On Physical Lines of Force/Modern derivations/The displacement current

On Physical Lines of Force → Modern derivations → The displacement current

What Maxwell had

Equation (112), from Part III, Prop. XIV, under the modest heading "To correct the equations (9) of electric currents for the effect due to the elasticity of the medium":

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

Provenance: the equation number is attested three independent ways; the body is a reconstruction in Maxwell's notation. See Sources.

The total current is the conduction current $p,q,r$ plus the rate of change of the displacement $f,g,h$. In modern notation:

$${𝐉}_{\text{total}}=𝐉+\frac{\partial 𝐃}{\partial t}$$

The second term is the displacement current. It is not a current of anything in the ordinary sense: no charge crosses the gap of a capacitor. It is a rate of change of the electric field that has the same magnetic effect a current would have.

Maxwell's route: elasticity

His argument occupies three lines and is worth having in front of you.

1. Equation (105) of Prop. XIII 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; particles in motion are a current; so it must be added to $p,q,r$.

His own phrase: "a variation of displacement is equivalent to a current."

And from the introductory pages of Part III, stated even before the propositions:

This displacement does not amount to a current, because when it has attained a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing.

The cell yields, and pushes back

The modern route: consistency

The argument textbooks give today has nothing to do with elasticity.

Charge is conserved. That is the continuity equation:

$$\mathrm{\nabla }\cdot 𝐉+\frac{\partial \rho }{\partial t}=0$$

Ampère's uncorrected law says $\mathrm{\nabla }\times 𝐇=𝐉$. Take the divergence; the left side vanishes identically; so the law requires $\mathrm{\nabla }\cdot 𝐉=0$, which contradicts continuity whenever charge accumulates.

Now use Gauss's law, $\rho =\mathrm{\nabla }\cdot 𝐃$:

$$\mathrm{\nabla }\cdot 𝐉+\frac{\partial }{\partial t}(\mathrm{\nabla }\cdot 𝐃)=\mathrm{\nabla }\cdot (𝐉+\frac{\partial 𝐃}{\partial t})=0$$

The combination $𝐉+\partial 𝐃/\partial t$ is divergence-free unconditionally. It is the only object that can sit on the right of Ampère's law without contradicting the conservation of charge.

The term is therefore not a hypothesis. Given Gauss's law and charge conservation, it is forced.

The capacitor

The variation of displacement is itself a current

The standard illustration, and the one that makes the necessity concrete.

A capacitor is charging. Current $I$ flows in the wires; nothing crosses the gap. Take a closed loop $C$ encircling one of the wires. Ampère's law in integral form says the circulation of $𝐇$ round $C$ is fixed by the current through any surface bounded by $C$. So take two:

surface	pierced by	uncorrected answer
$S_1$, flat, cutting the wire	the wire, carrying $I$	$I$
$S_2$, bulging out to pass between the plates	nothing at all	$0$

Two answers for one loop. The law as stated is inconsistent.

Now count the displacement term. Between the plates, with charge $Q$ on a plate of area $A$, the displacement is $D=Q/A$, so

$${\int }_{S_2}\frac{\partial 𝐃}{\partial t}\cdot d𝐒=A\frac{d}{dt}\left(\frac{Q}{A}\right)=\frac{dQ}{dt}=I$$

exactly the current missing from $S_2$. The two surfaces now agree, and they agree for every surface, because the total current density is divergence-free.

Two routes, one term

This deserves to be stated plainly, because it is the most remarkable fact about the paper.

	Maxwell, 1862	modern textbook
starting point	the cell substance is elastic	charge is conserved
motivation	to explain how rotation reaches the inside of a cell	to make Ampère's law self-consistent
the term obtained	$\partial 𝐃/\partial t$	$\partial 𝐃/\partial t$

Maxwell was patching a mechanical difficulty about spinning cells. He was not repairing a continuity equation — he does not raise the consistency problem at all. That the patch is precisely the term consistency requires is either remarkable luck or evidence that the mechanical analogy was tracking something structural. Historians have argued both; Daniel Siegel's Innovation in Maxwell's Electromagnetic Theory (1991) is the standard treatment and comes down on the side of the analogy doing real work.

Why everything depends on it

Without the term, the equations do not couple:

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

A changing magnetic field makes an electric field. A changing electric field makes nothing. The chain has one link and stops.

With the term:

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

Each field's change generates the other. Take the curl of one, substitute the other, and a wave equation appears. That is the next page, and it is light, radio, radar, and every wireless signal that has ever been sent.

Where the conventions differ

	Maxwell 1861	modern SI
displacement	$f,g,h$, with $R=-4\pi E^{2}h$	$𝐃=\epsilon 𝐄$; Maxwell's triple is $𝐃/4\pi$ up to sign
the sign	his own convention for the direction of displacement	fixed by $𝐃=\epsilon 𝐄$
the coefficient	$E$, an elasticity, with $\epsilon =1/E^2$	$\epsilon$, a permittivity
status of the term	a correction for elasticity	required by charge conservation

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Plain-language version: The missing term