Ampere: The Avis Phoenix of Electrodynamics
| Scientific Paper | |
|---|---|
| Title | Ampere: The Avis Phoenix of Electrodynamics |
| Read in full | Link to paper |
| Author(s) | Jorge A Guala-Valverde, Ricardo A Achilles |
| Keywords | Field, distant action, relative motion, torque |
| Published | 2008 |
| Journal | Apeiron |
| Volume | 15 |
| Number | 3 |
| No. of pages | 11 |
| Pages | 314-324 |
Read the full paper here
Abstract
Recent definitive experimentation unveils on Newtonian basis- standard field theory limitations in explaining homopolar torque production.
Overview
This short essay by Jorge A Guala-Valverde and Ricardo A Achilles, published in Apeiron in July 2008, argues that the homopolar (Faraday) motor is a decisive test case separating two rival force laws of Electrodynamics: André-Marie Ampère's 1827 law for the interaction of two current elements, and Hermann Grassmann's 1845 law, the ancestor of the modern Biot–Savart plus Lorentz Force treatment. The title's "Avis Phoenix" is the authors' image for Ampère's law rising again from the ashes of a century and a half of neglect.
The two laws are known to agree for the force exerted by a complete closed circuit on a current element, which is why the difference is normally dismissed as unobservable. The authors' claim is that homopolar machines break exactly that condition: the relevant interaction there is between a part of an ohmic circuit — the closing wire — and an individual Ampèrian magnetizing current element inside the magnet. In that configuration, they argue, the Grassmann–Biot–Savart route predicts zero reactive torque on the magnet, while experiments performed by the authors and collaborators between 1995 and 2002 show the magnet does turn. Their conclusion is that "we must incorporate Ampère's force law in energy-conversion physics" — a stronger claim than the usual dissident complaint about Newton's Third Law, since it is asserted to have direct experimental consequences.
The argument
The homopolar apparatus
The machine has three essential parts: a permanent magnet producing a uniform B field; a radial conductive bar (RB) free to rotate about a vertical conductive shaft, with sliding contacts on the shaft and on a metal ring at the magnet's rim; and a closing wire (CW), also on sliding contacts, completing the loop. The shaft is pointed at both ends and centred in a conical bearing so that rotation is nearly frictionless. A DC emf source is inserted in the shaft, driving an ohmic current IΩ.
Two configurations are compared. In Case 1 the magnet is at rest in the laboratory; with a centripetal current injected, the radial bar turns counter-clockwise and the closing wire turns clockwise. In Case 2 the radial bar is rigidly attached to the magnet; bar and magnet then rotate together counter-clockwise while the closing wire again turns clockwise.
The torque bookkeeping
For Case 1 the authors identify two active torques — τm,rb exerted by the magnet on the bar and τm,cw exerted by the magnet on the closing wire — which are experimentally equal and opposite, τm,rb = −τm,cw. Each wire then produces a reaction torque back on the magnet, τrb,m = −τm,rb and τcw,m = −τm,cw. The authors stress that the dominant interaction is magnet-on-wire; wire-on-wire coupling is negligible.
Case 2 is the argument's fulcrum. Bolting the bar to the magnet removes any relative motion between them, so that pair cannot function as a machine; the equal and opposite pair τrb,m and τm,rb now act on one rigid body and cancel. What remains is the magnet–closing-wire pair, still mechanically decoupled and still in relative motion. The authors therefore identify τcw,m — the torque exerted by the closing wire on the magnet — as the sole cause of the observed magnet rotation, a fact they say has been "ignored since Faraday's days."
Ampère's law versus Grassmann's
Ampère's force between elements IΩdlΩ and IAdlA is written in the paper as
d2FΩ,A = −d2FA,Ω = −(μ0/4π)(IΩdlΩIAdlA/r2)(2cos ε − 3cos α cos β),
a central force along the line joining the elements, and hence obedient to Newton's third law in its strong form. Grassmann's force, dFG = IΩdlΩ × B, is a cross product and therefore always perpendicular to the element it acts on.
The magnet is modelled, following Ampère, as a set of virtual peripheral magnetizing currents. The authors then compute the reactive torque on the magnet by taking the field dBΩ that the ohmic element creates at a virtual magnetizing element via Biot–Savart's law, and applying Grassmann's expression to that element. Because the resulting force is at right angles to IAdlA — which runs along the magnet's periphery — it has no component capable of turning the magnet about its own axis. Hence τGcw,m = τGrb,m = 0. This the authors call "a physical nonsense, in flagrant contradiction" with the experimental torque bookkeeping.
Ampère's law, being free of the orthogonality constraint, permits a tangential component on the virtual current element and so delivers a non-zero reactive torque, in agreement with the experiments. The authors note that where the interaction is that of a closed peripheral magnetizing current on a discrete circuit segment — the active torque on the radial bar — the two formulations agree, and the result is the elementary one given by James Clerk Maxwell:
τm,rb = ∫0R r × (IΩdr × B) = ½ R × (IΩR × B).
Wider claims
The closing section places the result in a Newtonian-electrodynamics lineage. The authors quote Andre K T Assis to the effect that Ampère's force was abandoned not on evidence but because it is incompatible with the Lorentz force on which Special Relativity was built, whereas Grassmann's force is compatible. They point to Weber's 1846 force law — expressed in charges and the zeroth, first and second time derivatives of their separation — as the charged-particle extension of Ampère's law, and to its use in relational mechanics as "the first rigorous implementation of Mach's Principle." They credit Thomas E Phipps with reviving Heinrich Hertz's Galilean-invariant form of Maxwell's Equations in total rather than partial time derivatives, a programme shared by Mario J Pinheiro, and endorse Assis's call for a Weber-based alternative to the magnetohydrodynamic equations of Plasma physics. J. D. Jackson's remark that fields "decouple conceptually the sources from the test bodies" is singled out as a misjudgement, since in homopolar machines the same B pattern behaves differently according to the motional state of its source.
Assessment
What is genuinely valuable here is the sharpness of the target. The Ampère–Grassmann equivalence theorem is real but conditional — it holds for the force of a closed circuit on an element — and the authors correctly identify that the homopolar geometry, where one asks for the torque exerted by a circuit segment back on the magnet's internal currents, sits outside the theorem's scope. Choosing an experiment on that boundary is exactly the right instinct, and the underlying laboratory work (published in the American Journal of Physics and Physica Scripta as well as Apeiron) is real, careful and reproducible.
The paper's arithmetic, so far as it goes, is correct. Equation (4) is the standard Ampère force law, with the correct (2cos ε − 3cos α cos β) angular factor and the correct μ0/4π prefactor. The Maxwell result for the bar is also right: for a radial bar of length R carrying current I in a uniform axial field B, the elementary torque is rIB dr, and integrating from 0 to R gives IBR2/2, which is exactly the magnitude of ½ R × (IR × B). There are no numerical results in the paper to mis-convert; it is an argument from geometry, not from measurement values.
The difficulties are in the inferential steps rather than the algebra. First, and most importantly, the zero-torque result attributed to the Grassmann–Biot–Savart route is obtained by modelling the magnet as purely peripheral Ampèrian current loops and then observing that a force perpendicular to a peripheral element cannot turn the magnet about its axis. That conclusion is an artefact of the model, not of Grassmann's law: a real magnetization distribution has bound currents throughout its volume, and the standard field treatment of magnet reaction torque proceeds through the total electromagnetic momentum and stress rather than through a hand-picked ring of virtual currents. The paper asserts the modelling choice in a single figure and never tests its sufficiency, so the claimed "physical nonsense" may be a nonsense of the idealization.
Second, the argument is framed as showing that field theory cannot explain the observed rotation, but no alternative quantitative prediction is offered — no computed value of τcw,m from the Ampère integral (9), and no comparison with a measured torque. Equation (11) sidesteps the integration entirely by invoking the experimentally-established equalities (1) and (3), which means the Ampère law is being used to argue that a non-zero result is permitted, not to predict its size. A discriminating experiment requires a number that the two laws disagree about; what is presented is a qualitative zero-versus-non-zero contrast that depends on the modelling assumption just noted. Cavalleri et al. (Phys. Rev. E 58, 2505) claimed a null result against exactly this class of Ampère-tension effects, and the paper cites the dispute without engaging its numbers.
Third, the Weber-force and Mach's-principle material in Section 4 is quotation and programme rather than argument; nothing in the homopolar analysis depends on it, and no reader should take it as having been supported by the experiment described. The paper is best read as a well-aimed conceptual challenge with a real experimental core, whose decisive step — the vanishing of the Grassmann reactive torque — rests on an idealization the authors do not defend.