Illustrated Physics
physics, drawn
Electromagnetism

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.
15 min read 5 figures Fields, not forcesWhat stays the same

Assumes: Where the energy of a field actually is · The angular momentum that is in nothing at all

Put a current loop in an electric field and leave it there. Nothing moves: the loop is clamped, the field is static, the current is steady, and the whole arrangement will sit unchanged indefinitely.

The electromagnetic field around it is not free of momentum. The momentum density of a field is proportional to E×B\mathbf{E}\times\mathbf{B}, and near the loop neither factor is zero and they are not parallel, so the integral over space is not zero either. Something in this entirely static arrangement has momentum.

That is not by itself alarming — a field can hold angular momentum with nothing turning, and this looks like the linear counterpart. What makes it a problem is a theorem.

The theorem that forces the answer

A closed system whose energy is not going anywhere cannot have net momentum. The argument is one line of relativity: the total momentum of a system is its total energy times the velocity of its centre of energy, divided by c2c^2. A static system’s centre of energy does not move. Therefore its total momentum is zero — exactly, not approximately.

So if the field has momentum, the matter must have an equal and opposite amount. And the matter is not moving.

A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.
Fig. 1 A square loop of a hundred square centimetres carrying twenty amps, sitting still in a megavolt per metre. The carriers going up the field cross a hundred kilovolts on the way, so those in one wire are more energetic than those in the other by that much per unit charge. The current is the same all the way round, so the same number pass per second in each — but they carry different energy, and momentum is energy times velocity over c². The two wires contribute unequally.

The resolution is in the carriers. Momentum is not mass times velocity; relativistically it is energy times velocity over c2c^2, and the energy of a carrier includes whatever electrical potential energy it has picked up. A carrier crossing the loop against the field arrives at the far side with less kinetic energy than it started with, or more, depending on direction — so the top wire’s carriers and the bottom wire’s carriers are not equivalent.

The current is the same everywhere, which fixes the number crossing per second. What is not fixed is the energy each carries, and it differs by exactly the potential drop across the loop. Multiply: the momentum in the top wire and the momentum in the bottom wire differ by the current times the area times the field, over c2c^2. That is mE/c2mE/c^2, with mm the magnetic moment — the same magnitude the field has, in the opposite direction.

The name it goes by is hidden momentum, and the name is fair. Nothing in the picture is moving as a whole, no part of the apparatus is drifting, and the momentum is real in the only sense that matters: leave it out and momentum is not conserved.

What “hidden” is hiding from

The name is standard and slightly misleading, so it is worth saying precisely what is and is not concealed.

Nothing is hidden from the equations. The mechanical momentum of a current-carrying loop is the sum over its carriers of energy times velocity over c2c^2, and that sum is not zero for the reason above; anyone who computed it would find it. What it is hidden from is the free-body diagram — the picture in which a body is a mass at a point, its momentum is that mass times its velocity, and internal structure is irrelevant.

That picture is a good one for almost everything, and its failure here is a failure of a definite kind. The momentum of a composite body is the sum of its parts’ momenta, and for ordinary matter the internal motions cancel because the parts move equally in both directions. Here they do not cancel, because the parts moving one way carry more energy than the parts moving the other, and the difference is electrical.

So hidden momentum is a warning about a habit rather than an exotic effect. A body’s momentum is not its mass times its velocity unless its internal motions are symmetric, and the moment there is a steady circulation in an external potential, they are not. The same statement in the energy channel is more familiar: a hot body is heavier than a cold one, and the extra mass is internal kinetic energy. Hidden momentum is that observation applied to the momentum channel, where it is much less familiar and equally true.

Why only the area

How much of it there is, and how little the shape matters. The hidden momentum of a loop of 100 cm² carrying 20 amps, against the electric field it sits in. The line is m×E/c² and the marked points are the momentum computed a quite different way — by summing the electric potential round each loop's own perimeter, for 4 shapes of the same enclosed area. They agree, which is the statement that the answer depends on the area and on nothing else about the wire. The size of the effect is the reason it went unnoticed for a century: at the largest field air will hold, about three megavolts a metre, the momentum is 6.7e-12 kilogram metres a second — the momentum of a gram moving at 6.7e-9 metres a second. It is not small in the way that matters, though: it is exactly the field's, and without it momentum is not conserved.
Fig. 2 The hidden momentum against the applied field, with the line from m×E/c² and the marked points computed a different way — by summing the electric potential round each loop’s own perimeter, for four shapes of the same enclosed area. They agree. The size of the effect is why it went unnoticed for a century: at the largest field air will hold, it is the momentum of a gram moving at seven nanometres a second.

The general expression is a line integral: the hidden momentum is the current times the integral of the electric potential round the circuit, over c2c^2. Written that way it is obviously a property of the whole loop and not obviously a property of its enclosed area.

That it comes out proportional to the area is a vector identity, and the figure checks it by brute force — summing the potential segment by segment round four quite different perimeters scaled to the same area and comparing with the closed form. The four agree. So a long thin loop and a compact one carrying the same current in the same field carry the same hidden momentum, provided they enclose the same area, and a loop shaped like an L carries the same as a square.

That is the same reduction that makes a current loop’s magnetic behaviour depend only on its moment: a loop of any shape behaves like a needle with a dipole moment equal to the current times the area. The hidden momentum inherits that, and it is worth noticing why — both are consequences of the current being divergence-free, which is what makes the awkward parts of a line integral round a closed loop turn into a statement about the surface it bounds.

The size is the other thing the figure is for. The numbers are minute: at a megavolt per metre, which is a third of what air will stand, a substantial loop carries about 10−1210^{-12} kilogram metres a second. That is a millionth of the momentum of a mosquito. It is nevertheless exactly the field’s, and the argument is not about size.

How this was found

The effect was not derived from first principles and then looked for. It was found because a paradox refused to go away.

Shockley and James, in 1967, considered a static arrangement of a magnet and charges and noticed that it had field momentum, and that switching the magnet off would therefore have to set something in motion — the field momentum has to go somewhere. If the system were genuinely at rest and had no other momentum, the whole thing would spontaneously start moving when the current was turned off, which is not allowed. Their paper’s title asked where the momentum was, and the answer took a few years to become clear.

What makes the resolution satisfying rather than an epicycle is that hidden momentum has consequences beyond fixing the ledger. It resolves the corresponding force paradox — a magnetic dipole in a non-uniform electric field would otherwise appear to feel a force with no reaction — and it is why the two common expressions for the force on a magnetic dipole disagree until the hidden term is put in. It is also why an object with an internal current has an inertia slightly different from what its rest energy alone would give.

There is a general observation buried in that history. The paradox appeared because a conservation law was applied to a subsystem — the field, and the visible motion — rather than to everything. The same trap catches the flux rule, where a law with unstated hypotheses is applied to a machine that violates them, and it caught the angular momentum of the Feynman disc too. In each case the fix is to widen the accounting rather than to amend the law.

The scale, and why nobody tripped over it

The numbers are worth dwelling on because they explain a hundred years of the effect going unnoticed, and because they show that being small is not the same as being negligible.

Take the largest values an experiment can reasonably reach: a field of a few megavolts a metre, which is where air breaks down, and a loop with a magnetic moment of a few tens of ampere metres squared, which is a substantial electromagnet. The hidden momentum is about 10−1110^{-11} kilogram metres a second. A milligram would have to be moving at ten nanometres a second to match it.

Nothing measures that directly, and nothing needs to. The effect is not detected; it is required, by an argument that would otherwise predict something absurd — a static object spontaneously accelerating when a switch is thrown. Its role is like that of the displacement current before Hertz: a term put in because leaving it out makes the equations inconsistent, believed on that ground alone for some time, and only later given an independent existence.

The comparison is not exact and the difference is instructive. The displacement current made a prediction — electromagnetic waves — that could be looked for and was found. Hidden momentum makes no comparable prediction, because it exactly cancels something else by construction. What it does is make a family of calculations come out right, and its evidence is that family rather than any single experiment.

The field’s side of the ledger

The power enters a resistor through its sides. A 0.5 metre length of copper wire 1.6 mm in diameter carrying 8 A. Its resistance, from ρL/A with ρ = 1.68·10⁻⁸ Ω·m, is 4.18 mΩ, so it drops 33.4 mV and dissipates 267 mW. At the surface the electric field is axial and equal to V/L = 66.8 mV/m, and the magnetic field is circumferential and equal to μ₀I/2πa = 2.00 mT. Their cross product divided by μ₀ therefore points radially inward everywhere on the surface, with magnitude 106 W/m², and integrated over the lateral area of 25.1 square centimetres it comes to 267 mW — exactly I²R. The energy does not flow along the wire. It flows in through the sides, out of the field in the space around the wire. And the same power crosses every coaxial surface drawn: at 2 times the radius the flux density is 53.2 W/m², over an area 2 times larger. The wire is drawn thick for legibility; the numbers are not.
Fig. 3 Where the power dissipated in a wire actually enters it: through the sides, from the field around it, rather than along the wire from the battery. The Poynting vector points radially inward everywhere on the surface, and its integral over the surface is exactly the dissipated power. The same vector, divided by c², is the momentum density in the argument above.

The quantity doing the work in both halves of this essay is the same one. The Poynting vector E×B/μ0\mathbf{E}\times\mathbf{B}/\mu_0 is the energy flux; divided by c2c^2, it is the momentum density. So an arrangement in which energy is flowing anywhere at all has momentum somewhere, and the static case is not an exception but the strangest instance: the energy flow in a static loop-and-field arrangement is a closed circulation, going round and round, delivering nothing.

That closed circulation is the thing that most invites disbelief and is the most straightforwardly true. In the figure above, energy really does flow into a resistive wire through its sides — from the field, which is being maintained by the source — and the flow is perfectly steady. There is nothing paradoxical in a steady flux that goes in a circle; a system with one has a non-zero momentum density everywhere and a total that depends on the geometry.

What the circulation does mean is that the field’s momentum, unlike its energy, cannot be localised to the sources. It is spread over the region where both fields are appreciable, which for a dipole in a uniform field is the whole neighbourhood, and the total is a property of the arrangement rather than of either part.

Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.
Fig. 4 The angular counterpart: a charge inside a solenoid, with angular momentum stored in the field and nothing rotating. Switch the solenoid off and the charge begins to turn, carrying away exactly what the field held. The linear case in this essay is the same accounting without the rotation, and it needs an extra term the angular case does not — because a charge is free to start moving and a clamped loop is not.

Putting the two beside each other is instructive because they differ in one respect. In the rotating case, switching off the field sets the charge turning and everything balances with no hidden term at all: the field’s angular momentum is handed over to visible rotation. In the linear case, switching off the current would set the whole arrangement moving unless something else already carried the opposite momentum — and the difference is that a magnetic dipole’s momentum is stored in the very current that makes it, so it is destroyed by the same act that releases the field’s.

The force that would otherwise have no reaction

There is a second paradox the same term resolves, and it is the one that matters for calculation rather than for tidiness.

Ask what force a magnetic dipole feels in a non-uniform field and two standard expressions are available, differing by a term involving the rate of change of the electric field. For a static arrangement they disagree, and the disagreement is not academic: one of them predicts a force on a stationary current loop in a static electric field gradient, with nothing to push back.

The resolution is that the loop’s momentum is changing even though the loop is not accelerating, because the hidden momentum depends on the field the loop sits in and the field varies from place to place. A rate of change of hidden momentum is a force in the ledger without being a force anybody applies, and once it is included the two expressions agree and the reaction is accounted for.

That is worth stating as a general point about mechanical bookkeeping. A body whose internal state carries momentum has an equation of motion with an extra term in it, and the term is invisible in the free-body diagram because it is not attached to any external agent. The same structure appears whenever an object has internal circulation — which is why the mechanics of a rotating fluid-filled shell is harder than the mechanics of a rigid one, and why a spinning body’s response to a torque is not the response of a mass to a force.

Where the momentum is not

The energy density of a point charge, and its total. Above: the energy density ½ε₀E² around an electron's charge, against radius, on logarithmic axes. It falls as the fourth power — a measured slope of −4.00 — because it is the square of an inverse-square field, so over the 2 decades drawn it changes by 8. Below: the energy in all the field outside a radius, obtained by integrating that density over spherical shells, which agrees with q²/8πε₀a to better than a part in a million. It falls only as 1/a — a measured slope of −1.00 — and so has no limit as the radius goes to zero. It passes the rest energy 81.9 fJ, which is 0.511 MeV, at 1.409 fm. The classical electron radius 2.818 fm is exactly twice that, because it is defined without the half; outside it the field holds half the rest energy. Either way the whole of a point charge's energy cannot be its field, and no radius makes the sum come out.
Fig. 5 The energy density of a point charge’s field against distance, and the total inside a radius. Both diverge as the radius shrinks, which is what makes “the field’s energy” a quantity that has to be handled with a stated cutoff. The momentum density in this essay is the same kind of object: an integral over space, dominated by the region where the fields overlap, with no meaningful localisation to any part of the apparatus.

It is tempting to ask where the field’s momentum is, and the question is less well posed than it sounds.

The energy density of a field is already a quantity with this difficulty. Bring in a point charge and the total energy in its own field diverges — the integral of E2E^2 near a point charge does not converge — so any statement about “the energy of the field” is a statement with a cutoff in it, chosen for physical reasons that are outside electromagnetism. The momentum density inherits the same character.

For the loop in a uniform field the total is finite and unambiguous, because the divergent parts belong to each source separately and cancel out of the cross term. What cannot be done is to say which cubic millimetre holds it. The density is spread over the whole region where both fields are appreciable, it points different ways in different places, and the total is a modest number left over from large cancellations. That is the ordinary situation for field momentum and it is why arguments about it are usually settled by computing a total rather than by localising anything.

Where the model stops

The loop is treated as a circuit with carriers of definite energy, and a real conductor is not so simple. The derivation supposes the current is carried by a fluid whose energy per carrier tracks the local potential, which is right for a superconducting loop or a beam of charges and is a caricature of conduction in a resistive metal, where the carriers are scattering constantly. The result survives, because it follows from the energy–momentum accounting rather than from the transport picture, but the mechanism drawn here is a model of the mechanism.

Everything is static, and that is doing all the work. The centre-of-energy theorem applies to a closed system in a steady state. Switch anything and the argument’s premise fails: during the transient, the centre of energy genuinely moves, momentum genuinely flows between the field and the matter, and the neat cancellation is a statement about the endpoints.

The field is taken as externally given and uniform. In reality the charges producing it are part of the system, and they feel the loop’s field in turn; a complete treatment has to include them, and the resulting bookkeeping is where the long-running arguments about the momentum of a field in matter live.

And nothing here is quantum-mechanical. The mechanical momentum of a current loop in a field is a classical relativistic effect, and the corresponding statements about an atom or a nucleus with a magnetic moment need a different treatment, in which the moment is not a circulating current at all.

What the pictures cannot show

The loop figure draws two wires with labels and cannot show what is actually unequal about them. The number of carriers per second is the same, the drift speed is nearly the same, and the difference is in the energy each carries — a part in 101510^{15} or so for realistic numbers. Nothing about the picture can be drawn to scale, and a figure that showed the imbalance visibly would be a figure about a quite different regime.

Nor does any figure here show the momentum density, which is the thing the theorem is an integral of. It is spread through the space around the loop, largest where both fields are appreciable, and it points in different directions in different places; the total is a cancellation of a large positive against a large negative, and a picture of the total says nothing about where it came from.

Where the ladder goes next

This ladder began with where the energy of a field actually is and continued with the angular momentum that is in nothing at all. This rung asks what a static object has to be carrying so the books balance. The rungs after it: the stress tensor, which is what makes “the force on a region” computable as a flux through its boundary rather than a sum over its contents; the momentum of light in a medium, where two expressions differing by the square of the refractive index have each been experimentally confirmed and the resolution is about which part of the system is being counted; and the self-energy of a charge, where the same accounting produces a factor of four thirds that took fifty years to remove.

The habit worth carrying away is about where to stop drawing the boundary. A conservation law applied to part of a system is not a conservation law. When one appears to fail, the first move is to ask what has been left outside the box — and the answer is sometimes a quantity carried by something that is visibly doing nothing at all.

Part 3 of 4

This essay is one argument about Field energy. The others:

Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges. The angular momentum that is in nothing at all Part 2 — A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry. What the plane between them carries. The stress transmitted across the plane halfway between two charges of 10 nC held 1 cm apart, against distance from the axis in units of the half-separation. For two like charges the field on that plane lies entirely in it — checked here rather than assumed — so the plane sees only the pressure across the lines, the stress is negative everywhere, and the two halves are pushed apart. For opposite charges the field on the plane is entirely perpendicular to it, the plane sees only the tension along the lines, the stress is positive, and the halves are pulled together. Faraday's two words for a field line, tension along it and pressure across it, are exactly these two curves; the whole content of the tensor is that they are the same quantity, ε₀E²/2, wearing two signs. The force is what is left after integrating either curve over the plane, and both integrals come to the same magnitude — the Coulomb force — which is the next figure. The force read off a surface that touches nothing Part 4 — Draw any closed surface through empty space, measure the field on it, and the sum of one expression over that surface is the total force on everything inside — whatever the contents are, and without knowing anything about them. The expression is Maxwell's stress tensor, and it turns Faraday's guess about tension along a field line into an exact statement.

Where this goes next

The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all. Where the energy of a field actually is A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W. Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges. The angular momentum that is in nothing at all A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry. Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment. The force that does no work A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Centre of massConservation lawsElectric fieldField energyFour-momentumMagnetic momentMomentumMomentum conservationPoynting vectorRelativity