Maxwell (and Heaviside): Primer

Project folder: Energy/Researchers/Maxwell (4 documents: Maxwell 1855-56 "On Faraday's Lines of Force"; Maxwell 1865 "A Dynamical Theory of the Electromagnetic Field"; Hunt 2012 "Oliver Heaviside: A first-rate oddity", Physics Today; Belrose 2014 "Oliver Heaviside: a first rate oddity, sage in solitude"). Built from: arbiter.md (final verdicts, including its "QE revision" and "Round 2" sections), physicist.md, mathematician.md, contrarian.md, the deep reads in deep/, plus my own reading of page images. Companion script: example.py (run command in its header). It reproduces every number in sections 3 to 5.

Source tags: [read] = from the document's text layer; [read: page image p.N] = from a rendered page of the PDF (N is the PDF page number, not the printed one); [web] = Internet source with URL; [memory] = recalled from training, with confidence high/medium/low.

Source note. All four PDFs have a text layer, but the two Maxwell scans are noisy OCR (optical character recognition) and the equations are garbled. The 1865 equations quoted below were read from page images (PDF pp.32, 33, 34, 47, 52 of dynamicaltheoryo00maxw.pdf); the sunlight estimate and the British Association coil figures were read from page images of PDF pp.53, 57, 59, 60 in a later deep read. The fluid-analogy derivations of the 1855-56 paper were later checked on page images in a deep read (arbiter, QE revision); this primer uses only their conclusions. The Heaviside history rests essentially on one secondary source (Hunt); Belrose mostly paraphrases Hunt. Experimental history (Hertz, Michelson-Morley, Aharonov-Bohm) is not in the folder and comes from [web] or [memory] as tagged.

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1. The idea in one paragraph

Michael Faraday pictured electric and magnetic effects as "lines of force" filling the space around magnets and wires, but he had no mathematics for them. James Clerk Maxwell gave them that mathematics. First (1855-56) he showed that the lines behave like the streamlines of an imaginary fluid, an analogy he insisted was only a notation. Then (1865) he wrote a complete set of equations for the field, added one new ingredient (the "displacement current", a current that exists wherever an electric field is changing, even in empty space), and found that the equations predict waves travelling at a speed he could compute from purely electrical measurements. That speed came out within a few percent of the measured speed of light, so he proposed that light is an electromagnetic wave. Twenty years later Oliver Heaviside, a self-taught telegraph engineer, rewrote Maxwell's twenty equations as the four compact vector equations every student now learns, and showed that energy flows through the field around a wire, not inside it like water in a pipe. Hertz then produced the predicted waves in the laboratory. This is the one project in the collection where the "outsider" ideas became the textbook; the interest lies in how they got there, what the original papers actually say, and where the alternative-science story about Heaviside is and is not accurate.

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2. What you need to know first

Vector calculus: divergence, curl, and the two big theorems. The divergence of a vector field, div F (also written ∇·F), measures how much the field "spreads out" from a point: positive at a source, negative at a sink, zero in a flow that neither gains nor loses fluid. The curl, curl F (∇×F), measures circulation around a point: how much a tiny paddle wheel would spin. The divergence theorem says the total outward flux through a closed surface equals the integral of div F over the enclosed volume; Stokes' theorem says the circulation of F around a closed loop equals the flux of curl F through any surface bounded by that loop. Two identities are used constantly: curl(grad f) = 0 for any scalar f, and div(curl F) = 0 for any vector F. Read: Griffiths, Introduction to Electrodynamics (4th ed.), Chapter 1; or Schey, Div, Grad, Curl, and All That.

Electrostatics and magnetostatics. Charges produce electric fields (Coulomb's law, equivalently Gauss's law: the flux of E out of a closed surface is proportional to the charge inside). Steady currents produce magnetic fields that circle them (Ampère's law: the circulation of B around a loop is proportional to the current through it). Read: Griffiths, Chapters 2 and 5.

Faraday's law of induction. A changing magnetic flux through a loop produces an electromotive force (EMF, the "push" around the loop, measured in volts) equal to minus the rate of change of the flux. This is how transformers and generators work. Read: Griffiths, Chapter 7.

Potentials. Because div B = 0, B can always be written as the curl of another vector, the vector potential A: B = curl A. The electric field is then E = −∂A/∂t − grad φ, where φ is the scalar (electric) potential. A is not unique: adding the gradient of any function to A (and a matching change to φ) gives the same E and B. This freedom is called "gauge freedom". Read: Griffiths, Chapter 10 (and Section 5.4 for the static vector potential).

Waves. The one-dimensional wave equation ∂²f/∂x² = (1/v²) ∂²f/∂t² has solutions f(x − vt) and f(x + vt): any shape moving at speed v. Writing a wave as a complex exponential exp(i(qx − nt)) (q the wavenumber, n the angular frequency) turns derivatives into multiplications, which is how damping and attenuation are usually found. Read: Griffiths, Chapter 9; any intro mechanics text on waves on a string.

Units: SI versus Gaussian. Modern engineering uses SI units (volts, amperes, with constants ε₀ and μ₀ in the equations). Nineteenth-century papers use "electrostatic" and "electromagnetic" unit systems in which factors of 4π appear in different places and the constants are absorbed. The ratio of the two unit systems' charge units has the dimensions of a speed; that ratio is the number Maxwell compares to the speed of light. Read: Griffiths, Appendix C (units).

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3. The textbook view

3.1 The four equations (Heaviside's form, in SI units)

Symbols: E electric field (V/m); D electric displacement (C/m²); B magnetic flux density (tesla); H magnetic field intensity (A/m); ρ_free free charge density (C/m³); J conduction current density (A/m²); t time. In a simple ("linear, isotropic") material D = εE and B = μH, with ε the permittivity and μ the permeability; in vacuum ε = ε₀ = 8.854 × 10⁻¹² F/m and μ = μ₀ = 1.2566 × 10⁻⁶ N/A². In a conductor Ohm's law is J = σE, σ being the conductivity (the inverse of the resistivity).

The term ∂D/∂t is Maxwell's displacement current. Without it, take the divergence of Ampère's law: div(curl H) = 0 forces div J = 0, which is false whenever charge piles up (a charging capacitor). With it, div J + ∂(div D)/∂t = div J + ∂ρ/∂t = 0: exactly the statement that charge is conserved. So the displacement current is what the equations need to be consistent with charge conservation.

3.2 Light as a wave

In empty space (no charge, no current), take the curl of Faraday's law and use the Ampère-Maxwell law:

curl curl E = −μ₀ ∂(curl H)/∂t = −μ₀ε₀ ∂²E/∂t².

With the identity curl curl E = grad(div E) − ∇²E and div E = 0, this becomes ∇²E = μ₀ε₀ ∂²E/∂t², a wave equation with speed

c = 1/√(μ₀ε₀) = 1/√(1.2566 × 10⁻⁶ × 8.854 × 10⁻¹²) = 2.998 × 10⁸ m/s.

That is the speed of light. (Since 1983 c is fixed at exactly 299,792,458 m/s by definition [web: https://physics.nist.gov/cgi-bin/cuu/Value?c].) The waves are transverse: E and B are perpendicular to the direction of travel. A longitudinal (along-the-direction) component is forbidden in empty space because div E = 0 and div B = 0. example.py Part A checks this arithmetic.

3.3 Energy in the field and the Poynting vector

Electric and magnetic fields store energy with density (energy per unit volume)

u = ½ E·D + ½ B·H (J/m³).

Take the vector identity div(E × H) = H·curl E − E·curl H, substitute the two curl equations, and you get Poynting's theorem:

−∂u/∂t = div S + J·E, with S = E × H.

In words: the field energy in a region falls either because energy flows out through the boundary (the flux of S, in W/m²) or because the field does work on charges (J·E, which in a resistor is heat). S is the Poynting vector, found by John Henry Poynting (abstract published January 1884) and independently by Heaviside shortly after [read: Hunt].

One subtlety the reader should keep: the theorem fixes only the total flux of S through a closed surface. Adding the curl of any vector field to S changes nothing measurable, because div(curl F) = 0. So "the energy at this point is flowing in this direction" is a convention, the standard and well-motivated one, not something the integral forces [arbiter, claims 3 and 6].

3.4 Potentials

B = curl A and E = −∂A/∂t − grad φ. Taking the curl of the second equation and using curl(grad φ) = 0 gives curl E = −∂B/∂t: Faraday's law falls out automatically. Hunt's sidebar gives exactly this derivation as Heaviside's "second circuital law" (with a typo: it prints "grad(−∂A/∂t)" where "curl" is meant; the result is unaffected) [read: Hunt]. Classically only E and B are measured, so A is fixed only up to gauge. In quantum mechanics the loop integral of A (the magnetic flux it encloses) shifts the phase of electrons that never enter the field: the Aharonov-Bohm effect, predicted 1959 and observed unambiguously by Tonomura in 1986 [web: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect].

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4. Where this researcher departs

Maxwell is unusual in this collection: his results are the textbook. "Departure" here means the places where what the 1865 paper actually says differs from the modern version in section 3. These are worth knowing because popular and alternative-science accounts often attribute to Maxwell things he did not write, or miss things he did.

4.1 Maxwell's twenty equations versus Heaviside's four

The 1865 paper lists twenty quantities and twenty equations, written out in x, y, z components [read: page image p.34]. A translation table (modern equivalents are approximate where the unit systems differ):

The equations, with the letters he gave them: (A) total current = conduction + rate of change of displacement; (B) magnetic force from the curl of the momentum; (C) curl of magnetic intensity gives 4π times total current; (D) the electromotive force, including a term for a moving conductor; (E) P = k f; (F) P = −ρ p; (G) e + div f = 0; (H) ∂e/∂t + div p = 0 [read: page image pp.32-34].

Three points of difference from section 3:

• Potentials first. Maxwell's E is built from A and Ψ, equation (D): P = μ(γ dy/dt − β dz/dt) − dF/dt − dΨ/dx [read: page image p.32]. The first term is the force on a moving conductor (today's v × B, with v the conductor's velocity); the rest is E = −∂A/∂t − grad φ. Heaviside eliminated A and Ψ and wrote everything in E and H. His reason was physical: the potentials "did not locate the energy correctly" and he regarded them as "quite distant from the real workings of the field" [read: Hunt]. The resulting dispute at the 1888 British Association meeting was nicknamed the "murder of Ψ" debate; during it FitzGerald wrote down what was "essentially a gauge condition" [read: Hunt]. The two forms carry the same classical physics; Heaviside's has no gauge freedom and no redundant equations.

• No quaternions in 1865. A common claim in alternative-science circles is that Heaviside and Gibbs "truncated" Maxwell's richer quaternion equations and threw physics away. The 1865 paper studied here is written in plain Cartesian components, not quaternions [read: page images pp.32-34]. Maxwell used quaternion notation alongside components in his 1873 Treatise [memory: high]. The equations' physical content does not depend on the notation; what is lost in eliminating potentials is only the gauge-dependent part of A, whose physically meaningful remainder (its loop integral) is still captured by B.

• The wave is described magnetically. Maxwell's plane wave "consists entirely of magnetic disturbances, the direction of magnetization being in the plane of the wave" [read: page image p.47]. The symmetric picture of perpendicular E and B oscillating together is a later restatement.

• Faraday had the idea first. Maxwell himself writes that "the propagation of transverse magnetic disturbances to the exclusion of normal ones is distinctly set forth by Professor Faraday in his 'Thoughts on Ray Vibrations'" (1846), and that Faraday's electromagnetic theory of light "is the same in substance" as his own, "except that in 1846 there were no data to calculate the velocity of propagation" [read: Dynamical Theory p.14]. So by Maxwell's own account Faraday has priority for the idea of light as electromagnetic vibration; what Maxwell added was the equations and the speed.

4.2 A sign error in the printed equations

The arbiter confirmed, from page images, a defect the mathematician found and the other two reviewers missed. I re-checked it on the same pages.

• (E) P = k f makes the displacement point along the electric field. Correct.
• (F) as printed, P = −ρ p [read: page image p.33], makes the conduction current point against the electric field. Ohm's law says it points along it: P = +ρ p.

Is this just a convention? No: Maxwell's own later derivation shows it carried through. On PDF p.52 he works out how light is absorbed in a conductor. Using (A), (E) and (F) he gets his equation (95):

d²G/dx² = +4πμ ( (1/k) d²G/dt² − (1/ρ) dG/dt ),

and with the trial solution G = e^(−px) cos(qx + nt) he finds p = 2πμn/(ρq), his (97) [read: page image p.52]. Here G is one component of the vector potential, x the direction of travel, q the wavenumber, n the angular frequency and p the attenuation rate.

example.py Part B substitutes the trial solution into both the printed equation and the version with the Ohm's-law sign. With phase qx + nt the crest moves toward −x. With the printed sign p > 0, so e^(−px) grows in the direction the wave travels: the wave is amplified. With the correct sign p < 0 and the wave decays. The size |p| is the same, so Maxwell's numerical estimate is unaffected, but the printed sign contradicts his own sentence on the same page, "the undulation is thereby weakened" [read: page image p.52]. In a toy case (all constants 1, ρ = 10) the printed equation increases the amplitude by a factor of 1.368 per wavelength travelled; the corrected one reduces it to 0.731.

(G) has a second, separate sign problem. With e as positive charge, the continuity equation (H) is right, and (G) should read e = +div f. Check: (C) requires the total current to have zero divergence, but as printed div p′ = div p + ∂(div f)/∂t = −∂e/∂t − ∂e/∂t = −2 ∂e/∂t. The (G) sign was later corrected in the Treatise [web: https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field]. Whether (F) was a typesetting slip or a convention in Maxwell's mind is not settled [arbiter, open questions].

Put in proportion: these are real internal inconsistencies, but they do not touch the displacement current, the wave speed, or the energy formula. Heaviside's four equations have matching signs, so they are in effect the repaired version [arbiter, claim 6].

4.3 The ether

Maxwell framed the theory in an "ethereal medium" that fills space [read]. He was also careful about it: he avoided his earlier mechanical vortex model, called words like "electric elasticity" "illustrative, not explanatory", and said "the conclusions arrived at in the present paper are independent of this hypothesis" [read: text layer, paras 73-75]. Only the claim that energy really resides in the field is meant literally (para 74). So the ether is an interpretation layered on the equations, not a premise of them. The physical evidence Maxwell gave for the medium was indirect, and one item is Faraday rotation: the plane of polarised light turns when the light passes through a transparent body along a magnetic field, Verdet found the rotation reversed in paramagnetic solutions, and Thomson argued from this that the medium must have some motion tied to the magnetisation [read: Dynamical Theory p.9].

There is one hidden assumption worth noticing. Equation (D) contains the velocity of a moving conductor, but velocity relative to what? The equations are not unchanged under a switch to a moving frame (they are not "Galilean invariant"), so implicitly they hold in one preferred frame. That puzzle was resolved by Lorentz and Einstein, not by Maxwell or Heaviside [mathematician; memory: high].

4.4 Heaviside's "real action is in the field"

Heaviside had "long been convinced that the real action lay not within a conducting wire but in the field around it", and S = E × H implied "the energy of an electric current does not flow within the wire like water in a pipe but instead passes through the surrounding field and enters the wire through its sides" [read: Hunt]. That is the textbook picture today (section 5 computes it). The departure is only in how strongly it is read: as a statement about where energy flows locally, it rests on the conventional choice of S (section 3.3). Belrose's antenna modelling with NEC-4 (Numerical Electromagnetics Code, a computer program that solves for currents on wire structures) is an application of the theory, not a test that could distinguish Heaviside's view from any other [arbiter, claim 6].

4.5 Cross-references to other projects in the collection (not verdicts)

Two choices made in these papers come up again in other projects in the collection. They are flagged here so you can recognise them later. Nothing in this section is a verdict on those projects.

• Field theory versus action at a distance. The 1865 paper opens by setting aside Weber's and Neumann's theory, in which charged particles act on each other directly across space with forces that depend on their relative velocity. Maxwell calls it "exceedingly ingenious, and wonderfully comprehensive", and gives one stated reason for not taking it as final: "the mechanical difficulties" of forces at a distance that depend on velocity. He chose instead to explain the actions through "the surrounding medium", that is, a field theory [read: Dynamical Theory pp.7-8]. He does not call Weber's theory wrong and says it "may yet be useful". Weber electrodynamics is a recurring alternative-science theme. Note the irony: Weber and Kohlrausch's measured ratio of units, v = 310,740,000 m/s, is the very number that gave Maxwell his speed of light [read: page image p.47].
• Longitudinal waves. Maxwell says his equations, "deduced from purely experimental evidence, show that transversal vibrations only can be propagated". Normal (longitudinal, that is, along the direction of travel) vibrations would appear only if one assigned a density to an "electric fluid", for which "we have, however, no evidence", so "both sciences [electromagnetism and optics] are at a loss when called on to affirm or deny the existence of normal vibrations" [read: text layer, Dynamical Theory pp.48-49]. Belrose adds one unsourced sentence: "Heaviside critized Helmholtz, because Helmholtz'e electromagnetic theory allowed for the existence of longitudinal rather than solely transverse wave propagation" [read: Belrose p.7]. (Helmholtz's theory had an adjustable constant; for values other than the one that reproduces Maxwell it allows longitudinal electric waves [memory: medium, not web-checked].) Belrose also quotes Arthur and Forfar that Hertz in 1890 published "a clarification of Maxwell's equations", "mathematically equivalent to Heaviside's version" but written out in components [read: Belrose p.7, second-hand]. Section 3.2 shows why the textbook equations allow no longitudinal wave in empty space: div E = 0 and div B = 0. This is the reference point for the scalar- or longitudinal-wave claims made elsewhere in the collection; whether those claims hold has to be judged in their own reviews.

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5. Worked example: energy enters a wire through its sides

Question. A straight copper wire of radius a = 1.0 mm carries a steady current I = 10 A. Take a length L = 1.0 m. Compute the Poynting flux S at the surface of the wire, integrate it over the side surface, and compare with the heat produced, I²R. This is the calculation behind Heaviside's claim that energy enters a wire through its sides.

Data: resistivity of copper ρ_Cu = 1.68 × 10⁻⁸ Ω·m [memory: high]; conduction-electron density of copper n_e = 8.5 × 10²⁸ m⁻³ [memory: high]; electron charge e = 1.602 × 10⁻¹⁹ C.

Step 1: resistance. Cross-section A = πa² = π × (1.0 × 10⁻³)² = 3.1416 × 10⁻⁶ m². R = ρ_Cu L / A = 1.68 × 10⁻⁸ × 1.0 / 3.1416 × 10⁻⁶ = 5.348 × 10⁻³ Ω.

Step 2: voltage and electric field. V = IR = 10 × 5.348 × 10⁻³ = 0.05348 V across the 1 m. Inside the wire the field is uniform and points along the wire: E = V/L = 0.05348 V/m. The component of E parallel to a surface is continuous across it (a consequence of Faraday's law with no changing B), so just outside the wire there is the same axial E = 0.05348 V/m. (There is also a radial E outside, from surface charges; it carries energy along the wire from the battery, but it does not contribute to the inward flow computed here.)

Step 3: magnetic field. Ampère's law around a circle of radius a: H × 2πa = I, so H = I/(2πa) = 10 / (2π × 1.0 × 10⁻³) = 1591.5 A/m, circling the wire.

Step 4: direction of S. E points along the wire (call it +z, the direction of the current). H circles the wire in the right-hand sense (the azimuthal direction, φ̂). The cross product ẑ × φ̂ = −r̂: S points radially inward, into the wire.

Step 5: magnitude of S. E and H are perpendicular, so |S| = EH = 0.05348 × 1591.5 = 85.11 W/m².

Step 6: total inflow. Side area 2πaL = 2π × 1.0 × 10⁻³ × 1.0 = 6.2832 × 10⁻³ m². P_in = 85.11 × 6.2832 × 10⁻³ = 0.5348 W.

Step 7: Joule heating. I²R = 10² × 5.348 × 10⁻³ = 0.5348 W.

Step 8: compare. The two agree exactly. It is not a numerical coincidence: symbolically, S × 2πaL = (IR/L) × (I/(2πa)) × 2πaL = I²R for any I, R, a and L. example.py Part C prints both numbers (ratio 1.000000000000) and the symbolic difference (0).

Step 9: contrast with the electrons. The drift speed of the conduction electrons is v_d = I/(n_e e A) = 10 / (8.5 × 10²⁸ × 1.602 × 10⁻¹⁹ × 3.1416 × 10⁻⁶) = 2.34 × 10⁻⁴ m/s, about 14 mm per minute. The electrons crawl; the energy arrives through the field.

What this shows and what it does not. It shows that the Poynting picture is self-consistent and gives the right answer: the energy dissipated in the wire equals the energy flowing into it through its surface. It does not prove that energy "really" travels through the space around the wire rather than along it, because a different choice of local flux (differing from S by a curl) would give the same total and is not ruled out by any measurement in a circuit [arbiter, claim 6]. Heaviside's view is the standard, natural choice; it is not an experimentally isolated fact. A useful side number: the magnetic energy density just outside the wire is ½μ₀H² = 1.59 J/m³, while the electric one is ½ε₀E² = 1.3 × 10⁻¹⁴ J/m³; in a low-voltage circuit almost all the stored field energy is magnetic.

5.2 Second worked example (or exercise): Maxwell's 1865 field strength of sunlight

Try this yourself first, then compare. example.py Part D does the same arithmetic.

What Maxwell wrote [read: page image p.53, printed p.505]. For a plane light wave he finds the energy flowing through unit area per second, W = P²/(8πμV), "so that P = √(8πμVW), where V is the velocity of light, and W is the energy communicated to unit of area by the light in a second". Here P is "the greatest value of the electromotive force" (the peak electric field) and μ is the magnetic coefficient of the medium, equal to 1 for air or the ether in his units. (The text layer drops the μ and reads P = √(8πVW); with μ = 1 the two are the same.) His input: "According to Pouillet's data, as calculated by Professor W. Thomson ..., the mechanical value of direct sunlight at the Earth is 83.4 foot-pounds per second per square foot." His result: "P = 60,000,000, or about 600 Daniell's cells per metre." (A Daniell cell is a zinc-copper battery of about 1 V.)

Step 1: convert the input to SI. 1 foot-pound (ft·lbf) = 1.3558 J and 1 ft² = 0.092903 m², so W = 83.4 × 1.3558 / 0.092903 = 1217 W/m². (The modern solar constant above the atmosphere is about 1361 W/m² [memory: high]; Pouillet's figure is about 11 % lower; some light is lost in the atmosphere before a ground-level instrument sees it [memory: medium].)

Step 2: understand Maxwell's units. He works in the "absolute electromagnetic" system with the metre, gram and second as base units [inferred from "metrical units" on the same page; deep read]. Energy is then counted in g·m²/s², and 1 J = 1000 of these, so W = 1.217 × 10⁶ in his units. For the electric field, 1 V/m = 10⁵ of his units [memory: medium; derived from 1 V = 10⁸ CGS electromagnetic units and the change of length unit from cm to m; not web-checked]. A check that this reading is right: his printed peak magnetic force is .193, and P/(μV) = 6 × 10⁷ / 3.1074 × 10⁸ = 0.193, so his P and H are in the same consistent system [read: page image p.53; calc].

Step 3: apply his formula with his own numbers. With μ = 1 and his V = 3.1074 × 10⁸ m/s (the Weber-Kohlrausch value): P = √(8π × 3.1074 × 10⁸ × 1.217 × 10⁶) = 9.75 × 10⁷ of his units = 975 V/m.

Step 4: the modern calculation. A plane wave of peak field E₀ carries the time-averaged energy flux S = E₀²/(2μ₀c), so E₀ = √(2 μ₀ c S) = √(2 × 1.2566 × 10⁻⁶ × 2.998 × 10⁸ × 1217) = 958 V/m (peak). The root-mean-square (RMS, the square root of the time average of E²) value is E₀/√2 = 677 V/m.

Step 5: compare. Maxwell's formula is the modern one: W = P²/(8πμV) in his units is the same as S = E₀²/(2μ₀c) in SI, and both relate a peak field to a time-averaged flux. So no factor of √2 (RMS versus peak) belongs in the comparison. Steps 3 and 4 differ only by √(V/c) = √(3.1074/2.998) = 1.018, because Maxwell's speed was 3.6 % high. The method is right.

Step 6: the discrepancy. Maxwell printed 6.0 × 10⁷ (600 V/m), not 9.75 × 10⁷. That is 1.62 times lower than his own formula gives from his own input (0.62 of it), equivalent to using W = 461 W/m² instead of 1217 W/m². It is not √2 (that would be 0.71), and it is not an OCR or typesetting slip in one digit, because his magnetic value .193 matches the printed 6 × 10⁷. The cause is not identified; a slip in converting foot-pounds or square feet, or a different W than the one he quotes, are possibilities that nothing on the page settles [arbiter, Round 2]. His values at the Sun's surface (about 13,000 cells per metre, magnetic 4.13) also look about ten times too small: the Sun's surface is about 215 solar radii from the Earth [memory: high], so the field there should be about 215 times larger, about 129,000 cells per metre, whereas his ratio is only about 21.5 [calc; arbiter, Round 2].

What this shows. A real-world number from the theory: direct sunlight has a peak electric field of roughly 1 kV/m, reversing about 10¹⁵ times a second (Maxwell: "more than a thousand million million times in a second"). It also shows, again, that a reader with a pencil can check a founding paper: the formula holds up, the printed arithmetic does not, and the reason is unknown.

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6. Evidence table

Verdicts are the arbiter's final verdicts, using the review scale.

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7. What a learner should take away

What is solid. The field equations, the displacement current, the identification of light as an electromagnetic wave, field energy and the Poynting theorem, and Heaviside's vector form are among the best-tested results in physics. Maxwell's central logical move in 1865 was sound: two numbers obtained by completely independent methods (a capacitor discharged through a galvanometer; a rotating mirror) agreed, and he said so in hedged language, "agrees sufficiently well" and "seems to show" [read: page image p.47].

How strong the 1865 evidence really was. The electrical value was 3.65 % above the modern c; the light values Maxwell quoted ranged from −0.60 % (Foucault) to +5.03 % (Fizeau), and Fizeau and Foucault disagreed with each other by 5.7 % (example.py Part A). The agreement was suggestive, not decisive. The same paper gives a yardstick for 1860s laboratory error. Maxwell calculated the self-inductance of the British Association's standard-resistance coil from its dimensions as L = 430,165 "metres" (his units; about 43 mH in SI [memory: high]). A least-squares fit to sixteen spinning-coil experiments gave 456,748 (6.2 % higher), and a "rough experiment" with his electric balance gave 410,000 (4.7 % lower) [read: page images pp.59-60]. He judged the calculation "probably much more accurate than either of the other determinations", but gave no error estimates [read: page image p.60]. So 5-6 % between calculation and measurement was normal for good work then, and a 5 % agreement between v and the speed of light could not by itself have proved the identity. The decisive test came 22 years later, with Hertz. That is a model of how a bold idea should be treated: proposed with the evidence stated honestly, then tested directly.

What was wrong, and why it is instructive. - The printed sign errors in (F) and (G) show that even a founding paper can carry mistakes that a careful reader with a pencil can find, and that nobody's authority substitutes for checking. They also show the difference between a fatal error and a repairable one: flip two signs and everything works. - On the same page as the sign error, Maxwell reports a failed prediction of his own theory: if gold had the same resistance at light frequencies as for ordinary currents, a gold leaf measured by Hockin should pass only 10⁻⁵⁰ of incident light, yet he found that between 1/500 and 1/1000 of green light gets through [read: page image p.52]. He published the discrepancy, after noting that much of the light goes through holes and cracks but enough passes through the gold itself "to give a strong green hue", and wrote plainly: "This result cannot be reconciled with the electromagnetic theory of light, unless we suppose that there is less loss of energy when the electromotive forces are reversed with the rapidity of the vibrations of light than when they act for sensible times" [read: page image p.53]. His escape route was the right one: the conductivity of metals depends on frequency, explained later by electron models of metals (Drude, around 1900) [memory: medium, not web-checked]. - His sunlight field strength (section 5.2) is 1.6 times smaller than his own formula and input give, and his Sun-surface values look about ten times too small. The formula is correct and matches the modern one; the printed arithmetic is not, for a reason nobody in this review could find [arbiter, Round 2]. An anomaly honestly reported pointed to new physics. - The ether was a reasonable hypothesis that Maxwell himself kept out of his equations. When experiment found no ether wind, the equations survived untouched.

The Heaviside story, told accurately. It is a genuine case of an outsider obstructed by an official: Preece, head of the Post Office telegraph engineers, declared Heaviside's loading paper "worthless and blocked it", and loading later worked commercially [read: Hunt]. But not every rejection was suppression. The Royal Society referee who rejected his third operator-calculus paper objected to his handling of divergent series, a real mathematical gap that was later filled (Bromwich 1916 and others; Laplace transform methods) [read: Belrose; date memory: high]. Heaviside's reply, "Shall I refuse my dinner because I do not fully understand the process of digestion?" [read: Hunt], is a fine line but not a proof. And the establishment did recognise him: Fellow of the Royal Society in 1891, a government pension in 1896 [read: Hunt]. Distrust of institutions is best served by keeping these episodes distinct.

What is genuinely open (and mostly conceptual). - Where field energy "is" and which way it "flows" locally, beyond the totals the theory fixes. - How physically real the vector potential is, beyond its loop integral. - Whether the (F) sign was a printer's slip or Maxwell's convention. - Where Maxwell's factor of 1.6 in the sunlight field, and the factor of ten in his Sun-surface values, came from (check Thomson's 1854 paper for the foot-pound convention, and another printing of the 1865 paper). - Whether anyone other than Hertz repeated his experiments in air in 1888-89, with a documented source (Hunt's article does not say; his book The Maxwellians may). - The Electrician episode, which rests on Biggs's later hint and needs archival sources.

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8. Check yourself

1. Why does Ampère's law without the displacement current contradict charge conservation for a charging capacitor? Use div(curl H) = 0.
2. Using μ₀ = 1.2566 × 10⁻⁶ N/A² and ε₀ = 8.854 × 10⁻¹² F/m, compute 1/√(μ₀ε₀). By what percentage did Weber and Kohlrausch's v = 310,740,000 m/s differ from it?
3. Repeat the worked example for a = 0.5 mm, everything else the same. What are R, S at the surface, and the total inflow? Does the inflow still equal I²R?
4. In Maxwell's trial solution G = e^(−px) cos(qx + nt) with n, q > 0, which way does the wave travel? If p > 0, does the amplitude grow or shrink along the direction of travel?
5. Why can adding curl F to the Poynting vector S (for any smooth F) never be detected by measuring the total power flowing through a closed surface?

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9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Cambridge University Press). Chapter 7 (displacement current and Maxwell's equations), Chapter 8 (Poynting's theorem; includes the power-into-a-wire calculation), Chapter 9 (EM waves), Chapter 10 (potentials and gauge) [memory: high for chapter topics]. - R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. II, Chapter 27 "Field Energy and Field Momentum", which discusses the energy flowing into a wire and the ambiguity in locating field energy (free online at feynmanlectures.caltech.edu) [memory: high]. - E. M. Purcell and D. J. Morin, Electricity and Magnetism, 3rd ed. (Cambridge University Press): a gentler alternative to Griffiths. - B. J. Hunt, The Maxwellians (Cornell University Press, 1991): the history of how FitzGerald, Lodge and Heaviside turned Maxwell's theory into its modern form [memory: high]. - P. J. Nahin, Oliver Heaviside: Sage in Solitude (IEEE Press, 1988): the standard biography; Belrose's subtitle borrows its title [memory: high]. - J. W. Arthur, "The Evolution of Maxwell's Equations from 1862 to the Present Day", IEEE Antennas and Propagation Magazine (2013): discusses the sign questions in the 1865 equations [memory: medium; cited via the Wikipedia article above].

The researchers' own key works and the documents in this folder. - J. C. Maxwell, "On Faraday's Lines of Force", Transactions of the Cambridge Philosophical Society 10 (1855-56) (MaxwellOnFaradaysLines.pdf). - J. C. Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society 155 (1865) (dynamicaltheoryo00maxw.pdf). Read paras 1-8 on PDF pp.7-9 (Weber, the evidence for the medium), para 20 on PDF p.14 (Faraday's priority), paras 73-75 (on the medium), the equations on PDF pp.32-34, Part VI on PDF pp.47-49, the absorption section on PDF p.52, the sunlight estimate on PDF p.53, and the coil calculation on PDF pp.59-60. - J. C. Maxwell, A Treatise on Electricity and Magnetism (1873) [memory: high]. - O. Heaviside, Electromagnetic Theory, 3 vols. (1893-1912) [memory: high]. - B. J. Hunt, "Oliver Heaviside: A first-rate oddity", Physics Today 65(11), 48 (2012) (HeavisideOliver-Hunt.12.pdf). - J. S. Belrose, "Oliver Heaviside: a first rate oddity, sage in solitude" (2014) (Belrose-MUL-2014-Vol2-Mar_Apr-005.pdf).

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Answers

1. Taking the divergence of curl H = J gives 0 = div J. But while a capacitor charges, current flows into a plate and charge accumulates there, so div J = −∂ρ/∂t ≠ 0. With the displacement current, 0 = div J + ∂(div D)/∂t = div J + ∂ρ/∂t, which is exactly charge conservation.

2. μ₀ε₀ = 1.2566 × 10⁻⁶ × 8.854 × 10⁻¹² = 1.1126 × 10⁻¹⁷ s²/m², so 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s. Then 310,740,000 / 299,792,458 = 1.0365: Weber and Kohlrausch's value was 3.65 % high.

3. A = π(0.5 × 10⁻³)² = 7.854 × 10⁻⁷ m², so R = 1.68 × 10⁻⁸ / 7.854 × 10⁻⁷ = 0.02139 Ω (four times larger). E = IR/L = 0.2139 V/m. H = 10/(2π × 0.5 × 10⁻³) = 3183 A/m. S = EH = 680.9 W/m². Side area = 2π × 0.5 × 10⁻³ × 1 = 3.1416 × 10⁻³ m². Inflow = 2.139 W = I²R = 100 × 0.02139. Yes; the symbolic identity in Step 8 holds for any radius.

4. A crest has constant phase qx + nt, so x = −(n/q)t: the wave moves toward −x. With p > 0, e^(−px) gets larger as x becomes more negative, so the amplitude grows along the direction of travel. That is what Maxwell's printed sign gives, and why it is wrong for an absorbing conductor.

5. By the divergence theorem, the total flux of curl F through a closed surface equals the volume integral of div(curl F), which is zero for any smooth F. So S and S + curl F give the same total power through every closed surface, and every energy-balance measurement (which counts power in or out of a closed region) gives the same result for both.