Primer: Kennard, "On Unipolar Induction" (1917)

Study document for E. H. Kennard, "On Unipolar Induction: Another Experiment and its Significance as Evidence for the Existence of the Aether", Philosophical Magazine (6) 33, 179-190 (1917; signed University of Minnesota, August 1916).

Built from arbiter.md (the verdicts used here, including its section 9 "QE revision"), the three reviews, the deep reads, sites.md and research/. No new verdicts are given. Source tags: [read] = in the paper or a saved document (with page); [web] = internet source with URL; [memory] = recalled from training, with confidence; [calc] = recomputed in example.py.

Page numbers. "p.1" to "p.12" are pages of the PDF in the project folder; "p.179" to "p.190" are the printed journal pages. PDF p.N = journal p.(N + 178), so p.10 = p.188. Barnett's 1919 AIEE report (research/files/AIEE-1919-report-electromagnetic-induction.pdf) is cited by its printed journal page, which is the PDF page + 1494 (journal p.1511 = PDF p.17). The contrarian review cites the same report by line of its extracted text ("p.735-745"); those are not page numbers.

Source note. The whole paper was read. Pages 1-9 were deep read against page images. Pages 10-12 were read in the cleaner Internet Archive OCR (research/files/Kennard-1917-PhilMag-33-179-IA-OCR.txt), and the Section 7 equations on p.10 were then checked against the page image (QE deep read D3). After the quality check (QE), four more deep reads were done: the boundary-value problem behind Kennard's calculated charge (D1), the sign convention of the results table (D2), the Section 7 equations (D3), and Kennard's statements on Hertz, Barnett and Fehrle (D4). Some related works were never obtained: Pegram's independent 1917 experiment (Phys. Rev. 10, 591), Fehrle 1913, Kennard's 1913 Phys. Rev. paper and Phys. Zeits. replies, and the 1922 National Research Council (NRC) bulletin. Statements about them are second-hand and labelled so. Two OCR traps matter for the numbers: the calibration deflection is 20.5 mm (the OCR has 26.5), and the solenoid-alone average is +1.2 mm (the OCR has "-12") [read: page images pp.7-8].

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1. The idea in one paragraph

When you spin a metal disk or cylinder near a magnet, the electrons in the metal are pushed sideways by the magnetic field, and a voltage appears between the axis and the rim. This is the Faraday disk, or "unipolar" (homopolar) generator. In 1916 people still disagreed about what the motion is measured relative to. One school said the magnet's field lines are carried around with the magnet like spokes, so only motion of the metal relative to the magnet counts. The other school (Lorentz) said the field is a state of a stationary medium, the aether, and the electrons feel a force because they move through that medium, whatever the magnet does. Kennard built a small capacitor (two nested metal tubes) inside a coil, arranged so that he could spin either one or both. He found that the capacitor charges up when it spins together with the coil, and that spinning the coil alone does nothing. That is the opposite of what the "lines rotate with the magnet" picture predicts and exactly what Lorentz predicts. He then argued that the only thing left for the electrons to move relative to was Lorentz's stationary aether. The experimental part is now textbook physics. The aether conclusion is an interpretation that later physics rephrased without an aether.

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2. What you need to know first

(a) The Lorentz force and motional EMF. A charge q moving with velocity v through a magnetic field B feels a force F = q v x B (SI units), where x is the vector cross product: perpendicular to both v and B, with size |v||B| sin(angle). Dividing by q gives a force per unit charge, which acts on charges like an electric field. Summing (integrating) it along a moving wire gives the "motional EMF" (electromotive force: a voltage that drives charge). Read: D. J. Griffiths, Introduction to Electrodynamics, chapter 5 (Lorentz force) and chapter 7 (motional EMF, Faraday's law) [memory, high confidence on chapter titles].

(b) The Faraday disk and "exceptions to the flux rule". A conducting disk spinning in a uniform axial field develops a voltage between centre and rim, V = ωBR²/2 (ω = angular speed in rad/s, R = disk radius). The usual "flux rule" (EMF = minus the rate of change of flux through a circuit) gets this case right only if you choose the circuit carefully. That is why the disk has puzzled people since Faraday (1831). Read: R. Feynman, Feynman Lectures on Physics, Vol. II, chapter 17, section "Exceptions to the flux rule" [memory, high].

(c) Electrostatics of conductors, capacitance and Laplace's equation. In empty space with no charge, the electric potential φ satisfies Laplace's equation, ∇²φ = 0. Given the potentials on all the metal surfaces, the solution is unique, and Gauss's law then gives the charge on each surface. For two coaxial cylinders of radii a < b and length L (ignoring the ends), the capacitance is C = L / (2 ln(b/a)) in Gaussian units (C = 2πε₀L / ln(b/a) in SI). Read: Griffiths chapter 2 (electrostatics) and chapter 3 (Laplace's equation, uniqueness, separation of variables) [memory, high].

(d) Gaussian units. Kennard uses the older cgs systems, which still appear in many physics texts. Field B is in gauss (1 tesla = 10,000 gauss), flux in maxwells (1 weber = 10⁸ maxwell), potential in statvolts (1 statvolt ≈ 300 volts), charge in electrostatic units (e.s.u.), and capacitance in centimetres (1 cm ≈ 1.11 pF). In Gaussian units the Lorentz force is written F = (q/c) v x B, with c the speed of light. That is why Kennard writes (1/c)[v x B]. Read: J. D. Jackson, Classical Electrodynamics, appendix on units [memory, high].

(e) Divergence and why it matters here. For a vector field f, the divergence div f = ∂f_x/∂x + ∂f_y/∂y + ∂f_z/∂z measures how much "comes out" of a small volume. Gauss's law says div E = 4πρ (Gaussian units; ρ = charge density), so in empty space with no charge div E = 0. Kennard's main theoretical argument (Section 7) is a divergence calculation. Read: Griffiths chapter 1 [memory, high].

(f) Bessel functions (only for the condenser calculation). When you solve Laplace's equation in a cylinder by separation of variables, the radial part satisfies Bessel's equation. Its solutions, J₀ and a second solution that behaves like log r near the axis, play the role that sin and cos play in a rectangular box. You do not need them to follow the main argument, only to check Kennard's printed solution. Read: M. L. Boas, Mathematical Methods in the Physical Sciences, chapter on series solutions and Bessel functions [memory, high on topic, not chapter number].

(g) The aether and special relativity, in one paragraph. Nineteenth-century physics pictured light and electric and magnetic fields as states of a medium, the aether. Lorentz's version (around 1895-1904) made the aether perfectly stationary, with electrons moving through it. Einstein's special relativity (1905) gave the same predictions for experiments in uniformly moving (inertial) frames without a preferred aether frame. The two are empirically equivalent for such experiments, which is why the dispute was partly a matter of interpretation [memory, high]. Special relativity is about uniform translation. Rotation is detectable by local experiments in both pictures (gyroscopes, the Foucault pendulum, the Sagnac effect) [memory, high]. Read: Griffiths chapter 12 (electrodynamics and relativity) [memory, high].

(h) Hertz's theory, the third candidate. Heinrich Hertz (1890) wrote the equations of electrodynamics for moving bodies on the assumption that the aether inside matter is carried along completely with the matter, so a body is never in motion relative to the aether it contains [memory, high; Hertz's paper was not read in this review]. For a spinning condenser this means its electrons do not move through the aether, so nothing pushes them radially. Kennard mentions Hertz only once, in a parenthesis: "according to either Lorentz's or the moving-line theory (but not according to Hertz's) the two cylinders of the condenser should come to different potentials, and the inner one at least should be charged" [read: page image p.2]. So, as Kennard states it, Hertz's theory predicts no charge on a spinning condenser in an energised solenoid. He does not say what it predicts when the solenoid spins alone or when both spin. By 1916 Kennard treated it as out of favour ("there appeared to be little doubt of the correctness of Lorentz's theory", pp.2-3) without giving a reason.

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3. The textbook view

The force. In a frame where the magnetic field is B and a charge moves with velocity v, the charge feels

F = q(E + v x B) (SI), or F = q(E + (1/c) v x B) (Gaussian).

Here E is the electric field in that frame. The velocity v is measured in the frame where E and B are given, which is ideally an inertial frame. A laboratory on Earth is close enough: corrections from the Earth's rotation are of order (v/c)² ~ 10⁻¹² [read: Chyba, Hand & Chyba 2025, research/files].

A spinning conductor in an axial field. Let a metal body rotate rigidly with angular speed ω about the z-axis, inside a uniform field B = B ẑ (ẑ = unit vector along the axis). A point at distance r from the axis moves at speed ωr in the azimuthal direction, so

v x B = ωBr r̂ (r̂ = unit vector pointing radially outward).

The electrons are pushed radially. They pile up until the electrostatic field inside the metal balances this push: E = -v x B. Integrating from the axis gives the potential of a metal point at radius r relative to the axis:

φ(r) - φ(0) = ωBr²/2 = Φ(r)·n,

where Φ(r) = πr²B is the magnetic flux inside radius r, and n = ω/2π is the number of revolutions per second. This is Kennard's rule: "any point of the apparatus metallically connected to earth comes to a potential of Nn e.m.u., where n = rev. per sec. and N = flux encircled by the point in each revolution" [read: p.5].

Does the field rotate with the magnet? For a magnet or solenoid that is symmetric about its axis and carries steady currents, spinning it about that axis changes nothing about B at any point. The field pattern is the same whether or not the source spins. In the standard view, the phrase "the lines rotate" has no physical content for such a field, because nothing in a steady symmetric field marks one line from another. So:

• spinning the conductor in the field gives an EMF (the electrons move through B);
• spinning only the magnet gives nothing (the electrons are at rest, and B at each point is unchanged);
• spinning both together gives the same EMF as spinning the conductor alone.

Chyba, Hand and Chyba (2025) describe this as "a standard part of the electromagnetics literature", citing the Barnett-Kennard-Pegram experiments [read: research/files]. They have a stake in the result, because they use it to propose generating power from the Earth's rotation.

Why an open circuit is needed. In a closed loop (spinning disk plus stationary wires) only the total EMF around the loop is measured, and the "relative motion" and "Lorentz" pictures give the same total. Only by measuring charges on an open conductor can you tell where in the circuit the EMF is generated. Kennard says this himself: "Experimentally, the question can be answered only by observations upon open circuits" [read: p.1].

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4. Where Kennard departs, side by side

Kennard's experimental claims match the textbook exactly. Indeed, his experiment is one of the reasons the textbook says what it says. His departure is in the interpretation (claim 6), and in one older theory he set out to kill (claim 5).

The exact point of departure is the last step of Section 8 [read: pp.189-190, IA OCR]:

"We must suppose each electron in the condenser to experience a radial force proportional to its distance from the axis ... It cannot be due either to the radial acceleration, for that is proportional to the square of the distance, nor to a possible rotation of the electron about its axis, for that should be the same for all electrons, nor to their motion relative to other essential parts of the apparatus, for there is no such motion. Apparently the only remaining alternative is to ascribe the force to motion of the electrons relative to something which is not matter, and which does not share in the rotation of the condenser, and this something must be the stationary æther of Lorentz."

And the conclusion: the result "seems therefore to lend definite support to the existence of an electromagnetic ether", is "perhaps, the only low-frequency phenomenon which cannot easily be described in terms of 'action at a distance' between electrons and atoms", and is "a stumbling-block in the way of those ultra-relativitists who would abandon the conception of an ether altogether" [read: p.190].

Fair statement of what holds. Kennard is right that the electrons must move relative to something that does not rotate with the condenser, and that it is not the magnet. In 1916 Lorentz's aether was the most developed candidate. The textbook names the same "something" differently: the electromagnetic field described in an inertial frame. That frame does not rotate with the condenser, and the field pattern of an axisymmetric source is the same whether or not the source spins. Both descriptions predict identical charges, so no unipolar experiment can tell them apart (arbiter, claim 6c). Kennard's own 1912 paper had said his method "is not capable of distinguishing" Lorentz from Hertz [read: 1912 IA OCR, per Investigator]. His critic Barnett wrote in 1919 that these experiments are "not inconsistent with the principle of relativity of Einstein, which relates to translatory motion only", and that they prove "either (1) that the ether does not exist and that the tubes of induction do not move, or (2) that the ether exists" [read: Barnett 1919, Trans. AIEE, journal p.1511 (PDF p.17), research/files].

Two smaller slips in the elimination argument (arbiter, claim 6): - Centripetal acceleration is ω²r, which is linear in r, not proportional to "the square of the distance" as printed. The step survives for a different reason. An ω² effect is the same whichever way the condenser spins, while the measured charge reverses sign when the rotation reverses [calc, Step 10 of example.py]. - The list of alternatives leaves out "motion relative to the field in an inertial frame". Kennard sets aside "Rotation relative to the earth ... in some unknown way" as "improbable" [read: p.189]. But the Earth-fixed laboratory is very nearly an inertial frame, so that "improbable" option is in effect the modern answer.

The second departure: the Laplace argument (Section 7). Barnett (and Poincaré and Abraham) had argued that even open-circuit experiments could not disprove the moving-line theory. Kennard's rebuttal [read: p.188]: if the moving lines set up the force (1/c)[v x B] in the free aether, that force must be divergence-free there. But for rigid rotation div(v x B) = 2ωB ≠ 0 [calc, Step 9]. So the lines "do not act on the ether, but only on material bodies". The two assumptions Kennard attributes to Barnett are printed as [read: page image p.10, QE deep read D3]

(1) f = (1/c)[V x B] + e + E, (2) div(Kf) = 4πρ,

where, in Kennard's words, f = "total electromotive intensity", e = "electric force due to changes in the magnetic field" (which "vanishes in the present instance"), E = "electric force due to electrostatic causes alone", and ρ = "electric density" (charge per unit volume). V is the velocity of the moving lines and B the magnetic field. K is not defined on these pages; it is presumably the dielectric constant, as in the standard Gaussian equation div(KE) = 4πρ [memory, high]. The reductio as printed: "in free aether we know from electrostatics that div E = 0. Hence in free aether div f = (1/c) div [V x B], and may not vanish, which contradicts (2)." Note "may not vanish": Kennard calls the proof "simple" but does not print it, and Step 9 of the worked example supplies it. The equations are Kennard's summary of Barnett; Barnett's own wording was not read. This argument is valid against the specific version that pairs a line-borne force in free space with equation (2). The algebra alone does not rule out every relative-motion theory. The experiments do that, and Kennard's own hedged sentence puts the weight there: the theory "seems therefore to be definitely disposed of by the results obtained by Barnett, Fehrle, and the author" [read: page image p.11].

The argument is older than the 1917 experiment. Kennard's 1912 paper already says "E' in general fails to satisfy Laplace's Equation" [read: research/files/Kennard-1912-PhilMag-23-937-Unipolar-Induction_IA-OCR.txt, found by the Investigator and confirmed by QE]. Together with his 1912 statement that induction depends "on motion relative to the ether" (Section 6 below), this shows he reasoned from theory to the aether reading before the 1917 data existed.

A textbook correction Kennard got right [read: p.188]. Rotating a dynamo's armature is not the same as rotating its field magnets the other way. The total EMF is the same, but in the first case it is generated in the longitudinal conductors, and in the second "a large fraction of it is developed in the radial parts", so the charge distribution differs. This follows directly from the standard view.

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5. Worked example: from 25 amperes to a 15 mm electrometer deflection

This follows Kennard's prediction chain (Sections 4-5), recomputed, plus the checks the arbiter relied on. Run it yourself:

.venv/bin/python review/Kennard/example.py

Data [read: pp.3-7, page images]: solenoid 22.5 cm long with 1150 turns, current I = 25 A. Outer cylinder diameter 6.64 cm (radius R = 3.32 cm), length 20.5 cm. Inner cylinder mean diameter 1.54 cm (r = 0.77 cm), length 15.1 cm, with guard rings in line. Shaft diameter 1.27 cm. Speed about 2000 RPM, taken as n = 33 rev/s. Total capacitance of inner cylinder + wire + electrometer C = 138 e.s.u. Electrometer sensitivity 18,000 mm per volt.

Step 1. Field in the solenoid. Turns per cm = 1150/22.5 = 51.1. In Gaussian units with current in amperes, B = 4π(turns/cm)·I/10 = 4π × 51.1 × 25/10 = 1606 gauss (0.1606 T). Kennard: 1600. SI check: B = μ₀nI = (4π × 10⁻⁷)(1150/0.225)(25) = 0.1606 T. [calc]

Step 2. Flux through the outer cylinder. Φ = Bπ R² = 1606 × π × 3.32² = 55,600 maxwells. Kennard measured 55,400 with a ballistic galvanometer [read: p.7]. The 0.4% difference is the end-correction of a short solenoid. We use his 55,400 from here on. [calc]

Step 3. Voltage between outer cylinder and axis. V₀ = Φ·n = 55,400 × 33 = 1.83 × 10⁶ e.m.u. = 1.83 × 10⁶ × 10⁻⁸ V = 0.0183 V. Kennard: 0.0183 V. The same number comes from the Lorentz force directly: ωBR²/2 = (2π·33)(0.1606)(0.0332)²/2 = 0.0183 V. The rim moves at 6.9 m/s, so v/c ≈ 2.3 × 10⁻⁸. The effect is first order in v/c, and no relativistic correction is involved. [calc]

So a few hundredths of a volt are spread across a capacitor whose plates are 2.5 cm apart. Kennard needs an electrometer that turns 1 mV into an 18 mm spot movement.

Step 4. Charge on the inner cylinder. This is the hard part. Kennard states the rule for the rotating metal: "any point of the apparatus metallically connected to earth comes to a potential of Nn e.m. units" [read: p.5], which is Step 3's φ ∝ r². On that rule the outer cylinder sits at V₀, the inner cylinder (r = 0.77) at (0.77/3.32)² V₀ = 0.054 V₀, and the shaft at 0.037 V₀. Kennard solved Laplace's equation in the two cylindrical gaps (between outer and inner cylinder, and between inner cylinder and shaft) with Bessel-function series, and took the charge from Gauss's law [read: p.6]:

Q₁ = (-7.62 + 1.40) V₀ = -6.22 V₀ (charge in e.s.u. when V₀ is in statvolts),

"accurate to one-half per cent". The -7.62 is the charge on the outer face of the inner cylinder and +1.40 the charge on its inner face. Two small corrections are part of the chain: +0.03 V₀ for the amber rings that hold the inner cylinder (from Kennard's 1912-13 dielectric papers) and -0.02 V₀ for the connecting wire, which Kennard says "just about cancel" [read: pp.6-7]. The cancellation is asserted, not shown. Whether -6.22 V₀ actually follows from the Nn rule is the subject of Step 6, and the answer is no.

With V₀ = 0.0183/300 statvolt: Q₁ = -6.22 × 0.0183/300 = -0.114/300 e.s.u. = -3.79 × 10⁻⁴ e.s.u. [calc]

Step 5. Electrometer deflection. When the field is switched on with the electrometer insulated, the total charge on the inner cylinder + wire + electrometer is fixed. So the potential shifts by V = -ΔQ₁/C [read: p.4]:

V = 3.79 × 10⁻⁴ / 138 statvolt = 2.75 × 10⁻⁶ statvolt = 0.82 mV, deflection = 0.82 mV × 18,000 mm/V = 14.8-14.9 mm.

Kennard: 14.9 mm. His text prints "0.117 × 18,000 ÷ 138", but only 0.114 gives 14.9 mm (0.117 would give 15.3 mm), so 0.117 is a misprint [calc]. Measured: the four "both rotating" rows give |E| = 14.4, 15.6, 16.4, 18.0 mm, mean 16.1 mm, about 8% above prediction. Speed was not logged, which is why Kennard relied on a ratio instead (Step 7).

Step 6. Checking -6.22 V₀: the main open item, and why claim 2 was downgraded. The theoretical ratio 0.231 (Step 7) is 6.22 V₀ multiplied by calibration factors, so the claim that theory and experiment agree "to 5 per cent" stands or falls with this number. Three checks were made [calc, Steps 5, 6 and 11 of example.py; deep read D1].

• Kennard's printed outer-gap solution, evaluated numerically: V/V₀ = -0.259 + 1.067 ln r - cosh(0.327x)[0.0205 J₀(0.327r) + 0.0352 K₀(0.327r)] - cosh(0.292x)[0.0027 J₀(0.292r) - 0.0111 K₀(0.292r)], where x (cm) is the distance along the axis from the centre and K₀ is his second solution of Bessel's equation, J₀ ln z + z²/4 - ... (not the modern modified Bessel function). It satisfies Laplace's equation, gives V = 0.995 V₀ at the outer cylinder's centre, and gives a charge of -7.58 V₀ on the outer face of the inner cylinder (Kennard: -7.62). The formula is read correctly, and his -7.62 follows from it.
• But that solution holds the inner cylinder at about -0.53 V₀, not at the +0.054 V₀ his own Nn rule gives. Along the whole cylinder the printed solution gives -0.53 V₀ times the same end fall-off as the outer cylinder, and Kennard says his boundary values are in error by more than 0.003 V₀ only near the end plates [read: page image p.6]. So -0.53 is the boundary value he imposed, not a fitting error. A striking coincidence: the constant and log terms at r = 0.77 give -0.259 + 1.067 ln 0.77 = -0.5379, while the Nn rule gives +0.05379: the same digits, with the opposite sign and ten times larger. The printed log coefficient 1.067 is exactly what boundary values -0.5379 and 1.021 produce. A slipped sign and decimal point is one possible explanation, but it is a hypothesis, and it does not explain the inner-face term +1.40 V₀ (see below).
• A re-solution by finite differences. Laplace's equation is solved on a fine grid in each gap (example.py Step 11; the solver reproduces the exact coaxial-capacitor charge to 0.02%). The wall thickness of the inner tube, the guard-ring gaps and the end-plate positions are not printed, so they are assumed. Results:

The last row comes close to Kennard's total, but with a different split (inner face -1.41 V₀, not +1.40 V₀), and the deep reader notes it does not follow from Lorentz's theory as the deep reader understands it. For the inner face to be +1.40 V₀ the cylinder must sit about 0.037 V₀ above the shaft; the Nn rule gives 0.017 V₀ (which yields +0.65 V₀). No assignment tried reproduces both of Kennard's terms from his stated rule.

How to weigh this (arbiter, section 9.1). Two readings are open, and the paper does not let you choose: 1. Kennard used a boundary convention for the inner cylinder that is not in this paper. He may have defined it in his 1912-13 Phys. Zeits. and Phys. Rev. papers, which were not obtained. Then -6.22 V₀ may be right. 2. The printed solution contains a slip, or applies the rule inconsistently. Then the Lorentz prediction is about -4.0 to -4.4 V₀, the theoretical ratio about 0.15-0.16, and the measured 0.242 exceeds it by 50-60% (0.242/0.148 = 1.63) [calc].

An earlier version of this primer argued that "the experiment itself sides with Kennard's number", because 0.242 agrees with 0.231. The arbiter rejected that argument as partly circular, and it is withdrawn here. Using the measurement to choose the calculation assumes that Lorentz's theory is exactly right and that the apparatus has no unaccounted effects (the accidental 10,000 ohm resistance, the rejected data sets, the unlogged speed, the asserted cancellation of the amber-ring and wire corrections). On its own, the measurement cannot tell a wrong calculation from an unmodelled effect.

What follows for the verdicts. The arbiter downgraded the quantitative half of claim 2 from Established to Plausible, untested: the 5% agreement holds only if the boundary assignment that gives -6.2 V₀ is the right one, and no reader can check that from the paper. It is not "Tested, not supported", because the convention reading is not ruled out. If reading 2 is right, the data show an excess of about 50-60% over the Lorentz value computed with the paper's own rule. The arbiter records that as an anomaly worth reporting, without inferring a real effect from it. Claims 1, 3, 5 and 6 do not depend on this number: every assignment except an all-zero one gives a charge of several V₀ of the predicted sign when the condenser turns, and the moving-line theory predicts zero.

What would settle it (arbiter, section 9.1): read Kennard's Phys. Rev. 1, 355 (1913) and Phys. Zeits. papers for his convention; derive independently the potential of an insulated inner cylinder joined to a stationary electrometer through a sliding contact on the axis, then rerun Step 11 with that value; read Pegram's 1917 paper for an independent ratio; or repeat the experiment with the speed logged and an error budget.

Step 7. The speed-independent test ratio. Kennard divided the electrometer reading E by the reading G of a galvanometer fed from a small Faraday disk on the same shaft. Both scale with speed × field, so the ratio cancels them. Calibrating both instruments against a known 0.02 V source [read: p.8, page image]:

E/G (theory) = (20.5/62.2) × (0.1/0.1197) × (6.22/7.42) = 0.2308.

Here 20.5 and 62.2 mm are the calibration deflections; 0.1 is the fraction of the calibration voltage sent to the galvanometer; 0.1197 is the Faraday disk's flux as a fraction of the outer cylinder's; and 7.42 is the electrometer case's coefficient of influence. With the OCR misreading 26.5 the result would be 0.298, so the page image matters [calc].

Step 8. How well does it agree? Measured "both rotating" ratios: 0.237, 0.237, 0.236, 0.258. Mean 0.2420, standard deviation 0.0107, standard error 0.0053 (2.2%). The offset from 0.2308 is +4.8%, or 2.1 standard errors. Other groups: half speed 0.228 (-1.2%), alternative method 0.238 (+3.1%); the three pooled come to +2.2%. The 13 A and 25 A check groups gave 0.25 and 0.27, an 8% gap Kennard does not discuss [read: p.9; calc]. Solenoid alone (sign-weighted rows -0.4, +0.6, +1.1, +3.5 mm): mean +1.2 mm, standard error 0.83 mm, about 1.4 standard errors from zero and 7.5% of the 16.1 mm full effect [calc]. Its speed was not recorded, because the speed-monitor disk sits on the condenser shaft, which was not turning. The galvanometer readings in those rows are G ≈ 0 (+0.5, -0.3, -0.6, +0.2 mm row by row; reversing average 0) [read: page image p.8].

Noise of single readings. Each tabulated value comes from a triple of raw electrometer readings with the drift removed by (d₁ - 2d₂ + d₃)/6. The raw triples Kennard prints swing by tens of millimetres: +8.5, +4.8, +3.7 and +22.4, +13.8, +1.8 mm for two solenoid-alone rows, which reduce to 0.4 and 0.6 mm; and +5.1, -39.1, +3.3 and -12.6, +29.9, -21.5 mm for two "both" rows, which reduce to 14.4 and 15.6 mm [read: page image p.8; reduction checked in deep read D2]. So the drift removal does most of the work, and individual raw readings in the null rows move by up to about 20 mm. Kennard states no single-reading noise figure. An earlier solenoid-alone group "ranged from .8 to 2.7 mm., and gave a reversing effect of only +.2 mm." [read: p.8].

Sign of the +1.2 mm residual. Kennard never says which physical sense "+" means for the rotation S or the current I [read: p.8]. Assume S means the same sense in both blocks of the table (the natural reading of one table with one definition). In the "reversing average" convention the "both" rows average +16.1 mm. Turning the solenoid alone at +ω is, relative to the magnet system, the same as turning the condenser at -ω, so the moving-line theory predicts about -15 to -16 mm for the solenoid-alone rows. The observed +1.2 mm has the opposite sign and about 8% of that size. On this reading the residual is not a weak trace of the moving-line effect. This is an inference by the deep reader and the arbiter, resting on the one assumption about S; Kennard calls the result "nil" and does not discuss its sign. Whether the residual fits Su's end-effect proposal (Section 6) cannot be decided, because Su's predicted sign is not stated in what was read.

What the numbers teach. The competing theory predicts a ratio of zero for "both rotating" and about -15 mm for "solenoid alone". The measurements are 0.24 and about +1 mm. The discrimination is decisive whatever the correct theoretical charge (Step 6). The quantitative agreement with Lorentz is a separate question: on Kennard's number it is a few percent, on the Nn-rule number it is 50-60%, and the paper does not settle which. Kennard himself called the results "somewhat rough" [read: p.9].

Step 9. Kennard's divergence argument. With v = ω(-y, x, 0) and B = (0, 0, B), v x B = ωB(x, y, 0), so div(v x B) = 2ωB ≠ 0 [calc, SymPy]. A charge-free vacuum field must have zero divergence, so v x B cannot be such a field in free space.

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6. Evidence table

Related and later work kept in the record (from the Investigator; leads, not verdicts): - Barnett 1912-1913 called Kennard's 1912 experiment "inconclusive" and "based on incorrect theory", and wrote "Mr. Kennard's fallacies are quite apparent" [read: Barnett PDFs, research/files, as quoted by the Investigator]. That is a rival's characterisation. His technical objection (Phys. Zeits., Sept. 1912, p.803), that the 1912 null "might conceivably be due to the non-rotation of the solenoid" because only the iron core turned, was fair, and the 1917 design answers it. Kennard's answer, in his words: the objection "is valid, but seems decidedly weak", because it "assumes a radical difference to exist between induced and permanent magnetism, for which there is no other evidence; it even requires that the remanent magnetism shall not acquire the inducing properties of permanent magnetism until the magnetizing force is removed, for the remanence in the bar employed amounted to over 25 per cent" [read: page image p.2]. That is the experimenter's own view of an objection to his work. Kennard's concession in the same passage: the 1912 rotating magnet "was necessarily enclosed in a second metallic screen of small diameter, and this might conceivably act so as to cut off the effect" [read: page image p.2]. By his own account, then, the 1912 null could have been screening rather than absence of induction, which is the stated reason for the 1917 apparatus. Kennard says Barnett later maintained his view "with some heat" [read: p.9]. - Kennard 1912 (Phil. Mag. 23, 937): a steel bar rotated inside a stationary solenoid, with an outer cylinder watched for charge. Result 0.0 ± 0.24 divisions against about 37 predicted by the moving-line theory. Before quoting that figure, Kennard subtracted "an effect apparently due to electromagnetic induction when the magnetization was reversed" [read: research/files, 1912 IA OCR, lines 195-199; confirmed by QE]; Su (2005, below) builds on this subtraction. Kennard already concluded then that a rotating magnet "does not 'carry its force lines' around with it" and that induction depends "on motion relative to the ether" [web: https://archive.org/details/londonedinburg6231912lond; research/files]. The same paper already contains the Laplace argument: "E' in general fails to satisfy Laplace's Equation" [read: 1912 IA OCR, line 136]. So both his aether reading and his Section 7 argument predate the 1917 data. - Fehrle 1913 (Ann. d. Phys. 42, 1169) "reported confirmatory observations with a rather different arrangement" (solenoid rotation alone gives no charge). Kennard adds that the work "is marred by several results which must certainly be wrong: one of his results even contradicts Faraday's law for closed circuits!" [read: page image p.2]. Kennard gives no details, Fehrle's paper was not read, and Kennard is a party to the dispute, so this is his characterisation of a third party's work. Fehrle's reported finding agrees with Kennard's own conclusion, so the criticism is not aimed at an opposing result. - Su 2005 (arXiv physics/0512179) argues that a short, non-uniform solenoid could leave a small residual effect on a stationary capacitor, and builds on the effect Kennard subtracted in 1912 [read: research/files]. Su is the author of a rival "local-ether" theory. Kennard's 22.5 cm solenoid is short, and a 1.2 ± 0.8 mm null leaves room for a small effect, so this is an open flag for claim 3. The +1.2 mm residual has the opposite sign to the moving-line prediction (Section 5, Step 8); whether it has the sign Su's proposal needs is not known. - Against Kennard's interpretation: an unidentified abstract. A Semantic Scholar record attached to Kennard's 1912 title carries the abstract "We describe undisputable experiments that categorically disprove absolutistic Kennard-Bartlett-type interpretations of unipolar induction" [web: Google snippet, research/2026-10-04-crossref-semanticscholar.md]. It cannot be Kennard's own abstract; the paper it belongs to is not identified (probably in the Bartlett or Guala-Valverde line). This is that source's characterisation, and its stake is unknown. It bears on the absolute-rotation reading (claim 6), not on the charge measurement, and it has not been assessed. - Kholmetskii 2003 and a "calculational error". A. L. Kholmetskii, "One century later: Remarks on the Barnett experiment", Am. J. Phys. 71, 558 (2003; an erratum exists) [web: research/2026-10-04-google.md]. Chyba, Hand and Chyba (2025) write that "a subsequent attempted theoretical refutation of these results was shown to be incorrect due to calculational error" (their refs 22-24) [read: research/files, via research/2026-10-04-aps-journals.md]. Which paper they mean is unverified; neither Kholmetskii nor refs 22-24 were read, and Chyba et al. have a stake in Earth-rotation power. This is a modern re-analysis of the Barnett-type experiment with a disputed exchange around it. It cuts against anyone relying on a refutation of the Barnett-Kennard result, and it is not evidence on Kennard's numbers. - Pegram's 1917 "reminder". A snippet from A. I. Miller's essay "Unipolar induction: a case study of the interaction between science and technology" (1981; doi 10.1007/978-1-4684-0548-4_3) says that in 1917 Pegram reminded Barnett and Kennard of something; the text is truncated and the essay was not reached [web: Google snippet, research/2026-10-04-google.md]. It bears on the open question of priority for the co-rotation result. - A 1924 engineering thesis. A University of Wisconsin College of Engineering B.S. thesis volume (1924) lists Kennard 1917 and a 1913 Kennard paper in its bibliography, p.70 [web: https://www.google.com/books/edition/_/KalsAAAAMAAJ?gbpv=1&bsq=unipolar]. It was not read. It shows the paper was taken up in an engineering setting within seven years, which counts against any idea that the result vanished from the literature (as distinct from Kennard's own output; Section 7). - Baumgärtel and Maher 2022 (Sci. Rep.): some authors invoke Ampère's force for the Kennard/Bartlett/Barnett experiments, while others hold classical explanations suffice, "albeit a full re-analysis ... with due regard to the influence of the closing wire might be necessary" [read: research/files]. This bears on torque-type interpretations, not on the charge measurement. - Weber/Ampère "Galilean" electrodynamics (Wesley 1990, 1997; Assis & Thober 1994; Bartocci & Capria 1991) and the spinning-magnet literature (Bartlett et al. 1977; Guala-Valverde 2002; Müller 2014). These share Kennard's view that open-circuit results are not trivially relativistic. Mainstream sources often call them fringe, which is those sources' characterisation [per Investigator, sites.md]. - Earth-rotation power (Chyba & Hand 2016; Chyba, Hand & Chyba 2025): the same "the field does not co-rotate" principle applied to a conductor co-rotating with the Earth. Contested in a 2026 Comment/Reply, with a null reported by Veltkamp and Wijngaarden [per Investigator; not independently checked]. This is a live modern descendant, but not evidence either way on 1917. - Millis 1996 (NASA Breakthrough Propulsion) cites Kennard in a Mach's-principle context. That is a citation, not a test.

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7. What a learner should take away

Solid. - The force on a charge is q(v x B), with v measured in the frame where B is given. A steady, axisymmetric field does not "rotate with its magnet" in any measurable sense. Kennard's and Barnett's open-circuit experiments are among the original evidence for this, and the result has stood for over a century. - The design logic is excellent and worth copying. Pick a configuration (both spinning) where the rival theory predicts zero and yours predicts a definite number. Then build in reversals (current direction and rotation sense) so that drifts that do not flip with both cancel out. Finally, measure a ratio against a monitor that shares the same speed and field, so the uncontrolled variables drop out. - The order of magnitude is a nice self-check: 1600 gauss, 33 rev/s and a 3.3 cm radius give 18 mV. That is why the whole experiment lives at the millivolt level.

Wrong, and why it is instructive. - The aether inference. Kennard eliminated candidates for what the electrons move relative to and found one survivor. But he did not list "the field, as described in an inertial frame", which gives the same predictions without a material medium. An elimination argument is only as good as its list of candidates. Equally instructive: two theories that predict the same numbers for every experiment you can do cannot be told apart by those experiments, however strongly one feels. - The "square of the distance" sentence. Centripetal acceleration is ω²r, linear in r. The slip did not matter, because the sign reversal already rules out that cause. Checking each step's scaling is cheap and catches such slips. - Precision claims. A theory error of ±0.5% means little when single sets scatter by ±10-15% and two check groups differ by 8%. Always compare a claimed agreement against the data's own spread. Here the offset is 2.1 standard errors. Note that Kennard was candid about the roughness, the rejected sets and the abandoned run. - Unprinted boundary values. Kennard's -6.22 V₀ cannot be checked from the paper, because he did not print the potentials he imposed, and a re-solution using the rule he did print gives about -4.0 V₀ (Section 5, Step 6). A calculation that a reader cannot reproduce is a weak support for a "5 per cent" agreement, however careful its author. And do not use the measurement to vouch for the calculation it is supposed to test. - The Laplace argument defeats one specific formulation, not every possible relative-motion theory. A proof that refutes an opponent's stated assumptions is not a proof that no theory of that kind can work.

Genuinely open (from the arbiter's open questions): - The main open item: is -6.22 V₀ right? The printed solution reproduces -7.62 V₀ for the outer face, but only because it holds the inner cylinder at about -0.53 V₀. Kennard's own Nn rule gives +0.054 V₀ there and a total charge of about -4.0 to -4.4 V₀, which would put the measured ratio 50-60% above the Lorentz value (Section 5, Step 6). Either Kennard used a convention not stated in this paper, or the printed solution contains a slip. His 1913 Phys. Rev. paper and an independent derivation of the inner cylinder's potential would decide it. Until then the 5% agreement is "Plausible, untested" (arbiter, revised). - Which sign convention does S carry in the results table, and is the +1.2 mm residual really of the opposite sign to the moving-line prediction? - What was the "attempted theoretical refutation" Chyba et al. mention, and what did Kholmetskii 2003 argue? Which paper does the "categorically disprove" abstract belong to? - How large is any end-effect residual for a short solenoid (Su 2005)? Kennard's null is only about 10%. - What did Pegram (1917) measure, and who has priority for the co-rotation result? (Miller 1981 mentions a 1917 Pegram "reminder" to Barnett and Kennard; not read.) - Why does Kennard's unipolar line stop in 1917? He did not vanish. He co-wrote the 1922 NRC bulletin Electrodynamics of Moving Media, published the 1927 paper behind the "Kennard" uncertainty relation, and later did kinetic theory and Navy hydrodynamics at the David Taylor Model Basin [web: AIP finding aid, https://history.aip.org/ead/20010114.html, per Investigator]. Competing explanations the arbiter set out: (a) the experimental question was seen as settled, since Barnett accepted the result by 1919; (b) apparatus and circumstance: the solenoid failed and he left Minnesota; (c) the field moved on to relativity and quantum theory; (d) he changed his view on the aether, which is unknown either way; (e) suppression or blocking, for which no evidence was found: the dispute was carried on openly in Phil. Mag. and Phys. Rev., and the paper was abstracted in German journals in 1917. Records that would decide: his 1922 NRC chapter, his 1926 Cornell electricity-and-magnetism lecture notes (AIP AR 176, Box 4 Folder 4), and a Cornell memorial statement.

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8. Check yourself

1. A copper disk of radius 10 cm spins at 50 rev/s in a uniform 0.5 T axial field. What voltage appears between axis and rim? What is it if instead the magnet spins and the disk is still?
2. In Kennard's table, the "both rotating" deflections are +14.4, -15.6, +16.4, -18.0 mm for the four combinations of rotation sense and current direction. Why does it matter that the sign flips, and which of Kennard's arguments uses this?
3. Show that div(v x B) = 2ωB for rigid rotation in a uniform axial field. Then explain in one sentence why this alone does not refute every theory in which "only relative motion counts".
4. Kennard's measured ratio is 0.242 ± 0.005 (standard error) against a predicted 0.231. The moving-line theory predicts 0. Which comparison carries the scientific weight, and why?
5. Name an alternative to "the stationary aether of Lorentz" that Kennard's elimination list leaves out, and say why no unipolar experiment can choose between the two.

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9. Further reading

Kennard's own works - E. H. Kennard, "On Unipolar Induction: Another Experiment and its Significance as Evidence for the Existence of the Æther", Phil. Mag. (6) 33, 179-190 (1917). The project document; clean OCR in research/files. - E. H. Kennard, "Unipolar Induction", Phil. Mag. (6) 23, 937-941 (1912). The first experiment (research/files; https://archive.org/details/londonedinburg6231912lond). - E. H. Kennard, "The Effect of Dielectrics on Unipolar Induction", Phys. Rev. 1, 355-359 (1913), and Phys. Zeits. 13, 1155 (1912), 14, 256 (1913). His replies to Barnett. The 1917 paper's Section 7 footnote cites the Phys. Rev. paper ("May 1913, p. 355") as the full discussion [read: page image p.10]. Not obtained; these are the first place to look for the boundary convention behind -6.22 V₀. - W. F. G. Swann, J. T. Tate, H. Bateman, E. H. Kennard, Electrodynamics of Moving Media, NRC Bulletin (1922). Full view on HathiTrust per the Investigator; the key record for his later views.

The dispute and its sequel - S. J. Barnett, Phys. Rev. (I) 35, 323 (1912), and Phys. Rev. 2, 323 (1913) (research/files). - S. J. Barnett, "A Report on Electromagnetic Induction", Trans. AIEE (1919), journal pp.1495-1513 (research/files). - G. B. Pegram, "Unipolar Induction and Electron Theory", Phys. Rev. 10, 591 (1917). Not obtained. - Fehrle, Ann. d. Phys. 42, 1169 (1913). Not obtained. - A. L. Kholmetskii, "One century later: Remarks on the Barnett experiment", Am. J. Phys. 71, 558 (2003). Not obtained. - A. I. Miller, "Unipolar induction: a case study of the interaction between science and technology" (1981), doi 10.1007/978-1-4684-0548-4_3. Not obtained. - H. Hertz, "Über die Grundgleichungen der Elektrodynamik für bewegte Körper", Ann. d. Phys. (1890), reprinted in English in his Electric Waves [memory, medium on the bibliographic details]. Not read in this review. - K. T. McDonald, "Is Faraday's Disk Dynamo a Flux-Rule Exception?" A long historical bibliography of the 1910s-1920s debate (research/files). - C. F. Chyba, K. P. Hand, T. H. Chyba, Phys. Rev. Research 7, 013285 (2025) (research/files). - C. C. Su, arXiv physics/0512179 (2005); C. Baumgärtel and S. Maher, Sci. Rep. (2022) (research/files).

Mainstream textbooks - D. J. Griffiths, Introduction to Electrodynamics: chapters 3 (Laplace's equation), 5 (magnetostatics, Lorentz force), 7 (electrodynamics, motional EMF) and 12 (relativity) [memory, high]. - R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, chapter 17 (the laws of induction, including exceptions to the flux rule) [memory, high]. - E. M. Purcell and D. J. Morin, Electricity and Magnetism (3rd ed.): induction and the relativity of fields. - J. D. Jackson, Classical Electrodynamics: the appendix on units, for Gaussian-SI conversion.

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Answers

1. V = ωBR²/2 = (2π × 50)(0.5)(0.1)²/2 = 0.785 V. If only the magnet spins, the field at each point is unchanged and the disk's electrons are at rest, so zero. This is the content of Kennard's and Barnett's solenoid-alone null.
2. A real induction effect must reverse when either the current (field direction) or the rotation sense reverses. Spurious effects such as thermal drift, contact potentials or friction charging usually do not flip with both. Grouping readings in fours and keeping only the part that reverses with both removes them [read: p.5]. The sign reversal also rescues Kennard's elimination step: a centripetal (ω²) effect cannot reverse with rotation sense, so it cannot be the cause.
3. v x B = ωB(x, y, 0), so div = ωB + ωB = 2ωB. This shows that a line-borne force cannot act as a field in the charge-free aether while Gauss's law also holds there. A relative-motion theory in which the lines act only on material bodies escapes the argument, as Kennard concedes ("the lines do not act on the ether, but only on material bodies"). It is the experiments that rule that version out.
4. The comparison with zero. The rival theory predicts no charge at all when both spin, and the observed ratio is about 0.24, roughly 45 standard errors from zero. That comparison holds whatever the exact theoretical charge. The 5% agreement with 0.231 is weaker on two counts: at 2.1 standard errors it is only marginally consistent with the stated precision, and 0.231 itself rests on -6.22 V₀, which Kennard's own Nn rule does not reproduce (Section 5, Step 6; on that rule the theory gives about 0.15).
5. "Motion relative to the electromagnetic field as described in an inertial frame" (in practice, the nearly inertial Earth-fixed laboratory). Lorentz's aether theory and special relativity give the same Lorentz force, and hence the same charges, for this apparatus. Since the predictions are identical, no measurement of the charge can favour one over the other.