More on the Magnetic Force Between Two Currents
| Scientific Paper | |
|---|---|
| Title | More on the Magnetic Force Between Two Currents |
| Read in full | Link to paper |
| Author(s) | Jan Olof Jonson |
| Keywords | special relativity theory, velocity of light, Lorentz transformations, Lorentz force, Maxwell |
| Published | 2010 |
| Journal | Proceedings of the NPA |
| Volume | 7 |
| Number | 2 |
| No. of pages | 11 |
| Pages | 679-689 |
Read the full paper here
Abstract
In this author's 1997 paper, 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
This is a sequel to Jonson's 1997 paper in the Chinese Journal of Physics, "The Magnetic Force between Two Currents Explained using Only Coulomb's Law". The programme of that paper, restated and extended here, is eliminative: there is no magnetic force and no magnetic field, only Coulomb's law applied with strict attention to the finite velocity of light. Because the signal from each element of a current-carrying conductor reaches the observation point after a delay that depends on where along the conductor the element sits, the charge density an observer registers is not the rest density; and because the positive lattice ions do not move while the conduction electrons do, the retarded densities of the two species fail to cancel. The residue is a small net charge density on a wire that is electrically neutral at rest, and it is this residue, acting through the ordinary electrostatic force, that Jonson identifies with what is conventionally called magnetism.
The new work in this paper is a consistency check. Since the 1997 derivation used no relativity at all, Jonson asks whether adding the special relativity theory (SRT) would change the answer. His conclusion is that it would not: the SRT charge-density enhancement enters at order v2/c2, whereas the retardation terms enter at order v/c in each of the two conductors, so for drift velocities of the order of millimetres per second "the SRT becomes completely irrelevant". He then turns the same tools on three rival treatments of the same experiment — those of J. P. Wesley, Domina Eberle Spencer and Keele — and argues that all of them either import Ampère's law as an empirical postulate or, in Keele's case, misderive it. The departure from the mainstream account is therefore not a small one: Jonson wants the Lorentz force law and the magnetic term of Maxwell's equations withdrawn as fundamental statements, retaining them at best as approximations.
The argument
Retarded charge densities
The starting point is Coulomb's law in differential form for charge densities rather than point charges, d2F/dx1dx2 = ρ1ρ2uR/(4πε0R2). Jonson then writes the density registered by an observer for the moving electrons of the emitting ("sending") conductor as ρ1' = ρ1(1 − (v1/c)cos α), noting that this expression "or its equivalent" is standard and is given by Jackson and by Resnick. What he claims as new is that the same delay operates a second time at the receiving conductor, giving ρ2' = ρ2(1 − (v2/c)cos β), with a further factor cos γ when the two circuits do not lie in one plane. Subtracting the stationary ion densities leaves net densities proportional to (v/c)cos α and (v/c)cos β respectively.
Multiplying the two residues, using I = λv and ε0μ0 = 1/c2, gives the paper's central result, his Eq. (10):
d2F/dx1dx2 = (μ0I1I2/4πR2) cos α cos β cos γ uR
The permeability μ0 is not assumed; it appears only because 1/(4πε0c2) = μ0/4π. The force is purely central, directed along uR, and is second order in v/c overall because each conductor contributes one factor.
Adding special relativity
Sections 2.2.1–2.2.8 redo the calculation with the Lorentz transformation applied to both the charge densities (γ-factor enhancement) and the separation vector (R → (γvx, y, z), i.e. length contraction along the current). Electrons and ions must be transformed separately, since only the electrons move. The resulting Eq. (21) is considerably more elaborate, but on discarding terms above the lowest order in v/c it collapses back to Eq. (10) exactly. This is the paper's stated answer: SRT does not disturb the 1997 result at conductor-current velocities.
A digression (2.2.2–2.2.3) argues that the second postulate was never clearly restricted to inertial frames, that Einstein took both positions on non-linear motion, and that the difficulty dissolves if the Lorentz transformation is read as intrinsically differential — dx2 + dy2 + dz2 − c2dt2 = const. — so that curved paths can be built from successive transformations between line elements. Jonson takes this to extend the theory to rotation (as in the Sagnac effect) and proposes renaming it the "Updated Relativity Theory".
Ampère's law and its users
Jonson reads Ampère's 1820s memoirs as a search for "any combination of terms that are able to satisfy certain qualitative experimental evidence", noting that the memoir he cites contains no numerical data at all and comparing the procedure to least-squares fitting of a polynomial. He writes the general ansatz as Force ∝ (1/rn)i1i2ds1ds2(sin α sin β cos γ − k cos α cos β) and states that Ampère's derivations gave n = 2 and k = 3/2.
Wesley is credited with the most serious attempt to compare Ampère's law with the Pappas–Moyssides measurements on the Ampère bridge, but is said to have simplified the law by setting cos γ = 1. Jonson's assessment of the comparison is candid: Wesley reproduces the measured force levels reasonably but gets the slope wrong, while Jonson's own retarded-Coulomb curve has "almost exactly the correct slope, but the level is 0.42 times the measurement results". He attributes the missing factor to the calibration of ammeters, which are scaled on the very force law he is rejecting, and argues that agreement in slope should count for more than agreement in level.
The charge against Keele
Keele had derived Ampère's law by applying the Lorentz transformation to Coulomb's law, obtaining for the field of a uniformly moving charge an expression equivalent to Resnick's, and then — after binomial expansion and subtraction of the static Coulomb term — the residual force f = kqsqtv2(0.5 − 1.5cos2θ)r/r3c2, his Eq. (31), which recast for continuous currents reproduces the Wesley form of Ampère's law. Jonson objects that a factor γ is missing from the separation vector in the numerator, and that the correct result is his Eq. (38), f = kqsqt(v2/c2)(−1.5cos2θ)(x,y,z)/r3 — the isotropic 0.5 term being absent. On that basis he concludes that "Ampère's law cannot be treated as a logical consequence of applying the Lorentz transformation to Coulomb's law" and must remain empirical.
He adds a separate objection to the standard textbook demonstration: one is told that in the frame co-moving with the electrons there is no magnetic field, so the whole force is electric — but in that frame the positive ions are moving and therefore constitute a current of their own, so the magnetic term does not in fact vanish anywhere.
Assessment
The attractive feature of the programme is its economy. Coulomb's law plus finite propagation speed is a smaller set of assumptions than field, force law and constitutive relations, and the appearance of μ0 in Eq. (10) as nothing but 1/(4πε0c2) is a genuinely clean piece of bookkeeping rather than a fitted constant. Jonson is also unusually honest about his own numbers: the 0.42 discrepancy is stated in the text rather than buried, and the objection that the co-moving frame still contains a current of ions is a fair criticism of a demonstration that is often presented too glibly.
The difficulties are nonetheless severe, and several of them are arithmetical.
Eq. (10) gives the wrong force between parallel wires. This is checkable in a few lines. Take two long straight parallel currents separated by d, so γ = 0 and cos α = cos β = u/R with u the axial offset and R2 = u2 + d2. The axial components cancel by symmetry; the transverse component per unit length is (μ0I1I2d/4π)∫u2du/R5, and ∫u2du/(u2+d2)5/2 over the whole line is 2/3d2. The result is μ0I1I2/6πd — exactly one third of the measured μ0I1I2/2πd that defines the ampere. Running the same integral on Ampère's kernel 2cos ε − 3cos α cos β returns μ0I1I2/2πd correctly. The shortfall is not a rounding matter: Jonson's angular kernel contains no sin α sin β cos γ term at all, so two current elements lying side by side and parallel exert no force on one another whatever. That the residual factor he reports on the Ampère bridge, 0.42, is of the same order as this missing factor of about 3 is suggestive. The instrument-calibration explanation is doing work that the derivation should be doing, and it is not independently testable, because it can be invoked to absorb any constant discrepancy.
The value k = 3/2 does not belong in the equation he writes it in. In the form Jonson gives, (sin α sin β cos γ − k cos α cos β), Ampère's constant is k = 1/2. The figure 3/2 belongs to the alternative writing (cos ε − (3/2)cos α cos β), where ε is the angle between the two elements; the two are related by cos ε = cos α cos β + sin α sin β cos γ. Jonson's own Eq. (27), quoted from Wesley as 2(ds1·ds2)/r3 − 3(ds1·r)(ds2·r)/r5, expands to 2(sin α sin β cos γ − ½ cos α cos β), so his Eqs. (26) and (27) contradict each other by a factor of three in the second term while both are said to be Ampère's law.
The error attributed to Keele appears to be Jonson's. The field of a uniformly moving charge is E = kq(1 − β2)r/[r3(1 − β2sin2ψ)3/2] — the Heaviside expression, standard since 1888. Expanding to order β2 gives 1 − β2 + 1.5β2sin2ψ, and subtracting the static term leaves β2(1.5sin2ψ − 1) = β2(0.5 − 1.5cos2ψ). That is Keele's Eq. (31) exactly. Jonson's Eq. (38) has lost the isotropic 0.5, which is the contribution of the (1 − β2) numerator he accuses Keele of neglecting. Since the whole rejection of "Ampère's law as a consequence of relativity" rests on this step, the conclusion of Section 3.4 is not established by the paper.
Retardation for uniform velocity. The deeper problem is one the paper never addresses. In standard electrodynamics the retarded (Liénard–Wiechert) field of a charge in uniform motion points at the charge's instantaneous, not retarded, position: the first-order retardation correction to the density is exactly cancelled by a corresponding term in the field expansion. Jonson keeps the first and drops the second, which is why his kernel is central and incomplete. A model that treats delay while omitting the induction term cannot be expected to recover the full force, and the factor-of-three deficit above is the visible sign of it.
Finally, the reinterpretation of the Lorentz transformation as "intrinsically differential" so that it applies along curves is not a departure from relativity but the ordinary treatment of instantaneously co-moving frames, and is already how accelerated motion is handled; it is offered here as a repair to a difficulty that the standard theory does not have. The paper's negative claims about Ampère's law being merely empirical are, by contrast, largely a historical point rather than a physical one, and on that point Jonson is on firm ground: Ampère did fit his exponents and coefficients to qualitative observations.