Electromagnetic WavesDisplacement Current

Maxwell's argument starts with something very ordinary: a capacitor being charged. Recall Ampère's circuital law from Chapter 4. Around any closed loop,

where i(t) is the current passing through a surface whose rim is the loop. In Fig. 8.1 a parallel plate capacitor C sits in a circuit carrying a current i(t) that changes with time. To find the magnetic field at a point P outside the capacitor, take a circular loop of radius r around the wire, its plane at right angles to the wire. By symmetry B runs along the loop and has the same size everywhere on it, so the left side of (8.1) is B(2πr):

But Ampère's law allows any surface with that loop as its rim, and Fig. 8.1 offers three: (a) a flat disc, (b) a pot that bulges towards the capacitor and has its bottom between the plates, and (c) a tiffin box without a lid, whose flat bottom S also sits between the plates. Try all three, then try Maxwell's version of the law.

The wire pierces the flat disc, so the full current i passes through it and (8.2) gives a magnetic field at P. The pot and the tiffin box touch no wire at all: no charge flows between the plates, so by Ampère's law B at P would be zero. Calculated one way there is a field at P; calculated another way there is none. The field at P cannot depend on which imaginary surface we draw, so Ampère's law must be missing something. The missing term has to give the same B at P for every surface. And something does pass through the bottom of the pot and the tiffin box: the electric field between the plates.

The Missing Term

Look closely at the tiffin box. Its flat bottom S lies between the plates, parallel to them. If each plate has area A and carries charge Q, the field between the plates is E = (Q/A)/ε₀ (Chapter 2). This field is perpendicular to S, has the same size over the area A of the plates, and is zero outside it. So the electric flux through S is

which simplifies to Φ_E = Q/ε₀Qε₀Q/A.

While the capacitor charges, Q grows and so does Φ_E. The current in the wire is i = dQ/dt, so

Multiply the rate of change of flux by ε₀ and you get the current i in the wirezerothe charge Q:

This is exactly the missing term. Add to the current carried by conductors through a surface a second term, ε₀ times the rate of change of electric flux through the same surface, and the total comes out the same, i, for the disc, the pot and the tiffin box alike. There is then no contradiction: B at P is the same whichever surface we use, and B at P just outside the plates equals B at a point M just inside them, as it should.

The Ampère–Maxwell Law

The current carried by moving charges is called the conduction current, i_c. The new term of (8.4) comes from a changing electric field (once called the "electric displacement", an old name still heard sometimes), so it is called the displacement current, or Maxwell's displacement current:

Maxwell's generalisation is that the source of a magnetic field is not just the conduction current, but also the rate of change of electric field. The total current through a surface is

Apply it to the charging capacitor. In the wire, outside the plates, i_c = i0 and i_d = 0i. In the gap between the plates, i_c = 0i and i_d = i0.

The current does not stop at the plates: it carries on across the gap as a displacement current of exactly the same size. In every respect the displacement current has the same physical effects as a conduction current. In some cases it is zero, for example in a wire with a steady current, where E does not change. In others, like the charging capacitor, the two kinds sit in different regions of space. Usually both are present together, because no material is a perfect conductor or a perfect insulator. And there can be large regions of space with no conduction current at all, only a displacement current due to changing electric fields. There we expect a magnetic field with no current-carrying wire anywhere near.

A Magnetic Field With No Wire

Is the displacement current only bookkeeping? No: it makes a magnetic field that can be measured, and the field between the plates (at M in Fig. 8.2) turns out to be the same as just outside them (at P). Take a capacitor with circular plates of radius R = 10 cm and a displacement current i_d = 0.5 A spread evenly over the plate area. A circular loop of radius r centred on the axis between the plates encloses the part of i_d that crosses its own area.

Between the plates the loop encloses only the fraction (r/R)² of the displacement current, so B makes a straight-line rise from zero on the axis. Outside, the whole 0.5 A is enclosed and B follows the 1/r fall of a wire's field. Ampère–Maxwell's law, applied to the loop of radius r, gives

At r = R the two agree: the field just inside the rim of the plates equals the field just outside it. The magnetic field lines between the plates are circles parallel to the plates, while E points straight across the gap, so here E and B are at right angles. Remember this; it will matter for waves.

Changing, Not Large

Now connect a capacitor to an ac supply, as in Chapter 7. The charge on the plates, and with it the field E between them, rises and falls like a sine wave. Is the magnetic field around the gap strongest when E is strongest?

The displacement current is a quarter of a cycle ahead of the electric field. Where E peaks, its graph is flat for an instant, dΦ_E/dt = 0, and there is no magnetic field at all. Where E passes through zero, it is changing fastest, and B is at its largest. A magnetic field is made by a changing electric field, not by a large one.

Fields That Make Each Other

The displacement current has, quite literally, far-reaching consequences. It makes the laws of electricity and magnetism much more symmetrical. Faraday's law says an emf is induced equal to the rate of change of magnetic flux. An emf between two points is the work done per unit charge in taking a charge between them, so an emf means an electric field. Faraday's law can therefore be read as: a magnetic field changing with time gives rise to an electric field. The Ampère–Maxwell law supplies the mirror image: an electric field changing with time gives rise to a magnetic field. Changing electric and magnetic fields give rise to each other, and that is why electromagnetic waves exist.

The symmetry is not perfect: no one has ever found a magnetic monopole, a source of magnetic field the way a charge is a source of electric field. The four laws together are Maxwell's equations in vacuum:

Match each law to what it says.

Two of the laws say where fields come from when nothing changes; the other two say how a change in one field makes the other. Add the Lorentz force, which says how fields push on charges, and you have all of classical electromagnetism.

Speed Round: Displacement Current

; in a charging capacitor i_d in the gap equals the current in the wires. Get five right to move on.