Faraday Pressure and Energy Conservation

Scientific Paper
Title	Faraday Pressure and Energy Conservation
Read in full	Link to paper
Author(s)	Stewart Ian Wells
Keywords	Faraday, energy, conservation laws, angular momentum, electric field
Published	2009
No. of pages	13

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Abstract

This sequel to 'Dual Dilemma from Faraday's Law' is an explication of the underlying principles of electrodynamics as they pertain to the classical conservation laws. Specifically, the analysis follows the seeming non-conservation of angular momentum resulting from counter-torque applied to a charged wheel by an induced electric field circulation, while simultaneously answering the need for energy conservation. In this context, a magnetic 'field pressure' is introduced to account for work demands on mechanical sources in electrodynamic systems. The problem of radiation from accelerated charges is also critically examined, and new formulations are derived to describe the phenomenon. The problem of the seeming non-compliance of the Lorentz force with the third law of motion is resolved, and the paper concludes with a brief refutation of the notion of 'field momentum'.

Overview

This paper is the sequel to Stewart Ian Wells' Dual Dilemma from Faraday's Law (NPA, 2008), and continues what he calls "the quest for an electrodynamics which scrupulously adheres to the laws of momentum and energy conservation in all situations." The earlier paper had proposed a "reactive ether mechanism" to keep induction in compliance with Newton's third law; the present one adds a second ingredient — a transverse magnetic "field pressure", explicitly credited to Faraday — to close a gap in the work-energy balance.

Wells' quarrel with orthodox electrodynamics is not with its predictions but with its account of interaction. He observes that "in the history of electromagnetism, there has never been a universally agreed upon, self-consistent, description of the precise mechanism by which interaction is accomplished", and that physics has concentrated on how fields act on charges while saying almost nothing about how charges act back on fields, or about what one field source "knows" of another. He takes up the question posed in Cynthia Kolb Whitney's NPA paper — "What do electromagnetic systems 'feel'?" — and answers it by giving the ether a load-bearing role: it is the ether, not any field, that absorbs the momentum that otherwise goes missing. The systems he analyses are, he stresses, "highly idealized"; the induced forces are "practically miniscule", the arguments "theoretical, and are not in themselves expected to be confirmable by direct laboratory tests." Quantities are written in scalar notation throughout.

The argument

Interaction, and the traction hypothesis

Most physicists, Wells writes, assume the action of one body's field on another is answered by the equal and opposite action of the other's field on the one. He belongs to an older school which holds this to be "only half of the interaction": not only does a field exert a force on a charge, but the charge exerts a force on the field, ordinarily communicated back to the discrete source where the flux lines terminate. When the field is induced — an electric circulation about a changing magnetic flux, with no ponderable source — there is nowhere for that reaction to terminate. Wells' answer is "field traction": the force on the field appears as a longitudinal compression in the ether, which absorbs the required momentum.

He is careful to note that traction also occurs for static fields, but cancels there. If two static charges drive each other apart, the traction of the first charge's field against the ether exactly counterbalances the traction of the second's, and no net momentum is left in the ether. Only when a charge reacts to an electrodynamically induced field — one not seated in a ponderable source — is there a net transfer.

The energy-conservation puzzle

The core argument is a two-path thought experiment on a pair of charged wheels sharing a common axis. In state A the wheels are far apart along the axis: the primary is at rest, the secondary already spinning at ω₂. Path one: move the primary close, then accelerate it to ω₁, equal and opposite to the secondary. During that acceleration the expanding primary magnetic field induces an electric circulation which does work on the secondary. Path two: accelerate the primary first, then move it into proximity — and now the work appears explicitly, as the labour of pushing one magnetic field against an opposing one. Wells notes that Coulomb forces can be removed from the picture by replacing the wheels with pairs of concentric, oppositely charged, counter-rotating spheres, so that the Coulomb fields vanish while the magnetic fields still oppose.

Energy is conserved along the second path but apparently not the first, where the only work done is that of generating the primary's field. Taking the system out along one path and back along the other would then violate the work-energy theorem — which Wells treats as "a corollary to Kirchoff's Law", that any isolated system carried around a closed path of energy states cannot return to its initial state having gained or lost energy.

The missing load, he reasons from internal clues, must be proportional both to the secondary's angular velocity ω₂ and to the primary's acceleration — that is, to the secondary's field magnitude and the primary's field change rate. He identifies it with the transverse pressure between field lines postulated by Faraday alongside their longitudinal tension. The lines themselves "may be a mere conceptual convenience, but the 'Faraday pressure' can certainly be regarded as real". Its rule is simple: when the two magnetic fields oppose, the secondary resists the primary's acceleration; when they align, the secondary assists it. Notably, this compressive action does not show up in the conventional field energy densities B²/2μ₀ and ε₀E²/2, nor as any "compression potential" — it is simply a measure of work done in the system.

Counter-intuitive consequences

Wells lists the peculiar features his model entails, and does not soften them. A vertical force on the primary yields rotational kinetic energy in the secondary rather than vertical kinetic energy. Applying enough torque to the secondary to hold its angular velocity constant would entirely relieve the primary torque, yet the primary would keep accelerating. A rapidly spinning, highly charged secondary receives greater torque from the induced field than the primary wheel does. The extra resistance felt by the primary is not real inertia but a "'virtual inertia'". The applied torque, though impressed on the primary, is expressed against the ether, which acquires the "missing" angular momentum — and, Wells is explicit, that ether momentum answers not the secondary's gain but the neutral member's. Because the ether and the neutral have immense inertia, their kinetic energy acquisition is negligible, so "work done on the primary is actually pared into the secondary energy": the torque is applied wholly to the primary and yet the energy appears elsewhere.

The conundrum of the initially stationary primary

Wells then presses his own model to breaking point. If the primary starts at rest (ω₁ = 0) while the secondary spins, then the input power τ₁ω₁ is zero, yet any finite angular acceleration produces a finite induced field E₁' and hence finite power delivered to the secondary. There is no accounting for that power short of admitting infinite primary torque — and a genuinely infinite moment of inertia in the primary is impossible, "if it could, no motion could ever be imparted to the system."

His resolution is that the physics had been described faultily: E₁' had been "naively correlated with the magnitude of α₁ alone." An induced electric field cannot spring instantaneously into existence at finite magnitude, because that implies an infinite dE/dt and hence an infinite curl of B. He therefore writes a general flux equation in which the total flux Φ_M contains both the term integrating the primary acceleration and a term in the second time derivative of the flux itself, plus a constant background from the secondary's rotation. A term whose second derivative is the negative of itself is required to forestall the catastrophe — which forces sinusoidal solutions. Fixing the coefficients by the physical requirement that both E₁' and the induced flux vanish at t = 0 yields, for constant angular acceleration, a flux with a sin t term and an induced field with a (1 − cos t) form. Wells calls the appearance of a periodic term out of a purely constant acceleration "rather perplexing", but points out that it is not sustained by a constant applied torque — the required torque itself carries the cosine — and remarks that "the result in fact lends some credence to the claim that uniformly accelerated charges can emit radiation."

Judging periodic solutions alone to be "theoretically intolerable", he then seeks non-periodic ones built from e^±t, subject to the same boundary conditions, and offers α₁ = α₀ sinh(t/t₀) as "one conspicuous solution", with a matching cosh-form flux and sinh-form induced field. The accompanying torque function contains a "virtual moment of inertia" I₁' — "the total resistance actually felt by the accelerating agency" — made of two parts: the ponderable inertia of the primary plus the reaction of its own charge against its own induced field, and the reaction of the charged secondary transmitted back through magnetic field pressure. He notes the result extrapolates to rectilinear acceleration of a charge.

The Lorentz force and the dismissal of field momentum

Wells then turns the traction hypothesis on the old Ampère–Weber versus Biot–Savart–Lorentz dispute. The Lorentz term Qv×B is not in general a central force, so two charges travelling in parallel experience a net magnetic torque unless their line of separation is strictly perpendicular to the motion — a difficulty he says also afflicts special relativity, since a torque present in one frame would vanish in the bodies' rest frame. If a field transformation of the general form E' = γ(v×B) holds, then a charge crossing a magnetic field meets an equivalent electric field, which engages in ether traction like any other; the counter-torque absorbed by the ether supplies the missing angular momentum.

The paper closes by rejecting "field momentum" outright. Take a charged wheel spinning clockwise: the magnetic field is said to carry the angular momentum associated with the extra torque needed to overcome the reactance of Faraday's law. Reverse both the sign of the charge and the sense of rotation and the field momentum must reverse — "yet the latter field is identical to the former field in both magnitude and direction!" Wells calls this "absurdity alone" sufficient grounds for dismissal. His second objection is that a secondary of overwhelming moment of inertia still acquires angular momentum from the primary's induced field while contributing nothing appreciable to the total magnetic field, so no construction of field momentum can answer it. "It is the ether, which after all, must carry the opposing momentum." The afterword draws the conclusion: the phenomena demonstrate the need for a fixed ether, with which special relativity is incompatible; unless Einstein's later "Lorentz invariant" ether of the 1920 Leyden address can be verified, Wells suggests the special theory "might best be discarded altogether."

Assessment

The paper's real strength is its refusal to let a bookkeeping discrepancy pass. Wells identifies a genuine soft spot in textbook treatments — the reaction to the force an induced field exerts on a charge has no obvious seat, because the field has no source charge — and he pursues it with unusual honesty. The two-path Kirchhoff argument is a legitimate way to expose an energy defect, the substitution of concentric counter-rotating charged spheres to null the Coulomb field is a clean piece of idealisation, and the "field pressure" rule (opposing fields resist, aligned fields assist) is at least a definite, falsifiable-in-principle statement. Most creditably, Wells does not hide behind his own construction: the section headed "A Final Conundrum" is him attacking his own model at its weakest point, and he flags the sinusoidal solution as "so highly counter-intuitive as to seem outright wrong."

The difficulties, though, are substantial. The first is that the discrepancy motivating the whole enterprise is a discrepancy in Wells' accounting, not in Maxwell's. In standard electrodynamics the two paths of his thought experiment do balance, because the field-interaction energy term — the cross term in the total magnetic energy, which is exactly the term that makes work necessary to push like poles together — is the same quantity along both routes. Wells notices that his "compressive action does not appear in the conventional field energy densities", and treats that as evidence the conventional densities are incomplete; the alternative reading is that his ledger has counted the interaction energy once on one path and not at all on the other. Nothing in the paper computes the total field energy of the two-wheel configuration and shows it differing between states, which is what the claim would require.

The second is that the ether is asked to do everything and constrained by nothing. It has "effectively infinite mass and moment of inertia", absorbs whatever momentum the analysis cannot otherwise place, and is invoked afresh at each impasse: for induced fields, then for the Lorentz torque, then in place of field momentum. Because it has no equation of motion, no density and no observable signature in the paper, it cannot fail — and a mechanism that cannot fail cannot be tested. This matters most in the closing section: field momentum is not a free hypothesis but is fixed by the Poynting vector, and the "absurdity" Wells identifies turns on the sign of the momentum density ε₀E×B, which depends on the electric field as well as the magnetic — reversing the charge does reverse E, so the two cases he says are identical are not identical in the quantity that actually carries the momentum. The critique lands only against a version of the field-momentum construct that assigns momentum to B alone.

Third, the mathematics of the flux equation is asserted more than derived. The claim that a finite E' at t = 0 entails an infinite ∇×B is a statement about idealised step functions, not about physically realisable accelerations; the "term whose second derivative is the negative of itself" is introduced to avoid a singularity created by that idealisation, and the constants k and l are described only as "dimensional constants needed to maintain the respective terms in units of webers" — that is, they are fixed by dimensional necessity rather than by any physical measurement, which leaves the resulting sin and cosh solutions underdetermined. Wells' suggestion that the periodic term "lends some credence to the claim that uniformly accelerated charges can emit radiation" is offered tentatively, and rightly so: it does not engage the actual literature on that question, and no radiated power is computed. Nor, throughout, is any number produced that could be compared with a measurement — a limitation Wells concedes at the outset when he says the effects are not expected to be confirmable in the laboratory. Taken on its own terms the paper is an internally serious attempt to make electrodynamics mechanically self-consistent; taken as a challenge to Maxwell and to special relativity, it rests on an energy defect that the standard theory does not actually have.

See also

• Stewart Ian Wells
• Cynthia Kolb Whitney
• Electrodynamics
• Aether
• Lorentz Force
• Maxwell's Equations
• Newton's Third Law
• Michael Faraday
• Special Relativity
• Andre K T Assis