The field that points where the charge is now

Everything in electromagnetism that is drawn as a static field is a lie about timing. The field at a point was not set by the charge as it is; it was set by the charge as it was, a distance divided by $c$ ago. For a charge sitting still the distinction is empty. For a charge moving, it ought to matter enormously — and for a charge moving steadily it turns out not to matter at all, in a way that is worth understanding before the case where it does.

The picture that has no time in it #

A static field is drawn with lines radiating from the charge, and the drawing carries no information about when anything happened.

Field lines are a choice of representation rather than a physical object, and for a stationary charge the choice costs nothing: the lines are radial, the field is the inverse square, and there is no time in the picture at all. Everything peculiar in this essay comes from asking what those lines do when the charge has moved — and the answer requires the picture to acquire a history.

The field is nevertheless a real object with a state at every point, which is what the second rung of the field ladder is about, and the timing is a property of that state rather than of the drawing.

Maxwell’s equations are local and their solutions are retarded: the field at $(\mathbf{r}, t)$ depends on the source at $(\mathbf{r}', t - |\mathbf{r} - \mathbf{r}'|/c)$ and on nothing later. The term Maxwell added to make the set consistent is the one that makes the solutions propagate, and its speed is the one the equations contain.

Uniform motion, and the extrapolation #

Now set the charge moving at a constant velocity and ask where the lines point.

Nothing has outrun light, and no conservation law has been troubled: the flux through any closed surface still counts the charge inside it, at every instant, however the charge is moving. The field at each point genuinely was determined by the charge as it was earlier; it is simply that the resulting arrangement happens to be radial from where the charge has got to. The reason is that the motion was predictable: an unaccelerated charge’s position at any later time is a linear extrapolation of its position and velocity at any earlier one, and the retarded solution encodes both.

The right way to say it is that the field carries no information the charge did not already have. If the charge were to be deflected, the news of the deflection would travel outward at $c$ and the extrapolation would be revealed as an extrapolation — which is exactly what the opening figure shows.

The flattening #

The same field is not spherically symmetric. It is crowded into the plane transverse to the motion and thinned fore and aft.

The distortion is the whole of the Lorentz factor. The transverse field is $\gamma$ times the static one and the forward field is $\gamma^2$ weaker, so a fast charge carries a field squashed into a pancake perpendicular to its motion. At $\gamma = 10$ the field ahead is a hundredth of what it would be at rest and the field to the side is ten times larger, which is why a relativistic bunch interacts almost entirely sideways.

The flattening is not a separate phenomenon; it is the same field seen from a frame that is moving, and it can be derived either from the retarded solution or from a Lorentz transformation of the static field, with identical results. Its practical consequence is that a fast particle passing an atom delivers a sharp transverse kick rather than a slow squeeze, which is why energy loss in matter goes into transverse excitations and why the loss rises logarithmically at high energy.

And it radiates nothing. The whole of this section describes a field that falls as $1/r^2$ and carries no energy away, however fast the charge is going.

How large the discrepancy would have been #

It is worth putting a number on the thing that does not happen, because the extrapolation is easy to accept as unremarkable once it has been stated.

An electron drifting along a wire moves at a fraction of a millimetre per second, so the retarded and present positions differ by nothing measurable and the question does not arise. An electron in a storage ring at 3 GeV moves at 0.999999985 c, and the light reaching a detector ten metres away left the electron 33 nanoseconds earlier, during which the electron travelled ten metres — a full beam-line’s length. If the field pointed at the retarded position, the apparent direction of the source would be wrong by ninety degrees.

It is not wrong at all, for the straight sections. The field of the coasting electron points at the electron, and the only light that arrives from anywhere else is the light emitted in the bending magnets, where the motion stopped being predictable. An accelerator’s beam-line is, quite literally, an instrument that separates the two pieces of this essay by geometry: the radiation comes off tangentially at the bends, and the electron’s own field goes with the electron.

The same arithmetic explains a much older observation. If gravity’s attraction pointed at the retarded position of the Sun rather than at the present one, the Earth’s orbit would gain energy and spiral outward on a timescale of a few hundred years. Laplace used exactly that to argue that gravity propagates at least millions of times faster than light. The argument is wrong for the reason above — the field of a uniformly moving source extrapolates, in general relativity as in electromagnetism — and the resolution is not that gravity is fast but that a predictable motion needs no news.

The kink #

Change the motion and the extrapolation fails. Consider a charge that has been moving steadily and is brought to rest over a short interval $\Delta t$.

Outside a sphere of radius $ct$ the news has not arrived, so the field there is the field of the charge that never stopped — radial from the extrapolated position, out ahead. Inside, it is the field of a charge at rest, radial from where the charge actually is. The two do not line up, and a field line cannot simply stop in empty space, because that would be a place where charge is.

So the two must be joined, across a shell of thickness $c\,\Delta t$, and the join is transverse to the radius. That transverse piece is the radiation.

Multiply that ratio by the radial field $q/4\pi\epsilon_0 r^2$ and the transverse field is

$$E_\perp = \frac{q\,a\sin\theta}{4\pi\epsilon_0 c^2 r}.$$

Integrating the energy flux of that over a sphere gives Larmor’s formula. There is no wave equation anywhere in the argument: the $\sin\theta$, the $1/r$ and the factor $c^2$ are all read off a picture of two static fields that do not line up.

A charge that changes its motion must glow, and the radiated power has a doughnut shape with a null along the acceleration. That null is the kink seen from far away: the transverse field carries the radiation, so nothing is radiated in the direction the charge is being pushed — which is the one direction in which the kink has no transverse component to offer.

The $1/r$ is the part worth dwelling on. The radial field falls as $1/r^2$ because it is spread over a sphere; the transverse field falls only as $1/r$ because the offset it comes from grows with $r$ — the extrapolated position moves further from the true one the longer ago the news left — while the shell thickness does not. A quantity falling as $1/r$ carries an energy flux falling as $1/r^2$, which times the area of a sphere is a constant. That is what it means to radiate: the energy gets away.

Where radiation takes over #

The two fields have different powers of distance, so which one dominates depends on where the observer is.

That number explains why radiation is invisible in ordinary life. A charged object accelerating at one gravity is surrounded, out to a light-year, by a field that is essentially electrostatic. Making an antenna is therefore not a matter of moving a charge a long way but of changing its motion quickly: the ratio contains the acceleration and neither the speed nor the distance travelled.

A static field falls as the inverse square because the same flux crosses larger and larger spheres. A radiation field does not: it falls as $1/r$, so the energy it carries falls as the inverse square and the total through any sphere is the same. That is the difference between a field that stays with its charge and one that has left — and it is why radiation reaches across a universe while a Coulomb field does not.

What is left behind #

Once the shell has passed, the radiation is an independent object.

The independence is worth stating plainly. Stop the charge, remove it, annihilate it — the pulse already emitted travels on regardless, carrying energy and momentum that are no longer anywhere near the source. That is the strongest argument in this collection that the field is a thing rather than a bookkeeping device for forces between distant charges: a bookkeeping device cannot outlive the entries it was keeping track of.

Three consequences that are usually met separately #

The two-piece structure explains several results that are ordinarily taught as unrelated.

Why the sky is polarised. Sunlight sets the electrons in air molecules oscillating, and each oscillating electron radiates with the sin²θ pattern above: nothing along its own line of oscillation, most at right angles to it. Looking ninety degrees from the Sun, only one of the two possible oscillation directions can send light towards the observer, so the light is polarised — a consequence of the null in the doughnut rather than of anything about air.

Why an accelerating charge in a gravitational field is a puzzle. A charge held stationary on a table is, by the equivalence principle, accelerating; a charge in free fall is not. Which of them radiates depends on who is asked, and the resolution turns on the fact that “radiation” is defined by the field far away rather than locally — which is exactly the distinction this essay’s two pieces make, and which no local measurement can settle.

And why the energy comes from where it does. The radiated energy is not taken from the field’s stored energy near the charge; it is supplied by whatever did the accelerating, through the work done against the recoil. Setting up that bookkeeping honestly requires the self-force the model breaks on, which is the next rung.

The spectrum a kink has #

The geometric construction gives the size of the radiated pulse and it also gives its colour, which is a second result out of the same picture and is worth extracting because it explains a spectrum that is otherwise pulled out of a calculation.

The transverse field is confined to a shell of thickness $c\,\Delta t$, so an observer standing still records a pulse lasting $\Delta t$ and nothing before or after. What frequencies are in such a pulse is a question about a Fourier transform, and the answer is the standard one: a pulse of duration $\Delta t$ has a flat spectrum up to a frequency of order $1/\Delta t$, and falls away above it.

So the radiation from a single sudden deflection is white — equal energy in every frequency interval — out to a cutoff set by how quickly the deflection happened, and nothing beyond. Counting photons rather than energy, that means the number per unit frequency falls as one over the frequency, which is the characteristic shape of a bremsstrahlung spectrum.

That shape is visible in every X-ray tube. Electrons accelerated through a voltage strike a target, are deflected violently by the nuclei in it, and emit a broad continuum with a sharp upper edge — and the edge is where a single electron’s whole kinetic energy has gone into one photon, which is the shortest pulse it could possibly make. Below the edge the continuum is roughly flat in energy per unit frequency, exactly as the pulse-duration argument says.

The same reasoning run backwards is how the deflection is measured. Given a spectrum with a cutoff, the cutoff is the reciprocal of the interaction time, so an observed high-frequency limit is a measurement of how abruptly the charge changed direction — which for a nuclear collision is a measurement of how close it came.

What makes this worth including here is that no wave equation appeared in it either. The shape of the spectrum came from the duration of a geometric feature, and the duration came from how long the motion took to change.

The exception: uniform motion that does radiate #

This essay’s central claim — that a charge in uniform motion radiates nothing — has one exception, and it is instructive because it fails exactly where the extrapolation argument fails.

Put the charge in a transparent medium rather than in vacuum. The field’s disturbances now travel at $c/n$ rather than $c$, and if the charge moves faster than that — which is entirely legal, since it is still below $c$ — the charge outruns its own news.

The extrapolation argument then has nothing to work with. It relied on the retarded field arriving from where the charge was and happening to assemble into something radial from where the charge is; that assembly requires the contributions from different retarded times to arrive at a point one after another in an orderly way. When the charge is faster than the disturbances it makes, contributions from two different earlier moments arrive at the same point at the same time, and the interference between them is constructive on a cone.

The result is Cherenkov radiation: a cone of light trailing the charge, at a half-angle whose cosine is the reciprocal of the charge’s speed in units of the medium’s light speed. Nothing accelerates at any point. The charge moves in a perfectly straight line at a perfectly constant speed and radiates continuously, and the energy comes from its kinetic energy, which falls.

The blue glow around a reactor core is that radiation, from electrons emitted by fission products moving faster than light in water. And the effect is an instrument: since the cone’s angle depends only on the speed, photographing the cone measures the speed, which combined with a momentum from a magnetic field gives a mass and therefore an identification. Detectors doing exactly that are standard in particle physics.

The exception is worth keeping in view because it identifies what the main claim actually rests on. It is not that uniform motion is special; it is that in vacuum a uniformly moving charge cannot outrun the field it makes, so the field can always be assembled into an extrapolation. Take away that guarantee and the conclusion goes with it.

Where the model stops #

The charge is a point. The field of a point charge diverges at the charge, its energy is infinite, and the force it exerts on itself is therefore undefined. Radiation reaction — the recoil on a charge from its own emission — requires that self-force, and the classical treatment of it produces an equation with runaway solutions and pre-acceleration. Classical electrodynamics is not consistent for a point charge, and the inconsistency lives exactly here.

The kink construction assumes a small change. Drawing the shell as a clean join between two static fields is exact only when the velocity change is small compared with $c$ and the shell is thin. For a relativistic deflection the pattern is beamed forward and the sin²θ becomes a much sharper function, which is what synchrotron radiation is.

The near field has been ignored throughout. Between the electrostatic zone and the radiation zone there is an intermediate region with a term falling as $1/r^2$ that is not the velocity field, and antenna design lives largely in it.

And nothing here is quantum. The field is continuous, the energy radiated is a continuous function of the acceleration, and there is no smallest amount. That description fails when the energy radiated in one period approaches $\hbar\omega$, and it is why the classical treatment of an electron in an atom predicts a collapse that does not happen.

What the pictures cannot show #

Every figure here draws field lines, and the lines are the very representation the site has an essay warning about. They are used because the kink argument is about the lines’ continuity — a field line cannot end in empty space — and that statement is a statement about the representation which happens to encode a true statement about the divergence of the field. It is a rare case of the drawing carrying the argument, and it should not be taken as licence.

Nor can any of these drawings show the timing, which is the subject. The shell is drawn as a circle at one instant; what is happening is that it expands at $c$, and the field inside it is quietly becoming the answer to a question that was asked earlier. A still picture of a causal structure is a contradiction in terms, and the reader has to supply the motion.

Where this ladder goes next #

What has been established is that the retarded solution has two pieces with different characters. One belongs to the charge, extrapolates its motion, falls as $1/r^2$ and carries nothing away. The other belongs to the change in the motion, falls as $1/r$, is transverse, and never comes back.

The habit worth carrying away is the question that separated them. Ask what the field would have been if nothing had changed, and look at the difference. That subtraction is the radiation, and doing it geometrically — rather than solving a wave equation — is available in far more places than it is used: it produces the Larmor formula here, it produces the bremsstrahlung spectrum from a sudden deflection, and it is the cleanest way to see why a charge in uniform motion, however fast, emits nothing at all.

What is left on this ladder is the self-force: what the field a charge has made does back to the charge that made it, why the classical answer requires a third derivative of position, and what that failure was eventually replaced by.