Action-at-a-Distance: A Key to Homopolar Induction
| Scientific Paper | |
|---|---|
| Title | Action-at-a-Distance: A Key to Homopolar Induction |
| Read in full | Link to paper |
| Author(s) | Jorge A Guala-Valverde, Ricardo A Achilles |
| Keywords | Homopolar Induction |
| Published | 2007 |
| Journal | Apeiron |
| Volume | 14 |
| Number | 3 |
| No. of pages | 15 |
| Pages | 169-183 |
Read the full paper here
Abstract
The crucial character of relative motion and Ampeers force law in interpreting homopolar induction was pointed out in recent experimentation performed by us. H. Montgomery suggests the compatibility of that experimental results with Maxwell's field theory. With the purpose of elucidating the applicable rationale this article identifies three independent energy-conversion mechanisms definable within the basic homopolar-machine frame and, hinging on a specially developed finite-element software, introduces an Amperian analysis of associated electro-and ponderomotive effects.
Overview
This 2007 Apeiron paper by R. Achilles and J. Guala-Valverde of the Applied Physics Group at Confluencia Tech University and the Fundación J. Palacios in Neuquén, Argentina, is a follow-up to the authors' long experimental campaign on the homopolar (Faraday-disk) generator. Their earlier experiments had shown that what matters in homopolar induction is the relative motion between the machine's material parts — magnet, disk and external conductor — rather than motion with respect to an abstract field filling space. H. Montgomery had replied that those results are equally compatible with Maxwell's field theory. The present paper is the authors' answer: rather than argue verbally, they build a finite-element numerical model of the machine from Ampère's original force law and compare its predictions, part by part, with those obtained from the Grassmann (Biot–Savart–Lorentz) law that underlies field theory.
The departure from the mainstream account is a matter of ontology as much as of arithmetic. Standard treatments assign the magnetic field B to the space surrounding a magnet, treat it as "constant" and unattached to the magnet body, and compute forces on moving charges via the Lorentz Force. Achilles and Guala-Valverde instead treat the magnet's magnetisation as an equivalent ring current made of real current elements, and let those elements interact directly and instantaneously with the disk's current elements through Ampère's Newtonian force law. The result is a machine in which the magnet is a full participant: it receives reaction torques and develops its own electromotive forces. In the Grassmann description it receives neither, and the authors argue this is where field theory quietly fails Newton's third law, energy conservation and the Faraday–Lenz law at once.
The argument
Grassmann versus Ampère
The paper opens by setting the two competing elementary force laws side by side. Ampère's 1822 expression gives the force between two current elements imdm and indn separated by rmn as proportional to (2 cos ε − 3 cos α cos β)/r2mn, where ε is the angle between the two elements and α, β are the angles each makes with the line joining them; a negative value means attraction. This force lies along the line joining the elements, so it obeys Newton's third law element by element. Grassmann's 1845 law, by contrast, gives each element a force built from a double vector product — the element crossed into the field created by its partner — so the two forces are generally not collinear and not equal and opposite. The authors note that this unsymmetrical law, generalised to free charges, is precisely what became the magnetic term of the modern Lorentz force, and that its adoption is why Ampère's law "became in such scenario disregarded".
The clearest contrast is drawn in a polar plot (their Figure 2) of the force on an element carried parallel to itself around a circle centred on a second, stationary element, sampled every 10° of the element-distance angle α. Grassmann's elementary force keeps an invariant horizontal direction throughout, which is the visible signature of its non-Newtonian character. Ampère's forces, referred to the Grassmann magnitude, are: attractive and doubled at α = 90°; zero at α = cos−1(2/3)1/2 = 35.3°; and repulsive with half the α = 90° magnitude at α = 0°, where the Grassmann force vanishes altogether. The paper concedes that integrating Grassmann's law around closed circuits cancels the unbalanced components, so that at the whole-circuit level the third law is recovered; but it insists the element-level differences are what "fire up existing homopolar-induction controversies", and points to the longitudinal forces confirmed by Ampère's bridge and, more recently, by wire-traction and breakage measurements and rail buckling in electromagnetic guns, work associated with Peter Graneau and others.
Three machines inside one machine
The paper's structural contribution is to dissect the homopolar frame — a conducting disk on a shaft, a coaxial cylindrical magnet supplying axial B, and an external conductor touching the disk at an inner and an outer radius — into three distinct energy-conversion mechanisms. H1 is the magnet–disk machine, which carries the greatest mutual inductance and the highest conversion rate. H2 is the magnet–conductor machine. E1, the disk–conductor pair, is a weaker electrodynamic machine able to operate only as a motor. Anchoring any two parts together disables the corresponding mechanism and hands torque and EMF production to the remaining subsets — which is exactly the experimental protocol the authors had used in their earlier work, now given a systematic name.
For H1 the two operating modes are treated separately. In motor mode (M-mode) a centripetal current im is injected into the disk from a circuit anchored to the magnet, and a tangential force turns the disk. In generator mode (G-mode) the disk is turned mechanically at angular speed ωm, and the radial EMF follows from the integral of ωm B along the radius. The model is formulated in terms of three currents: the magnet's spin-orientation equivalent ring current in, the injected radial current im, and the tangential charge-displacement current im' representing the disk's rotating free charge.
The finite-element computation
Because Ampère detected the proportionality Force ∝ Current2, the authors can normalise magnet and disk to unit radius. The magnet periphery is divided into 360 circumferential elements of π/180 length; the disk radius is divided into 80 increments. The element separation is r2mn = m2 + 1 − 2m cos γ, and the M-mode and G-mode force increments follow from the Ampère expression with the appropriate angle substitutions. The equivalent magnet current in is obtained from the field at the centre of a current loop, B = μ0in/2R; the displacement current im' is estimated from half the highest disk charge density the dielectric strength of air (3.0 × 106 N C−1) will permit, carried at the local rotational speed.
The analysis is deliberately coplanar — defensible, the authors argue, for the confined-field machines industry cares about, and governed by the same radial and tangential Ampère components as the axially separated case. Coplanarity does, however, create a singularity at m = 1, n = 0, where rmn, Δm and Δn all vanish together. Where Paul Wesley had removed it with current-density arguments, Achilles and Guala-Valverde evaluate the limit from each side with Δm = Δn, giving elementary attractions of magnitude 3μ0imin/4π in M-mode and μ0im'in/4π in G-mode at the n = 0+ side, with matching forces of the opposite sense at n = 0−. These turn out to be the largest elementary forces in the machine. Net torques and EMFs are then obtained by integrating the tangential components; the identity m sin α = cos α' = cos β, which follows from the geometry, is what guarantees that elementary and global torques satisfy Newton's third law in both modes. The whole scheme was implemented as a Fortran program, "H-Mac", with a graphical display of the incremental force distribution.
Results
The test case is a 1.0 T magnet with a 20 cm radius, 2 mm thick copper disk turning counterclockwise at 50 r/s for generator operation, and injected with 10 A of centripetal current for motor operation, evaluated in open-circuit G-mode and blocked-rotor M-mode — the standard conditions for testing conventional machines. The plotted force distributions show no tangential forces in the open-circuit generator (they appear once the external circuit is closed) and no radial forces in the blocked-rotor motor (they appear, producing counter-EMF, once the disk is released). Ampèrian generator radial forces and motor tangential forces and torques balance across disk and magnet, and both the G-mode magnetizing and M-mode demagnetizing magnet EMFs act to hold machine voltage and speed constant, in accord with Faraday–Lenz. Under Grassmann's law, by contrast, the magnet feels no force, no torque and no EMF in either mode — which, the authors conclude, "precludes the compliance of Newton's third law, energy conservation and Faraday-Lenz law by this latter theory".
Two engineering consequences are drawn from the force maps. Because the peak forces sit at the m = 1, γ = 0 singularity, making the disk radius larger than the magnet radius should raise machine power sharply by exploiting the m > 1 region. And because the force falls steeply away from the singularity in both the radial and peripheral directions, annular disks with multiple radial circuits — multiplying the number of singularities — are proposed as a route to a high-power homopolar machine. The paper closes by condensing the machine's physics into a Novak concept map, in which the tangential-force / magnet-EMF / magnet-current loop is simply absent for the Grassmann formulation, and by flagging a dynamic model with load resistance, inductance, inertia and friction as future work.
Assessment
The genuinely valuable part of this paper is the taxonomy. Splitting the homopolar frame into H1, H2 and E1 and noting that anchoring two parts kills the corresponding mechanism gives a clean vocabulary for a device whose literature has been muddled for a century by arguments over "whether the field rotates". It also converts the authors' earlier bench results into a testable bookkeeping scheme rather than a debating point. The numerical implementation is likewise a real step: rather than asserting that Ampère's law predicts a reaction on the magnet, they compute where on the magnet's periphery and on the disk radius the forces sit, and the resulting design suggestions — oversized disks, annular multi-circuit disks — are falsifiable by anyone willing to build them. That is a more constructive use of Ampère's law than most of the action-at-a-distance-versus-field polemic.
The difficulties are serious, and most of them concentrate on the singularity. The paper's headline claims — that the largest forces in the machine sit at m = 1, γ = 0, and that machine power can be radically increased by exploiting that region — rest entirely on a limit the authors themselves acknowledge is undefined, and which they evaluate by the ad hoc assumption that Δm = Δn as both tend to zero. The result of such a limit generally depends on the path taken, and no argument is offered that this particular path is physically preferred; Wesley's current-density treatment, dismissed in a clause, exists precisely because a filamentary idealisation breaks down where the separation falls below the conductor dimensions. Since these elementary forces are described as "the uppermost" and as exerting "an important impact on machine power calculations", a large part of the quantitative content of the paper inherits that arbitrariness. Similarly, the disk displacement current is fixed by taking half the maximum surface charge the dielectric strength of air will support — a bound, not a measurement, and one with no evident connection to the actual charge distribution in a rotating copper disk.
The central theoretical claim also needs qualification. The paper concedes, correctly, that Grassmann's law integrated over closed circuits satisfies Newton's third law; the asserted failure of energy conservation therefore does not follow from the element-level asymmetry alone. What the authors have actually shown is narrower: that in their model, where the magnet is represented by an equivalent ring current and the Grassmann calculation is stopped before the magnet's own circuit is closed, no reaction appears on the magnet. Standard field theory answers this by locating the reaction in the field's momentum and stress rather than in a co-located current element, and the paper does not engage that answer — Montgomery's objection, which is the stated occasion for the article, is never actually addressed on its own terms. Nor is the equivalence of a magnetised body to a filamentary peripheral loop defended; for the ferromagnetic magnet used, that equivalence holds for the exterior field but not necessarily for the near-singularity region on which everything here depends.
Finally, the paper stops short of the comparison that would settle matters. Ampère's and Grassmann's laws agree on all closed-circuit net quantities, so the two theories are distinguishable only through the distribution of forces within the material — and the paper computes exactly that distribution but reports no measurement of it. The 1.0 T, 20 cm, 50 r/s test case produces numbers with no measured counterpart, and the promised torque on the magnet, which is the whole discriminating prediction, is here asserted from the model rather than weighed against a torsion-balance result. Given that the authors' own earlier experimental work is what made this line of argument interesting, that omission is the paper's largest missed opportunity. On its own terms — as a computational exercise showing what Ampèrian electrodynamics implies for a homopolar machine, and where the two force laws part company inside the metal — it is coherent and carefully set out.