delivering a steady voltage. It is the homopolar generator, and it is the simplest dynamo there is.

Now try three variants. Spin the disc and hold the magnet still: there is an emf. Hold the disc still and spin the magnet about its own axis: there is no emf. Spin both together, rigidly: there is an emf, the same one as spinning the disc alone.

The third result is the one that starts the arguments, because the obvious explanation of the first — the disc cuts field lines — implies that spinning the magnet with the disc should carry the lines round with the conductor and produce nothing. It produces the full emf. So people ask whether the field lines rotate with the magnet, and the question has been answered both ways in print.

The question has no content, and this essay’s framing says why. A field is a function that assigns a vector to each point of space at each instant. It has no parts with identities that persist through time, so asking whether a particular line has moved is asking about a feature of a drawing rather than about the field. Field lines are a choice, and this is where the choice does the most damage.

Compute instead, with no lines anywhere. The force per unit charge on a carrier in the disc is $\mathbf{v}\times\mathbf{B}$ with $\mathbf{v}$ the velocity of that carrier, plus any electric field. Spin the disc: the carriers move, $\mathbf{v}\times\mathbf{B}$ is radial, and integrating it from axis to rim gives the emf. Spin the magnet alone: the magnet is axially symmetric, so rotating it about its axis leaves the field at every point exactly what it was — the field is static, there is no induced electric field, the carriers are not moving, and there is nothing. Spin both: the carriers move exactly as in the first case and the field is unchanged as in the second, so the answer is the first case’s.

Three experiments, three correct predictions, and no statement anywhere about whether anything rotates. The puzzle was manufactured entirely by a picture, and the essay’s account of what a magnetic field is — one component of an object whose split depends on the observer — has no room for it.

The transformation as a generator #

The frame-dependence is not only an explanation; it is a piece of engineering, and the arithmetic is worth one section because the numbers are surprisingly large.

A conductor moving at $\mathbf{v}$ through a magnetic field has, in its own frame, an electric field $\mathbf{v}\times\mathbf{B}$ pushing its charges along. In the ground frame the same push is the magnetic force on the moving carriers. Either way, the emf per unit length is $vB$, and it needs no changing flux, no coil and no moving parts beyond the conductor itself.

In low Earth orbit that expression is not small. A spacecraft moves at 7.8 kilometres a second through a field of about thirty microtesla, which is 0.23 volts per metre. A conducting tether twenty kilometres long therefore develops several kilovolts between its ends, from nothing but its own motion.

It has been flown. A tethered satellite deployed from the Shuttle in 1996 reeled out nearly twenty kilometres of conductor and generated some three and a half kilovolts at half an ampere before the tether failed — more voltage than predicted, because the ionosphere turned out to close the circuit more readily than the models assumed. Electrodynamic tethers are now a serious proposal for deorbiting satellites: run the current the other way and the same $\mathbf{I}\times\mathbf{B}$ force becomes a drag, so a dead spacecraft can be brought down with no propellant at all.

The same expression is what a magnetohydrodynamic generator uses, with a hot ionised gas as the moving conductor and electrodes on the walls, and what a flowmeter uses to measure the speed of a conducting liquid in a pipe without touching it.

None of it is exotic and all of it is this essay’s subject. In the pipe’s frame there is a magnetic field and a moving fluid; in the fluid’s frame there is an electric field and no motion. The voltmeter reads the same number, and the choice of which description to use is the engineer’s rather than nature’s.

What has to be given up to get it #

The derivation is elementary and it is not free. Three things have to be accepted before the contraction argument can be run at all, and each of them is a departure from ordinary intuition about charge.

Charge is invariant and length is not. The whole calculation rests on the number of electrons in a stretch of wire being the same in both frames while the length of that stretch is not, so the density changes and the charge does not. That is an experimental fact of considerable precision — a hydrogen molecule is neutral to better than a part in $10^{21}$, despite its electrons moving at a hundredth of light speed and its protons barely moving at all — and if charge depended on speed even slightly, ordinary matter would not be neutral.

Simultaneity has to go first. Counting the charges in “a metre of wire” requires deciding when the two ends of the metre are measured, and the two frames disagree about that. The contraction argument is a shorthand for a careful statement about which events are being counted, and running it without noticing produces paradoxes of the standard kind.

The current has to be steady. For a wire carrying an unchanging current the argument is exact. For a switching current it is not, because the field in one frame at one time is built from what the charges were doing at earlier times and at different places, and those retardations do not transform as simply as a density does. The clean picture is a statement about a steady state.

What the picture cannot show #

The lattice drawing at the top of this page is at 0.6 c, and the real effect is at $10^{-13}$ c. Nothing that could be drawn would show the actual contraction: a spacing changed by five parts in $10^{26}$ is a shift of $10^{-36}$ metres on a lattice spacing, which is twenty orders of magnitude below the Planck length. The figure is an honest picture of the mechanism at a speed the mechanism never operates at, and it is the only kind of picture available. Every published version of this diagram makes the same compromise, and none of them says so.

The drawing also shows discrete charges in a row, which is a one-dimensional cartoon of a three-dimensional conductor. Nothing in the argument depends on the charges being discrete, and a serious treatment works with the four-current density rather than with countable objects. Drawing them as beads makes the contraction visible and imports a mental model — of electrons as small balls at definite places — that is wrong about metals for entirely separate reasons.

And the drift velocity is drawn as though every electron moved at the same speed. The actual motion of a conduction electron is a fast random walk at the Fermi velocity, around $10^6$ m/s, with a drift of a fraction of a millimetre per second superimposed on it. The drift is what carries the current and everything above is correct about it; the picture of an orderly procession is a fiction that survives because only the average matters.

What the picture cannot show is the field configuration itself changing between frames, because what transforms is not one field into another but the pair of them into a different pair. The electric and magnetic fields are components of a single object, and a boost mixes them the way a rotation mixes the components of a vector — so asking which field is “really” there is like asking which component of a vector is the real one. The answer depends on the frame, and the object does not.

The ladder from here #

Later rungs on this anchor: the transformation of the fields written out, with the parallel components unchanged and the perpendicular ones mixed; the two invariants and what each classifies; the electromagnetic field tensor, in which the six components become one object and Maxwell’s four equations become two; the four-potential and gauge freedom; and the retarded fields of a moving charge, where the transformation of a Coulomb field produces both the magnetic part and the radiation.

The neighbouring ladders are length contraction, which is the entire mechanism, the magnetic force, which is what is being re-derived, and the invariant interval, which is the same habit — asking what a transformation preserves — applied to spacetime rather than to fields.