On "Hidden Momentum" of Magnetic Dipoles
| Scientific Paper | |
|---|---|
| Title | On "Hidden Momentum" of Magnetic Dipoles |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | Hidden momentum, momentum and energy of electromagnetic field |
| Published | 2005 |
| Journal | Apeiron |
| Volume | 12 |
| Number | 3 |
| No. of pages | 21 |
Read the full paper here
Abstract
This paper examines the problem of "hidden" momentum for quasi-static macroscopic systems. We have analysed a number of particular physical problems and shown that introducing of "hidden" momentum is strongly required to fulfil the energymomentum conservation law.
Overview
Kholmetskii's paper, published in Apeiron in July 2005, intervenes in a long-running controversy in classical electrodynamics: whether a stationary magnetic dipole sitting in an electric field carries a "hidden" mechanical momentum E × μ / c2. The concept was invented by Shockley and James in 1967 to rescue relativity from a paradox of their own devising, and it has been contested ever since — Comay, among others, argued that it is an artefact. Kholmetskii's position is unusual among dissident writers on electrodynamics in that he defends "hidden" momentum rather than attacking it, while simultaneously arguing that the more familiar conservation law based on the potential momentum qA is of restricted validity.
His starting point is that the standard relation dPM/dt = −Σqi dAi/dt is derived only for an isolated system of mechanically free charged particles. He notes two independent limitations on it: it is not gauge-invariant, so "the law of conservation of the canonical momentum has a restricted meaning"; and it is not Lorentz-invariant, since its four-vector generalisation yields a time-like component dEM/dt = −Σ qi dφi/dt which disagrees with the energy conservation law for free charges by a factor of one half. Once conductors and insulators carrying bound charge enter the system, he argues, the relation must be supplemented by a hidden-momentum term, and only then do momentum and energy conservation both survive.
The argument
Polarised conductors versus bound charges on insulators
Kholmetskii separates two cases that are often conflated. When the magnetising body is a conductor, it acquires polarisation surface charge σp in the external field, and the correct momentum balance simply requires adding a surface integral over the conductors. For a small conducting magnetic dipole that integral evaluates to (μ × E)/c2, which earlier authors christened "hidden momentum". On Kholmetskii's reading this is "a matter of terminology solely": one can always apply the surface-integral form directly "to get correct physical results without any reference to 'hidden momentum'".
The genuinely difficult case is a dipole made of bound charges fixed on insulators, where no polarisation charge is available and the extra term cannot be absorbed into a surface integral. That is the configuration in which he thinks hidden momentum is unavoidable.
The Shockley–James paradox
The paper reworks the original 1967 thought experiment. Two counter-rotating, oppositely charged insulating disks of radius r0, with net magnetic moment μ along z, sit a distance R ≫ r0 from a charge Q at rest. Mutual friction slows the disks to a stop over a time τ, slowly enough that radiation is negligible. The decaying vector potential induces an azimuthal electric field at the charge, and integrating the resulting force gives the charge a mechanical momentum PQx = −QA(R), exactly the potential momentum the system held before the magnetic moment was annihilated.
Since the disks' axis does not move, the centre of mass of "disks + particle" would appear to shift — a violation of special relativity. Shockley and James resolved it by positing a pre-existing hidden momentum Ph = (E × μ)/c2 stored as mechanical stress in the charged disks, which is released as ordinary translational momentum when μ decays. Kholmetskii checks that Phx = QA(R) and that the centre of mass therefore stays put.
He then recalls the theorem of Aharonov, Pearle and Vaidman, established for a classical model of the neutron: from ∂μTμν = 0 for a finite static configuration it follows that the total momentum vanishes, P = 0, while the electromagnetic part is PEM = E × μ/c2; hence the mechanical part must be PM = μ × E/c2 and is to be attributed to the dipole alone. The mechanism is explicitly model-dependent: for counter-rotating charged insulating disks the external field sets up mechanical stresses, and a Lorentz transformation of the stress-energy tensor converts stress into momentum density.
Charge orbiting a solenoid
The first of the two new problems is worked out in an appendix. A charge q orbits a long solenoid at constant angular frequency ω. Kholmetskii computes the Lorentz force on the solenoid due to the moving charge, integrating the field of the charge over each current loop and then over the length of the solenoid, and obtains Fx = −qvA/R, with the y- and z-components vanishing. In vector form F = −q(ω × A), so that F = −d(qA)/dt = −dPA/dt.
The point of the calculation is the absurdity it exposes if hidden momentum is denied. The orbiting charge experiences no net force apart from the external constraint holding it in its circle; yet the solenoid does experience a force, and that force rotates with the particle. A forced motion of the solenoid can do work, and "this work becomes infinite for infinitely long rotation of the particle". Introducing a hidden momentum in the solenoid — arising from stresses induced in its charged cylinders by the electric field of the moving charge, whose direction rotates as the charge orbits — supplies the equal and opposite reaction and closes the leak.
Magnetic dipole and a long charged wire
The second problem is the inverse of Shockley–James. A neutral magnetic dipole μ (again two counter-rotating charged disks) and a very long uniformly charged wire of linear density λ start far apart at x0, with x0 ≪ L. The wire is moved slowly to a distance h from the dipole, then μ is annihilated over a time τ. Integrating the force on the wire gives (PMw)x = 0 and (PMw)y = λμ/2πε0c2h = μE/c2, where E = λ/2πε0h is the wire's field at the dipole.
The revealing step is the assembly stage. Because the wire's velocity v and its electric field E are both along x, the product v × E vanishes everywhere on the dipole; by the field transformations the magnetic field of the moving wire is zero at the dipole, so the Lorentz force on the dipole is zero. Nevertheless, as the wire approaches, its field at the dipole grows, the pressure gradient inside the disks grows with it, and the hidden momentum increases. "Thus, in spite of the null Lorentz force, the dipole recoils." A second external force with a non-vanishing y-component must be applied to hold it in place, balancing the compensating force F′ = λμv/2πε0c2x2 applied to the wire; the total momentum transmitted to the system during assembly is then zero, as required.
Assessment
The paper's strength is that it does not argue about hidden momentum in the abstract. It picks two configurations in which the conventional bookkeeping visibly fails and shows what has to be added, and in the solenoid case it does the tedious integration rather than asserting the result. The appendix derivation of F = −d(qA)/dt from the Biot–Savart field of the orbiting charge, loop by loop, is a real calculation with a clean closed form, and the "infinite work" argument is the kind of consistency test that is hard to wave away: any account that leaves a permanently forced solenoid with no reaction has an energy source it cannot name.
The second example is genuinely instructive, and its logic is symmetric with the first. Kholmetskii's observation that the dipole recoils while feeling no Lorentz force — because the stress it carries changes as the wire approaches — makes the physical content of hidden momentum concrete: it is not an extra postulate but the momentum content of an elastically stressed body, obtained from the stress-energy tensor exactly as relativity requires.
The difficulties are of two kinds. First, the paper's central positive claim rests entirely on the counter-rotating-disks model of a magnetic dipole, and Kholmetskii himself flags, following Aharonov and colleagues, that "the manifestation of hidden momentum is model dependent". A dipole realised as an Ampèrian current loop, or as the intrinsic moment of an electron, is not made of stressed insulating matter, and nothing in the paper shows how the stress-to-momentum conversion is to be carried out for such a system. Since the theorem quoted is a general statement about static configurations while the mechanism offered is specific, the two are not on the same footing.
Second, the paper's treatment of its own key equation is uneven. It establishes that dPM/dt = −Σqi dAi/dt is neither gauge- nor Lorentz-invariant and then continues to use it in the Coulomb gauge and the non-relativistic limit, without spelling out how much of the subsequent momentum bookkeeping inherits those restrictions. The concluding claim that for mechanically free charges the relation "signifies a violation of Newton's third law in electromagnetic interaction" is asserted rather than derived here, and it sits awkwardly beside the paper's own demonstration that action and reaction are recovered in the quasi-static cases — Kholmetskii notes the tension with the accepted resolution of the Lewis–Tolman paradox but does not resolve it. Similarly, the final appeal to Graham and Lahoz's 1980 Nature measurement of angular momentum in a vacuum electromagnetic field is offered as experimental support for the rotational counterpart of the effect; no experiment bearing on the translational case is cited, and none is proposed.
Within its stated scope — quasi-static macroscopic systems of stressed charged matter — the argument is careful and the conclusion defensible. The paper does not claim more than that, and its value lies in the two worked paradoxes rather than in any general reform of electrodynamics.