By the 1860s the four experimental laws of electromagnetism were in place — Gauss for \mathbf{E}, Gauss for \mathbf{B}, Faraday's law of induction, and Ampère's law linking a current to the magnetic field that circles it. Four laws, beautifully confirmed in the laboratory. And yet, quietly, they contradicted themselves. Point Ampère's law at the wire charging up an everyday capacitor and it returned two different answers for the same magnetic field, depending on nothing more than an imaginary surface you were free to draw however you liked. That is not a subtlety — that is a law breaking.

James Clerk Maxwell found the missing piece, and it is the single most consequential correction in the history of physics. His fix rests on one sentence, worth reading twice: a changing electric field is itself a source of magnetic field, exactly as a real current is. Add that one term — the displacement current — and the contradiction vanishes, charge is conserved automatically, and, as a free and staggering bonus, the equations start to predict waves of electric and magnetic field sailing through empty space at the speed of light. This page builds that one term from the paradox that demands it.

Ampère's law, and the surface that shouldn't matter

Ampère's law in integral form says that if you walk once around a closed loop C, adding up the magnetic field along the way, the total is set by the current threading through the loop:

Here I_{\text{enc}} is the current passing through any surface whose boundary is the loop C. That little word "any" is the whole story. A wire loop is the rim of infinitely many surfaces — a flat drumskin stretched across it, a shape bulging out like a soap bubble, anything at all — and the physics had better not care which one you pick. For a steady, unbroken current it never does: whatever surface you choose, the same current pierces it, because steady charge cannot pile up anywhere. The trouble starts the moment charge is allowed to accumulate.

The differential form, which you have already met in Maxwell's equations in differential form, writes the same content locally as \nabla\times\mathbf{B} = \mu_0\mathbf{J}. Take the divergence of both sides and a fatal flaw jumps out.

The capacitor paradox

Take the cleanest possible case of charge piling up: a parallel-plate capacitor in the middle of being charged. A steady current I runs down the wire, dumps positive charge onto the left plate, and drains it off the right plate — but no charge crosses the gap between the plates. Now draw an Amperian loop C around the wire, just to the left of the capacitor, and ask Ampère's law for the magnetic field on that loop. To evaluate I_{\text{enc}} we must pick a surface bounded by C:

• Choice A — the flat drumskin. Stretch a flat disk across the loop. The wire spears straight through it, carrying the full current I. So I_{\text{enc}} = I and \oint\mathbf{B}\cdot d\boldsymbol{\ell} = \mu_0 I.
• Choice B — the bulging bag. Now let the surface balloon out to the right, sliding between the capacitor plates before closing back onto the same loop. This surface is bounded by the very same C — but the wire never touches it. The current crosses into the gap and stops; nothing conducts across the vacuum between the plates. So I_{\text{enc}} = 0 and \oint\mathbf{B}\cdot d\boldsymbol{\ell} = 0.

Same loop, same magnetic field on it — and Ampère's law hands back \mu_0 I one way and 0 the other. It cannot be both. Reveal the figure step by step to see the two surfaces sharing one rim: the flat one the wire pierces, and the bulging one that threads the gap.

Maxwell's fix: the changing E-field carries the current

Look again at the bulging surface. No conduction current crosses it — but something else is happening there that is completely absent from the flat surface: the region between the plates holds an electric field, and while the capacitor charges, that field is growing. Maxwell's insight was to treat the changing field as if it were a current in its own right.

Make it quantitative. For a parallel-plate capacitor with plate area A carrying charge Q, the field in the gap is uniform:

The electric flux through the bulging surface (which captures the whole field between the plates) is therefore \Phi_E = E A = Q/\varepsilon_0. Differentiate in time, and the conduction current in the wire, I = dQ/dt, reappears exactly:

There it is. The quantity \varepsilon_0\, d\Phi_E/dt flowing through the bulging surface is numerically equal to the very current I that pierces the flat surface. Maxwell christened it the displacement current I_d, and added it to the enclosed current in Ampère's law. Now both surfaces agree: the flat one collects I of conduction current, the bulging one collects I_d = I of displacement current, and the magnetic field on the loop comes out the same either way.

Why the old law had to be wrong

The capacitor is the vivid picture, but there is a colder, more general reason the correction is forced. Take the divergence of the uncorrected law \nabla\times\mathbf{B} = \mu_0\mathbf{J}. The divergence of any curl is identically zero, so

But the continuity equation — the mathematical statement that charge is never created or destroyed — says \nabla\cdot\mathbf{J} = -\,\partial\rho/\partial t. The old law therefore demands \partial\rho/\partial t = 0 everywhere, i.e. that charge can never accumulate anywhere. That is exactly the assumption the charging capacitor violates. Now add Maxwell's term and use Gauss's law \nabla\cdot\mathbf{E} = \rho/\varepsilon_0:

The corrected source term is automatically divergence-free, so the corrected law is consistent with charge conservation for every configuration, steady or not. The displacement current is not a patch bolted on to fix one experiment — it is the unique term that makes electromagnetism obey its own conservation law.

Worked example — displacement current in a charging capacitor

A parallel-plate capacitor has circular plates of radius r = 5.0\ \text{cm} (so area A = \pi r^2 \approx 7.85\times 10^{-3}\ \text{m}^2) separated by a small vacuum gap. While it charges, the electric field between the plates rises at a steady rate dE/dt = 2.0\times 10^{12}\ \text{V/(m}\cdot\text{s)}. Find the displacement current, and check it against the conduction current in the wire.

Via the field. The displacement current is the flux term I_d = \varepsilon_0\, d\Phi_E/dt = \varepsilon_0 A\, dE/dt, with \varepsilon_0 = 8.85\times 10^{-12}\ \text{F/m}:

Via the circuit. Because Q = CV, the same current is I = dQ/dt = C\,dV/dt. For a parallel-plate capacitor C = \varepsilon_0 A/d and, with gap d, V = E d, so dV/dt = d\,(dE/dt). The gap d cancels:

The two routes give the identical \approx 0.14\ \text{A} — the displacement current between the plates matches the conduction current in the wire, exactly as the paradox demanded. Notice the answer is completely independent of the plate separation.

The payoff: electromagnetic waves

The displacement current would be worth the trouble just for mending Ampère's law. What makes it one of the great moments in science is what it unlocks. Faraday's law already said a changing magnetic field makes an electric field: \nabla\times\mathbf{E} = -\,\partial\mathbf{B}/\partial t. Maxwell's new term supplies the mirror image — a changing electric field makes a magnetic field. Put the two together in empty space, where \mathbf{J} = 0:

• a changing \mathbf{E} creates a \mathbf{B};
• that changing \mathbf{B} creates an \mathbf{E};
• which, changing in turn, creates a \mathbf{B} — and off it goes.

The fields bootstrap each other forward with no charges or currents anywhere: a self-sustaining ripple, an electromagnetic wave. Combine the two curl equations and the wave speed drops out of the constants of electrostatics and magnetostatics alone:

Maxwell computed that number and recognised it instantly as the measured speed of light. Without the displacement current there is no \partial\mathbf{E}/\partial t term, the chain never closes, and light is not part of electromagnetism at all. That tiny extra current between the capacitor plates is, quite literally, what makes light possible.