Peek's law of corona: a primer

Project folder: Researchers/Peek-Frank. It holds one document: F. W. Peek Jr., Dielectric Phenomena in High Voltage Engineering, 2nd edition, McGraw-Hill, New York, 1920 (a 325-page scan). Peek (1881-1933) was an engineer at General Electric (GE), first in Schenectady and from 1916 at the Pittsfield works, where he became Chief Engineer in 1931 [web: https://ethw.org/Archives:Transformers_at_Pittsfield,_part_2, a GE-history source; it gives his death year as both 1931 and 1933; Wikipedia gives 27 July 1933]. GE's Consulting Engineering Department ran the experiments in the book [read: preface]. GE made transmission equipment and insulation, so it had a commercial stake in the results; that applies to every GE-sourced comparison below.

Peek is not an outsider. His corona law was adopted by the profession, is named after him, and is still used to estimate when a conductor starts to glow. He was made an APS (American Physical Society) Fellow in 1922 and received the Franklin Institute's Levy Medal in 1926 [web: https://en.wikipedia.org/wiki/Frank_William_Peek]. This project is in the review because the book is the classic source of an engineering law, and the reasoning Peek built around it (a finite "rupturing energy" stored over a finite distance) is a good case study in how a correct formula and a partly empty explanation can travel together.

Source note (repeated from claims.md and arbiter.md). The scan has no text layer: 324 of 325 pages were read by OCR (optical character recognition), which garbles most equations and tables. Chapter III (visual corona, PDF pp.60-113), Chapter IV (spark-over, sphere gaps, impulse, PDF pp.115-153), Chapter V (the experimental Lines A and B on which the loss law is fitted, PDF pp.154-189), the end of Chapter VI (oil, impulse and energy, PDF pp.196-202), Chapter VII (solid insulation, PDF pp.202-227), Chapters VIII-IX (electron theory and practical formulas, PDF pp.227-251), the loss-data appendix (PDF pp.277-294) and the appendix pp.295-311 were read in depth from rendered page images; the deep reads of Chapters IV-VII and the loss appendix were added after the quality check (QE, Step 4b) and are weighed in the arbiter's "QE revision". The rest of the book (Chapters I-II, the oil chapter before p.196, the design cases of Chapter X) rests on page-by-page digests. Nobody in this review has read Peek's original papers (1911-1915, the 1927 Part IV paper), the 1929 third edition, the 1910 Hayden-Steinmetz paper, Fitch's 1913 thesis page images or Hartmann's 1984 paper in full; the Investigator saved no files. Every conclusion below rests on the book plus the cited web sources, not on [memory] alone, unless a [memory] tag says so.

Page numbers: "p.N" is the PDF page. In Chapter III the printed page number is 23 lower.

Source tags: [read] = from the book (the extracted text or a page image); [web] = Internet source with URL; [memory] = recalled from training, with confidence; [computed] = calculated for this primer in example.py.

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1. The idea in one paragraph

Air is a good insulator until the electric field in it gets strong enough, about 30 kilovolts per centimetre, to tear electrons off air molecules and start a chain reaction. Near a thin wire at high voltage the field is very strong right at the surface and falls off quickly with distance. Engineers around 1900 noticed that thin wires did not start to glow ("corona", a faint violet glow with a hiss and a power loss) when their surface field reached 30 kV/cm; they needed much more, and the thinner the wire the more they needed. Peek measured this carefully on polished wires from 1910 onward and found a simple rule: the surface field needed for glow is 30 kV/cm times (1 + 0.301 divided by the square root of the wire radius in cm), corrected for air density. He explained it by saying that air does break at 30 kV/cm, but only if that field extends over a finite thickness of air, because breakdown needs a definite amount of energy; so the field must be 30 kV/cm not at the surface but a little way out, 0.301 times the square root of the radius. The rule works to about 5 percent and is still used. The explanation, as he worded it, turns out to be a restatement of the rule (any curve can be described that way), and Peek himself said he had not worked out what the "energy" was. The real physics behind it, an electron avalanche that has to grow large enough within the thin high-field layer around the wire, was supplied later by others, and it gives his square-root shape.

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2. What you need to know first

The electric field of a line charge and a pair of wires. A long straight wire carrying charge lambda per unit length has field E = lambda / (2 pi epsilon0 x) at distance x from its axis (Gauss's law with a cylinder); epsilon0 is the permittivity of free space. For two parallel wires of radius r with centre spacing s, much larger than r, and voltage e from each wire to the midplane between them ("voltage to neutral"), the field near one wire is very nearly E(x) = e / (x ln(s/r)), and at the surface (x = r) it is g = e / (r ln(s/r)). The field falls off as 1/x, so for a thin wire it is strong only in a thin layer. Read: Griffiths, Introduction to Electrodynamics, Ch. 2 (electrostatics, Gauss's law) [memory, high for the chapter].

Peak and rms (root-mean-square) values. A 60 Hz sine-wave voltage with rms value V_rms swings to a peak of sqrt(2) V_rms = 1.41 V_rms. Insulation breaks on the peak. Peek quotes fields in kV/cm peak ("max.") and voltages often in kV rms ("effective"); his 21.1 or 21.2 kV/cm rms is 30 kV/cm peak divided by 1.41. Read: any introductory circuits text, a.c. chapter.

Gas discharge: ionization by collision (Townsend). A free electron in a field E gains energy between collisions with molecules. If it gains more than the ionization energy of the molecule, a collision can free a second electron; both are accelerated and the number grows exponentially along the path. John Townsend (about 1900-1910) described this with a coefficient alpha, the number of new electrons one electron creates per centimetre of travel; after a distance d one electron becomes e^(alpha d). A standard rough form is alpha = A p exp(-B p / E), where p is gas pressure and A, B are constants for the gas. In air, oxygen also captures electrons ("attachment"), so what matters is the net coefficient, ionization minus attachment, which is zero at a field near 27-30 kV/cm at sea-level density [web: https://arxiv.org/pdf/physics/0508109, per the Physicist]. That is the physical meaning of Peek's 30 kV/cm. Read: Kuffel, Zaengl and Kuffel, High Voltage Engineering: Fundamentals, the chapter on breakdown in gases; Raizer, Gas Discharge Physics.

Streamer criterion (Raether and Meek, about 1940). A single avalanche turns into a self-sustaining discharge (a streamer, seen as glow) when it reaches roughly 10^8 electrons, that is, when the integral of the net alpha along the path reaches K = 18-20 (e^18.4 is about 10^8) [web: https://arxiv.org/pdf/physics/0508109, per the Physicist and arbiter]. Around a thin wire the integral runs only over the thin layer where the field is high, which is why a thin wire needs a higher surface field. This is the idea Peek was groping for.

Paschen's law. For a uniform gap of length d at pressure p, the breakdown voltage depends only on the product p d. It has a minimum, about 327 V in air at p d = 0.567 torr cm (1 torr = 1 mm of mercury), and rises on both sides [web: https://en.wikipedia.org/wiki/Paschen%27s_law]. At very small gaps there is too little path for an avalanche to grow, so the average field needed (voltage divided by gap) becomes very large. Read: Raizer, the chapter on breakdown and Paschen's law.

Relative air density. Peek writes delta = 3.92 b / (273 + t), with b the barometer in cm of mercury and t in degrees C; delta = 1 at 25 C and 76 cm Hg [read: p.67]. Because collision ionization depends on the field divided by the number of molecules per unit volume, the breakdown field of a uniform gap scales with delta.

Fitting a straight line to data. Least squares picks the line that minimises the sum of squared misfits and uses every point. Peek used the "method of group averages": split the points into two groups, and take the slope from the difference of the group sums. Both are legitimate; least squares has smaller scatter in its estimates. Read: Taylor, An Introduction to Error Analysis, Ch. 8 (least-squares fitting) [memory, high for the chapter].

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3. The textbook view

Symbols used throughout: - r : conductor radius (cm). R : sphere radius, or outer-cylinder radius (cm). s or S : centre spacing of two wires (cm). - x : distance from the wire's axis (cm). a : distance out from the wire surface (cm). - e : voltage from wire to neutral (kV). e_v : the "visual critical voltage", at which glow starts. e_o : the "disruptive critical voltage", at which the surface field equals g_o delta (loss begins). - g : surface field, or "gradient" in Peek's word (kV/cm). g_v : surface field at visual corona onset. g_o : "strength of air" in a uniform field, about 30 kV/cm peak. - E_s : surface field in the avalanche model; E_o : field at which net ionization is zero. - delta : relative air density (above). f : frequency (Hz). p : power loss (kW per km or per mile) or, in Townsend's formula, gas pressure (torr); the context says which. - alpha : Townsend ionization coefficient (per cm). K : avalanche growth exponent needed for a streamer (about 18-20). - m_o, m_v : Peek's "irregularity factors" for rough or stranded conductors (1 for a polished wire, about 0.87 for stranded cable).

Field near a wire. E(x) = e / (x ln(s/r)), so g = e / (r ln(s/r)) at the surface. Peek uses exactly this [read: p.64]; his Table I reproduces from it to within 1.1 kV/cm [computed by the Mathematician from the printed voltages].

Uniform field. Between large parallel plates at 1 cm spacing at sea level, air breaks down at roughly 30 kV/cm peak, a little less at large gaps. A standard empirical formula for uniform-field air at normal temperature and pressure (after Schumann) is E = 24.22 + 6.08 / sqrt(d) kV/cm peak, with d the gap in cm, valid above about 1 mm [web: search summary citing https://pupman.com/listarchives/2006/Apr/msg00188.html and https://archive.nptel.ac.in/content/storage2/courses/108104048/lecture6/slide2.htm, which gives the variant 24.4 and 6.29; pages not opened]. Worked: at d = 1 cm, E = 24.22 + 6.08 = 30.3 kV/cm; at d = 10 cm, E = 24.22 + 6.08 / 3.162 = 26.1 kV/cm (26.4 with the variant constants). So "30 kV/cm" is right near 1 cm and falls by about 13-15 percent at 10 cm. Breakdown field scales with density: g_o is proportional to delta.

Non-uniform field, mainstream mechanism. In a 1/x field the net ionization coefficient is positive only inside a radius x_c where the field exceeds E_o. Corona starts when

integral from r to x_c of alpha_eff(E(x)) dx = K, about 18-20.

A thinner wire has a thinner high-field layer, so its surface field must climb higher before the integral reaches K. Section 5 shows this numerically and shows that, for a small excess over E_o, it gives exactly the form E_s = E_o (1 + constant / sqrt(r)). A 1984 paper by G. Hartmann ("Theoretical Evaluation of Peek's Law", IEEE Trans. Industry Applications IA-20, 1647-1651) derives the onset field from Meek's streamer criterion, with an ionization coefficient that depends on humidity, pressure and temperature, and compares it with experiments [web: https://ieeexplore.ieee.org/document/4504655; abstract and search summary only].

Time lag. Breakdown is not instantaneous. There is a statistical lag (waiting for a free electron to start an avalanche) and a formative lag (time for avalanches and streamers to cross the gap). A very short voltage pulse therefore needs a higher peak voltage to cause breakdown; the curve of breakdown voltage against time is called the volt-time characteristic [memory, high]. Sphere gaps are still an international standard for measuring peak impulse voltage, to about +/-3 percent (IEC 60052; IEC is the International Electrotechnical Commission, which publishes international electrical standards) [web: https://iteh.es/catalog/standards/sist/c2e5bb77-cee9-434c-8385-19e10adef442/sist-en-60052-2004, per the Physicist].

Direct versus alternating voltage. In air and oil, breakdown depends on the peak field, so d.c. breakdown equals the a.c. peak (1.41 times the a.c. rms) to a first approximation, with polarity differences from space charge in non-uniform fields [memory, high]. Solid insulation is stronger under d.c. because a.c. heats it (dielectric loss) and drives partial discharges in voids [memory, medium].

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4. Where Peek departs

Peek's starting points are sound physics, and he says so in the language of 1915-1920. He knew Townsend's collision ionization and wrote that "the electron plays the principal part in impact ionization" [read: p.231]. He knew from his own impulse tests that breakdown takes time and that shorter pulses need higher voltage [read: p.232]. He credited the observation that air seems stronger around small wires to Harris J. Ryan (1904) [read: p.64-65]. And he rejected the older explanation, a dense film of air condensed on the metal, by a real experiment: tungsten and aluminium wires of the same size glow at the same field, whereas a condensed film should be denser on the denser metal [read: p.71-72, p.232]. He also said plainly that a curve fit alone would not satisfy him: "A number of equations could readily be written which would fit these data. It was desired, however, to establish or build up a rational equation" [read: p.232].

Here is where the book and the later textbook account sit side by side.

The exact point of departure is equations (21)-(24) on p.72. Peek writes the surface field at onset, g_v = e_v / (r ln(S/r)), and the field a distance a out, g_o = e_v / ((r + a) ln(S/r)), says "one is also led to believe a = phi(r)", a function of r to be found, and then fits the data: g_v = 29.8 (1 + 0.301/sqrt(r)). Substituting back, he writes that "the experimental values bear out the above theory" [read: page image p.72]. Dividing the two field equations gives a = r (g_v/g_o - 1), which exists for any measured curve. So the "bears out" step is a test that cannot fail. What does carry content is narrower and is real: that one g_o, close to the uniform-field strength, and one simple a(r) fit all radii, spacings and densities, and that the same distance idea predicted, before data according to Peek, that very small gaps should be much stronger (p.67), which the sphere-gap tables then showed. For spheres the gradient starts to rise when the spacing falls below X = 0.54 sqrt(R), and that number, read from one column of Table XI, matches twice the sphere "energy distance" 0.27 sqrt(R) obtained from a different column [read: page image p.87-90]. The arbiter first called this "the one place where the idea does more than relabel the fit", then, after the Chapter IV deep read, weakened the remark: the sphere-gap law 27.23 delta (1 + 0.54/sqrt(delta R)) is itself an empirical fit (within 1.5 percent for 18 of 24 points, worst -3.8 percent; two sample runs 4-6 percent below it against a stated 2 percent), and, in the arbiter's words, "the 0.54 appears in both the sphere law and the wire law, and both are fits" [read: deep read pp.115-138, per arbiter QE revision].

Peek was candid about limits. For spheres he wrote that Fig. 48 shows the gradient at a = 0.27 sqrt(R) "is not exactly constant for different spacings", so g_o and the coefficient "must also be a function of X" [read: p.90]. He used three different constants for three geometries (31 for a wire in a cylinder, 29.8-30 for parallel wires, 27.2 for spheres) and suggested that 31 is "the true strength of air" because the cylinder's field is "balanced" [read: p.88]; the reviewers read the spread as a sign that g_o is partly a fitting constant of each geometry.

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5. Worked example: when does a thin wire glow, and why 1/sqrt(r)?

All numbers are reproduced by example.py (run command in its header). Step numbers match the script's.

Step 1. A 2 mm wire at 1 m spacing (by hand)

Take two polished wires of radius r = 0.1 cm, spacing s = 100 cm, sea-level air (delta = 1).

• ln(s/r) = ln(1000) = 6.908.
• Peek's law: g_v = 30 (1 + 0.301 / sqrt(0.1)) = 30 (1 + 0.952) = 58.6 kV/cm peak, almost twice 30 kV/cm.
• Onset voltage: e_v = g_v r ln(s/r) = 58.6 x 0.1 x 6.908 = 40.5 kV peak to neutral = 28.6 kV rms to neutral = 57.2 kV rms between the two wires.
• If you assumed air breaks at 30 kV/cm at the surface you would predict 30 x 0.1 x 6.908 / 1.41 = 14.7 kV rms to neutral, too low by half [computed]. That gap between the naive figure and the measured one is the observation Peek's law describes.

Step 2. The "energy-storage distance", and why it always works

• a = 0.301 sqrt(0.1) = 0.0952 cm.
• The field falls as 1/x, so at x = r + a it is g_v r / (r + a) = 58.6 x 0.1 / 0.1952 = 30.0 kV/cm. Exactly g_o. That is Peek's picture: air breaks at 30 kV/cm, but the field must reach 30 kV/cm a finite distance out.
• Now try a made-up law that has nothing to do with Peek's data, g = 30 (1 + 0.2 / r^0.8). At r = 0.01, 0.1 and 1 cm it implies a = r (g/30 - 1) = 0.080, 0.126 and 0.200 cm, and the field at each of those distances is again exactly 30.0 kV/cm [computed]. Any curve can be dressed this way. The picture is a re-description; the testable content is that the distance comes out as one simple 0.301 sqrt(r) for every wire, spacing and density.

Step 3. Do Peek's constants depend on which points he kept?

Peek's Table III gives 19 polished wires (radius 0.0098 to 0.464 cm) [read: page image p.74]. He discarded points "varying widely from the average straight line ... as probably in experimental error" without listing them or giving a rule [read: p.74]. The reviewers inferred from his Table IV that 10 points were kept and 9 dropped; kept and dropped points have overlapping misfits, so the selection is not explained by distance from the final line.

• Group averages on the 10 kept points: g_o = 29.84, slope 8.99, coefficient 0.301 (the book's numbers) [computed].
• Least squares on all 19 points: g_o = 30.39, slope 8.71, coefficient 0.287 [computed].
• Peek's law against all 19 points: rms misfit 3.5 percent, worst 6.2 percent [computed].
• At r = 0.01 cm Peek gives 119.5 kV/cm and the all-points line 117.5; at r = 1 cm, 38.8 and 39.1 [computed].

So the discards moved the constants by a few percent; they did not create the law. The honest statement is g_o = 29-30.4 and coefficient 0.286-0.312 depending on fitting recipe (arbiter), so quoting "0.301" to three figures overstates the precision. The Mathematician also found that, with the exponent left free, the data prefer about 0.41 rather than 1/2, but then g_o falls to about 22 kV/cm, away from the uniform-field value Peek's reasoning needs; exponent 1 is clearly rejected.

Step 4. An avalanche criterion, worked in closed form

Use Townsend's alpha = A p exp(-B p / E) with crude air constants A = 15 per cm per torr, B = 365 V per cm per torr (the reviewers' values; the form is standard, the exact constants were not confirmed on the web), p = 760 torr, and no attachment. Ask when an avalanche starting at the surface reaches e^K.

The field is E(x) = E_s r / x. Change the integration variable from x to E (dx = -E_s r dE / E^2):

integral from r to infinity of alpha dx = E_s r x integral from 0 to E_s of A p exp(-B p / E) / E^2 dE = (A/B) r E_s exp(-B p / E_s).

(The last step: the derivative of exp(-Bp/E) with respect to E is (Bp/E^2) exp(-Bp/E).) The script checks this against a brute-force numerical integral: 2.421 both ways at E_s = 60 kV/cm, r = 0.1 cm [computed].

Setting (A/B) r E_s exp(-B p / E_s) = K and solving for E_s:

(kV/cm peak) [computed; the arbiter's run at 743 torr gives the same ratios to within 0.01]

The rise from r = 1 cm to r = 0.01 cm is 2.93 (K = 10) or 3.32 (K = 18), against Peek's 3.08. So electron-avalanche physics, with nothing fitted, gives the right shape: a thinner wire needs a disproportionately higher surface field. It does not give the right level: it is 25-55 percent high, because these crude constants leave out attachment to oxygen, which lowers net ionization near 30 kV/cm. The arbiter's summary: "shape supported, absolute value not derived by anyone here".

The formula shows why in one line: the product r E_s exp(-Bp/E_s) must stay fixed. Make r 100 times smaller and E_s has to rise until the exponential makes up the factor of 100.

Step 5. Where the square root comes from

Now include attachment in the simplest way: let the net coefficient be zero at E_o and rise linearly just above it, alpha_eff = k (E - E_o), where k is a constant. The positive region runs from x = r out to x_c = r E_s / E_o. With X = E_s / E_o, SymPy gives

integral of alpha_eff dx = k r E_o (X ln X - X + 1).

For a small excess, X = 1 + eps, this is k r E_o eps^2 / 2. Setting it equal to K:

eps = sqrt(2K / (k E_o r)), so E_s = E_o (1 + constant / sqrt(r)),

and the distance over which the field exceeds E_o is a = r eps, which grows as sqrt(r) [computed; the Mathematician's derivation]. The 1/sqrt(r) form of Peek's law and his "distance 0.301 sqrt(r)" both drop out of a 1/x field with a threshold. If alpha_eff/delta depends only on E/delta (Townsend scaling), the same steps give E_s = E_o delta (1 + constant / sqrt(delta r)), which is Peek's density law, so his radius law and density law are one result, not two. Peek's distance was a real feature of the physics, described in words that did not yet have the avalanche mechanism behind them. The model does not fix the number 0.301; that needs measured values of k and K for air, which nobody in this review has checked against Peek's curve.

Step 6. Peek's own line example: what corona costs

Peek's worked case is a three-phase 60 Hz line, 100 miles, 1/0 stranded cable (r = 0.187 in), spacing 120 in, 100 F, barometer 28.85 in Hg, m_o = 0.87 [read: page image p.244].

• delta = 17.9 x 28.85 / (459 + 100) = 0.924 (the English-unit form of 3.92 b/(273 + t)).
• e_o = 123 m_o r delta log10(s/r) = 123 x 0.87 x 0.187 x 0.924 x 2.807 = 51.9 kV rms to neutral (book: 52.0) [computed]. The constant 123 is 21.1 x 2.54 x ln 10 = 123.4: g_o in rms, inches, and base-10 logarithms.
• Loss coefficient for three conductors: 3 x (390/delta)(f + 25) sqrt(r/s) x 10^-5 = 0.0425 kW per mile per kV^2 (book: 0.042) [computed].
• At 110 kV between lines (63.5 kV to neutral): p = 0.0425 x (63.5 - 51.9)^2 = 5.7 kW per mile, about 570 kW for the whole line in fair weather (book Table LXXIX: 5.7 per mile with its rounded constants) [computed; read: page image p.245]. At 90 kV between lines the line sits at e_o and the formula gives no loss; at 130 kV it gives 23 kW per mile.

Two cautions from the reviews. The stranded-cable factor m_o = 0.87 is simply the ratio of the apparent "g_o" for cable (about 25.7 kV/cm) to that for smooth wire, so for cable "g_o" includes surface roughness and is not the strength of air [read: p.171-172, per the Mathematician]. And the loss law itself is a fit to Peek's own two experimental lines, good to about 10 percent over that range (refit exponent on r/s: -0.498 +/- 0.021; residual scatter 7.4 percent) [computed by the Mathematician from Table LV, p.171]. The Chapter V deep read adds detail (arbiter, QE revision): the constant 20,500 came from 12 of 21 tests, nine dropped without a stated reason; least squares on all 21 gives exponent -0.496 and 10.1 percent scatter, so the sqrt(r/s) form is robust; A = 241 is 20,500 / 85, a splice of the indoor frequency fit and the outdoor spacing fit; the (f + 25) offset is fixed by indoor data only (outdoor intercepts drift from -22 to +13 cycles); Peek himself called the density correction "not satisfactory"; and on Line A, the line he studied most, the averaged measurement is about 13-15 percent above the formula [read: page images pp.154-189, 277-294, deep readers; arithmetic per arbiter].

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6. Evidence table

Verdicts are the arbiter's (Step 3), using the review's scale. "Single source" means the numbers come from one laboratory; it is a description, not a dismissal.

Fitch 1913, the one independent non-GE pressure study. Theodore T. Fitch's Johns Hopkins thesis, done in J. B. Whitehead's laboratory, measured onset at 5-108 cm Hg on nine conductor sizes. As returned by a fetch tool from the archive.org text (not checked on a page image), Fitch found Peek's original equation "didn't closely match", proposed g = 33.65 (1 + 0.00295 p / sqrt(r)) kV/cm with p in cm Hg, said it fitted "about as closely as the readings can be taken", and concluded that critical intensity depends "on the number and spacing of the molecules, in accord with the theory of secondary ionization"; carbon dioxide (CO2), about 1.5 times denser than air, gave negligible change [web: https://archive.org/stream/influenceofdensi00fitcrich/influenceofdensi00fitcrich_djvu.txt]. At p = 76 cm Hg Fitch's formula gives 109 kV/cm at r = 0.01 cm (Peek 119.5, 9% lower), 57.5 at 0.1 cm (Peek 58.2) and 41.2 at 1 cm (Peek 38.8, 6% higher) [arbiter, computed]. So it supports the type of law, rising as 1/sqrt(r), and differs from Peek's numbers by up to about 9 percent and in how the second term scales with pressure.

Anomalies kept in the record (Peek does not comment on them): for the thinnest wires (r = 0.0318 and 0.0573 cm) the 100 kc impulse spark-over is below the 60 Hz value (for example 100 vs 135 kV), while at 500 kc it is above [read: page image p.304]; the Contrarian's untested reading is that 60 Hz corona builds a protective space-charge sheath that a single pulse has no time to form. Oil under continuous 90 kc voltage holds only 0.39 of its 60 Hz gradient (67 vs 170 kV/cm), attributed by Peek to heating without a measurement [read: p.310]. Thin wires under d.c. "vibrate before spark-over" [read: p.295]. Corona-driven vibration on a larger scale: a pair of 20-mil (0.5 mm) steel wires 500 ft long, about 10 ft apart, "vibrated, starting with a hardly perceptible movement, which in a few minutes had an amplitude of several feet at the center of the span", one wire in its fundamental (period about 1 s) and the other in its third harmonic; it was repeated in the laboratory (Figs. 83-84), and a rotating wire looked bright for half a turn, which Peek read as a stroboscope of positive and negative corona at the 60 Hz supply frequency [read: p.113]. No date, place, voltage or mechanism is given, the 1 s period and the 60 Hz reading are not reconciled in the text, and it was never followed up in these pages. Cumulative impulse damage: oiled pressboard (3.2 mm, 100 kV 60-cycle strength) survived 100 kV impulses indefinitely but failed after 100 impulses at 140 kV, 16 at 150 kV, 2 at 155 kV and 1 at 165 kV; no experimenter, date or specimen count is given and no second source is named in the book or found (the "insulator fails on the tenth impulse" passage is Peek's hypothetical, not a reported test) [read: p.215, deep reader].

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7. What a learner should take away

What is solid. - The observation and the formula. Thin wires need a much higher surface field before they glow, and g_v = g_o delta (1 + c / sqrt(delta r)) with g_o about 30 kV/cm and c about 0.29-0.31 cm^(1/2) describes it to about 5 percent for smooth wires at 60 Hz near sea-level density. Spacing independence over a 42-fold range is a striking, clean result. - The engineering consequences: use larger effective conductor radius (bigger conductors, hollow tubes, aluminium's larger diameter, and later bundled conductors) to raise the onset voltage. Peek's own e_o formula tells you why bundling works, even though one sentence in the book says grouping conductors "decreases" the critical voltage, probably a misprint for "increases" (the word is broken "de-/creased" across a line) [read: p.246]. It contradicts the book's own split-conductor data: two wires at one potential rise from 31.5 kV touching to 35.8 kV at the best spacing; three wires in a triangle have a critical voltage 20-30 percent above one wire of the same total material (Peek), and Whitehead found 16 percent for three and 20 percent for four [read: p.97-98]. - Paschen behaviour at small gaps, the qualitative time lag under impulses, and the sphere gap as an impulse voltmeter. The impulse work of 1915 is the book's most durable contribution; Peek carried it into lightning generators of 3.6 MV (1928), 5 MV (1929) and 10 MV (1932) [web: ethw.org, GE-history source].

What is wrong, and why the mistake is instructive. - "The experimental values bear out the above theory" (p.72) is a test that cannot fail. Whenever a theory contains a free function that you then fit to the data, agreement with that data is not evidence for the theory. The lesson generalises far beyond corona: ask what result would have contradicted the explanation. Here the real tests were elsewhere: that one g_o close to the uniform-field strength works for all wires, and that the distance idea predicted strong small gaps and the sphere onset spacing. - Selecting data by an unstated rule. Peek dropped 9 of 19 points. It did not change his answer much (Step 3), which is lucky, not good method. Always report the rule and the all-points fit. - In-sample agreement. Peek's "generally less than 1 per cent" agreement in Tables VI-VII is real, but those tables were part of the data fitted [read: p.79]. - The corona "rupturing energy" stored over the distance 0.301 sqrt(r) was never defined, given units or computed; Peek wrote that "thus far the form of this rupturing energy has not been considered as it was unnecessary" [read: p.234]. A quantity that never appears in an equation is a label, not yet a theory. The later avalanche picture replaced it with something you can count: e-foldings of electron number. - But the book does quote energies in joules elsewhere, and an earlier draft of this primer (and two reviewers) wrongly said it never did. Peek cites a 1910 GE investigation by Hayden and Steinmetz: "the disruptive discharge through a dielectric requires not merely a sufficiently high voltage, but requires a definite minimum amount of energy", with "the disruptive energy of oil ... about 30 times greater than it is for air" [read: p.148]; and, for sphere gaps set to 30 kV at 60 cycles and hit by a 100 kV transient, oil "will not spark over if the energy applied to the circuit is less than 1.04 joules (4 amp.). For air the minimum energy is 0.45 joule (1.5 amp.)" [read: p.202]. Why this is not the same thing as Peek's rupturing energy: these joules are energy offered by the supply circuit, not energy stored in the air next to the gap. The curves actually plot the current let through (amperes); how amperes became joules is not given and does not fit (1/2) L i^2 with one inductance L (the current ratio squared is (4/1.5)^2 = 7.1, the joule ratio 1.04/0.45 = 2.3). A small sphere gap, treated as a capacitor of a few picofarads, stores only about 1/20 to 1/40 of those joules at these voltages [deep reader's estimate with an assumed 5 pF, unchecked]. The same curves are what you expect if the source simply cannot feed a growing spark. And the book's own oil-to-air ratios disagree: 30 (p.148), 2.3 (the joules) and about 4 (the energy distances 1.2 sqrt(r) for oil against 0.301 sqrt(r) for air). The lesson: a measured threshold (here, a supply-limited one, which the arbiter rates Plausible, untested) does not by itself confirm the mechanism someone attaches to it.

What is genuinely open. - Humidity. Peek says humidity has no effect. For visual onset on polished wire he prints no numbers; the only series is Chapter V's Table LXIII, ten tests on Line A's stranded cable showing no effect on disruptive onset at about 1-2 percent, but with every fully humid test below freezing, so humidity and temperature cannot be separated [read: deep read p.183-184, per arbiter]. Later literature reports that onset rises with humidity (water molecules capture electrons) and that loss near inception depends on relative humidity, approximately cubically [web: search summaries of https://typeset.io/topics/peek-s-law-2vkw38v0 and https://digital-library.theiet.org/content/journals/10.1049/piee.1971.0131; full texts not read]. How large the effect is on visual onset on polished wires at 60 Hz is unknown to this review. - The discard rule and the full data. Peek's 1911 "Law of Corona" Part I paper (77 pages, free on archive.org [web: per Investigator]) should hold all the points; nobody here has refitted them. For the loss law, the outdoor test sheets were transcribed from the appendix (pp.277-294), but two tests of Table LV (86 and 77) are not there, and several printed density values differ by 0.02-0.04 from those computed from the printed barometer readings; an independent refit is still to be done (arbiter). - The loss law far from 60 Hz. The 100 kHz Alexanderson check supports the frequency slope on one wire, with no stated accuracy and an onset voltage that does not match Peek's formula. A repeat on a second wire with the onset stated would settle it. - What Hayden and Steinmetz actually measured in 1910: how amperes were turned into joules, whether any time was measured, and what the "30 times" oil-to-air figure refers to. The paper (Proc. A.I.E.E., the Proceedings of the American Institute of Electrical Engineers, the AIEE, 1910) has not been obtained. - The corona-driven conductor vibration of p.113: what drives it, and why a 1 s mechanical period goes with a 60 Hz stroboscope reading. - Fitch's pressure scaling (linear in p in the second term) against Peek's sqrt(delta): which fits better over delta 0.07-1.3? - Did Peek ever specify the rupturing energy? His 1927 paper is titled "The Mechanism of Corona Formation and Loss" (Trans. AIEE 46:1009-1024; the AIEE is a forerunner of today's IEEE, the Institute of Electrical and Electronics Engineers); its abstract reports cathode-ray oscillograph measurements (quadratic loss above the visual voltage, no loss below it on polished wire, a "probability law" on cables) but only the abstract was read [web: https://ieeexplore.ieee.org/document/5061442]. The 1929 third edition is also unread. - The thin-wire 100 kc impulse anomaly (space charge or measurement error?) and why Whitehead's positive-wire d.c. values sit 4-6 percent below the a.c. peak when Peek's own tables show them equal.

The record. Peek's work does not go silent: papers through 1927, lightning research to his death in a car-train collision near Gaspe, Quebec, on 27 July 1933 [web: Wikipedia; the GE-history site ETHW gives 1931 and 1933 in one source]. Nothing in the record shows the corona work being rejected or blocked; the issue the reviewers found is the reverse, acceptance without anyone re-examining the data selection. The narrower silences, why the "energy distance" was never developed, where the humidity data are, the missing numbers in the Leadville and Hendricks comparisons, have ordinary candidate explanations (superseded by avalanche theory; a heuristic Peek moved past; material in unread papers or GE internal reports), and the deciding records are named: the 1911-1913 papers, the 1927 Part IV paper, the 1929 edition, Faccioli's Leadville report, the 1912 N.E.L.A. article and GE test files (GE is an interested party), plus, from the QE revision, Hayden and Steinmetz 1910, Rayner 1912 (J.I.E.E., the Journal of the Institution of Electrical Engineers, London), Alexanderson (Proc. I.R.E., Proceedings of the Institute of Radio Engineers, 1914), Watson 1910, Mershon 1908 and Chubb and Fortescue 1913 on sphere gaps. Evidence that the book stayed in use: engineering articles still applying its formulas in Electronics (1950) and Westinghouse Engineer (1951) [web: worldradiohistory.com, per Investigator], and a German report (Zeitschrift VDI, the journal of the Verein Deutscher Ingenieure, the Association of German Engineers, 16 July 1932) of Peek's 10 MV lightning generator and 18-metre sparks [web: Wireless Engineer abstract, per Investigator]. Other leads not yet obtained: 84 catalogued items under Peek's name at Union College (site blocked), a 1916 passage in which Charles Steinmetz "argued that F. W. Peek Jr.'s experiments on high-voltage lines ..." (a truncated snippet from Wrestling with Nature), the 1931 Stanford chair offer he declined, and his 19 US patents (1908-1932, all assigned to GE) [web: per Investigator, sites.md].

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8. Check yourself

1. A polished wire of radius 0.5 cm runs on a mountain line where delta = 0.8. Using Peek's law with g_o = 30, what surface field (kV/cm peak) starts visual corona? How does it compare with the sea-level value?
2. Explain in two or three sentences why Peek's statement that "the field equals g_o at a distance 0.301 sqrt(r)" cannot be contradicted by the same data used to find 0.301. What observation could have contradicted his picture?
3. Peek's smallest sphere gap was 0.00254 cm at sea level, and it sparked at 0.497 kV peak. Compute the average field and the product p d in torr cm. Which branch of the Paschen curve is this, and why is the "strength" so much higher than 30 kV/cm?
4. In the avalanche model of Section 5, Step 4, the onset condition is (A/B) r E_s exp(-Bp/E_s) = K. At r = 1 cm and K = 18 the solution is E_s = 57.0 kV/cm. Without solving again, explain why E_s must be higher at r = 0.01 cm, and which part of the expression makes the rise modest rather than a factor of 100.
5. Peek writes that d.c. breakdown in air is "41 per cent. higher than the effective a.c. critical voltage". His appendix tables show d.c. and a.c. breakdown voltages nearly equal. Are these statements in conflict?

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9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics: Ch. 2 for the line-charge field and Gauss's law. - E. Kuffel, W. S. Zaengl and J. Kuffel, High Voltage Engineering: Fundamentals (2nd ed., 2000): breakdown in gases, Townsend and streamer mechanisms, corona, sphere gaps and impulse measurement. - Yu. P. Raizer, Gas Discharge Physics (Springer, 1991): ionization coefficients, Paschen's law, avalanche-to-streamer transition. - J. R. Taylor, An Introduction to Error Analysis: Ch. 8 on least-squares fitting, to compare with Peek's group-average method. - G. Hartmann, "Theoretical Evaluation of Peek's Law", IEEE Trans. Industry Applications IA-20 (1984) 1647-1651 [web: https://ieeexplore.ieee.org/document/4504655]. - Protsenko and Trotsenko, "A Hundredth Anniversary of the Peek's Law for Estimating the Corona Inception Electric Field", Naukovi visti NTUU KPI, 2014 no. 6, pp.35-41 [web: https://kpi.ua/nv-abstract-14-6]. - Wikipedia, "Peek's law" (irregularity factors and Whitehead's stranding factors) [web: https://en.wikipedia.org/wiki/Peek%27s_law]; "Paschen's law" [web: https://en.wikipedia.org/wiki/Paschen%27s_law].

Peek's own key works. - Dielectric Phenomena in High Voltage Engineering (1915; 2nd ed. 1920, the book reviewed; 3rd ed. 1929). The 1915, 1920 and 1929 editions are full view on HathiTrust, the 3rd also on archive.org [web: per Investigator]. - "The Law of Corona and the Dielectric Strength of Air", Trans. AIEE, Part I 1911 (free on archive.org), Part II 1912, Part III 1913, Part IV "The Mechanism of Corona Formation and Loss", 1927 [web: per Investigator; IEEE Xplore 5061442]. - "The Effect of Transient Voltages on Dielectrics", Trans. AIEE 1915 (the full impulse report); "The Effect of High Continuous Voltages on Air, Oil and Solid Insulation", AIEE 1916 [read: p.69 footnote; web: ethw.org]. - Lightning and Other High-Voltage Phenomena (1926), full view on HathiTrust [web: per Investigator]. - Related contemporary work: T. T. Fitch, The Influence of Density of Gas on the Formation of Corona (Johns Hopkins thesis, 1913) [web: https://archive.org/details/influenceofdensi00fitcrich]; J. B. Whitehead, "The Disruptive Strength of Air", Trans. AIEE 1910.

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Answers

1. g_v = 30 x 0.8 x (1 + 0.301 / sqrt(0.8 x 0.5)) = 24 x (1 + 0.301 / 0.632) = 24 x 1.476 = 35.4 kV/cm. At sea level, 30 x (1 + 0.301 / sqrt(0.5)) = 30 x 1.426 = 42.8 kV/cm. The mountain value is 0.83 of the sea-level one, not 0.8: the density enters both outside and inside the square root, so the correction term grows slightly as the air thins.

2. Dividing g_v = e/(r ln(S/r)) by g_o = e/((r + a) ln(S/r)) gives a = r (g_v/g_o - 1). For any measured onset curve g_v(r), and any chosen g_o, that formula yields a distance at which the field equals g_o, so "agreement" is automatic (Step 2 of the worked example does it for a made-up law). The picture could have been contradicted if no single g_o near the uniform-field strength had worked for all radii and spacings, if the distance had needed a complicated form, or if small gaps (shorter than the distance) had not turned out stronger; the sphere tables, where the gradient starts to rise at X = 0.54 sqrt(R), matched it.

3. Field = 0.497 kV / 0.00254 cm = 196 kV/cm. p d = 760 torr x 0.00254 cm = 1.93 torr cm (0.193 cm Hg cm in Peek's units). This is on the right-hand (rising) side of the minimum at 0.57 torr cm, close to it; at such small p d an electron crosses the gap in only a few dozen mean free paths, so to reach the needed multiplication each collision must be very likely to ionize, which needs a high field. Breakdown voltage (about 0.5 kV) is near the Paschen minimum, but divided by a tiny gap it is a huge field. The arbiter notes a Townsend-form Paschen estimate of about 384 V and Peek's own table about 467 V, so Peek's measurement is about 30% above the idealised estimate and 6% above his own table, with an unstated uncertainty from setting a 25-micrometre gap by counting screw turns.

4. The left side is proportional to r. Make r 100 times smaller with E_s fixed and the integral falls 100-fold, far below K, so E_s must rise. The factor exp(-Bp/E_s) is very sensitive to E_s: with Bp = 365 x 760 = 277 kV/cm, raising E_s from 57 to 189 kV/cm raises exp(-Bp/E_s) from exp(-4.87) to exp(-1.47), a factor of about 30, and E_s itself contributes a factor of 3.3; together they make up the factor of 100. That is why the field rises about 3.3 times, not 100 times.

5. No. For a sine wave the peak is 1.41 (sqrt 2) times the rms ("effective") value. "D.c. is 41% higher than the a.c. effective value" means d.c. equals the a.c. peak. The appendix tables list a.c. in kV maximum (peak), so there the ratio appears as about 1.0. The real exceptions, visible in the book's own tables, are thin wires in cylinders (d.c. 1.2-1.5 times the a.c. peak), long needle gaps (0.92-0.93) and solid insulation (1.27-2.59).