The field a changing field does not make

The term that made light found that Ampère’s law fails the moment a current stops being steady, and that the term Maxwell added to repair it — a changing electric field contributing to the magnetic field as a current would — turns the four equations into a wave travelling at a speed fixed by two constants measured on a bench. The field that makes the other, and only while it is changing found Faraday’s half of the same story: a magnet beside a coil does nothing until it moves, because the electric field that drives the current is tied to the rate of change of the magnetic one.

Put those two side by side and they read like a mechanism. A changing magnetic field makes a curling electric field; that electric field, changing in turn, makes a curling magnetic field; and the pair leapfrog each other across empty space at the speed of light. That picture is in textbooks, in museum displays and in the first paragraph of most explanations of radio, and it makes the two time-derivative equations into statements of cause and effect.

The equations themselves do not say that. They relate the fields at one place and one time. Which field is the cause and which the effect is an interpretation laid on top, and the solution of the equations, written out by Oleg Jefimenko in 1966, shows that the interpretation is not needed and in an important sense not true. Every electric and magnetic field is caused by charges and currents, at the retarded time, and by nothing else.

Solving for the fields instead of relating them #

The two equations that are not laws of motion found that Gauss’s two laws are conditions on the fields at one instant rather than rules for how they evolve; the evolution is carried by the other two. That is one way to read the four. Another is to ask for the solution: given the charges and currents everywhere and for all time, what are the fields? The potentials that are not unique gave the standard answer in two stages. In the Lorenz gauge each potential obeys a wave equation with a source, and the solution that respects causality is the retarded integral, a sum over each piece of charge and current as it was when light left it.

Jefimenko’s step was to differentiate the retarded potentials and write the fields themselves as retarded integrals:

$$\mathbf E = \frac{1}{4\pi\varepsilon_0}\int\left[\frac{\rho}{R^2}\hat{\mathbf R} + \frac{\dot\rho}{cR}\hat{\mathbf R} - \frac{\dot{\mathbf J}}{c^2 R}\right]dV', \qquad \mathbf B = \frac{\mu_0}{4\pi}\int\left[\frac{\mathbf J}{R^2} + \frac{\dot{\mathbf J}}{cR}\right]\times\hat{\mathbf R}\,dV',$$

with every quantity in the brackets evaluated at the retarded time $t - R/c$, where $R$ is the distance from the source element to the point where the field is wanted.

Read the integrands. They contain charge density, current density and the rates of change of both. They do not contain $\mathbf E$ or $\mathbf B$. The fields at a point are fixed entirely by what the charges were doing on the observer’s past light cone — the set of places and times from which a signal travelling at $c$ would arrive here now. Nothing else enters, and in particular the field at a neighbouring point a moment earlier does not enter.

The first terms in each are familiar. $\rho\hat{\mathbf R}/R^2$ is Coulomb’s law and $\mathbf J\times\hat{\mathbf R}/R^2$ is Biot and Savart’s, each with the source seen as it was when its light left. What is new is the time derivatives, which carry the part of the field that falls off as $1/R$ and therefore reaches far — the radiation — and which vanish when nothing is changing. That is the whole of the generalisation.

A current switched on #

The simplest test is a long straight wire, electrically neutral, whose current rises from nothing to $I_0$ in a short time. Before the switch-on there is no field anywhere. Long after, the magnetic field is Biot and Savart’s, $\mu_0 I_0/2\pi r$, and there is no electric field. In between, Jefimenko’s integrals can be done in closed form.

At any moment the field occupies a cylinder whose radius is the distance light has travelled since the switch-on, and outside it there is nothing. Just inside the front the field is much larger than the steady value, and behind the front it settles back onto Biot and Savart’s curve from above. Close to the wire it has settled already; further out it is still settling. For an instantaneous switch-on the closed form is

$$B = \frac{\mu_0 I_0}{2\pi r}\,\frac{ct}{\sqrt{c^2t^2 - r^2}},$$

which is infinite at the front and tends to the steady field as $ct/r$ grows. The ramp in the figure rounds off the infinity without changing anything behind it.

Nothing in this is surprising once the integrals are written down. A point at distance $r$ from the wire, at time $t$, sees each piece of the wire as it was a time $R/c$ ago. The pieces near the foot of the perpendicular are seen as they were recently, and their current has been on for a while; the pieces further along are seen as they were longer ago; and beyond a certain distance along the wire the light left before the switch-on, and those pieces are seen carrying no current at all. The field is a sum over the part of the wire that has been heard from, and that part grows with time until, in the limit, it is the whole wire and the sum is Biot and Savart’s.

The two terms, separately #

The overshoot is what makes the decomposition worth drawing.

The first term — the retarded current divided by the square of the distance — rises smoothly from zero to the steady value as more of the wire is heard from. By itself it never exceeds the steady field. The second term — the rate of change of the current at the retarded time, divided by $c$ times the distance — is non-zero only where the switch-on is being seen, which at any moment is two points on the wire, one either side, receding as $\sqrt{c^2t^2 - r^2}$. It is large just after the front arrives, when those two points are close and the light from them arrives nearly perpendicular to the wire, and it decays as they recede.

For the instantaneous switch-on the two are

$$B_1 = \frac{\mu_0 I_0}{2\pi r}\sqrt{1 - \frac{r^2}{c^2t^2}}, \qquad B_2 = \frac{\mu_0 I_0}{2\pi}\,\frac{r}{ct\sqrt{c^2t^2 - r^2}},$$

and they add to the closed form above. The overshoot belongs entirely to the second. At one and a half light-metres the first term has reached 68 per cent of the steady value and the second contributes another 78; by four light-metres the first is at 97 per cent and the second has fallen to 7.

A textbook reading would attribute the overshoot to the displacement current — to the electric field that is changing near the wire as the front passes. Jefimenko’s form says it comes from the rate of change of the current in the wire, seen at the retarded time. The two readings give the same number, as they must, but only one of them names something that exists independently of the fields: the current is made of moving charges and can be measured with an ammeter; the displacement current is a name for a combination of the field it is supposed to explain.

Faraday’s law holds, and makes nothing #

The electric field is simpler still. The wire is neutral, so the charge terms vanish and only $-\dot{\mathbf J}/c^2R$ remains. For the instantaneous switch-on the current changes only at the two points where the switch-on is being seen, and the electric field comes from them alone:

$$E = -\frac{\mu_0 I_0 c}{2\pi}\,\frac{1}{\sqrt{c^2t^2 - r^2}}.$$

It points against the current, as Lenz’s rule requires, and at the wire itself it is the electric field that the circuit that fights its own change found opposing any change in a current: the field of self-inductance, here computed from a sum over the wire’s history rather than assumed. It dies away as the switch-on recedes, and when the current is steady it is zero.

Now ask whether Faraday’s law holds between these two separately computed fields. For this geometry it reads $\partial B/\partial t = \partial E/\partial r$, the rate of change of the magnetic field at a point equal to the radial gradient of the electric field there.

It holds, to the accuracy of the numerical differences, at every time. But look at how it was obtained. The magnetic field was an integral over the current and its rate of change. The electric field was an integral over the rate of change of the current. Neither integral contains the other field, and neither was computed from the other. The law holds because both fields are built from the same current history, and the current history obeys the conservation of charge — the same condition that the two equations that are not laws of motion found keeping Gauss’s laws consistent with the other two.

The rule that is two laws wearing one coat found that the flux rule for a moving circuit and the flux rule for a changing field are different laws that happen to give the same answer. The same caution applies one level down. So the equation that relates the fields is a constraint, a consistency condition between two things with a common cause, and not a mechanism by which one produces the other. It resembles the relation between the readings of two thermometers that are both in the same bath: they agree, and the agreement can be written as a law, and neither thermometer warms the other.

Switching it off again #

A current switched on and off again makes the structure plainer. Each change in the current sends out a front, and behind each front there is a transient. While the current is steady the electric field is zero and the magnetic field is settling towards Biot and Savart’s; after the second front it falls back towards zero, with an undershoot that mirrors the overshoot. The electric field is two pulses of opposite sign, one per change of the current, and nothing in between.

The causal statement is now hard to resist. Each pulse in both fields is tied to an event in the current — the switch-on, the switch-off — seen at the retarded time. If the switch-off is moved later, both the second magnetic transient and the second electric pulse move later with it, and nothing about the first pair changes. The fields are effects, and the events in the current are their causes. A source that is large compared with the distance light travels while it changes is heard as a sum over its parts, each at its own retarded time — the arithmetic when the source is not heard all at once found making a phased array and a diffraction grating the same calculation — and the wire here is the extreme case, a source infinitely long, heard one widening stretch at a time. The field that points where the charge is now found the same structure for a single charge: a field that keeps up with steady motion exactly, and a kink, carrying radiation outward, wherever the motion changed.

Which part of the history is heard #

The geometry behind all of this fits on one picture of the wire’s history.

The red curve is the intersection of the observer’s past light cone with the wire’s history. Every field at the observer is a sum along it. The magnetic field’s first term sums the current along the part of the curve where the current is on; the second term, and the whole electric field, come from the two points where the curve crosses the moment of switch-on. As time goes on the observer’s cone rises, the red curve lengthens, the two green points move outward along the wire, and the contributions from them weaken as their distance grows.

That is the precise content of “the field propagates outward at the speed of light”. Nothing is passed from one place in space to its neighbour. What moves outward is the boundary of the region of spacetime from which the switch-on can be seen.

Why the leapfrog picture survives anyway #

None of this makes the familiar reading wrong as a description of how the fields evolve. Given the fields everywhere at one instant, Faraday’s law and the corrected Ampère’s law tell how they change in the next, without reference to any distant source; that is how numerical simulations of electromagnetism work, stepping $\mathbf E$ and $\mathbf B$ forward in turn on a grid. Read that way, each field’s rate of change is set by the other’s curl, and a wave packet far from all charges does carry itself across space.

The two readings answer different questions. The local, step-by-step one says how the fields now determine the fields a moment from now, and it is complete for that purpose. Jefimenko’s says what the fields are, given everything that happened, and it is complete for that. The local reading becomes a causal story only by adding an assumption the equations do not make: that the field already present somewhere is a source in its own right. In a free wave, far from the antenna, the assumption does no harm, because the antenna’s history and the present field encode the same information. It does harm when it is used to explain where a field came from — when an induced electric field is said to be “produced by” a changing magnetic field rather than by the current whose change both reflect.

The distinction has practical consequences. A loop of wire round a long solenoid whose current is changing has an electromotive force in it, although the magnetic field outside the solenoid is negligible — small, as the field outside the solenoid, which is not zero found, but far too small to account for the voltage. In the leapfrog picture that is puzzling: the changing field is inside, the loop is outside, and the electric field at the loop seems to have been made by something that is not there. In Jefimenko’s picture it is not puzzling at all: the electric field at the loop is the retarded integral of the changing current in the solenoid’s windings, which the loop encircles.

What the integrals leave out #

They need all of history. Jefimenko’s equations give the fields from the sources on the past light cone, which is only possible if the fields carry nothing in from infinitely far away and infinitely long ago. A field that arrives from outside the system — sunlight, a radio wave from elsewhere — appears in them only as the field of its own distant sources, and in practice is added as a free solution. The solution that is thrown away discussed the advanced alternative, which uses the future light cone instead and is excluded only by a boundary condition.

They assume a vacuum. In a medium, the bound charges and currents of the material are sources too, and the integrals over them are usually replaced by a permittivity and a permeability. That replacement is itself a statement about the material’s response to the field, and so reintroduces fields on the right-hand side — legitimately, as an approximation to sums over the material’s own charges.

They are classical. In quantum electrodynamics the field has degrees of freedom of its own, the photons, and the vacuum has fluctuations no source accounts for. Jefimenko’s fields are the expectation values of the quantum fields in states produced by classical currents, which is exact for those states and silent about the rest.

The wire is idealised. An infinitely long neutral wire is a mathematical convenience: a real circuit closes on itself, its current is not switched on everywhere at once, and its wires carry surface charges that make the electric field inside them push the current along. Each changes the details of the transients. None reintroduces fields as sources.

Still open: whether causation belongs in the equations at all #

Whether Jefimenko’s reading is the causal reading or one of two equivalent ones has been argued in the physics-education literature since the 1990s, without a settled answer. One side holds that since the fields in a region are determined by their values on its boundary and inside it at an earlier time, the local equations are as causal as the retarded integrals, and that “the field causes the field” is a fair description of a free wave. The other holds that only the retarded integrals identify independent causes — things whose values could be set at will — and that fields, being determined by charges, cannot play that role.

Behind the dispute is a question that physics leaves mostly to philosophy: whether causation is a feature of the laws or of the use made of them. The laws relate quantities. Causation picks some quantities as the ones an experimenter could vary and others as the ones that respond. For electromagnetism the choice made by every laboratory is clear — the current is varied, by a switch or a generator, and the fields respond — and Jefimenko’s equations are the laws written in that experimenter’s terms. Whether there are situations, in the early universe or near a black hole, where no such choice is available and the fields must be treated as their own causes, is a question the equations do not answer. They hold either way.