Ampere: mathematician's review (PhD mathematician role)

Source: Assis & Chaib, Ampere's Electrodynamics (Apeiron 2015), the only document. It has a full text layer, so there is no source gap. I read these passages myself with pdftotext: Sec. 4.3, 7.1 and 7.5 (derivation of k), 8.2-8.3 (bridge), 10.5, 10.8, 14.1-14.2 (Grassmann, eqs 14.3-14.12), 20.4, 21.2, 22.2 (Weber, about two pages). I did not read the translation of the Theorie (Part VII) or Ampere's first paper (Part VI). Equation 7.26 and the 14.x vector equations were read from the text layer and were not garbled. [read]

I also ran my own numerical check (script /tmp/amp/chk.py, my own scratch code, not part of the book). It sums the force on one closed loop from another, using two skew circles discretised into 400 elements each. [computed]

Corrections to Step 1

1. Where n = 2 comes from. Step 1's claim 2 says k = -1/2 comes from the null experiment "and from k = (1-n)/2 with n = 2", as if n = 2 were part of the experiment. The null experiment gives only the relation k = (1-n)/2. Ampere took n = 2 in 1822 by analogy with gravity, electrostatics and magnetism, and he said explicitly that "there is no proof deduced directly from experiment" (quoted at Sec. 7.5) [read]. Savary (1823) then derived n = 2 and k = -1/2 separately, and the "law of similarity" equilibrium of 1825, repeated by Ettingshausen, Felici and Tricker, gave n = 2 directly (Sec. 9, 10.5) [read]. So claim 2 is a chain of three things: the null experiment fixes k given n; the 1822 value of n was assumed; the 1825 experiment later fixed n.
2. Step 1's "one-parameter family indexed by k" is wrong as written. facts.json says the formulas form "a one-parameter family indexed by k that all give the same force on closed circuits". That is false for the central 1/r^2 law. My numerical check, summed over two closed loops, gives (net force on loop 2, x component, with a sign and factor-2 normalisation fixed at k = -1/2):

Only k = -1/2 matches the field law. This is why closed-circuit experiments can fix k at all. The book's own claim 1.2 ("same net force on closed circuits") is true, but only at k = -1/2. The family with equal closed-circuit forces is a different one: it differs from Ampere's by a total-derivative (gradient) term. Grassmann's non-central law belongs to that family. I did not read the book's treatment of that family in detail. The overall normalisation constant of Ampere's law was not checked against the book; the factor of 2 I found is a convention. 3. Framing of the "self-force". The book says (Sec. 21.2) that the field theory explains the bridge "only if the bridge's force on itself is included". Step 1 repeated this as a defect. Mathematically there is no addition. The Lorentz force on any element uses the total field at that point, which includes every current in the circuit, the bridge itself included. That is the definition of the force law, so including it is not an extra step. See claim 3. 4. Two minor points. Step 1 says about 29,000 lines of text. I get 25,679 lines with -layout, which is immaterial. The book is carefully partisan, and Maxwell's praise of Ampere (Sec. 20.4) is real [read] but concerns the force being equal, opposite and central, not the law being final.

Mathematical audit of the derivation (shared by claims 1 and 2)

I checked the Sec. 7.5 derivation. In the torque integrand (eq. 7.26), the one-form is
(k-1) r^(-n-2) rho^2 sin^2(phi-phi') dr + r^(-n-1) rho sin^2(phi-phi') drho + r^(-n-1) rho^2 sin cos dphi.
Closure (exactness) requires d/drho of the dr coefficient to equal d/dr of the drho coefficient:
2(k-1) = -(n+1), which gives k = (1-n)/2. The dphi terms give the same condition. [computed by hand] This matches Ampere's own statement. At n = 2, k = -1/2.

Hidden assumptions in Ampere's step: - The variables r, rho, phi are treated as independent. This is legitimate once z is eliminated. - "Vanishes for every shape of the mobile circuit" is turned into "the form is exact". This is standard, but it needs the integral to vanish for all paths, not just closed ones. The experiment (a mobile conductor with sliding mercury contacts, so effectively an open circuit) supplies this. - It assumes the force is central.

A mainstream cross-check: in the field picture, the torque on a mobile element about the axis of a circular loop is I d(rho A_phi), an exact differential. This follows from axial symmetry and div B = 0 (the rho B_z and rho B_rho components are the derivatives of the flux function). So the null result is reproduced by Biot-Savart-Lorentz too. The null experiment therefore cannot separate Ampere from Grassmann. It does separate k = -1/2 from other central laws, as the table above shows. [derived]

Other checks: - Collinear elements, same sense: with k = -1/2, cos(epsilon) - (3/2) cos(alpha) cos(beta) = 1 - 3/2 = -1/2, so the force is repulsive. [computed] - Grassmann's critical angle arccos(sqrt(2/3)) = 35.26 degrees [computed], confirmed against eq. 14.12. - Units: the Biot-Savart expressions 14.9 and 14.10 are dimensionally correct. Applying d s x (d s' x r-hat) gives -[(ds.ds') r-hat - (ds.r-hat) ds'] (14.9). [read, re-derived]

Claim 1: central 1/r^2 force with k = -1/2, strong third law

• a. Internally consistent. Both forms of the law (sin-sin-cos and cos(epsilon) + h cos cos with h = k-1 = -3/2) are algebraically the same. It is central by construction, so it obeys the strong third law. The 2 cos(epsilon) - 3 cos(alpha) cos(beta) form is quoted by Grassmann himself (eq. 14.11) [read].
• b. Consistent with all closed-circuit force data, because it gives the same force as Biot-Savart-Lorentz on complete circuits (checked numerically above, and the book states the same). Consistent with the null-rotation, similarity and bridge experiments.
• c. Tested by Ampere 1820-26 with null-method equilibria; the similarity equilibrium by Ettingshausen, Felici and Tricker [read, Sec. 10.5]. The values are as the book reports them; I did not read the primary sources. The bridge was repeated later (see claim 2).
• d. As a replacement for the field picture it would force three changes: no momentum or energy in the field for static currents (but radiation and retardation need a field or a retarded law), no Lorentz force on a lone moving charge, and non-central force laws would be abandoned. For steady currents on closed circuits nothing changes, because the two laws agree.
• Verdict (split): the formula as a description of closed-circuit forces is Established. The claim that it is the unique correct elementary law is Historically superseded but instructive. The two laws differ only for open current elements, which are not separately measurable.

Claim 2: k fixed by experiment; collinear repulsion confirmed by the bridge

• a. Consistent (derivation above). With n = 2 as an assumption, the null experiment fixes k. The 1825 experiment fixes n, so the result does not depend on analogy. See Correction 1.
• b. The bridge: the book quotes Ampere and de la Rive's 1822 report that the wire "moved parallel to the division" away from the current's entry, whatever the current direction [read, Sec. 8.2]. Mainstream recognises the effect. It is also reproduced by the Lorentz force when the whole circuit's field is used (see claim 3).
• c. The book calls it "completely successful" in Ampere's words; I take the original report as a qualitative observation (the bridge moved). It has been "reproduced and discussed in the literature until modern days" per the book [read, Sec. 8.2, refs to Assis & Bueno]. Quantitative work: Moyssides (2014, Eur. Phys. J. Plus) reports measured velocity of a floating wire in a mercury trough agreeing within error with both Ampere's and Biot-Savart-Grassmann-Lorentz laws [web: https://link.springer.com/article/10.1140/epjp/i2014-14034-2]. I did not read the paper itself; the summary above comes from the search result abstract.
• d. If the repulsion between collinear elements is real in the strong form, a longitudinal force exists between elements of a single straight wire, contrary to Biot-Savart. The bridge cannot show this, because a measured net force on an open piece is also given by Lorentz.
• Verdict (split): the derivation and the observed bridge motion are Established. "The bridge confirms collinear repulsion as distinct from the field law" is Not testable as stated by this experiment, since both laws predict the observed net force.

Claim 3: field law gives zero collinear force and breaks strong third law; explains the bridge only by an added self-force

• a. The first half is mathematically correct. Eqs 14.7-14.10 give zero for collinear elements (ds' parallel to r-hat kills the cross product). Applying 14.9 to a pair gives forces that are not equal and opposite in general (eq. 14.10 vs 14.9, Sec. 20.3). Correct. The second half is a framing problem (Correction 3). The self-force is not "added"; the field of the whole circuit includes the bridge. On an open piece like the bridge, Biot-Savart-Lorentz self-force need not vanish, because third-law-violating pairs (leg and crossbar) do not cancel in a rigid open piece.
• b. Moyssides (2014) reports the Biot-Savart-Grassmann-Lorentz law predicts the longitudinal forces and agrees with the Ampere ones [web: Springer link above]; his abstract says that law "anticipates longitudinal forces", so the two laws are "in all aspects equivalent" in his words. This is one author's work, and I have not checked it. [web, medium confidence on how strong the result is]
• c. The book itself says the effect is only quantitatively explained with the self-term [read, Sec. 21.2]; the authors acknowledge the field law works if the self-field is included. The result of Graneau and others claiming a failure of Lorentz (e.g. wire-fragmentation experiments) was not examined by the book and not by me. Not replicated independently as far as I can tell.
• d. If the field law truly failed on open pieces, momentum conservation would have to be enforced by something other than field momentum. Mainstream already says field momentum carries the balance (book concedes this, Sec. mainstream_conflict in facts.json [read]). Nothing in mainstream changes.
• Verdict (split): "Field law gives zero collinear force and violates the strong third law" is Established (mathematics). "Bridge needs an ad hoc added self-force, so the field law is defective" is Tested, not supported as a discriminating claim, with the caveat that this rests on one author's report [web] and that my numerical bridge calculation was not done. I computed closed-loop forces only; I did not compute the bridge itself.

Claim 4: all magnetism reduces to currents; poles unnecessary; magnetic, electromagnetic, electrodynamic phenomena unified

• a. Consistent. The far-field equivalence of a small loop and a dipole, and of a solenoid and a bar magnet, is exact in the limit (standard result; Ampere's own treatment, Sec. 10.8 [read]). Bound-current equivalence (M gives J_b = curl M) is the modern version.
• b. Consistent with experiment as to forces and fields outside matter. It does not follow that magnets are classical molecular currents: electron spin moments are intrinsic and are not circulating charge. Bohr-van Leeuwen theorem: classical statistical mechanics gives no magnetisation, so magnetism of matter needs quantum mechanics [memory, high confidence; not web-checked since the verdict does not depend on it].
• c. The equivalences were tested by Ampere and by many since; they are routine. Nobody has found a monopole, but the theory does not forbid it.
• d. Nothing in mainstream changes for the weak form. The strong form (only classical currents) is what modern physics has replaced.
• Verdict (split): weak form (equivalence of dipole and loop, magnets described by bound currents) is Established. Strong form (all magnetism is classical molecular currents) is Historically superseded but instructive.

Claim 5: Weber's law is a better foundation than the field concept

• a. The book gives only a two-page summary (Sec. 22.2) and no derivation; this is an assertion. [read] Weber's potential-energy form conserves energy [web: https://en.wikipedia.org/wiki/Weber_electrodynamics, Weber showed this in 1869-71]. Helmholtz objected that it gives negative effective mass in some configurations and unbounded acceleration [web, same source; his claim that energy is violated is described there as incomplete]. The constant c in Weber's law is a velocity ratio (Weber-Kohlrausch); I did not verify its numerical relation to the speed of light.
• b. Weber's law has no retardation (it is instantaneous), so it has no electromagnetic waves. Hertz's 1887 radiation experiments are the standard counter-evidence [memory, high confidence]. It is not Lorentz invariant [memory, high confidence]. Whether it reproduces Faraday induction for moving circuits is accepted [web: Wikipedia above]; I did not verify the induction claim in the book.
• c. The book does not claim tests of Weber against Maxwell. It lists modern references by the Weber school (Assis, Wesley, Graneau, Tajmar); I examined none of them. Neutral tests: none that I know of; I found none in my search.
• d. If Weber were true, special relativity, retarded potentials and Maxwell's displacement current would need to be replaced for fast-changing fields. Mainstream replaced Weber by Maxwell/Lorentz in the 1880s [web: Wikipedia above].
• Verdict (split): "Weber unifies electrostatics, Ampere and induction for slow, closed-circuit phenomena" is Plausible, untested against Maxwell in any regime where they differ (nobody has run a test that the book reports). "Better foundation than the field concept" is Historically superseded but instructive, because it has no waves.

Summary table

Sources used: book text layer [read]; own numerics [computed]; https://link.springer.com/article/10.1140/epjp/i2014-14034-2 [web, abstract only]; https://en.wikipedia.org/wiki/Weber_electrodynamics [web]; Bohr-van Leeuwen and Hertz [memory, high].