The Magnetic Force Between Two Currents Further Analyzed Using Coulomb's Law And Special Relativity Theory
| Scientific Paper | |
|---|---|
| Title | The Magnetic Force Between Two Currents Further Analyzed Using Coulomb's Law And Special Relativity Theory |
| Read in full | Link to paper |
| Author(s) | Jan Olof Jonson |
| Keywords | special relativity theory, Coulomb's law, velocity of light, Lorentz transformations, Maxwell's Equations, retardation, Ampere's bridge, Ampere's law |
| Published | 2009 |
| No. of pages | 21 |
Read the full paper here
Abstract
In a paper 1997 a model capable of explaining the electromagnetic forces between electric currents in conductors, using only Coulomb's law, was proposed, and the results applied to experiments upon Ampere's bridge. The approach succeeded, due to a rigorous geometric analysis, thereby focusing upon the delay effects thanks to the velocity of light. Simultaneously, the Lorentz force failed completely to explain the behaviour of the force. The special relativity theory (SRT) was not being used and due to the very low velocities involved in the currents in conductors, the need was not felt.
However, since the SRT is widely recognized, a check to what extent that would change the results above seems very urgent to perform. That is also one of the main concerns of this paper. And the result is that the SRT does not affect the results, as far as low charge velocities are involved, as is the case with circuit currents. The effects of propagation delay are of higher order and supersede those of the SRT.
Since efforts have been made by other scientists to explain the results with Ampere's bridge, thereby using Ampere's law, this theory will be discussed further here. A model claiming that Ampere's law is a direct consequence of applying the special relativity theory straightforwardly upon Coulomb's law will also be analysed, but with negative result. What Ampere derived through deduction with respect to experimental results remains empirical. The Lorentz force is again unable to explain the measurement results, even though the invariance of Maxwell's equations during Lorentz transformations according to the predominant school supports that claim that Maxwell's equations also are consistent with reality.
Overview
Jan Olof Jonson's programme is reductive in the strictest sense: there is no magnetic force, only Coulomb's law applied with correct retardation. In a 1997 paper in the Chinese Journal of Physics he showed that the force between two current-carrying conductors can be recovered from the electrostatic law alone, provided the finite propagation time of the field is carried through the geometry rigorously, and he applied the result to the Ampère-bridge measurements of Pappas and Moyssides. That paper deliberately ignored special relativity, on the grounds that drift velocities in copper are of order millimetres per second and γ − 1 is therefore utterly negligible.
The present paper closes that gap and then widens the argument. It does three things. First, it redoes the 1997 calculation with the Lorentz transformation included at every step and shows that the answer is unchanged to lowest order in v/c — the retardation terms are of lower order than the relativistic ones and therefore dominate. Second, it examines rival attempts to explain the same Ampère-bridge data using Ampère's original force law, in particular those of J. P. Wesley and Domina Eberle Spencer. Third, and most sharply, it audits James Keele's claim that Ampère's law follows directly from applying special relativity to Coulomb's law, and finds a specific algebraic error in the derivation. Jonson's conclusion is that Ampère's law remains what Ampère made it — an empirical fit — and that the Lorentz force, with its two disparate terms and three vector fields, is a needlessly complicated and internally confused starting point.
The argument
Retarded Coulomb's law for a current
The starting point is Coulomb's law in differential form with charge densities, d2F/dx1dx2 = ρ1ρ2uR / 4πε0R2. Jonson then computes what an observer actually registers. For the sending charges the observed density is
ρ1' = ρ1(1 − (v1·R)cosω/cR),
which he attributes to standard treatments including Jackson's. His claimed novelty is the corresponding correction at the receiving point, ρ2' = ρ2(1 − (v2·R)cosω/cR), which "is not being treated by Jackson. Instead, it seems to constitute a discovery by this author." The factor cosω is added because the two conductors need not be coplanar — a generalisation of the 1997 paper, which assumed they were.
Subtracting the contribution of the immobile positive ions leaves net densities that are entirely first order in v/c: ρ1 = −ρ1(v1·R)cosω/cR, and similarly for the second conductor. Substituting these into Coulomb's law, and using μ0ε0 = 1/c2 and I = ρv, gives the paper's central formula
d2F/dx1dx2 = μ0I1I2(cosθcosψcos2ω)uR / 4πR2,
with θ, ψ the angles each current makes with the separation vector and ω the angle between the two current planes. The force so obtained is central — it lies along uR — which is the property that lets it produce the longitudinal repulsion seen in Ampère's bridge.
Adding special relativity, and finding it makes no difference
Jonson insists that electrons and ions be transformed separately, since only the electrons move: the electron density picks up a factor γ(v), the ion density does not, and consequently the two species are separated by different distance vectors, so the "net charge density" shortcut of the previous section is no longer available. He also Lorentz-transforms the geometric factor uR/R2 component by component, x → γx.
Before doing so he pauses over an objection to his own tool. Special relativity has been criticised as unable to handle the Sagnac effect and non-linear motion, and Jonson notes that Einstein himself took positions both for and against extending the second postulate beyond inertial frames. His remedy is to treat the Lorentz transformation as "intrinsically of differential nature": divide any curve into infinitesimal straight segments and apply successive transformations between line elements, so that the invariant becomes dx2 + dy2 + dz2 − c2dt2 = const. He proposes calling the result an "Updated Relativity Theory."
Assembling all the pieces gives an expression (Eq. 24) considerably more complicated than the retarded-Coulomb one. But every relativistic correction enters at order v2/c2 while the retardation terms enter at order v/c, so on discarding higher orders the result collapses back exactly to the formula above. The conclusion Jonson draws is blunt: "in the case of typical conductor currents, the velocity is of order mm/s and hence, the SRT becomes completely irrelevant."
Ampère's law as empirical fit
Reading Ampère's 1820s memoir directly — and Blondel's history of it — Jonson emphasises that Ampère assumed proportionality to both currents, to both element lengths, to some angular combination, and to r−n, then let experiment fix n = 2 and k = −½ to give the familiar
Force = i ids ds(sinθsinθ'cosω − ½cosθcosθ')/r2.
This accounts both for attraction between parallel currents and for the repulsion between collinear elements demonstrated in Ampère's mercury-trough "boat" experiment at Geneva in 1822. But Jonson notes that the memoir contains no numerical data, and that the derivation "reminds of the way a computer program is computing the coefficients of a polynomial that gives a RMS fitting." A law fitted qualitatively, before the electron was known, is open to displacement by a theory "that at least to a good approximation gives rise to the same results but has a higher epistemological quality."
Wesley, Spencer, and the fit to Ampère's bridge
Wesley applied Ampère's law — in a modified vector form, Jonson notes — to the Pappas and Moyssides bridge measurements, and reproduced the magnitude of the force reasonably well but with a visibly wrong slope. Jonson's own retarded-Coulomb result gives "almost exactly the correct slope" but a level 0.42 times the measured values. He argues that a constant multiplicative discrepancy is precisely what one should expect from instrument calibration: ammeters and voltmeters are themselves calibrated on the force law being disputed, so adopting a different fundamental law requires rescaling the instruments. Between the two failure modes he takes the slope to be decisive, since it reflects the spatial structure of the law rather than a scale factor. Spencer's version, he observes, is likewise not written as Ampère wrote it but in Wesley's vector form, with the transformation between them not adequately shown.
The error in Keele's derivation
Keele derived, from the Lorentz force applied to a moving charge, an electric-field expression es = kqsr/γ2r3(1 − (v2/c2)sin2θ)3/2, expanded binomially, subtracted the static Coulomb term, and obtained a force from which Ampère's law follows in vector form — the same expression Wesley uses. Jonson's objection has two parts. Philosophically, Keele "sticks to the Lorentz force, as if it were to be a 'simplest presumption'", whereas Coulomb's law needs only two vector functions where the Lorentz force needs three. Technically, Jonson reports a specific omission: the γ belonging inside the numerator distance vector, r' = γ(x,y,z), is missing, so that with r3 → (γ2x2 + y2 + z2)3/2 the correct result is
f = kqsqtγ(x,y,z)(v2/c2)(1 − 1.5cos2θ)/r3,
"a huge difference" from Keele's Eq. (31). The consequence Jonson draws is that Ampère's law is not a corollary of relativistic Coulomb electrostatics; that it coincides with the Lorentz force's electric term only to lowest order in v/c; and that beyond that order "they indeed diverge."
The charge against the Lorentz force
Two further objections are levelled. The first is a charge of circularity: the transformation rules for force components, Fx = Fx', Fy = Fy/γ, Fz = Fz/γ, are required in advance to preserve mechanics under the Lorentz transformation, and then imposed on the force law. "One may not beforehand prescribe the result!"
The second is the textbook derivation in which one transforms to the frame K of the drifting electrons so that the magnetic field vanishes and the whole force becomes electric. Jonson objects that this forgets the positive ions, which are moving in K and therefore constitute a current there, generating a magnetic term of their own — of opposite charge and opposite velocity. There is, he concludes, "no such system, where the second 'magnetic' term disappears." He extends the point to free electrons: overall neutrality means positive charges exist somewhere, and if they are at rest where the electrons move they are moving where the electrons are at rest. Charge conservation in a closed system likewise requires a return current, so "electrons in free space may be regarded as participants in a closed electric circuit."
Assessment
The paper's animating idea is a good one and older than Jonson: that the magnetic force may be a bookkeeping device for retarded electrostatics between charges in relative motion. What distinguishes Jonson's version is that he refuses to stop at the standard first-order retardation of the source and insists on the matching correction at the field point, which is what generates his central, longitudinal force. That is a real structural difference from the Biot–Savart–Lorentz treatment, and it is why his formula can produce the longitudinal repulsion that Ampère's boat and Pappas and Moyssides' bridge appear to show and that the Lorentz force does not. His honesty about the fit is also a point in his favour: he reports his own factor-of-0.42 discrepancy openly rather than burying it, and he states plainly which feature of the data — slope versus magnitude — he takes to be diagnostic and why. The Keele audit is likewise concrete: a named equation, a named missing factor, and the corrected result written out, so the claim can be checked rather than merely believed.
Against this, several steps are asserted rather than established. The calibration argument that disposes of the 0.42 factor is the most consequential: it is plausible in principle that instrument scaling depends on the assumed force law, but Jonson does not carry out the recalibration and show that it yields 1/0.42 ≈ 2.4, nor does he show that the same rescaling leaves the many other electromagnetic measurements that use the same instruments intact. Until that is done, a factor-of-two-and-a-half disagreement with the data is being retired by an argument rather than by a calculation. The "differential Lorentz transformation" of Section 2.2.3, offered to extend relativity to curved motion, is presented in a paragraph and asserted to "increase the support for the usage of the SRT"; it is not derived, and the invariance of dx2+dy2+dz2−c2dt2 along a worldline is in any case standard, not a novelty requiring a new name.
The claim that the Lorentz force "is unable to explain the measurement results" also carries more weight than the paper's evidence base can bear. The Ampère-bridge longitudinal-force experiments are precisely the ones that have been contested for four decades: Pappas and Moyssides' measurements have been attributed by critics to thermal expansion, mercury hydrodynamics and contact effects rather than to a genuine longitudinal electrodynamic force, and no independent replication has settled the matter. Jonson does not engage this literature, and a theory whose principal empirical support is a disputed result cannot claim the Lorentz force has failed. Meanwhile the measurements the Lorentz force is normally judged against — the Kaufmann–Bucherer–Neumann velocity-dependence of the electron force, the operating principles of every synchrotron and cyclotron, and the precision of g−2 — go unmentioned, and it is not shown that the retarded-Coulomb model reproduces them.
The critique of the standard "transform to the electron rest frame" argument is worth taking seriously in its own right, because the objection is real: the ions do become a current in K', and elementary textbook presentations often gloss over this. But the standard treatment does not depend on the magnetic field vanishing in some frame; it depends on the total four-force transforming correctly, which Jonson's circularity charge misidentifies as an assumption. The force-transformation rules he objects to being "prescribed" are consequences of the transformation of the four-momentum, derivable independently of any electromagnetic input. Finally, the paper is uneven as a piece of writing — the language is often approximate, the equation numbering repeats (two equations are numbered 28), and the argument doubles back on itself between Sections 3.2 and 3.5 — which makes its genuinely checkable claims harder to find than they deserve to be.