The Electromagnetic Force between Two Parallel Electric Currents of "Infinite" Length Attained Using Respectively Ampère's Law and Coulomb‟s Law Including a Relativistic Analysis
| Scientific Paper | |
|---|---|
| Title | The Electromagnetic Force between Two Parallel Electric Currents of "Infinite" Length Attained Using Respectively Ampère's Law and Coulomb‟s Law Including a Relativistic Analysis |
| Read in full | Link to paper |
| Author(s) | Jan Olof Jonson |
| Keywords | Electromagnetism, Ampère's law, Ampère's Bridge, Coulomb's law, Special Relativity, retardation |
| Published | 2010 |
| No. of pages | 19 |
Read the full paper here
Abstract
In this paper an analysis is being made of a physical electric circuit that might correspond to the famous case of two parallel electric conductors of "infinite length". The case, invented by Ampère, is not physically possible, of course. All electric currents must namely be guided back to their origin. Therefore one ought to present a physical circuit that contains as a part two conductors infinitesimally close to each other. It appeared that Ampère's bridge would be a suitable choice as a starting model, provided a pair of conductors is inserted inside Practically, it resembles an "eight", with two closed circuits coming close to each other along one branch. This is explained more rigorously in the text. A benefit is that a set of detailed computations on Ampère's bridge by this author may be used as a mathematical basis. A comparison between the results attained by using Ampère's law and Coulombs law respectively is being made. Further, the Lorentz Transformation of the Special Relativity Theory is being applied on the Coulomb result. The result is that also the latter method succeeds in predicting the force between two conductors. This result must further be chosen, since it has been shown elsewhere that the very definition of Ampère's law is devoid of logically consistent argument, whereas Coulomb's law constitutes a 'simplest possible assumption. The usage of Coulomb's law is completed with a relativistic analysis, relevant to the properties of the actual circuit, i.e. geometry, velocities.
Overview
Jonson's paper attacks the textbook staple of two infinite parallel wires attracting one another, and it attacks it first on physical rather than mathematical grounds. An infinite current does not exist: "All electric currents must namely be guided back to their origin." Any calculation performed on the idealisation is therefore a calculation on something that cannot be built, and Jonson's remedy is to replace it with a real closed circuit in which two of the branches happen to run alongside each other at a very small separation — a configuration he builds out of Ampère's bridge, giving a figure-of-eight of two closed circuits sharing one close-run branch.
Having built that circuit, he asks the question the paper is really about. The attraction between parallel currents is conventionally obtained either from Ampère's original force law or from the F = BiL rule based on the Lorentz Force; the two agree in this special case, which Jonson thinks has misled physicists "to believe that the two laws are also formally identical." His own position, argued across a series of earlier papers, is that Ampère's law is an ad hoc definition "devoid of logically consistent argument," and that Coulomb's Law — "the simplest possible assumption" — should be the sole cause of the force between charges, with the magnetic field abandoned altogether. The obstacle is that plain Coulomb's law gives the wrong sign. The paper's purpose is to remove that obstacle by adding the Special Relativity transformation of forces and the retardation of fields, and thereby to recover the attraction, and with it Ampère's law, as a consequence of Coulomb's law.
The argument
Which branches matter
Jonson reuses the detailed branch-by-branch integrals he had previously computed for Ampère's bridge, quoting them rather than re-deriving them. The bridge's branches are numbered, and the paper lists the Ampère-law forces between the pairs 10–5, 1–7, 1–5, 1–6, 8–9 and 2–9, each expressed in terms of the geometrical lengths L, M, N and the current I, with logarithmic and square-root terms. He also carries forward his earlier finding that the current (voltage) source itself contributes, giving a correction summing to about 0.51I2 for branches 5, 6 and 7.
The scaling argument is then simple and is the pivot of the whole construction. With L, M and N all of order half a metre to a metre, every contribution is of order unity except the one between the two closely spaced branches, which behaves as L/r12 when r12 ≪ L, M, N. The force is therefore dominated by that pair, and "a circuit, where two parallel conductors are situated infinitesimally close to each others may in a mathematical sense be regarded, as if these two conductors were the only ones in the circuit." That is Jonson's justification for the textbook idealisation — not an appeal to infinity, but a demonstration that the rest of the closed circuit contributes negligibly.
The Ampère result and the Coulomb paradox
Applying the infinitesimal limit to the two-parallel-branch expression, the surviving term gives dF ∝ −I2(L/2r12). The sign is negative for currents in the same direction, hence attractive, "in good accordance with experience." Jonson draws the conclusion that "the 'cross product Lorentz's law' is unnecessary," and adds that Ampère's law has the further merit of explaining the repulsion Ampère observed in the mercury basin between the poles and a floating copper boat, which the Lorentz force law cannot.
Then comes the difficulty he calls "a paradox." The corresponding Coulomb term — the "b-term" in his earlier notation — differs by a sign inside the square root, and the same infinitesimal analysis produces a positive, i.e. repulsive, force between parallel currents in the same direction. Jonson does not soften this: "This would seem to prove the unfeasibility to use Coulomb's law in the actual experimental situation, and, therefore disproving the whole theory. A the disproval of a theory in one sole case namely implies that the theory is false." His escape is that relativity has not yet been applied, and that the smallness of drift velocities makes one "easily a victim of the temptation not to investigate the effects of the SRT at all."
The four relativistic Coulomb contributions
Each conductor carries immobile positive ions and moving electrons, so four interactions must be summed. Jonson works in the standard configuration with frame K at rest with respect to the positive ions and the circuitry, and K′ comoving with the electrons of the first conductor at velocity v1; the electrons of the second move at v2, and the simplified case v1 = v2 is treated. He is explicit that only two things change the shape of the Coulomb expression: the Lorentz transformation and its derived expressions, and the delay in the retarded observation of fields generated by moving charges. The transverse force transforms as Fy = F′y/γ(v), following Resnick.
- Ion–ion. No velocities are involved, so this is plain Coulomb's law between thin linear charge elements, d2Fy = λ1λ2Δx1Δx2y/r3.
- Electron–electron. Both charge elements are length-contracted, the separation vector acquires a γ-scaled longitudinal component, the force must be transformed back from K′ to K, and two retardation factors appear because the signal from a moving element is delayed by direction while the field arrives at the moving element of the second conductor with a position-dependent delay. Series expansion for v/c ≪ 1 leaves a bracket of the form [1 + (3/2)(v1/c)2cos2α − (1/2)(v1/c)2 − (v1v2/c2)cos α cos β] multiplying the electrostatic term.
- Electrons of conductor 1 on ions of conductor 2. Obtained from the previous expression by dropping the second conductor's contraction and its delay term (the ions are at rest in K), and changing sign; the γ-division survives because the source electrons are at rest in K′.
- Ions of conductor 1 on electrons of conductor 2. No transformation from K′ is needed since the source ions already sit in K, but a delay factor (1 − v2cos β/c) appears because the target electrons recede from the observer, together with the sign change for opposite charge.
Result
Summing the four contributions for equal velocities gives
- d2Fy,total = −(λ1λ2Δx1Δx2y/r3)(v1v2/c2)cos α cos β,
equivalently expressible with the currents as I1I2/c2 divided by r3. The purely electrostatic part has cancelled between the four terms, leaving a residue of order (v/c)2 with the sign required. Jonson observes that this "has the same asymptotic properties as" the Ampère differential force law, "even though the coupling constants are different," and that the two would part company for unequal velocities and different directions — conditions Ampère could not have varied or measured, "not even the electron had been discovered." His conclusion: "Ampère's law appears as a consequence of applying the SRT to Coulomb's law in the case of two current carrying conductors. And since this law is sufficient, the Lorentz force is no more needed."
Assessment
Two things in this paper are genuinely good. The first is the insistence on a physically realisable circuit. Objecting that infinite parallel wires cannot exist is not pedantry: Jonson uses the objection constructively, and his branch-scaling argument — showing that the L/r12 divergence of the closely spaced pair swamps the order-unity contributions of the return path — is a real derivation of the idealisation rather than an appeal to it. The second is his honesty about the paradox. He states in plain terms that non-relativistic Coulomb's law gives the wrong sign, and that a single disproof falsifies a theory, before attempting the rescue. Many papers in this literature would have suppressed the intermediate result.
The programme itself is also not eccentric. That the magnetic force between currents can be recovered from electrostatics plus relativity is standard textbook physics, and the drift-velocity smallness is compensated exactly as Jonson describes, by the enormous compensating charge densities. What is distinctive is the claim that this makes both Ampère's law and the Lorentz force derivable rather than fundamental.
The difficulties are substantial. Most of the mathematical work is not in this paper: the branch integrals, the current-source corrections, the "b-term" and the disproof of Ampère's law are all quoted from earlier papers by the same author, several of them published only on the Natural Philosophy Alliance site, so the argument cannot be checked from this document alone. Several key steps are asserted rather than shown — that Ampère's law is "devoid of logically consistent argument," that efforts to relate it to the Lorentz force "have been disproved," and that Keele's earlier relativistic treatment of Coulomb's law was "incomplete." The final summation is performed only for the special case of equal electron velocities and, as Jonson admits, "after some boring steps" that are not displayed.
More seriously, the paper's headline conclusion does not follow from its own result. Jonson finds a total force of order (v1v2/c2)cos α cos β / r3 and says it has "the same asymptotic properties" as Ampère's law "even though the coupling constants are different." A force law that differs in its coupling constant is not the same force law; the SI definition of the ampere fixes that constant, so a mismatch is a quantitative discrepancy against the most precisely realised electromagnetic measurement there is, and it is passed over in one clause. Nor is any number given: no evaluation of the predicted force for a stated geometry, no comparison with the Pappas–Moyssides measurements on Ampère's bridge that Jonson elsewhere analyses, and no error estimate for the series expansion. The claim that the Lorentz force is "no more needed" also reaches well beyond the case treated — the Lorentz force governs charged-particle motion in cyclotrons, mass spectrometers and cathode-ray tubes, none of which reduce to two parallel currents, and none of which is discussed.
Finally, the sign of the surviving term deserves more scrutiny than it receives. Whether the four contributions cancel as claimed depends on the exact treatment of the retardation factors and on which of the (1 − v cos α/c) terms are kept to which order, and the text moves from the four bracketed expressions to the single-line total without showing the cancellation. Since the entire point of the exercise is that a sign came out wrong before relativity was applied, the step where the sign comes out right is precisely the one a reader most needs to see.