Fedderson-Daedalus: Primer

Source gap (read this first). The folder is named after Berend Wilhelm Feddersen, but none of Feddersen's own papers are in it. Both PDFs are Google Books scans. Step 0 has since produced OCR text for 405 of 406 and 26 of 27 pages, and the tables that matter were checked against rendered page images (two independent full reads of Trowbridge and Sabine; one full read of Rich). Tags [read: OCR] mean the OCR text only, not checked against the page image. The remaining gap is Feddersen's own papers.

• "Daedalus-Fedderson-109.pdf" is not Feddersen's work and not the journal Daedalus. It is the Proceedings of the American Academy of Arts and Sciences, vol. 25 (1890). Two papers in it matter most here: Trowbridge and Sabine, "Electrical Oscillations in Air" (pp. 109-123), and Trowbridge, "Motion of Atoms in Electrical Discharges" (pp. 192-194). Page numbers in this primer are journal pages unless marked "PDF p.". The same volume holds about twenty other papers; four are relevant background and appear in sections 3 and 7: Safford and Hofman on capacity at telephone frequencies, Nott on condenser dielectrics, A. E. Dolbear on flexed ring magnets, and a memoir of J. P. Joule. Nott, Dolbear and the Joule memoir have since been read in full from page images (second QE round) [read: deep read]; Safford and Hofman is still read only through the digest and OCR text.
• "Oscillatory_spark_discharges_between_unl.pdf" is D. L. Rich, "Oscillatory Spark Discharges between Unlike Metals", Physical Review 10, 140-155 (1917). It was a University of Michigan dissertation [read].
• Feddersen's own papers appeared in Poggendorff's Annalen between 1858 and 1862 (cited in the 1890 footnote [read: page image p.109]). Everything said about Feddersen himself rests on that citation and on [memory, high] or [web] https://en.wikipedia.org/wiki/Berend_Wilhelm_Feddersen. To get the full picture you would need those Annalen papers.

Tags: [read] = taken from the documents (page images unless marked OCR); [web] = Internet source with URL; [memory] = recalled from training, with a confidence level; [calc] = my own calculation, reproduced in example.py.

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1. The idea in one paragraph

Charge a Leyden jar (an early capacitor) and let it discharge across a small air gap. You see one bright spark. In the 1850s Feddersen photographed that spark in a fast-spinning mirror, which smears it out in time, and found that it is not one flash. It is a row of flashes: the current sloshes back and forth between the jar and the wire loop many times, like a mass bouncing on a spring, before it dies away. This "oscillatory discharge" is the root of spark radio. The folder holds two later studies built on that discovery. In 1890 Trowbridge and Sabine at Harvard measured the timing of the flashes very carefully, believed it was a few percent off the textbook formula, and blamed a kind of internal friction in the air ("dielectric viscosity"). Re-examined today, their own tables agree with the formula once a stray capacity they listed themselves is included, though the capacitance they quote cannot itself be checked from their printed condenser dimensions to better than about 15%, so the experiment was simply not accurate enough to show a few-percent departure either way. Trowbridge also asked whether metal atoms from the electrodes are thrown back and forth across the gap with each swing, and found they are not. In 1917 Rich, at Michigan, used the same photographic trick on gaps between two different metals (iron, bismuth, zinc, copper). He found the spark starts more easily in one direction than the other, so a gap of unlike metals acts a little like a one-way valve (a rectifier). From this he ranked the metals Fe, Bi, Zn, Cu.

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2. What you need to know first

Capacitors and inductors. A capacitor stores energy in an electric field: charge q on it gives voltage V = q/C, with C the capacitance in farads (F). An inductor (a coil or simply a loop of wire) stores energy in a magnetic field and resists changes of current: the voltage across it is L dI/dt, with L the inductance in henries (H) and I the current. Read: Griffiths, Introduction to Electrodynamics (chapters on electrostatics and electrodynamics; Faraday's law and inductance), or Purcell and Morin, Electricity and Magnetism.

Second-order linear ODEs and damping. The equation m x'' + b x' + k x = 0 (mass, friction, spring) describes a damped oscillator. If friction is small it rings at angular frequency close to sqrt(k/m), and its amplitude decays as exp(-b t / 2m). The series LCR circuit is the same equation with L in place of m, R in place of b and 1/C in place of k. Read: Boas, Mathematical Methods in the Physical Sciences (ODE chapter); Taylor, Classical Mechanics (oscillations chapter).

Old CGS (Gaussian) units. Nineteenth-century papers measure capacitance in "centimetres" (electrostatic units, ESU) and inductance in "centimetres" too (electromagnetic units, EMU). The two systems differ by a factor of the speed of light. Conversions: 1 cm of inductance = 1 nH; 1 cm of capacitance = 1.113 pF. If you plug CGS numbers into an SI formula without converting, you are off by a factor of c. Read: the units appendix of Jackson, Classical Electrodynamics, or the appendix on units in Griffiths.

The rotating mirror. If a plane mirror turns at angle θ, the reflected beam turns by 2θ. A mirror spinning f times per second sweeps a reflected spot across a screen at distance D at speed u = 2 × (2πf) × D. Events separated in time by Δt land Δx = u Δt apart on the photographic plate. This is how Feddersen, Trowbridge and Sabine turned a timing problem into a length measurement. (Standard optics; any intro optics text, e.g. Hecht, Optics, chapter on reflection.)

Electrical breakdown of air (sparks). Air insulates until the field is high enough (about 3 MV/m, i.e. 3 kV per mm, at sea level for gaps of a few mm [memory, high]) for free electrons to gain enough energy between collisions to ionise molecules. Each new electron does the same, giving an avalanche. Positive ions striking the negative electrode (the cathode) knock out more electrons; how efficiently they do this depends on the cathode metal and its surface. That is the one place the electrode material enters the breakdown condition [web] https://en.wikipedia.org/wiki/Paschen%27s_law. Once the spark forms, its channel is a hot, conducting plasma whose resistance changes rapidly with time and current. Read: Raizer, Gas Discharge Physics; for a gentle start, the Paschen's law article above.

The Doppler effect for light, and the work function. A light source moving towards you at speed v (much less than c) has its wavelength λ shortened by Δλ = λ v / c. The work function φ of a metal is the minimum energy needed to pull one electron out of its surface, a few electronvolts (eV) for most metals. Read: any intro modern-physics text, e.g. Krane, Modern Physics, or Serway, Moses and Moyer, Modern Physics (photoelectric effect chapter).

Dielectrics and polarisation. Putting an insulator between capacitor plates raises the capacitance by a factor K, the dielectric constant (relative permittivity), because the material's charges shift slightly and partly cancel the field. Some of that shifting is instant (electrons in atoms), some is slower (polar molecules rotating, charges drifting to impurities and grain boundaries). The slow part cannot keep up with fast changes, so K, and with it the capacitance, can be smaller at high frequency than when measured slowly. Read: Griffiths (chapter on electric fields in matter); Purcell and Morin (dielectrics chapter).

Counting statistics. If events occur at random at a steady average rate, the number counted in a fixed window follows a Poisson distribution, whose standard deviation is the square root of its mean. To ask whether two rates differ, divide the difference by the combined standard error. Read: Taylor, An Introduction to Error Analysis (the Poisson chapter).

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3. The textbook view

The series LCR circuit. A capacitor C, charged to voltage V0, is connected through an inductance L and a resistance R. Kirchhoff's voltage law around the loop, with q the charge on the capacitor and I = dq/dt the current, gives

L d²q/dt² + R dq/dt + q/C = 0.

Try q = exp(s t). Then L s² + R s + 1/C = 0, so

s = −R/(2L) ± sqrt( R²/(4L²) − 1/(LC) ).

• If R < 2 sqrt(L/C) the square root is imaginary and the charge oscillates: q(t) = q0 exp(−t/τ) cos(ω t), with decay time τ = 2L/R and angular frequency ω = sqrt(1/(LC) − R²/(4L²)).
• The ideal (lossless) period is T0 = 2π sqrt(LC). This is the formula William Thomson (Lord Kelvin) derived in 1853 [memory, high; not web-checked, no verdict depends on it] and Feddersen confirmed photographically.
• It is handy to define the characteristic impedance Z0 = sqrt(L/C) (in ohms) and the damping ratio ζ = R/(2 Z0). Then the actual period is T = T0 / sqrt(1 − ζ²), and the amplitude shrinks by a factor exp(−2πζ / sqrt(1 − ζ²)) every period.

So resistance does two things at once: it makes each swing slightly slower, and it makes the swings die away. You cannot have one without the other. Keep that in mind for section 5.

In 1890 units. Trowbridge and Sabine write the same formula as t = (2π / v) sqrt(LC), with L in EMU centimetres, C in ESU centimetres, and v = 3 × 10¹⁰ cm/s, "the ratio between the electrostatic units and the electromagnetic units" [read: page image p.110]. The 1/v is exactly the unit conversion, and v is the speed of light.

Sparks. A real spark gap is not a fixed resistor. Before breakdown it is an open switch. After breakdown its resistance falls as the channel heats, and near each current zero the channel can cool and stop conducting until the voltage rises enough to restrike. The energy lost in the gap is what eventually ends the train of flashes [memory, medium]. Air itself, as the insulator between capacitor plates, is a nearly perfect dielectric: relative permittivity about 1.0006 and negligible energy loss at kilohertz to megahertz frequencies [memory, medium; not web-checked].

Box: dielectric absorption in solids, and why air is different. Charge a glass or paper capacitor, short it briefly, then leave it open: the voltage creeps back up. Part of the charge had soaked into slow polarisation in the solid, and it comes out over seconds to hours. This is dielectric absorption (also called "soakage"); it is real, standard, and the reason the capacitance of a solid-dielectric condenser depends on how fast you charge it [memory, high]. The 1890 volume itself documents it three times: - Safford and Hofman find that the capacity of cable insulators "varied with rate" of charging, so slow galvanometer measurements do not predict telephone-frequency behaviour [read: digest, PDF pp.15-18]. - Nott, comparing condensers made with solid dielectrics (paraffined paper, hard rubber, mica; no air, nothing near megahertz), finds residual discharges of 2-27% of the full charge coming back in instalments over intervals of 1 to 28 minutes, and rising again after a wait (his No. 72: 0.1 at 2-minute intervals, then 1.2, 1.5 and 3.0 after 6, 22 and 28 minutes). Capacity rises with charging time, by 16-85% for leaky units and 1.6-5.5% for good ones between 1 ms and 30 s [read: deep read, page images PDF pp.287-296]. That is the textbook signature of dielectric absorption [memory, high], though Nott never uses the word. He adds: "When as short a time as 0.002 sec. was tried in charging Nos. 14 and 15, the discharge exceeded the charge, but the results were too irregular to warrant any definite conclusions" [read: page image PDF p.290, via the deep read]. He gives no numbers for it, and his one footnoted "apparent gain" (No. 9, 0.5%) is within scatter. Getting out more than you put in sounds like over-unity. The most likely reading, which is the reviewers' interpretation and not Nott's, is absorption: charge soaked in during earlier runs leaks out and is counted in the later discharge. A mismatch between the two galvanometer methods (charge measured one way, discharge another) is not excluded either. Nott, to his credit, drew no conclusion. - Trowbridge and Sabine's own starting point is that "the measurements of capacity by the ordinary slow methods form no criterion of the capacity of glass under electrical influences which last but three millionths of a second", which is why they switched to an air condenser [read: p.110].

All of this concerns solids (glass, paper, rubber, mica) and polar substances. Air is a thin gas of non-polar molecules (N₂, O₂) with K − 1 ≈ 5.4×10⁻⁴ and no measurable dispersion at MHz [memory, high]. Trowbridge and Sabine cited Boltzmann and Klemenčič, who found that √K matches the optical refractive index for simple gases but not for complicated molecules such as sulphur dioxide and ethyl bromide [read: p.121]. Those are polar molecules whose rotation cannot follow fast fields [memory, high]. Their own source therefore points against a rate-dependent capacity for air. The mistake was to carry a genuine effect in solids over to a gas where it is negligible.

Polarity and electrode material. In the Townsend picture of breakdown (named after J. S. Townsend), the sparking condition depends on the cathode through a "secondary emission coefficient" γ, the number of electrons released per ion hitting the cathode, which is "strongly dependent on the cathode material, surface roughness, impurities" [web] https://en.wikipedia.org/wiki/Paschen%27s_law. So a gap with different metals on each side is not symmetric, and the textbook allows it to break down more easily one way than the other. How large that effect is for clean versus oxidised metals in air at millimetre gaps is not something textbooks tabulate.

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4. Where these researchers depart

Two further details the reviewers uncovered:

• Rich's mixed wording. In some passages Rich's prose calls the favoured electrode the "anode" ("bismuth anode", "iron anode", "zinc anode"), but his table headings put that metal negative, which makes it the cathode, the electrode that emits electrons. Elsewhere he writes correctly that "the discharge can start more readily when iron is negative", and his summary says "the electrons are emitted from iron". So the usage is mixed, not uniformly wrong [read: deep read, 13 passages, Rich pp.142-155]. In every table the favoured half-cycle is the one where the higher-ranked metal is negative, so the tables are self-consistent and agree with the series. One link of the series, Bi versus Zn, has no data anywhere in the paper; Bi above Zn rests only on Guthe's separate coherer series [read: deep read]. Rich gives no gap length for any table, only a blanket "0.1 mm to 3 or 4 mm", and the Cu-Cu control has longer trains (about 4 sparks) than the Cu-Fe film (2-3), so the control was probably not run under matching conditions [read: p.148; deep read].
• Two arithmetic issues in Rich. The Cu-Fe average is printed 6.2 but 205/30 = 6.83 [read: p.147-148; calc]. The Cu-Zn "1.27" divides 14 trains by only the 11 half-cycles that had any train ("11 half-cycles only" footnote); over all 32 it is 0.44 [read: page image p.154; calc]. Both mistakes make the effect look smaller than his data show.

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5. Worked example: the 1890 period, checked from scratch

The goal: take Trowbridge and Sabine's own circuit constants and photographic distances (p.115-116 [read: page images]), compute what the textbook predicts, compute what they measured, and test the candidate explanations for the gap. Steps F and G add two shorter checks, one on Trowbridge's atoms and one on Rich's 1917 film. Every number below is reproduced by example.py (its sections A-I).

Step A. The ideal period

Their constants for fast oscillations: self-induction L = 39,700 cm (EMU) and capacity C = 5317.9 cm (ESU) [read: p.115].

Convert to SI: - L = 39,700 × 1 nH = 39.7 μH. - C = 5317.9 × 1.1127 pF = 5.917 nF.

Then T0 = 2π sqrt(LC) = 2π sqrt(39.7×10⁻⁶ × 5.917×10⁻⁹) = 2π × 4.847×10⁻⁷ s = 3.045 μs.

The 1890 way gives the same: (2π / 3×10¹⁰) × sqrt(39,700 × 5317.9) = 3.043 μs; the paper prints ".00000304" [read: p.115; calc]. With their slow-period inductance 41,090 the same formula gives 3.096 μs, matching their printed 3.10 [calc]. Good: their theory is correctly computed, and it confirms that the "capacity of wire, 200" listed on the same page was not included (we will come back to this).

Step B. What they measured

The mirror turned at f = 54.06 revolutions per second, and the plate was D = 259.7 cm from it [read: p.115]. The reflected spot sweeps at

u = 2 × 2π × 54.06 × 259.7 cm/s = 176,400 cm/s (about 1.76 km/s).

A band spacing of 0.529 cm on the plate therefore means 0.529 / 176,400 s = 3.00 μs. Doing this for every spacing in the "sample record" table reproduces all their printed periods (to within 0.01 μs) [calc]:

• Right terminal: 3.00, 3.24, 3.38, 3.06, 3.00, 3.11, 3.24, 2.90 μs
• Left terminal: 3.45, 3.25, 3.01, 2.85, 3.22, 3.12 μs

Mean of the 14 periods: 3.132 μs; standard deviation 0.18 μs; standard error of the mean 0.05 μs [calc].

Compare with Step A: the measured mean is 2.9% longer than 3.045 μs, about 2 standard errors. Each band was measured three or four times; for one band the repeat readings (.598, .595, .602, .592 cm) spread by only 0.7% [calc]. So the period-to-period scatter (2.85 to 3.45 μs, about ±6%) is real, not reading error.

That was one photograph. Their table on p.116 gives all six negatives. Leaving out the first interval of each row (a half-period on the right terminal, a start-up interval on the left), the full periods average 3.11 μs on the right terminal (33 values, standard deviation 0.13 μs) and 3.17 μs on the left (33 values, standard deviation 0.19 μs) [read: page image p.116, via the deep read; calc]. The scatter within a negative is 2-3 times the reading error (0.063 μs per setting), so it is real. But when the deep read asked whether the same up-and-down pattern repeats from negative to negative, which is what a property of the air would produce, the answer was no (analysis of variance across oscillation number, p = 0.53 right, 0.10 left), and successive intervals alternate long-short, which is the signature of setting error [calc in deep read]. Credit where due: their measurement was careful, and the swing-to-swing scatter is a real feature of the spark. The "periodicity" they built on it is not shown by their numbers.

Step C. Is a resistance in the spark enough?

The textbook's first suspect is the spark's resistance. Characteristic impedance: Z0 = sqrt(L/C) = sqrt(39.7×10⁻⁶ / 5.917×10⁻⁹) = 81.9 Ω.

• The wire's own resistance, 1.54 Ω [read: p.115], gives ζ = 1.54 / (2 × 81.9) = 0.0094 and lengthens the period by only 4 parts in 100,000. Negligible.
• To lengthen the period by 2.9% we need T/T0 = 1/sqrt(1 − ζ²) = 1.029, so ζ = 0.234 and R = 2 × 81.9 × 0.234 = 38 Ω. (For 5%, R = 50 Ω, the figure the mathematician gave.)
• But now check damping. With ζ = 0.234 the amplitude falls to exp(−2π × 0.234 / sqrt(1 − 0.234²)) = 0.22 of itself each period. After 8 periods it is 0.22⁸ ≈ 5×10⁻⁶ of the start [calc].

The plate shows 8 or 9 clearly measured periods. A spark carrying a few millionths of its starting current would not show. So a steady 38 Ω resistor cannot produce both the observed slowing and the observed number of oscillations. The original reviewers missed this; the arbiter has since withdrawn its "50 Ω is plausible" reasoning on these grounds. "It's just spark resistance" is too quick. A real spark is not a steady resistor: its voltage drop stays roughly constant as the current changes [memory, medium], so its effective resistance is low near peak current and high near current zero. That can shift individual swings in uneven ways without heavy exponential damping. Working that out properly needs a model of the arc, which nobody in these papers supplied.

Step D. The stray capacity they left out

On the same page they list "Capacity of wire, 200" (cm) [read: p.115], yet the theoretical time uses only 5317.9. Adding it:

[calc] Now compare the last row with the six-negative means from Step B: right terminal 3.11 μs against 3.10 μs (+0.3%), left terminal 3.17 μs (+2.4%). The wire's capacity is spread along the wire rather than sitting at the capacitor, so the full 200 cm is an upper bound, and the paper never says whether it was meant to be included. Even so, the stray capacity the authors themselves listed removes nearly all of the discrepancy on the right terminal and about half of it on the left (4.3% falls to 2.4%), without any new property of air. What is left (about 2% on the left terminal, plus the swing-to-swing scatter) is the size of effect a nonlinear spark can plausibly give.

But how good is 5317.9 itself? The paper also prints the condenser's dimensions: 19 zinc cylinders, height 30.47 cm (the OCR text reads 80.47, a misread digit), inner diameter 7.60 cm, outer 25.95 cm, about 0.5 cm apart [read: page image p.114 (PDF p.150), via the deep read]. Treat the 18 gaps between neighbouring cylinders as coaxial capacitors in parallel, each worth C = l / (2 ln(b/a)) in ESU centimetres (l the height, a and b the inner and outer radii of that gap; this is the authors' own formula). The sum is 4,511 cm, or 4,544 cm with Maxwell's small end correction [calc, example.py section I]. The quoted 5317.9 cm is 17% higher, and nothing in the text explains the difference (a zinc sheet about 0.75 mm thick would, but the paper gives no thickness; that is an inference only). If the geometric value were right, theory with the wire would be 2.87 μs and the measured 3.11 and 3.17 μs would exceed it by 8-10%. The quoted 5317.9 is the value they actually used (it reproduces their printed 3.04 and 3.10 μs), and it is not a fitted number, but it cannot be checked from their dimensions to better than about 15%. So Step D shows that their stated inputs need no new physics; it cannot show that the true deviation is zero. Honest summary: the 1890 experiment was not accurate enough to confirm or deny a departure of a few percent from T = 2π sqrt(LC).

Step E. Their test of the spark explanation

To rule out the spark, they photographed a glass Leyden jar with a short (0.4 cm) and a long (1.3 cm) spark and wrote that "no appreciable difference between the two plates" could be detected [read: page image p.120]. Their table (p.116) gives the full periods [read: page image p.116, via the deep read]:

• 0.4 cm spark: 3.22, 3.30, 3.44, 3.33, 3.37, 3.30, 3.36 μs; mean 3.33 μs.
• 1.3 cm spark: 3.32, 3.45, 3.37, 3.42, 3.30, 3.43, 3.42, 3.43, 3.42, 3.50 μs; mean 3.41 μs.

The difference is +0.075 μs, or +2.2%. Divide by the combined standard error (the Welch t statistic, which allows the two groups different spreads) and you get t = 2.3, roughly a 1-in-25 chance of a difference this large from scatter alone [calc]. That is fragile (one negative per spark length), but it is not "no difference", and it is the same size as the anomaly the test was meant to explain. The test was also run on glass, not on the air condenser where the anomaly was claimed.

Lesson from Steps A-E. Their observation of scatter holds up; their explanation was never needed. Before invoking new physics, put every measured element (including the ones you listed yourself) into the standard formula; check that your alternative explanation fits all the data (here: slowing and the number of oscillations), not just one number; and read your own control experiment's numbers before summarising it in words.

Step F. Trowbridge's atoms, in one line

If iron atoms crossed the 6 mm gap [read: p.193] in each half-period of a T = 3×10⁻⁷ s oscillation, their speed would be v = 0.006 m / 1.5×10⁻⁷ s = 4×10⁴ m/s. An iron atom (mass 55.8 atomic mass units = 9.27×10⁻²⁶ kg) moving that fast has kinetic energy ½ m v² = 463 eV, against about 1.3 eV of thermal energy at 10,000 K. The predicted Doppler shift at λ = 4000 Å is Δλ = λ v/c = 0.53 Å, which is the "whole tenth metre" (the old name for 1 Å) Trowbridge expected [read: p.193; calc]. For his slower 2.4 μs circuit the energy needed falls to 7 eV, which is not absurd on its own; there the deciding argument is collisions: in air at atmospheric pressure an ion collides billions of times a second, so its drift speed is limited to roughly mobility × field, of order 10²-10³ m/s even in strong fields [memory, medium; ion mobility not web-checked], giving shifts of 0.01 Å or less. So his null result was expected; one photograph pair, with no repeats, confirms it but adds little. A second page-image read confirms the periods (3.0×10⁻⁷ s and 2.4×10⁻⁶ s; the OCR's 8×10⁻⁷ is wrong), the 6 mm gap and the 20,000-lines-per-inch grating. His stated sensitivity reads "1/100 of a wave length" (about 85% confidence; 1/1000 is not fully excluded), and it is not clear whether he means a fraction of the wavelength or of a line's width; he gives no calculation and no wavelength, L, C or exposure data [read: page image PDF p.229, via the deep read]. Taken literally as a fraction of the wavelength it would be far too coarse to see a 0.5 Å shift. For the slower 2.4 μs circuit the expected shift is only about 0.07 Å [calc]. Trowbridge himself notes that ordinary convection "would take place too slowly to give a spectroscopic effect". Note what the test addresses: bulk motion of metal atoms, not whether electrons or ions carry the current. Credit: this energy argument was not new. Trowbridge says he was prompted by a published argument of J. J. Thomson (1890), so the decisive calculation was already in print when he did the experiment [read: p.193, per arbiter].

Step G. Rich's own film, checked against the LC formula

Rich (1917) gives his film circuit: a coil of coppered iron wire with "low frequency inductance 25 millihenrys", and a condenser of 48 window-glass panes whose "absorption was very bad", so that "the low frequency capacity ranged from an eighth to a half microfarad, depending on the duration of the charge". The film moved at about 42,000 mm/s, and successive sparks landed 4.1 mm apart [read: p.146].

• Time between sparks: 4.1 mm ÷ 42,000 mm/s = 97.6 μs.
• Successive sparks alternate between the electrodes [read: p.145], so each spacing is half a period: T = 195 μs (a frequency of about 5.1 kHz).
• The naive prediction with C = 0.125 μF is T = 2π sqrt(0.025 × 1.25×10⁻⁷) = 351 μs, a spacing of 7.4 mm; with 0.5 μF it is 702 μs and 14.8 mm.
• Turn it around: what capacitance fits the measured period? C_eff = T² / (4π² L) = (1.95×10⁻⁴)² / (4π² × 0.025) = 0.039 μF, three to thirteen times below his slow-method values. (If each spacing were a full period, C_eff would be 0.0097 μF.)

[calc] Resistance cannot close the gap, because damping only lengthens the period. The likely explanation (an interpretation, not in Rich's paper) is the box in section 3: the glass condenser's capacity at 5 kHz is well below its slow-charge value, helped by losses in the iron-wire coil. So the LC formula survives, but the capacitance does not stay at its slow-method value. This is a measured case of exactly the effect Trowbridge and Sabine invoked, in a solid dielectric, which is where it belongs.

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6. Evidence table

Verdicts are the arbiter's final verdicts, including its QE revision and Round 2. Round 2 changed no label; it weakened the basis of 2a and added caveats to 3.

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7. What a learner should take away

Solid. - The damped LC oscillation is real and was beautifully demonstrated by a purely optical trick. Trowbridge and Sabine's numbers, worked through in section 5, reproduce their own theoretical and measured periods to the hundredth of a microsecond. That is good 1890 experimental work. - Atoms are not shuttled across a spark gap at megahertz rates. An energy budget plus collisions settles it. - Rich's polarity asymmetry is large compared with his own symmetric Cu-Cu control and is not an artefact of his arithmetic (his slips go the other way). Electrode-dependent breakdown is allowed by standard discharge physics. - Capacitance measured slowly is not capacitance at high frequency, for solid dielectrics. Rich's own film (Step G) shows an effective capacity three to thirteen times below his slow-method value.

Wrong, and why it is instructive. - "Dielectric viscosity of air" is the classic case of reaching for new physics before exhausting ordinary circuit effects. The authors listed a stray capacity on the same page and left it out of their prediction; with it, the right-terminal mean agrees with theory to 0.3%. Include every element you measured; put a number on each alternative before rejecting it. - A second lesson from the same numbers: a 0.3% agreement is only as good as the inputs. Their quoted capacity is 17% above what their own printed dimensions give, so the honest conclusion is not "no deviation" but "this experiment could not tell". Before claiming a few-percent anomaly (or denying one), check that every input is known to better than a few percent. - They took a real effect (dielectric absorption in glass) and extended it to air, where it is negligible, even though the gas results they cited pointed the other way. A true phenomenon in one material is not evidence for it in another. - Their spark-length control showed a 2.2% shift and they reported "no appreciable difference". Read your own control numbers before summarising them in words. - Nott's "the discharge exceeded the charge" (section 3 box) is what an over-unity artefact looks like: most likely stored charge from earlier runs leaking out (dielectric absorption, which his own residual-discharge tables show over minutes), possibly a mismatch between his charge and discharge measurements. He gave no numbers and was right to call the results "too irregular" and draw no conclusion. - But the easy mainstream rebuttal ("it's just spark resistance") also fails a check: a constant resistance large enough to slow the swings by 3% would have killed them within a few periods. Sceptics owe the same arithmetic as believers. - In some passages Rich called the emitting electrode the "anode", while elsewhere he wrote "iron is negative". Electrons leave the cathode (negative electrode). Sign conventions are where many experimental papers trip, so always reconcile prose against table headings. - An explanation (electron emission, governed by work function) that predicts an order the data do not show is not confirmed by the data, even if the data are good.

Also in the folder: two stories worth knowing. - Dolbear's ring magnets (PDF pp.309-315; journal pp.271-277; presented 14 January 1891). A. E. Dolbear (the digest misreads him as "Dodge") flexed magnetised iron rings inside coils and saw galvanometer kicks of 30-40° that reversed when the ring was turned over; struck rings "sang" to each other through a telephone over about a yard; his magnetising coil gave about 1,300 ampere-turns (his estimate) [read: deep read]. It is a magnetic and acoustic paper, with nothing about wireless, electrostatics or Feddersen. He ran no control with an unmagnetised ring, the Earth's field mattered by his own Experiment XI, and his "sympathetic vibration" between magnets is not separated from plain mechanical or airborne coupling [read: deep read]. That much is ordinary: bending iron changes its magnetisation (the Villari or magnetostrictive effect), and a changing flux induces a current [memory, medium]. From it he went on to suggest that the ether outside a closed ring magnet carries waves that "would not necessarily give rise to electro-magnetic effects directly", and he tied this to the "Vortex Ring theory of atoms, of the properties of which I have so often spoken to the Academy" [read: OCR PDF pp.314-315; confirmed in the deep read]. He offers these explicitly as suggestions and proposes no test. Dolbear is a recognised name in later ether literature [memory, medium; not web-checked]. Here the measurements are ordinary and the speculation has no measurement behind it (arbiter: Not testable as stated). - Joule and the Royal Society (PDF pp.384-389). The volume's memoir of J. P. Joule says that when he presented the mechanical equivalent of heat, "some of the members of the Royal Society expressed the opinion that it was nonsense from beginning to end ... and as a body they refused to publish it in their Transactions", with William Thomson the only friendly voice; it appeared "in 1850, seven years after its presentation" [read: OCR PDF p.387; confirmed in a full deep read]. A second refusal is also recorded: in 1849 a British Association committee would not print the sentence "Friction consists in the conversion of mechanical power into heat" [read: deep read]. Joule's values for the mechanical equivalent went 838 (1843), 781.8 (1847), 772 (1849), 772.55 ft·lb per British thermal unit (1878) as he refined the method [read: deep read]. Caveats on the source: the memoir is unsigned, written 40-47 years after the events, cites only Roscoe for these quotations, and blends a refusal with a delay, so it is a secondary report. Its dates are muddled: it gives 21 August 1848 for the Cork paper, yet "seven years after" 1850 fits 1843, and the British Association met at Cork in 1843 [memory, high; not web-checked]. It also confuses the mechanical equivalent J with the later unit, the joule, and names none of Mayer, Colding or Helmholtz. Even so, this is a documented case of an establishment resisting a correct result. It is worth holding next to the 1890 "dielectric viscosity" story, which was also published by a respected establishment and was wrong. What decided both was not who held the view but whether the numbers survived checking.

Genuinely open. - Was Rich's asymmetry ever replicated or refuted? Nothing found, but the search was incomplete. - If not work function, what ranks iron above bismuth and zinc? Candidates: oxide films, metal-vapour supply, cathode-spot formation, thermal properties (suggestions, untested). - Where does Bi sit relative to Zn? No data shown. - Is there any departure from T = 2π sqrt(LC) at the few-percent level in a spark circuit, and what exactly is the swing-to-swing scatter in the 1890 periods? The 1890 data cannot say: their capacity is uncertain by about 15%. A modern repeat with a capacitance measured directly at the working frequency and a voltage-current trace of the spark would settle it. - Rich gives no gap length for any table, and his Cu-Cu control seems to have run under different conditions. Any repeat must record the gap for every film. - A cheap modern test of claims 4 and 5 is within reach of a hobbyist with care: a photodiode or fast camera, a signal generator driving a high-voltage supply, polished Fe, Bi, Zn and Cu electrodes at fixed gap, blinded polarity, a Cu-Cu control, many runs, and a repeat in an inert gas to separate surface-film effects from emission. Treat high voltage with real respect.

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8. Check yourself

1. Rich's sparks land 4.1 mm apart on film moving at 42,000 mm/s, and successive sparks alternate electrodes. With L = 25 mH, what effective capacitance does that imply, and why can extra resistance not explain the difference from his quoted 0.125 μF?
2. In an LCR circuit, why can you not lengthen the period by 3% using resistance alone without also changing something else you could see on the photograph?
3. If the rotating mirror had spun at 27 revolutions per second instead of 54, what plate spacing would a 3.0 μs period produce?
4. In Rich's Cu+Fe- half-cycles iron is the negative electrode. Is iron then the anode or the cathode, and which way do electrons leave the gap?
5. Rich's Cu-Zn column reads "1.27 trains per half-cycle (11 half-cycles only)" out of 32 observed, with 14 trains. Why is 1.27 the wrong number to compare with the other column's 4.77, and what is the right one?

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9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics (inductance, circuits, units appendix). - E. M. Purcell and D. J. Morin, Electricity and Magnetism (LCR circuits; an excellent appendix on CGS units). - J. R. Taylor, An Introduction to Error Analysis (Poisson statistics, comparing rates). - Yu. P. Raizer, Gas Discharge Physics (Townsend breakdown, secondary emission, sparks and arcs). - Paschen's law and the secondary emission coefficient: https://en.wikipedia.org/wiki/Paschen%27s_law - Work functions of the elements: https://en.wikipedia.org/wiki/Work_function - Feddersen biography: https://en.wikipedia.org/wiki/Berend_Wilhelm_Feddersen - History of spark-gap transmitters (where damped LC oscillation went next): https://en.wikipedia.org/wiki/Spark-gap_transmitter

The researchers' own work. - B. W. Feddersen, papers in Poggendorff's Annalen der Physik und Chemie vols. 103 (1858), 108 (1859), 112 and 113 (1861), 115 and 116 (1862), as cited by Trowbridge and Sabine [read: p.109]. Not in the folder; the first thing to obtain. - J. Trowbridge and W. C. Sabine, "Electrical Oscillations in Air", Proc. Amer. Acad. Arts Sci. 25, 109-123 (1890). In the folder (PDF pages 145-159). - J. Trowbridge, "Motion of Atoms in Electrical Discharges", Proc. Amer. Acad. Arts Sci. 25, 192-194 (1890). In the folder (PDF pages 228-230). - D. L. Rich, "Oscillatory Spark Discharges between Unlike Metals", Phys. Rev. 10, 140-155 (1917). In the folder. - Also in the 1890 volume (in the folder): F. H. Safford and G. U. G. Hofman, "Telephonic Specific Inductive Capacity" (PDF pp.15-18; digest and OCR only); C. Nott, "On the Comparison of some Electrical Condensers" (PDF pp.282-296; journal pp.244-258); A. E. Dolbear, "On Change of Form affecting a Magnetic Field" (PDF pp.309-315; journal pp.271-277); memoir of James Prescott Joule (PDF pp.384-389; journal pp.846-851). The last three were read in full from page images in the second QE round. - K. E. Guthe, coherer polarity series, Ann. d. Physik 4, 762 (1901), as cited by Rich [read: p.154].

Code. example.py in this folder reproduces section 5 (Steps A-G, with the geometry check of Step D in its section I) and the Rich statistics: .venv/bin/python review/Fedderson-Daedalus/example.py

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Answers

1. The spacing is 97.6 μs, a half-period, so T = 195 μs and C_eff = T²/(4π²L) = (1.95×10⁻⁴)²/(4π² × 0.025) ≈ 0.039 μF, about a third of the lowest quoted value. Resistance only makes the period longer (T = T0/sqrt(1 − ζ²)), but the observed period is shorter than the naive 351 μs. The capacitance itself must be smaller at 5 kHz: dielectric absorption in his glass panes.
2. Resistance both slows the swings (T = T0 / sqrt(1 − ζ²)) and makes them decay (amplitude × exp(−2πζ/sqrt(1 − ζ²)) per period). A 3% slowing needs ζ ≈ 0.23, which makes each swing about a fifth of the one before, so only two or three flashes would be visible, not eight or nine.
3. Sweep speed halves to about 88,200 cm/s, so spacing = 88,200 × 3.0×10⁻⁶ ≈ 0.26 cm (half the original 0.53 cm). Slower mirrors give smaller, harder-to-measure spacings, which is why they spun it fast.
4. The negative electrode is the cathode. Electrons leave the cathode (iron) and travel to the anode (copper). So "iron anode" in Rich's prose contradicts his own "Cu+Fe-" heading; the table, and his own summary that electrons leave iron more easily, both point to iron as cathode.
5. 1.27 = 14/11 divides only by the half-cycles that produced at least one train, while 4.77 = 143/30 divides by all half-cycles. On the same basis the first is 14/32 = 0.44, so the true asymmetry is about 11:1, larger than printed (Poisson z ≈ 10).