Maxwell's Displacement Current

Any surface, the same current

Ampere's circuital law, from the chapter Sources of the Magnetic Field, links the field round a closed loop to the current through it: \(\oint\vec B\cdot d\vec l = \mu_0I_\text{enc}\). The current is counted through a surface whose edge is the loop. Think of a soap film on a wire ring. For a steady current it does not matter which film you choose. A flat film and a bulging one on the same ring are crossed by the same current. Charge cannot pile up anywhere between them.

A capacitor breaks the rule

Now charge a capacitor from a cell through a resistor. While it charges, a current \(I\) flows in its leads. Draw a loop round the left lead. Take first a flat disc bounded by the loop, which the wire pierces: the current through it is \(I\). Take next a surface bounded by the same loop that bulges round the left plate and passes between the plates. No wire crosses the bulge, and no charge crosses the gap. The current through it is zero.

So the same loop, with the same magnetic field round it, would give \(\mu_0I\) through one surface and zero through the other. The law cannot be right as it stands. Something must happen between the plates that the law has missed. There is no current there, but there is an electric field, and while the capacitor charges, that field grows.

Maxwell's extra term

In the gap of a parallel-plate capacitor the field is \(E = q/\epsilon_0A\), as the chapter Capacitance and Dielectrics found. So the charge on a plate is \(q = \epsilon_0AE\). The product \(EA\) is the electric flux \(\Phi_E\) through the gap. So \(q = \epsilon_0\Phi_E\), and the current in the lead, \(dq/dt\), equals \(\epsilon_0\,d\Phi_E/dt\). A quantity in the gap matches the current in the wire exactly.

James Clerk Maxwell added this displacement current to the current in Ampere's law. The new law is the Ampere-Maxwell law. Now the flat disc counts the current \(I\) in the wire, and the bulge counts the displacement current \(I_d\) in the gap. The two are equal at every instant, so both surfaces give the same answer.

The displacement current has a direction as well as a size. It points along the change of \(\vec E\). While the capacitor charges, \(\vec E\) grows from the positive plate to the negative plate. So \(I_d\) runs across the gap in the same direction as the current arriving at the positive plate. While it discharges, \(\vec E\) shrinks, and \(I_d\) runs from the negative plate back to the positive one. Either way, the current never stops at the gap.

A magnetic field in the gap

Because the displacement current makes a magnetic field, a field circles the gap of a charging capacitor too. Take circular plates of radius \(R\), charged by a current \(I\). A loop of radius \(r < R\) in the gap encloses only the flux through its own area, a fraction \(r^2/R^2\) of the whole. Ampere-Maxwell then gives \(B\cdot2\pi r = \mu_0Ir^2/R^2\).

\[ \begin{gathered} B = \dfrac{\mu_0Ir}{2\pi R^2}\quad(r \le R) \\ B = \dfrac{\mu_0I}{2\pi r}\quad(r \ge R) \end{gathered} \]

Inside the gap the field grows in proportion to \(r\), like the field inside a solid wire. Beyond the edge it falls as \(1/r\), exactly like the field of the lead. For plates of radius \(10\,\text{cm}\) charged by \(2\,\text{A}\), the field at the edge is \(\mu_0I/2\pi R = 4\,\mu\text{T}\).

Four laws for every field

With the displacement current in place, four laws describe every electric and magnetic field. Together they are Maxwell's equations. Each says in one sentence what makes a field, or what a field cannot do.

Look at the last two laws side by side. A changing magnetic field makes an electric field. A changing electric field makes a magnetic field. Each change can keep the other going, with no charge or current nearby. Maxwell saw that the two together could travel through empty space as a wave. The next lesson follows that wave.