Of Maxwell’s four equations, three were already known in some form before him: Gauss’s law, the magnetic law and Faraday’s law. The fourth, Ampère’s, was known too, but it was incomplete. What Maxwell added to it was a single term, $\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}$. It looks like little. It was this term that turned electricity and magnetism into light.

Ampère’s law

In 1826 the Frenchman André-Marie Ampère found the rule linking a current to the magnetic field around it. In today’s language: if you go round a closed path around a wire and add up the component of the magnetic field along the path, the result is proportional to the current $I$ crossing any surface bounded by that path:

$$\oint\mathbf B\cdot d\boldsymbol\ell=\mu_0 I.$$

The word any is important. Think of the rim of a butterfly net: the net can be short or long, but the wire that passes through the rim also passes through the net. The current through the net does not depend on the shape of the net.

The capacitor that breaks the rule

Now put a capacitor in the circuit: two parallel metal plates separated by air or vacuum. While the capacitor charges, a current $I$ arrives along a wire at one plate, and another leaves the other plate. But between the plates no current flows: there is no wire, only empty space.

Take a circular path around the wire, before the capacitor. If the net is flat, cutting the wire, the current through it is $I$. If the net is long, like a bag that passes between the plates without touching the wire, the current through it is zero. Same path, same integral of the magnetic field, two different answers. Ampère’s law cannot be right in this case.

What changes between the plates

No charge passes between the plates, but something does change: the electric field. As the plate accumulates charge $Q$, the field between the plates, of area $A$, grows:

$$E=\frac{Q}{\varepsilon_0A}.$$

The flux of the electric field through the area of the plates is $\Phi_E=EA=\frac Q{\varepsilon_0}$, and it changes at the rate

$$\varepsilon_0\frac{d\Phi_E}{dt}=\frac{dQ}{dt}=I.$$

Exactly the current arriving along the wire! Maxwell proposed that a changing electric field acts like a current. He called it the displacement current, $I_d=\varepsilon_0\frac{d\Phi_E}{dt}$, and added it to Ampère’s law:

$$\boxed{\oint\mathbf B\cdot d\boldsymbol\ell=\mu_0\left(I+\varepsilon_0\frac{d\Phi_E}{dt}\right).}$$

Now the two nets agree. Through the flat net there is only the current $I$ in the wire. Through the long net there is no current, but there is the changing field between the plates, which contributes the same amount.

Why “displacement”?

The name comes from the 1861 mechanical model. Maxwell imagined that even in a vacuum there was a medium, the ether, whose electrical particles were displaced a little when the field changed, like stretched springs. That small displacement would be a real current. The ether does not exist, as Michelson, Morley and Einstein would show, but the term survived the model: nature obeys the equation, even though the mechanical explanation was wrong.

Charge is conserved

There is another reason the term must be there. In differential form, Ampère’s law without the correction says $\nabla\times\mathbf B=\mu_0\mathbf J$. But the divergence of a curl is always zero, so it would force $\nabla\cdot\mathbf J=0$: current could never accumulate anywhere, and no capacitor could charge. With Maxwell’s term,

$$0=\nabla\cdot(\nabla\times\mathbf B)=\mu_0\nabla\cdot\mathbf J+\mu_0\varepsilon_0\frac{\partial}{\partial t}(\nabla\cdot\mathbf E).$$

By Gauss’s law, $\nabla\cdot\mathbf E=\frac\rho{\varepsilon_0}$, and what is left is

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

This is the conservation of charge: the charge in a region only increases if current flows into it. Maxwell’s correction is exactly what is needed for the equations to respect this principle.

Exercise: a capacitor accumulates 6 microcoulombs in 2 milliseconds. What is the average displacement current between the plates?

The average current in the wire is $\frac{\Delta Q}{\Delta t}=\frac{6\times10^{-6}}{2\times10^{-3}}=3\times10^{-3}$ A, or 3 mA. Between the plates the average displacement current is the same, 3 mA, even though no electron crosses the space between them.

Exercise: why had nobody noticed the displacement current before Maxwell?

Because it is very small, except when the fields change very fast. The term is multiplied by $\varepsilon_0\approx8.85\times10^{-12}$. In a capacitor with plates 10 cm by 10 cm, a field growing by 1,000 V/m per second produces $I_d=8.85\times10^{-12}\cdot1{,}000\cdot0.01\approx9\times10^{-11}$ A, less than a tenth of a nanoampere. Only with oscillations of millions of times a second, as in Hertz’s experiments of 1887, does the effect become obvious.

Sources: James Clerk Maxwell, “On physical lines of force”, Philosophical Magazine 21 and 23 (1861–1862) · The Feynman Lectures on Physics, volume II, chapter 18 · Basil Mahon, The Man Who Changed Everything (Wiley, 2003), ch. 7.

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