A charge sitting still has a field that never changes, and a charge gliding at constant speed just carries its field along. Shake it, though, and a kink tears loose from the field and runs away at the speed of light: that kink is every radio wave and every ray of light there is.
Common mix-up: moving charges don’t radiate; accelerating charges do. Electrons drifting steadily through a straight wire carrying direct current send out no wave at all. The wiring in your walls radiates (a little) only because 60 Hz AC reverses direction 120 times a second, and every reversal is an acceleration.
Put a single charge in empty space. Its electric field reaches out in every direction, and if you draw it as field lines, they are straight spokes pointing away from the charge. Nothing about this picture changes from one moment to the next. A static field is a map, not a messenger: it tells any other charge how hard it would be pushed, but it doesn’t carry anything anywhere.
The strength falls with the square of the distance. The reason is geometry: the same bundle of lines spreads over a sphere whose area grows as r². Go twice as far and the lines are spread over four times the area, so the field is a quarter as strong. A balloon carrying one nanocoulomb (a typical rub on a sweater) produces about 9 volts per meter at one meter away, 2.2 volts per meter at two meters, and 0.09 volts per meter at ten. By the time you cross a room it has all but vanished. (Chapter 1 has the full story of the static field.)
Now let the charge glide past at a steady speed. The field simply comes along with it. At everyday speeds the lines still point straight out from wherever the charge is right now; nothing ripples, nothing breaks off. This isn’t a special rule, it’s forced on us by relativity. Ride alongside the charge at the same speed and you see an ordinary charge at rest. Since “at rest” and “moving steadily” are the same thing seen from two equally valid viewpoints, a steadily moving charge can’t do anything a resting one doesn’t. It can’t radiate.
So whatever makes a wave, it must involve a change of velocity. That is the whole secret of this chapter.
For a charge moving at constant velocity v, the exact field still points from the charge’s present position, which is surprising, since the information left from earlier positions. The two effects (light-travel delay and the motion during that delay) cancel exactly for uniform motion. What does change at high speed is the shape: the field gets squashed toward the plane perpendicular to the motion by the Lorentz factor γ = 1/√(1 − v²/c²). Sideways it is γ times stronger; straight ahead and behind it is γ² times weaker.
At 1% of light speed γ is 1.00005, so the squash is invisible. For an electron in CERN’s LEP collider at 104 GeV (the tunnel the LHC now uses), γ was about 200,000, and the field was a thin pancake. Still: no acceleration, no radiation. The pancake just rides along.
Here is the picture J.J. Thomson drew in 1904, and it’s still the clearest explanation there is. Take a charge at rest. Give it a short, sharp push, so it moves a little, and stop it again.
Close to the charge, the field lines now point out from the new position. Far away, they can’t, yet: no information travels faster than light, so a point ten meters off doesn’t learn about the push until 33.4 nanoseconds later. Out there the lines still point back to the old position, as if nothing happened. The field lines are continuous (they can’t just end in empty space), so between the “new” region and the “old” region each line has to jog sideways. That jog is a thin shell, as thick as the distance light covers during the push, and it expands outward at exactly the speed of light.
Look at what the jog does to the field. Inside the shell the lines are mostly sideways, across the direction of travel. That sideways field is the radiation. And here is the striking part: the shell carries on after the charge has stopped. The charge is sitting quietly again, but the news of its push is already a sphere of disturbance spreading through space, and it will keep going forever.
Shake the charge back and forth and you send out one kink per swing. They line up behind each other into a train of ripples, and that train is an electromagnetic wave. The spacing between ripples is the distance light covers during one shake: the wavelength, λ = c / f. Shake a charge 100 million times a second and the ripples are 3.00 meters apart; that’s FM radio. Shake it 2.4 billion times a second and they’re 12.5 centimeters apart: Wi-Fi.
The kink is also lopsided in a useful way. A charge shaken up and down produces kinks that are strongest out to the sides and zero straight above and below, because lines pointing along the shaking direction don’t get bent sideways at all. That sin θ factor is why a vertical antenna sends almost nothing straight up.
Let the charge accelerate for a short time Δt up to a small speed v = aΔt, then coast. A time t later, the kink shell sits at radius r = ct and is c Δt thick. Inside it, the lines come from a charge that has moved v t since the push; outside, from the old spot. Follow a line at angle θ to the motion: crossing the shell, it shifts sideways by v t sin θ while moving outward by only c Δt.
The field inside the shell points along that line, so the ratio of its sideways part to its outward part equals that ratio of distances: E⊥ / Er = v t sin θ / (c Δt) = a t sin θ / c = a r sin θ / c². The outward part is just the ordinary Coulomb field, q / (4π ε₀ r²). Multiply, and E⊥ = q a sin θ / (4π ε₀ c² r). One factor of r cancels, which is the whole reason radiation reaches so far.
The same ratio tells you where radiation beats the static field for a single shaken charge: when a r / c² > 1, that is, beyond r = c² / a. The instruments below use exactly this construction, line by line, with light slowed down so you can watch it.
Drag the charge around (mouse or finger), or pick a motion. Every line is drawn from where the charge was when that bit of field left it, so kinks run outward at the speed of light; slow light down to watch them. Sideways, radiating parts of the field glow orange. With the canvas focused, arrow keys nudge the charge and the space bar kicks it.
Power is shown relative to the default up-and-down shake and averaged over half a second. It goes as a²/c³: double the shaking frequency at the same size and acceleration rises fourfold, power sixteenfold. The charge is held below light speed, as any real charge must be; the drawing is the non-relativistic version of the field.
The static field fades as 1/r². The radiated field fades as 1/r. At first that sounds like a small difference. It isn’t. Go a thousand times farther away and the static field drops a millionfold, the radiated field only a thousandfold. Far from any source, the radiation is the only part left.
There is a deeper reason it has to be 1/r. Energy in a field goes as the field squared, so the energy flowing through each square meter of the radiation goes as 1/r². The sphere it passes through has area 4πr². Multiply them and the r’s cancel: the total power crossing any sphere is the same, no matter how big. The wave isn’t losing energy, just spreading it thinner. If the field fell any faster, energy would quietly vanish on the way out, and that doesn’t happen.
Real antennas are not single charges but dipoles: plus and minus sloshing back and forth in a rod. A dipole’s static field fades even faster, as 1/r³, because from far away the plus and minus nearly cancel. Around any antenna there are three layers. Close in, the 1/r³ “near field” dominates: energy stored around the antenna, surging out and back each cycle without leaving. Further out a 1/r² piece takes over briefly. Beyond about λ/2π, the 1/r radiation wins and never gives the lead back.
That boundary depends wildly on frequency. For an AM station at 1 megahertz, λ/2π is 48 meters, so the whole neighborhood around the mast sits in the near field. For FM at 100 megahertz it is 48 centimeters. For 2.4 gigahertz Wi-Fi it is 2 centimeters, so you are in the far field of your router almost everywhere. For a 60 hertz power line it is 795 kilometers: you are always in the near field of the grid, which is why power lines mostly make local electric and magnetic fields rather than broadcast waves.
Treat a 50 kilowatt FM transmitter as spreading its power evenly over a sphere (a simplification; real broadcast antennas focus it toward the horizon). At 10 kilometers the intensity is 50,000 ÷ (4π × 10,000²) ≈ 40 microwatts per square meter. That sounds like nothing, but your radio only needs a tiny fraction of it.
For an oscillating dipole of moment p₀ at angular frequency ω, with wavenumber k = ω/c = 2π/λ, the electric field is a sum of three pieces whose sizes go as 1/(kr)³, 1/(kr)², and 1/(kr), all multiplied by the same constant k³p₀/(4πε₀). They are equal at kr = 1, which is r = λ/2π. Inside, the first term dominates and the field is in step with the charges, like a static dipole that happens to be changing. Outside, the last term dominates; it is 90° out of step with the near field and is the only one that carries power away on average.
Engineers usually quote the “far field” as starting a few wavelengths out, and for big dish antennas at 2D²/λ (D is the dish diameter), because the pattern also has to settle down. Those are rules of thumb on top of the physics above.
Pick a frequency, then slide outward from an antenna. The three curves are the three parts of a dipole’s field, scaled so they meet at λ/2π. Watch which one rules where you’re standing.
Far from the source, the train of kinks straightens into a clean travelling wave, and it has a very particular shape. The electric field points across the direction of travel, never along it. A magnetic field rides with it, also across the direction of travel, and at right angles to the electric field. Point your fingers along E, curl them toward B, and your thumb points the way the wave is going.
The two fields rise and fall exactly in step: when E peaks, so does B; when E passes through zero, so does B. They are not two waves that happen to travel together. Each one is made by the other changing: a changing magnetic field makes an electric field (Faraday’s law), and a changing electric field makes a magnetic one (Maxwell’s addition). Chapter 3 shows how those two rules combine into a wave equation whose speed comes out as 1/√(μ₀ε₀) = 299,792,458 meters per second.
Their sizes are locked too: E = cB. In SI units that makes the magnetic field look tiny: a wave with an electric field of 1 volt per meter carries a magnetic field of just 3.34 nanotesla. But measured by the energy they hold, the two halves are exactly equal.
Which way the electric field points is the wave’s polarization. A vertical antenna shaking charge up and down sends out vertically polarized waves; turn it sideways and the waves turn with it. If you drive two crossed antennas a quarter-cycle apart, the field vector spins as the wave travels: circular polarization. GPS satellites use right-hand circular polarization so a receiver doesn’t care how it’s tilted. Sunlight is unpolarized, a jumble of all directions from countless independent atoms, but sunlight glancing off water or a road comes back mostly horizontally polarized, which is exactly what polarized sunglasses block.
Take a wave travelling along x: E = E₀ cos(kx − ωt) pointing along y, B = B₀ cos(kx − ωt) pointing along z. Faraday’s law says the curl of E equals −∂B/∂t. The z-component of the curl here is just ∂Ey/∂x = −kE₀ sin(kx − ωt). The right-hand side is −∂Bz/∂t = −ωB₀ sin(kx − ωt). Set them equal: kE₀ = ωB₀, so E₀/B₀ = ω/k, which is the wave speed, c.
The same law fixes the directions: for E along y and travel along +x, B has to be along +z. Flip B and the wave runs the other way. The energy density in the electric part is ½ε₀E², in the magnetic part B²/(2μ₀); substituting B = E/c and c² = 1/(μ₀ε₀) shows they are identical.
Set the electric field’s strength and the frequency, or pick a real-world case. Change the polarization and watch E (orange) and B (blue) turn together. The readouts are computed from E₀ with E = cB and I = ½cε₀E₀².
In 1897 Joseph Larmor worked out how much power an accelerating charge pours out. The answer is beautifully simple: it goes as the charge squared times the acceleration squared. Double the acceleration and you get four times the power. Because the kink field goes as sin θ, the power goes as sin² θ: a doughnut around the direction of shaking, with nothing along the axis.
For one electron the numbers are humblingly small. Shake an electron a micrometer up and down 100 million times a second and its peak acceleration is ω²x₀ ≈ 3.9 × 10¹¹ m/s², and its peak radiated power is about 9 × 10⁻³¹ watts. A radio station needs kilowatts. The trick is numbers and teamwork: an antenna carrying an amp of current has more than 10¹⁸ electrons passing each point every second, all sloshing in step. Their fields add, and since power goes as field squared, N electrons moving together radiate N² times one electron’s power, not N times.
The a² is behind some everyday sights. For a charge shaken at angular frequency ω, the acceleration is ω² times the swing, so the power goes as ω⁴. That’s why the sky is blue: sunlight shakes the electrons in air molecules, and blue light at 450 nanometers is re-radiated (700/450)⁴ ≈ 5.9 times more strongly than red at 700 nanometers (Chapter 11). It’s also how an X-ray tube works: electrons slammed to a stop in a metal target decelerate so violently that their kinks are X-rays, called bremsstrahlung, German for “braking radiation”. And it’s why particle accelerators that bend electrons in circles glow fiercely (synchrotron light): circling is acceleration, even at constant speed.
Every antenna is this formula put to work. A rod driven by an alternating current is a dipole, its electrons accelerating back and forth billions of times a second. The best length is about half a wavelength, which lets the current build up into a strong standing wave along the rod; Chapter 7 takes that apart.
The energy flow through a small patch of a sphere is cε₀Erad², with Erad = q a sin θ/(4πε₀c²r). That gives power per unit solid angle q²a² sin²θ / (16π²ε₀c³). Integrate sin²θ over the whole sphere (the result is 8π/3) and you get Larmor’s P = q²a²/(6πε₀c³). For an electron the constant works out to 5.7 × 10⁻⁵⁴ watts per (m/s²)².
For a dipole p = p₀ cos ωt the acceleration term becomes ω²p₀, and averaging cos² over a cycle gives P = p₀²ω⁴ / (12πε₀c³). For a half-wave dipole antenna this is usually written as P = ½ I₀² Rrad, with a radiation resistance of about 73 ohms: the power leaving as waves looks to the transmitter like heat in a 73-ohm resistor. Feeding 100 watts takes a peak current of √(2 × 100 / 73) ≈ 1.66 amps at the antenna’s centre.
At speeds close to c, Larmor’s formula picks up powers of γ (Liénard’s generalization), and the radiation beams forward into a cone about 1/γ wide. That beaming is why synchrotron light comes out as a narrow, intense searchlight.
A wave carries energy, and you can say exactly where it’s going. John Henry Poynting showed in 1884 that the energy flow is E × B / μ₀: a vector at right angles to both fields, pointing along the direction of travel, whose size is the power per square meter. Averaged over a cycle, for a wave with peak field E₀, that comes to half of c ε₀ E₀².
Sunlight is the best example there is. Just above Earth’s atmosphere, the Sun delivers 1,361 watts on every square meter facing it, a value called the total solar irradiance. That number follows straight from the inverse-square law: the Sun puts out 3.828 × 10²⁶ watts, and spread over a sphere 149.6 million kilometers in radius, that’s 1,361 W/m².
Run the Poynting relation backwards and you can find the fields inside that sunlight. E₀ = √(2I / cε₀) = √(2 × 1,361 / (299,792,458 × 8.854 × 10⁻¹²)) ≈ 1,013 volts per meter. That is the peak of an equivalent single wave; real sunlight is a broad jumble of frequencies and polarizations, and the root-mean-square field is about 716 V/m. The matching magnetic field is E₀/c ≈ 3.4 microtesla, about a tenth of Earth’s own steady magnetic field, but swinging back and forth hundreds of trillions of times a second.
Compare that with the radio around you. A Wi-Fi router putting out 0.1 watts, 3 meters away, gives a peak field near 0.8 V/m (again treating it as radiating evenly). A phone transmitting 1 watt, 1 meter away: about 7.7 V/m. Sunlight’s field is over a hundred times stronger than either, and it is not dangerous for the same reason radio isn’t: what matters for breaking molecules is the energy of each photon, not the field strength (Chapter 6).
The energy stored per cubic meter in a wave is u = ½ε₀E² + B²/(2μ₀). With B = E/c the two halves are equal, so u = ε₀E². That energy moves at c, so the flow is S = uc = cε₀E², and averaging cos² over a cycle supplies the ½.
A wave carries momentum too: an intensity I delivers a pressure I/c on a surface that absorbs it, twice that on a mirror. For sunlight that’s 1,361 / 299,792,458 ≈ 4.5 micropascals absorbed, 9.1 micropascals reflected. Tiny, but it is what pushes solar sails and comet dust tails, and engineers must account for it when predicting spacecraft orbits.
Every source of electromagnetic waves is, at bottom, charge being accelerated. Only the speed of the shaking changes.
| Source | What accelerates | Frequency | Wavelength |
|---|---|---|---|
| Power line | Electrons in the wires reversing 120 times a second | 60 Hz | ≈5,000 km |
| AM radio mast | Electrons surging up and down a tall steel tower | 1 MHz | 300 m |
| FM antenna | Electrons in a rod about 1.5 m long | 100 MHz | 3.00 m |
| Wi-Fi antenna | Electrons in a few-centimeter trace on a circuit board | 2.4 GHz | 12.5 cm |
| Microwave oven | Electrons whirling in a magnetron’s magnetic field | 2.45 GHz | 12.2 cm |
| Your skin | Charged atoms jostling in warm molecules (peak at body heat) | ≈32 THz | ≈9.3 µm |
| Sodium lamp | An electron changing state in an atom (a quantum jump; the classical picture only goes so far) | 509 THz | 589 nm |
| X-ray tube (100 kV) | Electrons braking hard in a tungsten target | up to 2.4 × 10¹⁹ Hz | down to 12 pm |