Part III — Prop. XIV

On Physical Lines of Force → Part III → Prop. XIV

Maxwell's statement

Prop. XIV. — To correct the equations (9) of electric currents for the effect due to the elasticity of the medium.

Printed on p. 19.

That is the whole of it. The most consequential term in nineteenth-century physics arrives under the heading of a correction.

The argument, which takes three lines

1. Equation (105) relates the electromotive force to the displacement: $R=-4\pi E^{2}h$.
2. Differentiate with respect to time. A varying electromotive force produces a varying displacement.
3. A variation of displacement is particles in motion, and particles in motion are a current. So it must be added to $p,q,r$ in the equations of currents.

Maxwell's phrase, from the surrounding prose: "a variation of displacement is equivalent to a current."

Equation (112)

The corrected equations of currents:

$$4\pi (p+\frac{df}{dt})=\frac{d\gamma }{dy}-\frac{d\beta }{dz}\mathrm{(}\mathrm{112}\mathrm{)}$$

$$4\pi (q+\frac{dg}{dt})=\frac{d\alpha }{dz}-\frac{d\gamma }{dx}$$

$$4\pi (r+\frac{dh}{dt})=\frac{d\beta }{dx}-\frac{d\alpha }{dy}$$

where $f,g,h$ are the components of the electric displacement and $p,q,r$ the conduction current as before.

Provenance: the number (112) is attested three independent ways — Wikisource's transcription labels the three corrected current equations (112); the prose on p. 19 says "Differentiating (112) with respect to x, y, and z respectively"; and secondary sources identify (112) as where the displacement current was added. The body above is a reconstruction in Maxwell's notation. See Sources.

Equation (113)

Differentiating (112) with respect to $x$, $y$, $z$ respectively and adding gives the equation of continuity for free electricity:

$$e+\frac{df}{dx}+\frac{dg}{dy}+\frac{dh}{dz}=0\mathrm{(}\mathrm{113}\mathrm{)}$$

Provenance: the number (113) is attested; the body is a reconstruction. See Sources.

In modern notation this is $\mathrm{\nabla }\cdot 𝐃=\rho$ up to Maxwell's sign convention and his factor of $4\pi$ — Gauss's law for electricity, the last of the four. With it, every one of the modern Maxwell equations is now present somewhere in this paper.

Why the correction was necessary, though Maxwell does not say so

He arrives at the displacement current from elasticity. There is a second route, which he does not take here, and which is how the term is usually motivated today.

The uncorrected equation (9) says $\mathrm{\nabla }\times 𝐇=4\pi 𝐉$. Take the divergence of both sides. The divergence of a curl is identically zero, so the equation demands

$$\mathrm{\nabla }\cdot 𝐉=0$$

— electric charge can never accumulate anywhere. That is true for steady currents in closed circuits and false the moment anything charges up.

The variation of displacement is itself a current

The standard illustration is a capacitor being charged. Current flows in the wires; no charge crosses the gap. Take a closed loop round one wire, and ask what current passes through a surface bounded by that loop:

• a flat surface cutting the wire is pierced by the full conduction current;
• a surface bulging out to pass between the plates is pierced by nothing at all.

Both must give the same circulation of magnetic force, because the circulation depends only on the loop. Uncorrected, the equation gives two different answers. Corrected, the changing displacement between the plates supplies exactly what the second surface was missing, and the two agree.

Maxwell got the right term for a reason that has nothing to do with this argument. He was patching an elastic medium, not repairing a continuity equation. That the patch turns out to be precisely the term consistency requires is either a considerable piece of luck or evidence that the mechanical analogy was tracking something real. Historians have argued both ways; Daniel Siegel's monograph is the standard treatment.

What the term makes possible

Everything.

Without it, a changing magnetic field makes an electric field (equation 54), but a changing electric field makes nothing. The chain stops. With it, the two equations close on each other:

$$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t},\mathrm{\nabla }\times 𝐇=4\pi 𝐉+\frac{\partial 𝐃}{\partial t}$$

A changing magnetic field produces a changing electric field, which produces a changing magnetic field, and the disturbance propagates. Take the curl of one and substitute the other and a wave equation appears — which is Prop. XVI, and which is light.

Radio, radar, every wireless transmission and every optical fibre depends on the term Maxwell added in this proposition to fix a difficulty about spinning cells.

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Next: Prop. XV · Previous: Prop. XIII

Modern derivation: The displacement current · Gauss's law · Plain-language version: The missing term