The voltage a moving magnet adds to an insulator

Maxwell’s four equations in empty space are complete: given the charges and currents, they fix the fields. Inside matter they are not, because matter responds to the fields by polarising and magnetising, and the response has to be supplied separately. The usual supply is two lines. The electric displacement is the permittivity times the field, $\mathbf D = \varepsilon\varepsilon_0\mathbf E$, and the magnetic field strength is the flux density divided by the permeability, $\mathbf H = \mathbf B/\mu\mu_0$. The field the matter takes away found what the first of those means: an insulator’s molecules stretch in an applied field and their dipoles cancel part of it.

Those two lines carry an assumption that is easy not to notice. They describe how a material responds to fields in the frame in which it is at rest. A glass block on a bench is at rest in the laboratory and nothing more needs saying. A material that moves — a rotor, a stream of liquid, a cylinder spinning on an axle — is at rest in no single frame the laboratory uses, and the question of what its constitutive relations become when it moves is a question about how electric and magnetic fields look to observers in relative motion. That is the question special relativity answers, and in 1908 Hermann Minkowski wrote down the answer for moving matter. In 1913 Marjorie Wilson and Harold Wilson tested it, with a cylinder of wax full of steel balls spun in a magnetic field, and the test came down on a detail that no theory before relativity had any reason to include.

Every piece of the wall is moving through the field #

The first figure sets up the experiment. A thick-walled tube of insulating material turns about its own axis, and a magnetic field points along that axis. Nothing is applied electrically: there is no battery, no electrode with a voltage on it, no electric field in the laboratory to start with. But every piece of the wall moves sideways through the magnetic field at a speed proportional to its distance from the axis, and a charge moving at velocity $\mathbf v$ through a field $\mathbf B$ feels the force $q\,\mathbf v\times\mathbf B$. With the field out of the page and the cylinder turning anticlockwise, that force points straight outward, along the radius, and grows from the inner wall to the outer.

If the cylinder were metal, the story would end in a sentence. Electrons in the metal would be pushed inward until the electric field of their displaced charge exactly cancelled the push, the outer surface left positive, and a voltmeter connected by sliding contacts between the inner and outer surfaces would read $\omega B(b^2-a^2)/2$, where $a$ and $b$ are the two radii. That is Faraday’s disc, the homopolar generator he built in 1831, which the rule that is two laws wearing one coat found to be a case where the flux rule and the Lorentz force have to be applied with care, because the circuit’s shape is not changing and the voltage still appears.

An insulator has no free electrons to push. What it has is bound charge: electrons tied to their molecules, which can be displaced a little relative to the nuclei, and in a magnetic material a population of magnetic moments as well. The push $q\,\mathbf v\times\mathbf B$ stretches each molecule along the radius, and the insulator becomes polarised, as it would in an ordinary electric field. The polarisation leaves a sheet of bound positive charge on the outer surface and negative on the inner, and that charge makes a voltage between the surfaces. The question is how much.

The answer that seemed obvious in 1904 #

Before relativity there was a natural way to calculate it, and Hendrik Lorentz’s electron theory supplied it. The material’s charges are moving at $\mathbf v$; each feels the force it would feel in an electric field $\mathbf E+\mathbf v\times\mathbf B$; so the material polarises exactly as an insulator at rest would in that combined field. The polarisation is $\varepsilon_0(\varepsilon-1)(\mathbf E+\mathbf v\times\mathbf B)$, the displacement is $\varepsilon_0\mathbf E$ plus the polarisation, and the rest is bookkeeping. A moving magnetisation, in this account, has no electrical consequences at all: a magnetic moment carried along is still only a magnetic moment.

If the two surfaces are connected only to an electrometer, so that no free charge can flow between them, the displacement inside the material has nothing to begin or end on and must vanish. Setting it to zero gives the laboratory field across the wall, $E_r = -\frac{\varepsilon-1}{\varepsilon}\,\omega rB$: the same shape as the metal’s but a fraction $(\varepsilon-1)/\varepsilon$ of its size. A material with a large permittivity polarises almost as completely as a metal, and gives almost the full voltage; one with $\varepsilon$ close to 1 hardly polarises, and gives almost none.

Harold Wilson spun a cylinder of ebonite in a magnetic field in 1904 to test this, and found it. The measurement also ruled out the alternative then current, in which a moving body was supposed to carry the electromagnetic field along with it entirely, and so would behave much more like a metal. Lorentz’s factor won. It is worth noticing, though, what kind of material ebonite is. It is an insulator with a modest permittivity and no magnetism to speak of; its permeability is 1. That detail decided less than it appeared to.

The same calculation done in the material’s own frame #

Relativity changes the starting point. The constitutive relations $\mathbf D=\varepsilon\varepsilon_0\mathbf E$ and $\mathbf B=\mu\mu_0\mathbf H$ are statements about a material at rest, and they can be trusted only in the frame in which the material is at rest. So the correct procedure is to go to that frame, apply them there, and transform the results back to the laboratory. Magnetism is electricity seen sideways found the rule for the first step: an observer moving at $\mathbf v$ through a magnetic field $\mathbf B$ measures an electric field $\mathbf E' = \mathbf E+\mathbf v\times\mathbf B$, to first order in $v/c$. Six numbers, one object found that the same rule applies to $\mathbf D$ and $\mathbf H$, because they too are the parts of a single object seen from different frames: $\mathbf D' = \mathbf D+\mathbf v\times\mathbf H/c^2$ and $\mathbf H' = \mathbf H-\mathbf v\times\mathbf D$.

Doing the three steps — transform $\mathbf E$ and $\mathbf B$ into the material’s frame, apply the material’s own relations there, transform $\mathbf D$ and $\mathbf H$ back — gives the displacement in the laboratory:

$$\mathbf D = \varepsilon\varepsilon_0\,\mathbf E + \varepsilon_0\,\frac{\varepsilon\mu-1}{\mu}\,\mathbf v\times\mathbf B .$$

The figures check this formula against an exact Lorentz transformation of all four fields, with the full factor $\gamma$ and no approximation, at the speeds of the spinning wall; they agree to better than a part in a billion, and the difference that remains is of order $v^2/c^2$, about $10^{-14}$ at these speeds. Lorentz’s electron theory had the same structure with $\varepsilon-1$ in place of $(\varepsilon\mu-1)/\mu$. For a non-magnetic material, $\mu = 1$, the two coincide exactly. For a magnetic one they do not.

Three answers for one cylinder #

With the electrometer drawing no charge, the displacement in the wall must still vanish, and the laboratory field becomes $E_r = -\frac{\varepsilon\mu-1}{\varepsilon\mu}\,\omega rB$. The second figure draws the three accounts side by side for a cylinder of an insulator with $\varepsilon = 6$ and $\mu = 3$. The metal’s field runs straight up from the inner wall to the outer, reaching about 6 volts per metre; relativity’s is 94 per cent of it, Lorentz’s 83 per cent. Integrated across the two-centimetre wall they give 94, 89 and 79 millivolts.

One feature of the figure is worth pausing on, because it holds for all three. The field grows in proportion to the radius, and a radial field proportional to $r$ has a uniform divergence. By Gauss’s law, then, the spinning wall carries a uniform density of charge throughout its bulk as seen in the laboratory — here the charge of sixteen electrons in every cubic millimetre — with an equal and opposite charge on its two surfaces. Nothing has been added to the cylinder; it is electrically neutral overall and neutral in its own frame. The volume charge is a laboratory description of a polarisation that is not uniform, and it is a reminder that “how much charge is here” for a moving material depends on who is asking, as charge and current are one thing found for a current-carrying wire.

The test needed an insulator that was also a magnet #

The third figure is the reason the 1904 experiment could not decide between relativity and Lorentz’s theory, and the reason the 1913 one could. Plotted against the material’s permeability, Lorentz’s prediction is a flat line: magnetism plays no part in it. Relativity’s starts at the same value when $\mu = 1$ and rises from there, approaching the metal’s full voltage as the permeability grows. Ebonite sits at the left edge of the figure, where the curves touch.

To separate them, the Wilsons needed an insulator with a permeability well above 1, and no such material was to hand. Insulating magnetic materials of the modern kind — ferrites — were decades away. So they made one, by embedding a large number of small steel spheres in paraffin wax. Each sphere is a conductor and a strong magnetic material, but the spheres do not touch, so the composite as a whole conducts nothing; its permittivity is raised by the conducting spheres and its permeability by their iron. Spun in a magnetic field between electrodes on its inner and outer surfaces, the composite gave a charge on the electrometer that agreed with the relativistic factor and not with Lorentz’s.

It is a small difference to have hung a theory on. For the values drawn here it is thirteen per cent, and the measured quantity is a charge of a few picocoulombs on an instrument that must not let any of it leak away. A modern repetition published in 2001 by John Hertzberg and colleagues, with a magnetic insulator made for the purpose in place of steel and wax and electronics the Wilsons did not have, confirmed the relativistic prediction with much smaller uncertainty.

What the extra term is #

The relativistic coefficient comes apart neatly, and the fourth figure draws the split: $(\varepsilon\mu-1)/\mu = (\varepsilon-1) + (\mu-1)/\mu$. The first part is Lorentz’s, the material’s bound charges pushed sideways by the force they feel as they move. The second is new, and it belongs to the material’s magnetisation.

A magnetised material is full of small magnetic dipoles, and the magnetisation of the spinning wall is set by the field along the axis. In the material’s own frame each dipole is a pure magnetic moment $\mathbf m$. Seen from a frame in which it moves at $\mathbf v$, the same object has an electric dipole moment as well, $\mathbf p = \mathbf v\times\mathbf m/c^2$. That is not a peculiarity of steel spheres. It follows from the transformation of the fields of any current loop: a loop of current carried sideways has its two sides contracted by different amounts and its charges no longer balance, so a neutral moving current loop is, to first order, an electric dipole. The spinning wall’s magnetisation is a density of such loops all moving round the axis, and their electric dipoles add a polarisation pointing along $\mathbf v\times\mathbf B$, the same direction as Lorentz’s.

The extra term is never larger than 1, however strong the magnetism, while Lorentz’s part grows with the permittivity. That is why the effect is easy to compute and hard to see. It is also why it was not obvious in advance: a theory that did not know a moving magnet is an electric dipole would never have looked for a dependence on permeability, because nothing in its picture of the material connected magnetism to polarisation.

The same dipole, met three other ways #

The electric dipole of a moving magnetic moment turns up wherever magnetism and motion meet, and it has been met under other names before.

The electron in an atom is one. An electron moving through the electric field of a nucleus carries its spin’s magnetic moment with it, and so carries an electric dipole $\mathbf v\times\boldsymbol\mu/c^2$ that sits in the nucleus’s electric field with an energy of its own. Viewed from the electron’s frame, the same energy is the interaction of its magnetic moment with the magnetic field of the nucleus moving past it. That energy is the spin–orbit coupling that splits atomic lines into doublets, and the spin that runs ahead of its own motion found that getting its size right needs a further relativistic correction, Thomas’s factor of one half, which comes from the electron’s frame turning as it goes round.

The second is the reverse case, a magnetic dipole that does not move sitting in an electric field. The momentum of something that is not moving found that such a system carries a momentum $\mathbf m\times\mathbf E/c^2$ in its electromagnetic field while nothing in it moves, and that the mechanical parts carry an equal and opposite “hidden” momentum that keeps the centre of energy still. Hidden momentum and the electric dipole of a moving magnet are the same relativistic fact seen from two sides: one is what a magnet in an electric field owes, the other what a magnet in motion shows.

The third is the experiment run backwards. Wilhelm Röntgen showed in 1888 that a disc of insulator spun between the plates of a charged capacitor deflects a magnetic needle: a moving polarisation is a current, and makes a magnetic field. Alexander Eichenwald measured the effect quantitatively in 1903. It is the companion of the Wilsons’ result: there a moving magnetisation looked like a polarisation; here a moving polarisation looks like a magnetisation, $\mathbf M = \mathbf P\times\mathbf v$. Both terms are needed for Maxwell’s equations in a moving medium to transform correctly between frames, and the pair together are what Minkowski’s relations encode.

Why the moving-medium equations needed relativity at all #

The fifth figure gives the sizes for a cylinder of laboratory scale. The voltage is proportional to the rate of turning and to the field, as the Faraday disc’s is; at fifty turns a second in half a tesla it reaches 89 millivolts on the relativistic account and 79 on Lorentz’s. Ten millivolts is not a small voltage. The difficulty is that it comes from an insulator. The cylinder, faced with metal on its two surfaces, is a capacitor of about fifty picofarads, and the whole signal is four picocoulombs of charge that any leakage path, any contact potential at the sliding contacts and any stray field from the magnet’s own electrical surroundings will compete with.

What the experiment tests is not a small correction to a well-known effect. The velocities involved are about ten metres per second, one thirty-millionth of $c$, and the relativistic correction to time or length at that speed is a part in $10^{15}$. Yet the extra term in the polarisation is not small at all: it is of order one in units of the effect being measured. The reason is that the effect being measured is itself first order in $v/c$, and the transformation of the fields has first-order terms in it. Relativity at these speeds does not show itself as tiny corrections to clocks. It shows itself in the structure of electromagnetism, which, as magnetism is electricity seen sideways argued from a wire, is already relativistic at the speed of a snail.

That is also why Minkowski’s version of the moving-medium equations was not a matter of taste. Before 1905 several sets of equations for moving bodies were on offer — Hertz’s, Lorentz’s, Cohn’s — and each fitted the experiments then done. They differed in what a moving magnetisation did and in how the fields in a medium transformed, and those were questions that had no principled answer until the transformation of the fields was known. Once it was, the constitutive relations of moving matter followed from those of matter at rest, with nothing left to choose. The spinning cylinder is a case where the choice that relativity forces could be checked against the one that seemed natural without it.

Where the first-order formula stops #

The formula for the displacement in a moving medium was derived to first order in $v/c$, and the full relations of Minkowski hold at any speed. At the speeds of a spinning laboratory cylinder the difference is a few parts in $10^{15}$ and the figures check that the first-order form reproduces the exact transformation to that order. The real limits are elsewhere.

The constitutive relations themselves assume a linear material that responds instantly and the same way in every direction. A rotating material is not quite that. Its parts are accelerating towards the axis, and the relations were written for a material in uniform motion; for slow rotation the material is treated as moving uniformly at each point with the local velocity, which is an approximation whose errors are of order $\omega a/c$ times the effect, utterly negligible here but not zero. Steel spheres in wax are also not a uniform material on the scale of a sphere, and the effective permittivity and permeability of such a composite are averages whose values had to be measured on samples, which added uncertainty of its own to the 1913 result.

There is a subtler point about the rotating frame. The argument goes to the frame of each small piece of the material, applies its relations there and comes back. No single inertial frame is the rest frame of the whole cylinder, and the electrodes and sliding contacts move too. The analysis survives because the electrometer is stationary and measures a potential difference between two conductors in a field that, in the laboratory, is static; the moving parts of the circuit run along the surfaces, where $\mathbf v\times\mathbf B$ has no component, so they add no electromotive force of their own. A circuit arranged differently — leads running radially and rotating with the cylinder — would read something else, which is the same lesson Faraday’s disc teaches about what a voltmeter in a moving circuit measures.

A smooth material with no molecules in it #

The figures draw the fields and voltages of a smooth material in steady rotation. They do not show the molecules being stretched or the magnetic moments being carried, and the picture of a moving current loop with unequal contractions on its two sides, which is the honest mechanism for the moving dipole, is not drawn at all; it is the content of a Lorentz transformation, not of a diagram of the cylinder. Nor do the figures show the experiment’s real difficulties: the contact potentials at the brushes, the charge leaking through the wax, the stray capacitance of the leads. Those decided how hard the measurement was, and the figures would look the same whether the Wilsons had succeeded or not.

Still open: what force a moving medium feels #

The constitutive relations of moving matter are settled; the force and momentum that light exerts inside a material are not entirely. The momentum light carries into glass found the long dispute between Minkowski’s and Abraham’s forms of the momentum of light in a medium, which turns on how the momentum is divided between field and matter rather than on any disagreement about the fields. The same division appears in a magnetised material moving in an electric field, and experiments with rotating and translating magnetoelectric media continue to test which division the forces on real materials follow. The spinning cylinder measures fields and voltages, which every consistent theory agrees on; what moves, and with what push, is where the remaining arguments are.

The habit worth carrying away is to ask in which frame a material law was measured. A spinning insulator in a magnetic field holds a voltage between its surfaces, and its size is the fraction $(\varepsilon\mu-1)/\varepsilon\mu$ of a metal’s because a moving magnetisation is also an electric polarisation — 89 millivolts against 79 for $\varepsilon = 6$ and $\mu = 3$ — where a theory that applies the material’s law in the laboratory finds $(\varepsilon-1)/\varepsilon$ and misses the magnet entirely. Maxwell’s equations need no change for moving matter; the material’s response has to be written where the material is at rest.