Ampere's force between current elements: a primer

Project folder: Researchers/Ampere. It holds one document: A. K. T. Assis and J. P. M. C. Chaib, Ampere's Electrodynamics (Apeiron, Montreal, 2015), a 548-page history and analysis plus an English translation of Ampere's 1826 Theorie. The PDF has a full text layer.

Source note (repeated from claims.md and arbiter.md). No pages were unreadable, but the book was read selectively. It was covered by page-by-page digests (32 of 33 chunks; PDF pp.481-519, the end of Ampere's Notes, the conclusion and the redrawn figures, has no digest) and by in-depth reads (all in deep/Amperes-Electrodynamics-28a544.md) of Ch. 21 (the bridge and the self-rotating magnet, PDF pp.266-281), Ch. 22 (Weber, PDF pp.282-289), Ch. 6.3-7.5 (the Despretz null and the 1822 derivation of k, PDF pp.117-142), Part VI (Ampere's first paper, autumn 1820, PDF pp.290-323), the opening of the Theorie translation (Parts VII-VIII to Sec. 29.1, PDF pp.324-349) and the Theorie passages on ether, electric fluids and vis viva (PDF pp.420-465). The rest of the Theorie translation, including Ampere's own statement of the four equilibrium cases (Sec. 29.2-29.11) and Note 31.1, was read only at digest level. What we say about Ampere's own words comes from the book's quotations and commentary. For this revision, the passages on the 1821 Despretz experiment (PDF pp.119-120), Savary's 1823 work (PDF pp.154-156), the bent wire (PDF pp.165-167) and the similarity experiment (PDF pp.190-192) were checked against the extracted text. The authors are open supporters of Ampere's and Weber's electrodynamics, so the book takes sides as well as reporting history. The outside papers (Moyssides 2014, Graneau, Wesley, Peoglos, the counter-rotation papers) were seen only as abstracts, search snippets or citations.

Page numbers: "PDF p.N" is the page of the PDF file; the book's printed page number is 3 lower.

Source tags: [read] = from the book; [web] = Internet source with URL; [memory] = recalled, with confidence; [computed] = calculated for this primer (see example.py).

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1. The idea in one paragraph

In 1820 Oersted found that an electric current deflects a compass needle. Within weeks, Andre-Marie Ampere showed that two current-carrying wires also push and pull on each other, and he set out to find the "Newtonian" law behind this: a force between two tiny pieces of current ("current elements") that, like gravity, acts straight along the line joining them, weakens with distance, and is equal and opposite on the two pieces. By 1822 he had such a law, with one number in it, called k, fixed by clever experiments in which nothing moves. His law made a strange prediction: two pieces of current flowing in the same direction along the same straight line should repel each other, so a current should try to stretch its own wire. He tested this with a floating U-shaped wire on mercury (the "bridge" or "hairpin" experiment), and the wire moved as predicted. Today's textbooks use a different law, from Biot, Savart, Grassmann and Lorentz, built on the magnetic field. That law gives no force between in-line elements and, element by element, breaks Newton's third law. The book argues that Ampere's law is therefore the better one. This primer shows that the two laws give exactly the same force on any piece of a complete circuit, so the bridge cannot choose between them. The dispute is about how to split forces among elements and where momentum is stored, not about anything a steady-current experiment has measured.

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2. What you need to know first

Vectors, dot and cross products. For vectors a and b at angle phi, a . b = |a||b| cos(phi) (a number) and a x b has size |a||b| sin(phi), at right angles to both. The triple product a x (b x c) = b(a . c) - c(a . b) ("BAC minus CAB") is used once below. Read: Griffiths, Introduction to Electrodynamics, Ch. 1.

Line integrals around closed loops. A wire is a curve. Its current is split into short pieces ds (a small vector along the wire, in the direction of the current). The force on a whole circuit is a sum (integral) over its pieces, written with a circle on the integral sign when the curve closes on itself. One fact matters a lot here: if a quantity is a total derivative along the loop (the change of some function f as you move along the curve), then summing it around a closed loop gives zero, because you end where you started. Such a quantity is called an exact differential. Read: Griffiths Ch. 1 (line integrals, the gradient theorem).

The magnetic field and the Lorentz force. A steady current I' in a small piece ds' creates a magnetic field at distance r given by the Biot-Savart law. A current I in a piece ds placed in a field B feels the force I ds x B (the Lorentz force on a current). Read: Griffiths Ch. 5.

Newton's third law, weak and strong. The weak form says the force of A on B is equal in size and opposite in direction to the force of B on A. The strong form adds that both forces lie along the line joining A and B (they are "central"). Gravity and Coulomb's law obey the strong form. When forces obey it, the total momentum and the total angular momentum of the bodies are both conserved. Read: Taylor, Classical Mechanics, Ch. 1.

Momentum stored in fields. In Maxwell's theory the electromagnetic field itself can carry momentum (density epsilon0 E x B, where epsilon0 is the vacuum permittivity and E the electric field). Light pushing on a mirror (radiation pressure) is the simplest measured example. If the forces between pieces of matter are not equal and opposite, the books are balanced by a change in field momentum. One caution for this primer: for steady currents in complete circuits you do not need field momentum at all. The net forces between whole circuits already balance under both laws discussed below (Section 3), and the total torque on a closed steady system is zero [memory, high]. Field momentum matters for changing currents and for radiation. Read: Griffiths Ch. 8; Feynman, Lectures on Physics Vol. II, Ch. 27.

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3. The textbook view

Symbols used throughout: - I, I' : the currents in the two elements, in amperes (A). - ds, ds' : the element vectors (length in metres, direction of current). ds, ds' are their lengths. - r : the vector from the centre of ds' to the centre of ds; r = |r|; r-hat = r/r, the unit vector along it. - mu0 : the vacuum permeability, 4 pi x 10^-7 N/A^2, so mu0/(4 pi) = 10^-7 N/A^2. - All forces below are the force on ds exerted by ds', in newtons (N).

Biot-Savart field of an element:

dB = (mu0 / 4 pi) I' ds' x r-hat / r^2

Force on another element (Lorentz): dF = I ds x dB, which gives the Grassmann force (Grassmann wrote it down in 1845 [read, Ch. 14]):

dF_G = (mu0 / 4 pi) (I I' / r^2) ds x (ds' x r-hat)

Using BAC-CAB this is (mu0/4 pi)(I I'/r^2)[ ds' (ds . r-hat) - r-hat (ds . ds') ].

Three facts about it, all easy to check by hand: 1. If ds' is parallel to r-hat (the source element points at the other one), ds' x r-hat = 0 and the force is zero. In particular, two elements in a line exert no force on each other. 2. The force on ds is always perpendicular to ds (it is ds x something). An element is never pushed along its own length. 3. It is not symmetric: the force of ds on ds' is in general not minus the force of ds' on ds. Element pairs can violate even the weak third law. The mainstream answer is that isolated steady current elements do not exist (charge would pile up at their ends), that for complete circuits the third law holds for the net forces, and that any imbalance between pieces is carried by field momentum [memory, high].

The textbook result for whole circuits. Summing over a complete loop 1 acting on a complete loop 2, the first term of the BAC-CAB form drops out (it is a total derivative around loop 2), leaving

F_on 2 = -(mu0 / 4 pi) I1 I2 (closed double integral) r-hat (ds1 . ds2) / r^2,

which is symmetric and obeys Newton's third law. Every closed-circuit measurement since the 1820s agrees with this [memory, high].

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4. Where Ampere departs

Ampere wanted the force between two elements to be central and to obey the strong third law, like gravity. The book gives his 1822 law in the form [read, eqs. 2.1-2.6]:

force = (i i' ds ds' / r^n) (sin alpha sin beta cos gamma + k cos alpha cos beta), with n = 2, k = -1/2

Here i, i' are currents in Ampere's own units, alpha and beta are the angles each element makes with the line joining them, and gamma is the angle between the two planes that contain that line and each element. A positive value means attraction. An equivalent form is (cos epsilon + h cos theta cos theta') with epsilon the angle between the two elements, theta and theta' the angles each makes with r-hat, and h = k - 1 = -3/2 [read, eq. 2.3]. In SI units and vectors the book writes it as [read, eq. 2.16]:

dF_A = -(mu0 / 4 pi)(I I' / r^2) [ 2 (ds . ds') - 3 (r-hat . ds)(r-hat . ds') ] r-hat

(conversion: Ampere's i i' corresponds to (mu0/2 pi) I I' in SI [read, eq. 2.21]).

Side by side:

Where k and n came from. Ampere did not measure the force between two elements; nobody can. He used "null" experiments, in which something that could move stays still, and asked what the law must be for nothing to move. The constants were fixed in steps, and the order matters:

1. A failed prediction, January 1821. At that time Ampere believed k = 0. On 15 January 1821 he predicted that a small compass needle, with one long wire running horizontally below it and another running vertically beside it (same current, same distance c), would turn. He ran the experiment with Cesar-Mansuete Despretz on 20 January 1821. The needle did not move: Biot and Savart's theory was right and his prediction was wrong. He set that memoir aside, stopped publishing for about two years, and did not mention the experiment until 1823 [read, PDF pp.119-120, 122]. The book shows that his complete law gives a torque from one long straight wire inclined at angle b to the horizontal proportional to [(1 - k) + (1 + 2k) sin^2 b]/c [read, eq. 6.7, PDF p.119]. So the horizontal wire (b = 0) gives (1 - k)/c, the vertical wire (b = 90 degrees) gives (2 + k)/c in the opposite sense, and the net is (1 + 2k)/c. Zero net torque means k = -1/2. Had Ampere done this calculation, he could have had k a year early [read, PDF p.120]. Note that eq. 6.7 is the authors' reconstruction; "in 1821 Ampere did not perform any of these generic calculations" [read, PDF p.120]. (One sentence on PDF p.119, that Ampere "did not doubt" the constants should be non-zero, sits oddly beside the statement that he then held k = 0; the text does not resolve this.) Part D of the worked example checks eq. 6.7 numerically.
2. The "no continuous rotation" experiment, 1822. A mobile conductor, joined to the circuit through sliding mercury contacts and free only to turn about the axis of a fixed circular current, never turns continuously [read, Sec. 7.5]. This was another failed prediction: still using k = 0, Ampere expected (in an unpublished manuscript, chemise 206 bis) that a closed astatic coil would rotate indefinitely. He performed the experiment in March 1822, found no continuous rotation, analysed it between March and June, and presented the analysis to the Academy on 10 June 1822 without mentioning the failed prediction [read, PDF pp.132-135]. (A different experiment, on 27 March 1822, gave a positive result: continuous rotation driven by wires alone [read, PDF p.128].) Requiring the torque from his law to vanish for every shape of mobile conductor forces the torque integrand to be an exact differential, which gives k = (1 - n)/2, i.e. n - 1 + 2k = 0 [read, Sec. 7.5; re-derived by the mathematician reviewer]. With n = 2, k = -1/2. But in 1822 n = 2 was not measured. Ampere wrote that "there is no proof deduced directly from experiment" for it, because the experiments behind it used magnets rather than two conductors and extend to element pairs only "by an extension which cannot be considered as a complete proof" [read, PDF p.141]. A May 1822 experiment showed n > 1, so k < 0 [read, Sec. 8.3].
3. Savary, 1823. Ampere's student Felix Savary added a second null result. Gay-Lussac and Welter had found (1820) that a ring-shaped magnet exerts no force outside itself; Ampere made the all-electric version, a wire coiled into a closed ring (a toroid), and found it did not act on a mobile conductor [read, PDF pp.154-156]. Savary computed that such a ring exerts a force proportional to kn + 1, so kn + 1 = 0. Combined with n - 1 + 2k = 0 this gives 2k^2 - k - 1 = 0, with two solutions: (k, n) = (1, -1) or (-1/2, 2). The bridge (k < 0) and the n > 1 experiment pick the second [read, eqs. 9.2-9.7, PDF p.155]. So by 1823, n = 2 no longer rested on analogy alone.
4. The "similarity" equilibrium was never performed by Ampere. In 1825-26 he described a fourth null experiment (three similar circles in geometric proportion) that would test n = 2 directly, and listed it among the four "cases of equilibrium" from which the Theorie deduces the law. But at the end of the Theorie he wrote: "I have not had time to build the instruments shown in figure 4 ... the experiments for which they are intended have not yet been done" [read, PDF p.191]. Variants were first run by Albert von Ettingshausen (1878) and Riccardo Felici (1882), and later by Tricker with solenoids; all found n = 2 [read, PDF pp.191-192].

The lesson: the book's (and the Theorie's) phrase "uniquely deduced from experience" is partly a reconstruction. The value k = -1/2 does rest on several independent null results. The exponent n = 2 was asserted in 1822, derived by Savary in 1823, and measured directly only in 1878-1882.

Ampere's own words point both ways. In his first paper (autumn 1820) he wrote that he "was soon convinced that this law could not be found by experiment", because experiments cannot be done on infinitesimal pieces of current, and he adopted a hypothesis of "sufficient probability" [read, PDF pp.305, 307]. In the 1826 Theorie (Sec. 29.1, a passage copied from his 10 June 1822 paper) he says he "relied solely on experimentation" and that the formula will stand whatever the physical cause [read, PDF pp.346-347]. His method there ("method B") is to find equilibria of a mobile conductor that do not depend on the shape of the fixed conductors, then work out what element law makes that true; he admits it is "of much more restricted application" than direct measurement [read, PDF p.348]. It also assumes from the start that the force is central and equal-and-opposite. So "uniquely from experience" is a methodological programme resting on assumptions, not a pure experimental output. Two further cautions: the tidy list of "four cases of equilibrium" on PDF pp.326-336 is the translators' summary, not Ampere's words (his own statement, Sec. 29.2 onward, was not read in depth here) [read, PDF p.326]; and the 1820 papers (Part VI of the book) are entirely qualitative. They give no k, no n and no measured force; the torsion balance is described but no data are reported, some experiments failed at first (the Earth acting on a current, an anomalous helix result, poor mercury contacts), and his statement that currents attract in vacuum was probably never tested (the translators found no such experiment) [read, PDF pp.296 fn 26, 306-313]. "Ampere measured k" is therefore wrong.

The bridge. A negative k means in-line elements repel. Until mid-1822 Ampere had held k = 0. His one hesitation pointed the other way: Gay-Lussac and Thenard's unpublished observation of two wires attracting just before a spark would have implied k > 0, and he explained it away (as an effect of air pressure) [read, PDF pp.91, 148]. So the result was a surprise, and he tested it with de la Rive in Geneva in September 1822. A contemporary journal report (not Ampere's own words, though the book introduces it as his description) says a U-shaped wire floating on two mercury pools "moved parallel to the division, so as to elongate the current" [read, PDF p.148]. The book's argument is that the field law gives zero in-line force, so it can explain the bridge only if one includes "the force exerted by the bridge on itself", which (they say) violates the third law. The authors concede in the same passage that, with that term included, the field law does match the measured net force [read, PDF p.271]. Ch. 21 gives no numbers for the measured force or the size of the self-term; it points to the authors' own papers and books [read, PDF p.271, footnotes 11-12].

The magnet that spins on its own axis. In November-December 1821 Ampere made a bar magnet float upright in mercury and spin about its own axis, with current entering at its top and flowing out sideways through the mercury to a copper ring. Faraday had tried and thought it impossible. Ampere reported it on 7 January 1822 [read, PDF pp.126-127, 271]. A modern version is the "homopolar motor": battery, screw, small magnet, wire touching the magnet's side. Faraday and Biot explained it by the magnet's own pole pushing on the current inside the magnet; textbooks today say the same thing in field language (the magnet's B acting on the radial current inside it) [read, PDF pp.272-273, 278]. The book says this is a "self-torque" that breaks Newton's third law, and that the real driving torque comes from the current in the mercury or external wire, which must feel an equal and opposite reaction torque. Ampere predicted that the mercury would turn the opposite way; the book cites Davy (1823) and several modern demonstrations in which a free external wire counter-rotates [read, PDF pp.275-280]. No torque or speed values are given in the book. Whether Davy's 1823 experiment actually tested this prediction is unknown: the only confirming quote is from a bulletin written by Ampere himself in the third person [read, PDF p.277]. Note also that the third-law argument in Sec. 21.3.6 is the authors' reconstruction of what Ampere "would" say (the heading puts "Ampere" in quotes) [read, PDF p.278].

What the calculation says (from the deep read of Ch. 21) [computed]: for steady, closed currents the field theory also gives zero total torque on the whole system (magnet + mercury + wire + battery), so it too predicts the counter-rotation. The textbook term "magnet's B on the internal current" is exactly the force Ampere's own law gives on that current from the magnet's closed bound currents (Part B below). The two accounts of the torque on the magnet differ only by the circuit's interaction with itself. For i = 1 A, B about 0.5 T and magnet radius 6 mm, the driving torque is about i B a^2/2 = 9 x 10^-6 N m, and those self-terms are about (mu0/4 pi) i^2 a = 6 x 10^-10 N m, ten thousand times smaller. The "self-torque" is a way of dividing up internal forces, not a net torque on an isolated system.

Ampere's energy argument. Ampere argued that forces depending only on distance (such as forces between static magnetic poles, which obey Coulomb's law) cannot keep a body turning against friction: after one turn everything is back where it started, so no net work can have been done. Continuous rotation, as in Faraday's 1821 rotations and Ampere's own motor, therefore shows that the source is moving electricity, which draws its energy from the battery. He used this against Faraday's and Biot's picture of a current-carrying wire as a set of little transverse magnets, that is, against models with fixed poles [read, PDF pp.426, 456-457, 459-460; read, digest PDF pp.248-251; read, PDF p.270]. This is an early use of what we now call energy conservation (then "vis viva", living force, proportional to mass times speed squared). In modern terms: if forces depend only on the distances between fixed points, they come from a potential energy that returns to its starting value after each turn, so a rigid body cannot speed up turn after turn [memory, high]. That is correct physics. The argument cuts both ways: Ampere also shows (PDF p.460) that two rigid closed circuits can never keep each other turning. Faraday's rotations and Ampere's own motor escape this because the sliding mercury contact means the circuit is not rigid in his sense ("parts invariably connected to one another") [read, PDF pp.429 fn 267, 460].

Was Ampere a pure action-at-a-distance theorist? Not quite. His formula is a direct, central force between two current elements, and he has no field in Maxwell's sense. But he did not think the force was ultimate: he called it "elementary" only in the sense that chemists call a substance simple when they cannot yet decompose it [read, PDF p.425]. He preferred a mechanism in which two electric fluids, moving rapidly through the wire and repeatedly combining and separating, produce the force through inverse-square forces between their molecules; he admitted he had not done the calculation, for which "data are required, which are still missing" [read, PDF pp.426-428]. "Ether", for him, was his name for the neutral fluid formed when the two electricities combine (his own footnote 218) [read, PDF p.420]. In his closing summary he wrote that, "without our being allowed to reject explanations based on the reaction of the ether set in motion by electric currents, there is no need, up to now, to resort to them" [read, PDF p.462]. So he neither adopted nor ruled out an ether, and he said the formula stands either way [read, PDF p.428]. (This resolves an apparent contradiction in the page digests, one of which compressed the p.462 sentence to "ether theories are unnecessary".) The book's "pure action at a distance" framing is right about the law and too strong about Ampere's beliefs. No verdict depends on this.

The exact point of departure. Both laws take the same input and the same closed-circuit data. They differ only in one assumption about something no experiment isolates: what the force between two isolated elements is. Ampere chose "central and strong-third-law". Grassmann chose "the Lorentz force on the element from the Biot-Savart field". Maxwell already noted that closed-circuit experiments cannot settle the element law without such an extra assumption [web: https://en.wikipedia.org/wiki/Amp%C3%A8re's_force_law].

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5. Worked example

We compute three things, each building on the last, and then two short historical checks. Part A is done by hand. Parts B to E sum thousands of elements, so they are done by example.py (run command in its header; Parts A-E here are its PARTs 1-5); every number below came from that run [computed].

Part A. Two single elements (by hand)

Take I = I' = 10 A, ds = ds' = 1 mm = 10^-3 m, distance r = 1 cm = 10^-2 m. The common prefactor is

P = (mu0/4 pi) I I' ds ds' / r^2 = 10^-7 x 10 x 10 x 10^-3 x 10^-3 / 10^-4 = 10^-7 N.

A1. Side by side (both along +x, ds at distance r along y). Then ds . ds' = ds ds', and both elements are perpendicular to r-hat, so the cos theta terms are zero. - Ampere: bracket = 2(1) - 3(0)(0) = 2. Force = -2P r-hat: an attraction of 2 x 10^-7 N. - Grassmann: ds'(ds . r-hat) = 0, so the force is -P r-hat: an attraction of 1 x 10^-7 N. The two laws differ by a factor of 2 for this pair. That is no contradiction: single pairs are not measurable.

A2. In line (both along +x, ds at distance r along x). Now ds . ds' = ds ds', and r-hat . ds = ds, r-hat . ds' = ds'. - Ampere: bracket = 2 - 3 = -1. Force = +P r-hat: a repulsion of 1 x 10^-7 N. (In Ampere's form, cos epsilon + h cos theta cos theta' = 1 - 3/2 = -1/2; with Ampere's old guess k = 0 it would be 1 - 1 = 0.) - Grassmann: ds' x r-hat = 0, so the force is zero.

A3. A "T" (ds' along +x at the origin, ds along +y at (r, 0, 0)). - Ampere: ds . ds' = 0 and r-hat . ds = 0, so the force is zero both ways. The third law holds trivially. - Grassmann on ds (from ds'): ds' is along r-hat, so zero. - Grassmann on ds' (from ds): the unit vector now points from ds to ds', i.e. -x. ds x (-x-hat) = y-hat x (-x-hat) = +z-hat; then x-hat x z-hat = -y-hat. The force is 10^-7 N in the -y direction. So ds feels nothing and ds' feels 10^-7 N: the weak third law fails for this pair. This is the "primitive couple" Ampere objected to [read, Sec. 20.2-20.3].

Part B. A closed loop acting on one element (the null-experiment situation)

Now let the source be a complete circular loop: radius 5 cm, 10 A, in the xy-plane, centred at the origin. The test element (1 mm, 10 A) sits at (3, 4, 5) cm and points along (1, 2, 2)/3. Summing 20,000 loop elements:

Two lessons. 1. Only k = -1/2 stops a closed circuit from pushing an element along its own length. That is exactly what the null experiments detect: a push along the mobile wire would make it rotate. The book reports Ampere's own version of this result: the net action of a closed circuit on an element is perpendicular to the element [read, Sec. 10.1]. The along-element force grows in step with (k + 1/2), as the table shows. 2. At k = -1/2 the two laws give the same force on an element from a complete circuit, not just the same direction. This is a general result, not a coincidence of this geometry: the difference between the two element laws is a total derivative along the source circuit, which sums to zero around a closed loop [memory, high; checked numerically here for one geometry; the physicist reviewer found the same along-element result].

Part C. Two closed loops, and why closed circuits cannot fix k by themselves

Two coaxial loops, both of radius 5 cm, 2 cm apart, 10 A each in the same sense (1,500 elements each):

• Grassmann sum: F_z = -2.742 x 10^-4 N (the minus sign means the loops attract).
• Ampere (k = -1/2) sum: -2.742 x 10^-4 N (agrees to 3 parts in 10^12).
• Exact textbook formula with complete elliptic integrals K and E (from differentiating the mutual inductance): -2.7418 x 10^-4 N. The element sum agrees to 2.5 parts in 10^7, so the code is right.

Now change k but keep Ampere's prefactor (mu0/2 pi):

A tilted, off-centre second loop gives the same ratios in all three force components. So on closed loops, changing k only rescales the force by (2/3)(2 + k). It does not change its direction or how it depends on geometry. In 1822, when the overall constant was not known independently, a loop-on-loop experiment could not have told k = 0 from k = -1/2. This settles a disagreement among the reviewers: the mathematician was right that the force is not k-independent at fixed prefactor; the physicist and contrarian were right that closed-loop data cannot pin down k; and the arbiter's (2 + k)/3 factor is the same result with a different normalisation (the book's SI prefactor, eq. 2.16, is the one used here [read]).

What this means for the bridge

Put Parts B and C together. Treat the bridge as part of one closed circuit (battery, leads, mercury, bridge). By Part B, the force on each bridge element from the whole circuit is the same in both laws. Add these up over the bridge: - In Ampere's law, the bridge's force on itself is zero (central, equal-and-opposite pairs inside a rigid piece cancel), so "whole circuit" = "rest of the circuit". - In the Lorentz law, the bridge's own field is part of the field at each bridge element. Leaving it out would simply be using the law wrongly.

So the net force on the bridge is the same number in both laws, for every geometry. In symbols, writing F_X(source -> bridge) for the force law X gives:

F_Lorentz(whole circuit -> bridge) = F_Ampere(whole circuit -> bridge) = F_Ampere(rest of circuit -> bridge) + 0.

"Field law including the bridge's self-term" equals "Ampere's law without it", exactly. The deep read of Ch. 21 checked the element-by-element identity independently with a 200,000-segment closed tilted ellipse acting on one element: both laws gave (1.28957, -0.89818, -0.66334) x 10^-3 in its units, equal to every printed digit [computed, deep read]. This is why the arbiter changed the verdict on "the field law fails the bridge quantitatively" to Refuted by calculation: no experiment on the net force of a rigid bridge can separate the laws, and Moyssides' 2014 reported agreement (https://link.springer.com/article/10.1140/epjp/i2014-14034-2, abstract only) is no longer needed. The book itself concedes that the field law matches the measured force when the self-term is included [read, PDF p.271]. Caveat: for a thin wire the self-terms grow without limit (logarithmically) as the wire radius goes to zero, so a real numerical calculation must use finite wire thickness and the actual spread of current in the mercury. The equality still holds for any closed, steady current distribution; this primer did not do the full bridge calculation. The book's "self-force" complaint is about bookkeeping (which pairs you count), not about a measured number.

Part D. The 1821 Despretz null experiment, checked (example.py PART 4)

Model the compass needle as Ampere did: a small current loop (radius 1 mm) whose plane is perpendicular to the needle's axis. Put a long straight wire (+-200 m) at c = 5 cm from it, in the vertical plane of the needle's axis, tilted by angle b from the horizontal. Sum Ampere's force with a chosen k over all pairs of elements and take the torque about the vertical thread. The book's eq. 6.7 predicts torque(b)/torque(0) = 1 + [(1 + 2k)/(1 - k)] sin^2 b.

The tiny shortfalls (about 3 parts in 10,000) come from the wire's finite length. The Grassmann (textbook) law gives no dependence on b (ratio 0.9997 at 90 degrees, same finite-length error), as it must: a long straight wire's field at distance c does not depend on how the wire is tilted in that plane.

Now the experiment. The horizontal and vertical wires carry the same current and push the needle in opposite senses [read, PDF p.120]. In units of 1/c, the net torque is (2 + k) - (1 - k) = 1 + 2k. - With Ampere's 1821 belief, k = 0: net = 1, so the needle should turn. It did not. - With k = -1/2: net = 0, as observed.

So the null result Ampere took as a defeat in 1821 actually pins down k = -1/2, the same value he found a year later from a different experiment. That agreement between independent null results is strong evidence that k = -1/2 is the right value within Ampere's family of laws. It does not favour Ampere over the textbook law: Biot and Savart's theory predicted the same null, which is exactly why it embarrassed Ampere in 1821.

Part E. Biot and Savart's bent wire: two errors that cancelled (example.py PART 5)

In 1820 Biot and Savart bent a long vertical wire at a point M into a V, each arm making angle i with the horizontal and leaning away from a magnetic needle at distance c from M (i = 90 degrees is a straight wire). They reported that the action on the needle was proportional to the angle i, and they inferred that each wire element acts on a magnetic pole with a force proportional to sin(omega)/r^2, where omega is the angle between the element and the line to the pole [read, PDF pp.165-167]. That element law is today's Biot-Savart law. But integrating it along the bent wire does not give a result proportional to i:

You can check this by hand. A semi-infinite wire that starts at M produces at the needle a field (mu0 I/(4 pi c)) cot(theta/2), where theta is the angle at M between the wire and the line from M to the needle. Here theta = 180 degrees - i, and cot(90 degrees - i/2) = tan(i/2). Two arms give (mu0 I/(2 pi c)) tan(i/2), which at i = 90 degrees is the familiar straight-wire field mu0 I/(2 pi c).

Savary calculated tan(i/2) in 1823. Biot and Savart then repeated their measurement more carefully, and in Biot's 1824 textbook they reported tan(i/2) while keeping the same element law [read, PDF pp.165-166]. So, as the book puts it, they made two mistakes: they first misread the measurement (i instead of tan(i/2)), and then they wrongly thought sin(omega)/r^2 integrates to i. The two errors cancelled, and the element law survived [read, PDF p.167]. The lesson for this whole subject: an integrated measurement on one geometry can never prove an element law, and an element law must be checked by doing the integral, not by guessing it.

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6. Evidence table

Verdicts are the arbiter's final verdicts, including its QE revision and its Round 2 review (arbiter.md). Round 2 changed no verdicts; it corrected dates and sharpened wording for claims 2, 5 and 6, as reflected below. Claim numbers follow the revised claims.md.

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7. What a learner should take away

What is solid. - Ampere's experimental programme was excellent. Null experiments (nothing moves) are the most reliable kind of measurement, because they need no calibrated force gauge. From them he got the right forces for every closed circuit, decades before Maxwell. Three independent null results (Despretz 1821, no rotation 1822, the closed toroid 1823) all point to k = -1/2 (Parts B and D). - His law and the textbook law agree on every force on every piece of a complete steady circuit (Part B). His prediction that a current pushes along its own conductor, and the bridge's motion, are real. So is the magnet that spins on its own axis, which he was the first to achieve. - The equivalence of a current loop and a magnet (claim 5) is in every textbook today. - His energy argument (distance-only forces cannot keep anything turning against friction, so continuous rotation needs a battery-driven current) is sound and ahead of its time.

What is wrong, and why the mistake is instructive. - The book's inference "the field law gives zero in-line force, so it cannot explain the bridge without an illegitimate self-force" does not hold. The field at a point includes every current, including the one you are standing in. The same goes for the spinning magnet: the "self-torque" is a split of internal forces, and the total torque on the whole closed system is zero in both theories. The lesson: when two theories agree on all closed circuits, an argument that one of them "fails" usually rests on comparing partial sums that neither theory claims are measurable. - "The bridge confirmed k = -1/2" overstates it. The bridge confirmed the sign (in-line repulsion). k came from the null experiments, and n = 2 was asserted in 1822, derived in 1823 and measured only in 1878-1882. Ampere never built his own similarity apparatus, so the claim that the law was "uniquely deduced from experience" is partly a later reconstruction; in 1820 Ampere himself said the element law "could not be found by experiment", and his later method assumes central, equal-and-opposite forces from the start. Keep track of which experiment fixed which number, and of who is speaking (Ampere or his translators). - Biot and Savart got the right element law through two errors that cancelled (Part E). An element law cannot be read off one integrated measurement; you have to do the integral. - Weber's law as a replacement for fields fails because it has no waves; Hertz's radio waves are the test it cannot pass. Note that the book itself only asserts Weber's merits (Ch. 22 has no formula or test); the case for and against Weber has to be read elsewhere.

What is genuinely open. - Whether any measurable quantity differs between the two element laws. Part B suggests that for steady, closed currents the answer is no, since the force on every element is identical. Differences could appear only where currents are not closed or not steady (fast transients, charge build-up), which is where Graneau's exploding-wire experiments sit. Those were run mainly by advocates and need a neutral, controlled test with thermal and magnetic-pressure effects accounted for [arbiter]. - Whether Davy's 1823 mercury experiment really tested Ampere's reaction-torque prediction, and what the cited modern counter-rotation papers measured (none were read here). - How Ampere himself states and justifies the four equilibrium cases (Theorie Sec. 29.2-29.11) and Note 31.1, which this review has not read in depth. (What he believed about the origin of the force is now settled from the text: a preferred electric-fluid mechanism, ether not excluded, formula independent of either; Section 4.) - Whether "strong third law for matter" or "field momentum" is the better picture is partly a matter of interpretation for steady currents. For radiation it is not: light carries measurable momentum.

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8. Check yourself

1. Using Ampere's law in the form cos epsilon + h cos theta cos theta', compute the force factor for two in-line elements with currents in the same direction when k = 0 and when k = -1/2. Which one predicts that the bridge moves?
2. In Part A3, which element feels a force under the Grassmann law, and how big is it? Why does the mainstream view not see this as a breakdown of momentum conservation?
3. Part C shows that the closed-loop force scales as (2/3)(2 + k). Why could Ampere not use two closed coils to measure k, and what kind of experiment did he use instead?
4. Explain in two or three sentences why Ampere's law and the Lorentz law predict the same net force on the floating bridge.
5. In the 1821 Despretz experiment, use eq. 6.7 to find the torques from the horizontal and vertical wires (in units of 1/c) for k = 0 and for k = -1/2. Which value explains why the needle did not move? Does this result favour Ampere's law over the Biot-Savart law?

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9. Further reading

Mainstream - D. J. Griffiths, Introduction to Electrodynamics (4th ed.), Ch. 1 (vector calculus), Ch. 5 (Biot-Savart law, Lorentz force), Ch. 8 (momentum in fields). - R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, Ch. 27 (field energy and field momentum). - J. C. Maxwell, A Treatise on Electricity and Magnetism (1873), Vol. 2, the chapter on Ampere's investigation of the mutual action of currents, where he praises Ampere and explains why closed-circuit data cannot fix the element law [memory, high; book Sec. 20.4 quotes his praise [read]]. - E. T. Whittaker, A History of the Theories of Aether and Electricity, Vol. 1, for the history of Ampere, Grassmann and Weber [memory, high]. - Wikipedia, "Ampere's force law": https://en.wikipedia.org/wiki/Amp%C3%A8re's_force_law ; "Weber electrodynamics": https://en.wikipedia.org/wiki/Weber_electrodynamics

The researchers' side - A. K. T. Assis and J. P. M. C. Chaib, Ampere's Electrodynamics (Apeiron, 2015): the source document. Key sections: 2.1 (the law), Ch. 24-25 (Ampere's first paper, 1820), Sec. 29.1 (his method), Sec. 29.26 (his closing propositions on fluids and ether), Ch. 6 (the 1821 Despretz experiment, eq. 6.7), 7.2.3 (the spinning magnet), 7.5 (deriving k), 8.2-8.3 (bridge and n > 1), 9.1 and 9.6-9.7 (Savary; the bent wire), 10.1 (closed circuit on an element), 10.5 (the similarity experiment), 14.1 (Grassmann), 21.2-21.3 (bridge and spinning-magnet explanations). - A. K. T. Assis and J. P. M. C. Chaib, American Journal of Physics 80, 990-995 (2012), DOI 10.1119/1.4746698, on Ampere's motor; the source of the book's Sec. 21.3 [read, PDF p.271 fn 13]: https://www.ifi.unicamp.br/~assis/Am-J-Phys-V80-p990-995(2012).pdf [web, via the Ch. 21 deep read]. - J. A. Bueno and A. K. T. Assis, Inductance and Force Calculations in Electrical Circuits (2001), cited in the book for the quantitative bridge analysis [read, Sec. 21.2 footnotes]. - P. Graneau and N. Graneau, Newtonian Electrodynamics (1996), the main modern case for Ampere's law and longitudinal forces [read, cited in the book]. - P. G. Moyssides (2014), Eur. Phys. J. Plus, floating-wire measurements compared with both laws: https://link.springer.com/article/10.1140/epjp/i2014-14034-2 (abstract only seen).

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Answers

1. In-line, same direction: epsilon = theta = theta' = 0, so the factor is 1 + h = k. For k = 0 it is 0 (no force, the bridge should not move); for k = -1/2 it is -1/2, a repulsion. Only k = -1/2 (more generally, any negative k) predicts the bridge motion.
2. ds' (the element lying along the line joining them) feels 10^-7 N in the -y direction; ds feels nothing. Single steady current elements cannot exist on their own (charge would pile up at their ends), and for complete circuits the net forces obey the third law. Any imbalance between parts is carried by momentum in the electromagnetic field, which is measured directly in radiation pressure.
3. Changing k only multiplies the closed-loop force by a constant, which could not be told apart from a change in the overall constant of the law (unknown independently in 1822). Ampere instead used null experiments with a mobile, open conductor joined by sliding mercury contacts. There, any k other than -1/2 gives a force along the conductor and hence a rotation, which was not seen (Part B).
4. The bridge is part of a closed circuit. For each bridge element, both laws give the same force from the whole closed circuit (Part B). Summing over the bridge gives the same net force. In Ampere's law the bridge-on-bridge part is zero; in the Lorentz law it is not, but it is part of the definition, not an extra assumption.
5. Horizontal wire: (1 - k); vertical wire: (2 + k), in the opposite sense. For k = 0: 1 and 2, net 1, so the needle should turn. For k = -1/2: 3/2 and 3/2, net 0, so it stays still, as observed. It does not favour Ampere over Biot-Savart: the textbook law also predicts no motion (a long straight wire's torque does not depend on its tilt, Part D). It only fixes k within Ampere's family of laws.