The Faraday Induction Law and Field Transformations in Special Relativity
| Scientific Paper | |
|---|---|
| Title | The Faraday Induction Law and Field Transformations in Special Relativity |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | Faraday induction law, transformation of electromagnetic field, successive Lorentz transformations |
| Published | 2003 |
| Journal | Apeiron |
| Volume | 10 |
| Number | 2 |
| No. of pages | 17 |
Read the full paper here
Abstract
The non-invariance of the Faraday induction law, revealed in [1] through calculation of an e.m.f. along a mathematical line, is further analyzed for integration over a conducting closed circuit. The principal difference of a conductor from a mathematical line is the appearance of internal electromagnetic fields induced by rearranged conduction electrons. In our analysis we distinguish two general cases: 1- the internal electromagnetic fields from the conduction electrons contribute an induced e.m.f.; 2 - the internal fields do not give such a contribution. Case 2 makes a conducting circuit similar to a mathematical line, where the Faraday law is always correct, while the Einstein relativity principle is violated. However, in such a case the violation of special relativity occurs not for a hypothetical model problem, but in physical reality.
Overview
Alexander L. Kholmetskii, of Belarus State University, follows up an earlier Apeiron paper in which he argued that Faraday's flux rule, ε = −d/dt ∫SB·dS, does not follow from the corresponding Maxwell equation once the surface S(t) and its bounding contour Γ(t) are themselves in motion, because the total time derivative of the flux is not the same as the integral of ∂B/∂t. In that paper he tested the flux rule for Lorentz invariance using a purely geometrical "mathematical line" and found it not invariant. The present paper asks what happens when the contour is a real conductor.
The physical difference, Kholmetskii says, is that a conductor contains mobile charges. External fields rearrange the conduction electrons, and the rearranged electrons produce internal fields which may be negligible outside the conductor but are significant inside it. He splits the problem in two. In case 1 the internal fields contribute to the electromotive force; in case 2 they do not, and the circuit then behaves like the mathematical line. His conclusion is stark and stated twice: because Faraday's law is empirically established and is not Lorentz-invariant, "the special theory of relativity, in its application to electromagnetism, was disproved by Faraday as long as several decades before its creation."
The argument
Case 1: internal fields contribute to the e.m.f.
A rectangular conducting loop A-B-C-D sits inside a charged flat condenser FC, with side AB acting as a sliding bridge moving at velocity u along the plates and the remaining sides connected through sliding contacts to a voltmeter. In the laboratory frame K the conduction electrons in the stationary segments BC, CD and DA rearrange until the resultant internal field vanishes, ER = Eext + Eint = 0; since the transformations are homogeneous, E′ = B′ = 0 in those segments for every inertial observer, and they contribute nothing anywhere.
In the moving bridge AB the rearranged electrons produce Eint = −γuE along y and a magnetic field Bint = −γuEu/c2 along z, with γu = 1/√(1 − u2/c2). The residual force per electron gives an e.m.f.
ε0 = lE(1 − 1/γu) ≈ lEu2/2c2,
where l is the length of AB. Kholmetskii then repeats the calculation in a frame K0 in which the whole apparatus moves at v along x, transforming the fields and using the relativistic composition u′ = (u + v)/(1 + uv/c2). To order c−2 the result is the same, ε = ε0 — in agreement with the relativistic transformation of e.m.f. But the magnetic flux through ABCD in K0 has time derivative dΦ/dt ≈ −uvlE/c2, which taken with the opposite sign does not match ε0. So in this configuration relativity survives and the Faraday law fails.
Case 2: internal fields do not contribute
This is the configuration Kholmetskii regards as decisive. An elongated conducting loop has its upper lead AB inside a charged flat condenser, its vertical wires entering through tiny holes C and D in the lower plate so that the field distortion is negligible. Frame K1 is attached to the loop and K2 to the condenser; in a third frame K0 the loop moves at v along x while the condenser moves with V{v,u} in the xy-plane, so that relative to the loop the condenser moves only along y.
Here the internal field from redistributed electrons does not contribute along AB, the velocity of AB is parallel to its own axis so magnetic forces along it vanish, and the magnetic forces in the two vertical sides cancel because those sides have equal velocity in any frame. The circuit is thus "similar to a mathematical line."
Transforming the condenser's electrostatic field into K0 through an intermediate rotated frame gives, to order c−2, E0x = −Euv/2c2, E0y ≈ E(1 + v2/2c2), B0x = B0y = 0 and B0z ≈ Ev/c2. A magnetic field therefore exists inside the condenser in K0, and the area ABDC between the lower plate and the loop shrinks with time, giving
dΦ/dt ≈ −EuvL/c2.
Computing the e.m.f. directly from ε = ∮(E + v×B)·dl requires one further subtlety that Kholmetskii treats carefully: because the condenser's x axis contracts only along the direction of V, the plates appear in K0 tilted by a small angle φ ≈ −uv/2c2, so the segments DB and AC are not equal. Adding the contributions gives ε ≈ EuvL/c2 = −dΦ/dt. Faraday's law holds in K0, and the e.m.f. is non-zero.
Transforming instead from K0 to the loop frame K1, the magnetic field vanishes, B1z ≈ 0, while E acquires the x-component E1x = −Euv/2c2. Kholmetskii identifies this tilt with the Thomas-Wigner rotation between K1 and K2 for the given motion diagram, Ω ≈ uv/2c2 = −φ: an observer in K1 simply sees the condenser spatially rotated, and rotating a charged condenser induces no e.m.f. in a loop threading it.
The contradiction
So an e.m.f. exists in K0 and is absent in K1, in full accordance with the flux rule in both frames, since the flux exists and changes in K0 and disappears in K1. Kholmetskii's objection is that this is not merely a frame-dependent quantity but a frame-dependent fact about a physical system: "a current in the loop A-B-C-D cannot exist in one inertial frame and be absent in another inertial frame." He stresses that this reveals no mathematical imperfection in relativity — the field transformations were used throughout and were used correctly — but rather "a simple fact that the empirically discovered Faraday induction law is not Lorentz-invariant." Explaining the non-invariance within an ether theory adopting absolute space is deferred to a separate paper. He thanks Walter Potzel, Thomas E Phipps, George Galeczki, Oleg Missevitch, Vladimir Onoochin and Victor Evdokimov for discussions.
Assessment
The paper's distinctive strength is that it does not attack relativity from outside. Every field transformation, every velocity composition and the Thomas-Wigner rotation are taken from standard sources (Cullwick, Møller) and applied consistently to order c−2. The choice of geometry is careful and well motivated: Kholmetskii isolates precisely the case in which a real conductor stops behaving like a conductor for the purposes of the line integral, so that the earlier "mathematical line" result can be transferred to a physically realisable apparatus. The recognition that the condenser plates appear tilted in K0 by φ ≈ −uv/2c2, and that this exactly compensates in the loop frame as a Thomas-Wigner rotation, is a genuinely nice piece of relativistic bookkeeping which many treatments would have missed. Case 1 is also handled fairly: he reports that there relativity holds and the flux rule fails, rather than presenting only the result he wants.
The central difficulty is the interpretive step, and it is a large one. What Kholmetskii computes is the electromotive force around a loop; what he asserts is that a current would exist in one frame and not another. Those are not the same statement. The e.m.f. is defined as a line integral of (E + v×B) taken over a contour at a fixed time in a chosen frame, and a surface of simultaneity is exactly the object that changes between frames. In case 2 the two frames differ by a relative velocity v and the contour has spatial extent L; the disagreement he obtains is of order EuvL/c2, precisely the size one expects from relativity of simultaneity along a contour of length L. Kholmetskii's own analysis in fact exhibits the mechanism — the segments DB and AC are unequal in K0 because the plates are tilted — but he treats this as an incidental correction rather than as a sign that the quantity being compared is not frame-independent to begin with. The observable that must agree between frames is the reading of the voltmeter, and the paper does not carry the calculation through to a current in a resistive circuit, an accumulated charge, or any other quantity that could be recorded and later compared.
Second, the argument turns entirely on second-order terms in u'v/c2 in a configuration involving three frames, a rotated intermediate frame, sliding contacts and holes in a condenser plate. Approximations are consistently taken "to the adopted accuracy of calculations", including the remark that the length contraction of AB "is not significant". In an argument whose entire content lies at order c−2, a discarded c−2 term is not obviously negligible, and the paper does not bound what has been dropped. No numerical estimate of the predicted e.m.f. is given, and no experiment is proposed, so the claim cannot be settled by measurement as presented.
Third, the framing overreaches the result. Faraday's flux rule is, in standard treatments, not a fundamental law but a theorem valid under conditions — the same conditions Kholmetskii himself identifies in his opening equation, where he shows that d/dt∫B·dS ≠ ∫∂B/∂t·dS for a moving surface. Having established that the flux rule and the Maxwell equation are inequivalent for moving circuits, the natural reading is that the flux rule has limited scope; Kholmetskii instead keeps the flux rule fixed and concludes against relativity. The Maxwell-Lorentz equations themselves, which are Lorentz-covariant by construction, are never shown to fail anywhere in the paper. Set against the accumulated agreement of relativistic electrodynamics with measurement — particle accelerator kinematics, the anomalous magnetic moment of the Electron computed in Quantum Electrodynamics, the γ-scaling of muon lifetimes — a disproof resting on the definition of e.m.f. around a moving loop is a very thin thread. The promised sequel treating the problem in an absolute-space ether theory would have to reproduce all of that agreement as well.