The angular momentum that is in nothing at all

Mount a ring of charge on a disc that can turn freely. Down the axis, not touching, put a solenoid carrying a current. Nothing moves. There is a radial electric field from the ring, a magnetic field confined to the inside of the solenoid, and a total angular momentum of, by any ordinary accounting, zero.

Now switch the solenoid off.

The disc turns. The collapsing flux drives a circumferential electric field round the ring, the ring feels a torque, and when everything has settled the disc is rotating with an angular momentum of $q\Phi/2\pi$. Nothing outside the apparatus was touched. Either angular momentum was created, or it was there before the switch was thrown — and it was not in the matter, because nothing was moving.

The rung below, and what it established #

The energy of a capacitor can be booked in two ways: as work done moving charge, or as an integral of $\tfrac12\epsilon_0 E^2$ over the volume between the plates. Both give the same number.

That agreement is suggestive and not decisive. Two accountings that always agree are two accountings, and an economist would say the question of which is real does not arise. The angular momentum case is decisive, and the reason is worth naming before the arithmetic: here the two accountings disagree about the initial state, and the experiment resolves them.

What is in the apparatus before the switch #

The angular momentum density of an electromagnetic field is

$$\boldsymbol{\ell} = \epsilon_0\,\mathbf{r} \times (\mathbf{E} \times \mathbf{B}),$$

which is $\mathbf{r}$ crossed into the momentum density, exactly as it would be for matter. The momentum density $\epsilon_0 \mathbf{E} \times \mathbf{B}$ is the Poynting vector divided by $c^2$ — the same object that describes energy flow.

In the apparatus, $\mathbf{E}$ is radial from the ring and $\mathbf{B}$ is axial inside the solenoid. Their cross product is azimuthal — it circulates — and crossing $\mathbf{r}$ into it gives an axial angular momentum density. Integrating over the region where both fields are non-zero gives $q\Phi/2\pi$.

The ring sits where the magnetic field is zero, and that is the fact which makes the result surprising without making it wrong. A long solenoid’s field is uniform inside and very nearly nothing outside, so a ring placed around the outside is in a region with no magnetic field to speak of — and yet it acquires angular momentum when the current is switched off. Nothing local explains it, which is the difficulty; what explains it is a flux the ring encloses rather than a field the ring is sitting in.

What it depends on #

The stored quantity is $q\Phi/2\pi$: the charge, and the magnetic flux the ring encloses. It contains neither field strength at the ring.

That is the same structure as Gauss’s law, in which the flux through a surface counts what is enclosed and ignores where it is, and it should provoke the same warning. A quantity that depends only on an enclosed flux is not a quantity localised at the place the flux is measured. The angular momentum density is spread through the space between the solenoid and the ring, in the region where both fields are non-zero — which is a region the ring is at the edge of and the solenoid is at the centre of, and which contains no matter at all.

The numbers, and why nobody has noticed #

The apparatus above holds 3.2 × 10⁻¹⁰ kg m²/s, from a microcoulomb of charge and two milliwebers of flux. That is small — a gram-scale disc a few centimetres across would end up turning at a fraction of a revolution per minute — and it is the reason the effect is a thought experiment in most textbooks rather than a demonstration.

It has nevertheless been done. Graham and Lahoz built a version in 1980 with a cylindrical capacitor in an axial magnetic field, suspended as a torsion pendulum, and measured the impulse when the field was switched. The result agreed with $q\Phi/2\pi$ to the accuracy of the apparatus. Feynman, who set the problem in his lectures and did not resolve it there, was writing about something that was subsequently measured.

The smallness is worth putting in perspective by comparison with the energy. The field energy of the same solenoid is of order a joule, easily enough to notice; the angular momentum is a ten-billionth of an SI unit, which is not. There is no deep reason for the disparity — it is a matter of the units, and of the factor of $c^2$ that separates the momentum density from the energy density.

One electromotive force can be separated into its two mechanisms — a circuit that moves, and a field that changes — and only the second is at work here, because nothing in the apparatus moves until the field has already begun to collapse. The quantity transferred depends only on the change in the flux and not at all on how quickly the change was made, which is why the answer contains $\Phi$ and no time. Switch off slowly or quickly and the same angular momentum arrives.

The transfer, watched #

The strongest form of the argument is not the initial-state calculation but the accounting during the switch.

A changing flux drives a circumferential electric field, and that is the one law in the subject with a rate of change in it — and it is really two laws wearing one coat, of which only the transformer half applies here, since nothing in the apparatus moves.

A changing magnetic field makes an electric one, and the shape of what it makes is the whole answer. The induced electric field goes round rather than out — it circulates — which is precisely the shape needed to torque a ring rather than to push it across the room. So the transfer is not mysterious once the field is allowed to hold the angular momentum in the first place; what is mysterious is only where it was beforehand.

The torque on the ring is the charge times the induced field times the radius, and the induced field is $-(1/2\pi R)\,d\Phi/dt$, so the radius cancels and the torque is $-(q/2\pi)\,d\Phi/dt$. Integrating over the switch-off gives exactly $q\Phi/2\pi$, whatever the shape of the switching curve.

That is the calculation the opening figure runs, and its two curves come from different places: one from the field’s own density integral, one from a torque integrated in time. They sum to a constant because angular momentum is conserved, and the constant is the initial field value.

What the disc ends up with is angular momentum of the entirely ordinary sort: a mass distribution and a rate of turning. The conserved quantity is the same one that survives a change of shape, so the transfer has to balance — and it is worth checking that it does, because a bookkeeping scheme that produced angular momentum from nowhere would be worse than the puzzle it was invented to solve.

Where the momentum sits, and the objection to it #

The obvious objection is that the region between the solenoid and the ring has zero magnetic field, so it cannot hold an angular momentum whose density contains $\mathbf{B}$.

It is true and it does not help. The trap is the familiar one about reading a field line as a thing in a place: the density is defined pointwise, its integral is what the conservation law is about, and where a diagram shows nothing there may still be something to integrate. A long solenoid’s field is not exactly zero outside — the field lines that go up the inside come back somewhere — and the integral of the angular momentum density over the outside region is what the calculation returns. In a genuinely infinite solenoid the return field is spread over infinite space at vanishing strength, and the integral remains finite and equal to $q\Phi/2\pi$, which is the sort of limit that has to be taken carefully once and then trusted.

There is a second and more interesting way to see it, which does not depend on the return field at all, and which is worth a paragraph because it changes what the answer is about: the vector potential. The angular momentum is $q$ times the circulation of $\mathbf{A}$ round the ring divided by $2\pi$, and $\mathbf{A}$ is non-zero outside a solenoid even where $\mathbf{B}$ is not. That is the same quantity that appears in the Aharonov–Bohm effect, and the two results are the same fact: a charged particle outside a solenoid is affected by the flux inside it, mechanically here and in the phase of its wavefunction there.

The temptation at that point is to say the vector potential is therefore the real object and the fields are derived. It is worth resisting for the same reason the field-line warning exists. The vector potential is not unique — a gradient can be added to it anywhere without changing anything measurable — and what appears in both results is its circulation round a closed loop, which is gauge-invariant and equal to the enclosed flux. So the honest statement is neither that the field is here nor that the potential is here, but that a loop integral is the quantity with physical content, and no smaller object is.

The same thing without a paradox #

Field momentum is not an exotic quantity summoned to rescue this apparatus. It is measured routinely.

Nobody finds the travelling version strange. Light pushes, and the push is the field’s momentum being delivered — the identical $\varepsilon_0\,\mathbf{E}\times\mathbf{B}$, moving this time rather than sitting still in a static arrangement. The static case feels paradoxical only because nothing appears to be happening, and the resolution is that the same quantity is present whether or not anything is happening.

What makes the disc surprising is only that the fields are static. The same object appears in every travelling wave, where the two fields are perpendicular to each other and to the direction of travel and their cross product points the way the energy is going. A beam of light obviously has momentum because it obviously goes somewhere; a pair of crossed static fields obviously has nothing to do with motion. The formula does not distinguish the two cases, and the experiment agrees with the formula.

Two more configurations that hold something #

The disc is the famous case and it is not the only one, and the others are worth listing because they make the general rule visible.

A charge beside a magnet holds linear momentum. Put a point charge next to a bar magnet, both at rest. The crossed fields have a non-zero $\mathbf{E} \times \mathbf{B}$ integrated over space, so the pair holds linear momentum, in a direction perpendicular to both the field of the charge and the moment of the magnet. Move either one away slowly and the momentum has to go somewhere; it goes into the matter, and the two objects recoil in opposite directions with no force ever having acted between them along the line joining them.

A capacitor in a magnetic field holds it too, which is the same statement with the charge spread over two plates, and this version can be built on a bench. The measurement is difficult because the momentum is tiny, and it has been done.

What matters for every one of these configurations is the flux enclosed, and not the field where anything material is sitting. That is why the answer for the disc contained a flux and no local field strength at all, and it is the honest statement of what “the angular momentum is in the field” means: not in the field at the ring, but in the arrangement as a whole.

The pattern is that any configuration with both an electric and a magnetic field present in the same region holds momentum, angular momentum, or both. That is nearly every configuration in the subject, and it goes unnoticed because the stored amounts are small and because nothing usually changes.

The torque a beam of light exerts #

The disc apparatus was measured once, in 1980, with difficulty. The same claim about field angular momentum had been confirmed forty-four years earlier in a form that is easier to arrange, and the experiment is worth describing because it makes the quantity concrete.

Circularly polarised light carries angular momentum along its direction of travel. In the classical description this comes straight out of the density above: the electric field rotates rather than oscillating in a plane, so $\mathbf{E}\times\mathbf{B}$ has a component that circulates about the beam axis. In the photon description it is $\pm\hbar$ per photon, with the sign given by the handedness.

A quantity of angular momentum arriving per second is a torque, and it can be caught. Beth, in 1936, suspended a half-wave plate on a fine quartz fibre and passed circularly polarised light up through it. A half-wave plate reverses the handedness, so each photon leaves with the opposite angular momentum from the one it arrived with, and the plate must absorb the difference — $2\hbar$ per photon.

The numbers explain why this is a hard experiment rather than a demonstration. The angular momentum flux in a beam of power $P$ is $P/\omega$, so a watt of green light carries about $3\times10^{-16}$ newton-metres. Beth reflected the beam back through the plate a second time to double it, used a resonant torsion pendulum to accumulate the effect over many periods, and reversed the handedness at the pendulum’s own frequency so the signal was a driven oscillation rather than a static deflection. The measured torque agreed with the prediction in magnitude and in sign.

There is a second kind of optical angular momentum, found much later. A beam whose phase front is a helix rather than a plane carries orbital angular momentum of $\ell\hbar$ per photon, where $\ell$ is how many times the phase wraps round the axis — and unlike the polarisation contribution it has no upper limit. Beams with $\ell$ in the hundreds have been made. Focused onto a small absorbing particle, such a beam spins it, which is an optical spanner in the same sense that a focused Gaussian beam is optical tweezers.

Both are the same $\epsilon_0\,\mathbf{r}\times(\mathbf{E}\times\mathbf{B})$ integrated over a beam instead of over a static apparatus. What the disc paradox adds is only that the integral does not need anything to be travelling.

The momentum that is in the matter after all #

The limitation named above under hidden momentum deserves working through, because it is the one place where naive field bookkeeping gives a wrong answer, and the wrongness is instructive.

Take a small current loop — a magnetic moment — and place it in a uniform electric field. The fields overlap, so by the argument of this essay there is momentum in the field, and the integral gives $-\epsilon_0\mu_0\,\mathbf{m}\times\mathbf{E}$, which is not zero.

Nothing is moving, and nothing is going anywhere. So a completely static arrangement appears to have a net linear momentum — and unlike the angular momentum case, that is a contradiction rather than a surprise. A system whose energy distribution is not moving must have zero total momentum; that is a theorem, and it follows from the same relativistic accounting that makes energy and momentum two parts of one object.

The resolution is that the loop’s own carriers carry momentum that does not cancel. Charges moving in the direction of the electric field are being accelerated and those moving against it decelerated, so at any instant the ones going one way round are faster than the ones going the other. In a non-relativistic treatment that makes no difference: the current is the same everywhere round the loop, so the slower carriers must be denser, and the two effects cancel in the momentum. Relativistically they do not, because momentum is $\gamma m v$ rather than $mv$, and the mismatch leaves a net mechanical momentum in the wire equal and opposite to the field’s.

Shockley and James found this in 1967 and it is called hidden momentum — hidden because it is carried by matter that is visibly not going anywhere. The total is zero, the theorem is satisfied, and the field momentum is entirely real; it is simply not the whole account.

What this changes about the essay’s main argument is nothing, and it is worth saying why. The disc holds angular momentum, and the charge on its ring is static — there is no current, so there are no carriers to carry a hidden anything. That is why the configuration is the clean one and why Feynman chose it. But the general lesson is a caution about the method rather than about the result: a field integral is a correct account of what the field holds and is never by itself a complete account of the system, and the check that catches the omission is a conservation theorem rather than a more careful integral.

Where the model stops #

Hidden momentum is real and is not this. A current loop in an electric field has a linear momentum that is not in the field but in the charge carriers, arising because they are faster where the potential is lower. Any complete accounting of a system with currents and fields must include it, and leaving it out produces a violation of the centre-of-mass theorem. The angular momentum case above is clean because the ring’s charge is static, and the general case is not.

The switch-off is treated as slow. Everything above assumes the disc’s rotation stays negligible during the transfer and that no radiation leaves. A fast switch radiates, and the radiated field carries away angular momentum of its own, so the disc gets less than the full $q\Phi/2\pi$.

The solenoid has been idealised. A real one has ends, a return field, a resistance and an inductance, and switching it off dissipates a stored magnetic energy that has nothing to do with the angular momentum. Separating the two is straightforward and it is not automatic.

And the field’s angular momentum is not always a rotation. Light can carry angular momentum in two distinguishable ways, one from polarisation and one from a helical phase front, and the classical density above does not naturally separate them. The separation is well defined for a paraxial beam and controversial in general, which is a live subject rather than a settled one.

What the pictures cannot show #

The angular-momentum figure plots two curves against time and gives no indication of where the field’s share is. It is spread through a volume, weighted by $r \times (E \times B)$, and most of it is close to the solenoid where the return field is strongest — which is nowhere near the ring that ends up turning. No plot against time carries that.

Nor can any figure here show the initial state as anything but static. That is the whole difficulty: the drawing of the apparatus before the switch is a drawing of nothing happening, and the claim is that something is nevertheless stored. A picture cannot distinguish a stationary state with angular momentum from a stationary state without one, and the only way to tell is to change something and watch.

Where this ladder goes next #

The rung below established that the field can be given an energy consistently. This one establishes something stronger: that the field’s conserved quantities are not merely a consistent alternative accounting but the only accounting that works, because there is a configuration in which the matter’s books are empty and the total is not.

The habit worth carrying away is a test for whether a description is a description or a device. Look for a state in which the two accountings disagree. Where the field and the action-at-a-distance pictures always agree, the choice between them is aesthetic. Where a quantity has to be somewhere during a delay — during a switch-off, during a propagation time, between an emission and an absorption — only the local description can say where, and the other one has a conservation law failing for the duration.

What is left on this ladder is the stress in the field: the fact that the same object that carries energy and momentum also carries a tension along the lines and a pressure across them, from which the force between two magnets can be computed by integrating over a surface that encloses neither of them.