Radiation from a moving charge

Drag anywhere on the picture: the charge follows (never faster than 0.98 c). Stop, and watch the kink run out.

Experiments

Start here

A charge (the red dot) moves in the plane of the screen, and the white lines are its electric field, worked out exactly from how it has moved. Light is slow here: the bar at the bottom left of the picture shows how far it goes in a second.

Drag anywhere on the picture and the charge follows your pointer, at up to 0.98 of the speed of light. Then stop. Or pick a motion from the side panel and let it run.

Look for the kink: every change of speed sends a ring out at the speed of light, and the sideways field in that ring (coloured orange and blue) is the radiation. Below the picture: which way the charge is radiating now, and how much. The formula is at the end, under How it works.

1. Start and stop: the kink is the radiation

The opening set-up. The charge waits a second, speeds up to $\beta=v/c=0.6$ in 0.15 s, coasts 10 units, stops in 0.15 s, waits, and comes back. Each change of speed sends a shell out at $c$ (the ring of orange and blue: watch it grow by one scale bar a second). Inside the shell, the lines are those of the charge as it moves now. Outside, nothing has heard yet: the lines still point from where the charge would be had it kept its old velocity. Field lines cannot break, so in the shell they run sideways to join up, and that sideways field is the radiation: J. J. Thomson's picture of 1904. Make kinks of your own: drag the charge and stop it; or press Sudden stop for one stop at a time, the charge coasting in from the left and stopping in the middle.

2. At rest, or slow: Coulomb's field keeps up

The field lines go straight out, evenly spaced: Coulomb's field. Drag the charge slowly, much slower than light, and the lines keep up with it, nearly straight: a slow charge carries its Coulomb field along, and radiates almost nothing (the strip chart stays near zero). Flick it, and the kink is back.

3. Inside, the new field; outside, the old

The probe (the yellow dot: drag it) shows where the charge was when it sent the light now arriving at the probe (the ring marked "then") and the path of that light. The velocity field at the probe points away from the blue cross, which is where the charge would be now had it kept the velocity it had then. The read-out gives both parts of the field there, the peak of the acceleration part over the last 3 seconds, and how long ago the light left, $t-t_r$. The arrows are the velocity field (blue), the acceleration field (orange) and their sum (white), all to one scale. Until the shell reaches the probe the cross runs on, as if the charge had never stopped; when the shell has passed, the cross sits on the stopped charge.

4. Far away, the kink wins

Thomson's argument: a stop that takes a time $\tau$ makes a shell $c\tau$ thick, across which a line must step sideways by $vt\sin\theta=vR\sin\theta/c$, the distance the charge would have gone on. So $E_\perp/E_r\approx vR\sin\theta/c^2\tau$, which with $a=v/\tau$ is $$E_\perp\approx\frac{q}{4\pi\epsilon_0}\,\frac{a\sin\theta}{c^2R}:$$ the radiation falls off as $1/R$, Coulomb's field as $1/R^2$. The probe sits 8 units above the stop and keeps the peak of $|E_\mathrm{acc}|$ over the last 3 seconds: 0.118 there, 7.5 times the Coulomb field of 0.0156 ($1/8^2$: the read-outs are in units where $q/4\pi\epsilon_0=1$). Thomson's estimate, with the largest deceleration of this smooth stop, $a=1.5v/\tau=60$ units/s², gives $a/c^2R=0.117$. Drag it to 4 units above the charge and wait for the next stop: the peak doubles, to 0.236, and the Coulomb field quadruples, to 0.0625.

5. Uniform motion: lines from where it is now

A charge coasting at $\beta=0.8$ forever. (The view follows it; the dots are fixed in the lab.) Its field lines are straight, and they point from where the charge is now, not from where it sent the light now arriving (the probe's ring): that is what the $\hat n-\vec\beta$ of the velocity field does. The field is squashed into a pancake (Heaviside, 1888), $$E=\frac{q}{4\pi\epsilon_0}\,\frac{1-\beta^2}{R^2\left(1-\beta^2\sin^2\psi\right)^{3/2}}$$ at distance $R$ and angle $\psi$ from the velocity: across the motion $\gamma$ times Coulomb's field, along it $1/\gamma^2$ times. The light fronts, sent out at regular intervals and each centred where it was sent, crowd together ahead of the charge.

6. Shake it: dipole radiation

The charge goes up and down once a second, at up to $\beta=0.1$. The colour is the radiation field (orange one way round the charge, blue the other): waves of length $\lambda=cT=8$ units run outward, strongest sideways. The field lines wiggle as they pass. The polar plot is Larmor's $\sin^2\theta$, with $\theta$ measured from the acceleration.

7. Nothing along the axis

The colour is now the energy flux, $|E_\mathrm{rad}|^2$. Above and below the charge it is dark, because $\hat n\times(\hat n\times\vec a)=0$ when $\hat n$ is along $\vec a$, and the probe there reads $|E_\mathrm{acc}|=0$ while its velocity field does not. Drag the probe round the charge, keeping its distance, and watch the peak of $|E_\mathrm{acc}|$ grow as $\sin\theta$.

8. A fast dipole leans forward

Now up to $\beta=0.8$, with a period of 2 s. The lobes of the polar plot lean towards the velocity, up while the charge rises and down as it falls: the $1/\kappa^5$ in $dP/d\Omega$ throws the radiation forward. The waves are no longer sine waves. The exact radiated power, Liénard's formula (the strip chart's solid line), climbs far above Larmor's slow-charge formula (dashed): for motion along a line the ratio is $\gamma^6$, shown in the read-out as P/P_Larmor.

9. Round and round: cyclotron to synchrotron

Circular motion at $\beta=0.9$. The energy flux is a spiral: the charge throws its radiation forward, into a cone about $1/\gamma$ wide, and sweeps it round like a lighthouse; a distant observer sees a flash once a turn. In the plane, $dP/d\Omega\propto(\cos\theta-\beta)^2/(1-\beta\cos\theta)^5$ at an angle $\theta$ from the velocity: half power at 0.147 rad, none at $\cos\theta=\beta$, 0.451 rad (and $1/\gamma=0.436$). This is synchrotron light. Slide $\beta$ down to 0.3 and the brightness up to 1: the beam relaxes into the two-armed spiral of a rotating dipole (cyclotron radiation), dark where the line of sight ran along the acceleration. There are only 8 field lines because each winds round many times: far out, the sideways radiation field outweighs the outward Coulomb field, so the lines creep outward.

10. How much it radiates: Larmor and Liénard

The strip chart shows the radiated power by Liénard's formula $$P=\frac{q^2\gamma^6}{6\pi\epsilon_0c^3}\left(a^2-|\vec\beta\times\vec a|^2\right)$$ (solid) and by Larmor's, $q^2a^2/6\pi\epsilon_0c^3$ (dashed), which is right only when $v\ll c$. In circular motion $\vec\beta\perp\vec a$ and the ratio is $\gamma^4$: 3.84 at $\beta=0.7$. Change $\beta$ and watch P/P_Larmor in the read-out; along a line the ratio is $\gamma^6$ (try the oscillation).

11. Constant proper acceleration

The charge arrives at $0.9c$, slows down with a constant proper acceleration $g$ (the one it would feel), turns round and leaves: $x=x_0-\frac{c^2}{g}\left(\sqrt{1+(gt/c)^2}-1\right)$. Its acceleration in the lab, $g/\gamma^3$, changes all the while, yet Liénard's power stays at $\frac{2}{3}\frac{q^2g^2}{4\pi\epsilon_0c^3}$: the flat top of the strip chart (0.0833 here). Whether such a charge radiates was argued over for half a century; seen from the lab, it does.

12. A flash as it turns

A charge at $\beta=0.9$ is turned through 60° in a quarter of a second, as a magnet would turn it. Its radiation leaves as a single flash, beamed forward over the directions it pointed while it turned. A nucleus deflects a fast electron in much the same way (changing its speed too): that is bremsstrahlung, "braking radiation", the X-rays of an X-ray tube.

13. The velocity field and the acceleration field

Lines of the velocity field alone (blue), with the acceleration field in colour. The velocity field points from where the charge would be had it kept its retarded velocity, so its lines barely notice the shells. Neither part alone obeys Gauss's law, $\oint\vec E\cdot d\vec A=q/\epsilon_0$: the flux of the velocity part through a sphere crossed by a shell is not $q/\epsilon_0$, and the acceleration part makes up the difference.

14. Your turn

The charge follows your pointer, smoothed so that its acceleration never jumps, and held below $0.98c$. Flick it and stop; shake it for waves; swing it round in a circle and watch the polar plot point along the velocity; lower $c$ and everything turns relativistic. The arrow keys move it too.

How it works

A point charge $q$ moves in the plane of the screen, and you see its electric field in that plane, computed exactly. News of the charge travels at $c$, so the field at $\vec r$ now depends on where the charge was at the retarded time $t_r$, when it sent out the light arriving now: $|\vec r-\vec w(t_r)|=c(t-t_r)$. Liénard (1898) and Wiechert (1900) found the exact field of a point charge on any path: $$\vec E=\frac{q}{4\pi\epsilon_0}\,\frac{(\hat n-\vec\beta)(1-\beta^2)}{\kappa^3R^2}+\frac{q}{4\pi\epsilon_0}\,\frac{\hat n\times((\hat n-\vec\beta)\times\dot{\vec\beta})}{c\,\kappa^3R}$$ with everything on the right taken at $t_r$: $R=|\vec r-\vec w(t_r)|$, $\hat n$ the direction from that retarded position, $\vec\beta=\vec v/c$ and $\kappa=1-\hat n\cdot\vec\beta$. The first term, the velocity field, falls off as $1/R^2$. The second, the acceleration field, falls off only as $1/R$ and is perpendicular to $\hat n$: it is the radiation, and it is what the colour shows (or its square, the energy flux).

$c$ is a slider, in units per second (the view is 20 units across its shorter side), and the charge your hand drags is smoothed and held below $0.98c$. The field lines leave the charge evenly spaced in angle in its own rest frame. They are a slice through a three-dimensional field, so their spacing is only a rough guide to its strength. The polar plot under the picture is $dP/d\Omega$, the power radiated per unit solid angle, by direction in the plane; the strip chart is the total power by Liénard's formula (solid) and Larmor's (dashed), see experiment 10.

What is exact, and what is not. The fields are the exact Liénard–Wiechert fields of the path the charge follows. The path is stored 240 times a second, with quintics between the samples that match their positions, velocities and accelerations. Your hand's motion is smoothed into such a path. The colours are computed on a grid coarser than the screen, and when the field is very busy (after wild shaking) the field lines are cut short so that the picture keeps moving. $dP/d\Omega$ is the power per unit solid angle per unit of the charge's time; a distant observer whom the charge approaches receives it in less time, and so sees it brighter still. The numbers are in units where $q=1$ and $1/4\pi\epsilon_0=1$ (so the Coulomb field is $1/R^2$), with lengths in units and times in seconds.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.