The term that made light

By 1860 electromagnetism was four statements, each with its own experimental history and none of them obviously connected to the others. Charge makes a diverging electric field. There are no magnetic charges. A changing magnetic field makes a circulating electric field. A current makes a circulating magnetic field. Between them they accounted for everything that had been measured, and one of them was wrong.

That is not a subtlety at the edge of the theory. It is a straightforward contradiction, available in any laboratory with a battery and two metal plates, and it says that the fourth statement is incomplete rather than approximate. What is missing is the thing that is happening between the plates and nowhere else in the circuit: an electric field that is growing.

Repairing it costs one term #

The repair is to add to the enclosed current a quantity built from the changing electric flux:

$$\oint\mathbf{B}\cdot\mathrm{d}\boldsymbol\ell = \mu_0\left(I_{\text{enc}} + \varepsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t}\right).$$

For the bag-shaped surface, the second term is exactly what the first term is for the flat one. The figure computes both from the geometry — the flux between the plates is $(V/d)A$, so $\varepsilon_0$ times its rate of change is $\varepsilon_0A/d$ times $\mathrm{d}V/\mathrm{d}t$, which is the capacitance times the rate of change of voltage, which is the current in the wire. The two are not approximately equal; they are the same product of the same three quantities.

Where the flux is is between the plates, traced from the charge on them: its flux through any surface stretched between them is $EA$, and the term added to Ampère’s law is $\varepsilon_0$ times how fast that changes. The name displacement current is Maxwell’s, from a mechanical model of the ether in which something really did move. The model is gone and the name has stayed, which is unfortunate, because nothing about the term is a current: no charge crosses the gap, and the quantity is a rate of change of a field.

Two things about the repair are worth separating. It was required — the law was inconsistent without it, and Maxwell arrived at it by demanding consistency with the conservation of charge rather than by fitting an experiment. And it was, at the time, entirely unmeasurable: the term is $\varepsilon_0\,\mathrm{d}E/\mathrm{d}t$, and $\varepsilon_0$ is $8.85\times10^{-12}$, so at any frequency available in 1861 the effect is nothing at all. Maxwell added a term nobody could detect to fix a problem nobody was complaining about.

The law before the repair is exact for a steady current, and that is precisely why the trouble is invisible in it. Take four different paths round one wire: three of them return the same 12.566 µT·m and the fourth, enclosing nothing, returns zero. Nothing there is wrong and nothing hints at what is coming — a steady current has the same value through every surface bounded by a given loop, so the ambiguity that breaks the law cannot arise.

What four consistent equations then allow #

Once the term is in, the four statements have a symmetry they did not have before: a changing magnetic field makes a circulating electric field, and a changing electric field makes a circulating magnetic field. That is a closed loop of causation, and closed loops of causation support waves.

Half of it was already known: a changing magnetic flux drives an electric circulation, which Faraday established in 1831 and which is the basis of every generator and transformer built since. Maxwell’s term is the mirror image of that, and the mirror image is what closes the cycle. With only one of the two, a disturbance in the fields dies away; with both, each field’s collapse builds the other, and the disturbance propagates.

Taking the curl of one equation and substituting the other gives, in vacuum, the wave equation for each field separately, with a speed

$$c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}.$$

Neither constant has anything to do with light. $\varepsilon_0$ comes from measuring the force between two charges — or, better, the capacitance of a known geometry. $\mu_0$ comes from measuring the force between two currents. Both are laboratory electrical quantities, obtainable with a balance and a battery in a windowless room.

The coincidence is the argument, and Maxwell put it in one sentence in 1862: the velocity is so nearly that of light that the inference can scarcely be avoided — light consists, he wrote, in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena. It is one of the very few occasions in physics where an entire phenomenon was identified on the strength of a number matching to two significant figures.

What the wave looks like #

The solution the equations support has a definite structure, and every feature of it is forced rather than chosen.

Both fields are transverse, because a longitudinal component would have a divergence and there is no charge to supply one. They are perpendicular to each other, because each is the curl of the other’s rate of change. They are in phase, because the two equations relate a derivative of one field to a derivative of the other in a way that a common sinusoid satisfies with no phase offset — an assertion easy to check by substitution and easy to get wrong by analogy with a circuit. And the ratio of their magnitudes is $c$, which is why the magnetic part of a light wave is numerically tiny and why almost all of light’s interaction with matter is electric.

The energy, and where it goes #

A wave that carries no matter still carries energy, and the equations say how much. The energy density is $\tfrac12\varepsilon_0E^2+B^2/2\mu_0$, and because $B=E/c$ those two terms are exactly equal: a light wave carries half its energy in the electric field and half in the magnetic one, at every instant.

The flow is given by the Poynting vector $\mathbf{E}\times\mathbf{B}/\mu_0$, which points along the propagation direction and whose magnitude is the intensity. That expression has a consequence nobody expects on first meeting it: it applies to static fields too, so a charged capacitor sitting next to a magnet has energy circulating round it for ever. That is not a paradox and it is not observable in isolation; it is a reminder that the field is being treated as the thing that holds the energy, which is a commitment the field picture makes and the force-at-a-distance picture does not.

The momentum follows from the same expression divided by $c$, which is why light exerts a pressure and why a perfectly reflecting sail feels twice the push of a perfectly black one.

The other combination of the same two constants #

Two constants can be combined in two independent ways, and the essay has used only one of them. The other is just as consequential and is met far less often.

Dividing rather than multiplying gives

$$Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = 376.73\ \text{ohms},$$

a quantity with the units of resistance and no material anywhere in it. It is the impedance of free space, and it is the ratio of the electric to the magnetic field in a plane wave — which is why the two amplitudes were locked together above, and where the factor came from.

Its role is what an impedance’s role always is: it says how much field goes with how much power. A wave carrying a stated intensity has an electric field fixed by dividing by $Z_0$ and taking a square root, so the kilovolt per metre quoted for sunlight earlier in this essay is not an independent number but a consequence of the intensity and this constant.

The practical importance is in antennas. An antenna is a device for handing power between a circuit, whose impedance is set by its components, and free space, whose impedance is 377 ohms — and a mismatch reflects power rather than radiating it, exactly as a mismatch between two lengths of cable does. Most of the difficulty in antenna design is arranging that match over a useful bandwidth, and the number being matched to is this one.

It also explains a fact about the two fields that would otherwise look like an accident. Because the impedance is 377 ohms rather than, say, a milliohm, a given power flux comes with a large electric field and a small magnetic one, and the electric part therefore does nearly all of the interacting with matter. Had the constants come out the other way round, optics would be a subject about magnetic fields.

Two constants, two combinations: one is a speed and identified light; the other is an impedance and decides how light couples to everything. Both were measured with batteries and balances before anybody suspected either had anything to do with the other.

The coincidence that became a definition #

The chain of reasoning in this essay ran from two measured electrical constants to a speed, and compared that speed with a measurement of light. The modern arrangement of units has inverted the chain completely, and the inversion is worth following because it is a good example of what a definition is for.

In 1983 the metre was redefined: the speed of light in vacuum was declared to be exactly 299,792,458 metres per second, and the metre became whatever length makes that true. So the speed of light stopped being a measured quantity. It cannot be measured, because measuring it would be measuring the definition of the metre against itself.

At the same time the ampere was defined by fixing the force between two current-carrying wires, which made $\mu_0$ exactly $4\pi\times10^{-7}$ by construction — and therefore, through the relation this essay derives, made $\varepsilon_0$ exact as well. For thirty-six years all three of the quantities in Maxwell’s coincidence were defined rather than measured, and the coincidence had become an identity in the unit system.

The 2019 redefinition changed which of them is fixed. The ampere is now defined by fixing the elementary charge, so $\mu_0$ is no longer exact: it has to be measured, and it is measured through the fine-structure constant to about a part in $10^{10}$. Its value is still $4\pi\times10^{-7}$ to within that uncertainty, which is why nothing in engineering noticed.

What survives untouched is $c^2 = 1/\varepsilon_0\mu_0$, which is a relation between the constants rather than a statement about their values, and which no choice of units can affect. That is the durable content of Maxwell’s result: not the number, which has been a definition and a measurement in turn, but the statement that the constant of electrostatics and the constant of magnetostatics multiply to the reciprocal of the square of a speed — and that the speed is light’s.

The pattern is worth naming, because it recurs whenever a physical constant is measured very well. A quantity discovered as a coincidence becomes, once it is trusted, a way of transferring precision between two things that are hard to measure separately — and eventually the relation is promoted to a definition and the measurement is retired.

The confirmation, twenty-two years later #

Maxwell died in 1879 with the theory unconfirmed by anything except the coincidence of two numbers. The confirmation, when it came, was direct and was designed to be: Hertz set out in 1886 to make an electromagnetic wave with a circuit and detect it with another circuit some distance away, which is a thing no previous theory suggested was possible.

The transmitter was a spark gap across an induction coil; the detector was a loop of wire with a gap of its own, in which a faint spark appeared when the loop was tuned to the same frequency. That alone showed action at a distance through air. What made the experiment decisive was what Hertz did next.

The trick that turned a detector into a ruler was reflection. Hertz bounced his waves off a zinc sheet at the end of the room so the outgoing and returning waves added to a standing pattern, then walked the detector along the room looking for the places where no spark appeared. Those are nodes, and the distance between them is half a wavelength. With the wavelength measured directly and the frequency known from the circuit, the speed followed — and it was the speed of light.

He then showed that the waves reflect off metal, refract through a prism of pitch, are polarised by a grid of parallel wires, and form interference patterns. Every one of those is a property of light, demonstrated at a wavelength of about 60 cm, with apparatus a metre across.

He demonstrated polarisation with a grid of parallel wires, and the transmitted intensity followed $\cos^2\theta$ in the angle between the grid and the wave. That the waves could be polarised at all establishes that they are transverse, which is what the equations require: a longitudinal electromagnetic wave would need a divergence, and in a vacuum there is no charge to supply one.

After that the identification was not an inference from a coincidence but a list of matching behaviours.

Asked what use it might be, Hertz said none whatever: “It’s of no use… this is just an experiment that proves Maestro Maxwell was right.” That judgement is quoted for its irony and it was a perfectly reasonable reading of the situation in 1888.

Four equations, and the twenty they replaced #

The set is called Maxwell’s equations and it is not what Maxwell wrote. His 1865 paper has twenty equations in twenty unknowns, written out component by component, with the potentials as the primary quantities and a scalar equation for the charge continuity among them. There are no vector operators in it, because the notation did not yet exist in usable form.

The compression to four was done by Heaviside in 1884, working alone and mostly unpaid, who invented enough vector calculus to do it and threw away the potentials as unphysical scaffolding. What is taught everywhere as Maxwell’s equations is Heaviside’s arrangement of Maxwell’s physics, and the fact that they can be written in four lines is a fact about the notation rather than about nature — the same content in the language of relativity is two equations, and in the language of differential forms is arguably one.

That history has a moral worth keeping. The reason the displacement current is hard to spot in the original is that it is buried in a mechanical model of the ether involving rotating vortex cells and idle wheels between them, which Maxwell believed in as scaffolding and which was abandoned within a generation. The term survived the model that produced it, which is the ordinary fate of a good result obtained from a bad picture.

What it costs #

The four equations are linear, and matter is not. Two beams of light in vacuum pass through one another without interacting at all — a fact so familiar it is invisible, and one that fails for waves of almost every other kind. In matter the linearity is approximate and breaks at high intensity, which is the whole subject of nonlinear optics.

They say nothing about sources. The equations relate fields to charges and currents and do not say what a charge is, why it comes in units, or how it responds to a field; that last requires the separate force law. A complete theory needs both, and the pair is not consistent for a point charge, which is where classical electromagnetism ends.

The wave has no smallest amplitude. The classical field can be as weak as one likes, and it cannot: light arrives in indivisible lumps whose energy depends on the frequency and not on the intensity. The equations on this page describe the average behaviour of enormous numbers of those, and they describe it superbly — but the photoelectric effect is invisible to them, and so is everything else about how light is emitted and absorbed.

Vacuum is an idealisation with a measured permittivity. Quantum electrodynamics gives empty space a polarisability, so $\varepsilon_0$ is not quite the constant of a theory in which nothing is there. The correction is tiny and it is the reason the phrase “the vacuum” carries a definite article in modern physics.

The one place the repair can be seen directly #

Because the added term is $\varepsilon_0\,\mathrm{d}E/\mathrm{d}t$ and $\varepsilon_0$ is so small, the displacement current is negligible in ordinary circuits at ordinary frequencies. There is one arrangement in which it is not merely visible but dominant, and it is worth naming because it converts an argument about consistency into an experiment.

Inside a capacitor being charged, the conduction current is exactly zero and the displacement current is the whole of it. So the magnetic field between the plates is entirely due to the new term, and it is measurable: for a capacitor of radius $R$ charged at a steady rate, the field at radius $r$ inside the gap rises linearly to $\mu_0 I/2\pi R$ at the rim, which is precisely the profile a solid wire of the same radius carrying the same current would have. The gap behaves magnetically exactly as though the wire continued through it.

That is an odd and rather beautiful result, and it is the sharpest possible statement of what the term does. It is not repairing an accounting error at the edges; it makes an empty space with a changing field indistinguishable, as far as any magnetic measurement is concerned, from a conductor carrying a current.

What the picture cannot show #

The plane wave drawn above is infinite in extent and monochromatic, which no real light is. A beam of finite width diffracts; a pulse of finite length contains a range of frequencies; and the neat picture of two perpendicular sinusoids is the limiting case of both. Nothing in the figure hints at either, which is why the transition from this picture to a real beam is where most of the difficulty in optics lives.

The figure also draws the fields as arrows at points along a line, which suggests that the wave is a thing travelling along that line. It is a plane wave: at every instant the field has the drawn value across an entire infinite plane perpendicular to the direction of travel.

A travelling wave at two instants is what any one component of the field is doing: the two snapshots differ by a shift and by nothing else, which is the definition of a travelling solution. And the reason the electromagnetic case took so long to accept is contained in that sentence — a shift of what was the question nobody could answer, and the answer turned out to be nothing at all.

Drawing that is not possible, and drawing what is drawn instead is the standard compromise and the standard source of confusion about what a “ray” of light is.

The ladder from here #

Later rungs on this anchor: the derivation of the wave equation from the four in full, with the vector identities and boundary conditions written out; energy, momentum and the Poynting theorem as a conservation law rather than an expression; waves in conductors, where the field decays into a skin depth; waves in dielectrics, where the speed drops to $c/n$ and refraction becomes a statement about that; polarisation as the choice of transverse direction; and the potentials, which is the form in which the equations become two rather than four and in which relativity makes them one.

The neighbouring ladders are induction, which is the half of the cycle Faraday found, what a wave is, which the solution turns out to be an instance of, and the photon, which is what the wave is made of and which none of this predicts.