Illustrated Physics
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Radiating charge — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.

    A charge that turns must glow

    An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

    part 1 · astrophysics
  2. The self-force matters at 6.27 × 10⁻²⁴ s, and nowhere a charge has ever been. The ratio of the radiation reaction to the applied force, which is τ divided by the time the force takes to change, on a logarithmic axis. τ = μ₀q²/6πmc = 6.266e-24 s for an electron, and light crosses 1.879 femtometres in that time — two thirds of the classical electron radius. a 3 GHz accelerating cavity: 1.9e-14; a 500 nm optical field: 3.8e-9; an electron orbiting a proton, ground state: 4.1e-8; an X-ray at 0.1 nm: 1.9e-5; over a classical electron radius: 6.7e-1. The largest of them, "over a classical electron radius", is still 1.5e+0 times too slow. So the correction is never large for any force anybody can apply, and the only regime where it would be is one in which the charge's own structure has already made the whole description meaningless.

    The force a charge exerts on itself

    Larmor's formula says how much an accelerating charge radiates and says nothing about who pays. Conservation says the charge does, so there is a force on it — and the equation that force produces has a free particle accelerating for ever with nothing pushing it, or else beginning to move before it is pushed. Both solutions are absurd, and the interval over which they are absurd is smaller than the electron the equation was written for.

    part 2 · astrophysics
  3. The power at stake, which is none. The power an electron radiates by Larmor's formula, against its acceleration, with 5 cases marked. a charge on a table: 9.8e+0 m/s², 5.49e-52 W; a laboratory centrifuge: 1.0e+6 m/s², 5.71e-42 W; a proton at the LHC: 1.9e+16 m/s², 2.06e-21 W; an electron in a linac: 2.0e+19 m/s², 2.28e-15 W; an electron in a hydrogen atom: 9.0e+22 m/s², 4.62e-8 W. The slope is two, measured on the drawn line. A charge held at one gravity radiates 5.49e-52 watts, which over the whole age of the universe comes to 2.39e-34 joules — far less than one photon of any kind. So the question of whether it radiates is not an experimental question about a charge on a table, and never has been. Every number here is a straight line on logarithmic axes with an exponent the figure measures.

    Whether a charge on a table glows

    The equivalence principle says a charge at rest in a gravitational field is a charge accelerating in empty space, and an accelerating charge radiates. Nothing is supplying the energy. The argument has run for eighty years, and its resolution is that radiation is not something a single observer can define.

    part 3 · astrophysics
  4. Radiated while the push holds, paid for when it stops. The power a charge radiates, the power the radiation reaction force takes from its motion, and the rate of change of the Schott term mτ a·v, through a push that rises over the first 20 per cent of its duration, holds steady, and falls away over the last 20, in units of mτa₀² where a₀ is the steady acceleration. Radiated power is a², the reaction force's take is −ȧv, and the Schott rate is found by differencing a·v along the trajectory; at every instant the first equals the sum of the other two. While the push is steady the reaction force is exactly zero and the charge still radiates at the full rate, all of it drawn from the Schott term. When the push stops, ȧ is large and negative while the charge is moving fast, and the reaction force takes 0.775 units, more than the 0.750 radiated over the whole push. The difference is what it handed back while the push was starting: then the charge is still slow, the reaction force points along the rising acceleration, and it does 0.025 units of work on the charge instead of taking any. The totals agree to a part in a hundred thousand.

    The bill that arrives when the pushing stops

    A charge accelerating steadily radiates at the full Larmor rate while the radiation reaction force on it is exactly zero, so for as long as the push holds, nothing about the charge's motion pays a single watt. The energy is lent by the field that travels with the charge, the loan is called the Schott term, and it is repaid the moment the acceleration changes.

    part 4 · astrophysics
  5. One turn of a circling charge, as it arrives. The radiation field reaching a distant observer in the plane of a charge's circular orbit, over one full turn, for speeds of 0.2c, 0.6c, 0.9c, each scaled to its own peak and plotted against arrival time as a fraction of the orbital period. At 0.2c the flash arrives 20.9 per cent of a turn wide at half height, having been emitted over 25.4 per cent; at 0.6c the flash arrives 6.0 per cent of a turn wide at half height, having been emitted over 14.2 per cent; at 0.9c the flash arrives 0.7 per cent of a turn wide at half height, having been emitted over 6.3 per cent. The slowest charge sends something close to a sine wave, with a trough 0.44 of its peak; the fastest sends a spike whose peak, divided by its speed, is 100 times a slow charge's, and a trough of 0.0028 of that peak. The field integrates to zero over every turn, as a radiation field must.

    The flash a circling charge sends once a turn

    A slow electron going round a magnet broadcasts one radio frequency. The same electron at nearly the speed of light, on a circle of the same kind, sends X-rays. Nothing about the mechanism changes between the two. The charge chases its own light, and the chase compresses one power of γ worth of emission into three.

    part 5 · astrophysics
  6. Every colour a bound electron can meet. The light-scattering cross-section of one electron bound with a resonance at 2.105 eV (589 nm) and damped only by its own radiation, in units of the Thomson cross-section, against the frequency of the light in units of the resonance, on logarithmic axes spanning nine decades of frequency and twenty-five of cross-section. Far below the resonance it rises with slope 4.001, Rayleigh's fourth power: at a hundredth of the resonance it is 1.0 × 10⁻⁸ times the Thomson value. At the resonance it spikes to 2.5 × 10¹⁵ times the Thomson value, which is 3λ²/2π = 1.66 × 10⁻¹³ m² exactly, over a width of 2.0 × 10⁻⁸ of the resonant frequency. Above it the cross-section settles to the Thomson value, 6.652 × 10⁻²⁹ m² — 1.0002 times it at a hundred times the resonance — and stays within 10 per cent of it from 4.6 times the resonant frequency, 9.8 eV, to 29 keV, 3.5 decades, until the photon energy approaches the electron's rest energy, where the Klein–Nishina factor for a free electron takes it down to 0.431 of the Thomson value at 511 keV and 0.077 at 10 MeV.

    The cross-section that forgets the colour

    An electron on a spring, shaken by light, scatters it — weakly and very blue below its resonance, enormously at it, and above it by exactly the same amount whatever the colour. That flat stretch is the Thomson cross-section, 6.65 × 10⁻²⁹ m², and it is why an X-ray picture of a crystal is a map of electrons and why the corona is white.

    part 6 · astrophysics
  7. The cone a charge leaves behind in water. A charged particle crossing water at 0.99 of the speed of light in vacuum, which is 1.317 times the speed of light in water, drawn at one instant. From each point it passed, its field set off a spherical wavelet travelling at c/n; the wavelets from earlier points have grown larger. Because the particle outruns them, they all touch one cone with its tip at the particle, at an angle θ = arccos(1/nβ) = 40.6° to the path — the light of Cherenkov radiation, which travels out perpendicular to the cone. No point on the path is special and the particle never accelerates: the radiation is a property of the field moving faster than the medium can carry it, the electromagnetic counterpart of a sonic boom or a boat's wake.

    Why the glow of a fast charge is blue

    A charge moving in a straight line at constant speed does not radiate — in a vacuum. In water, if it moves faster than light does in water, it glows, and the glow is blue. The cone of the light is the familiar half of the story. The other half is a count: Frank and Tamm's formula says each centimetre of water yields about two hundred visible photons, in numbers that rise as the inverse square of the wavelength, which is why a reactor pool shines blue and why the light keeps getting stronger into the ultraviolet until water stops being transparent. The charge never accelerates. What radiates is the water.

    part 7 · astrophysics
  8. Light from a charge that crosses into glass at a steady speed. The transition radiation emitted as an ultrarelativistic charge crosses from vacuum into a dense medium, per unit solid angle, against the angle to the path in units of 1/γ, for photons at three frequencies: one tenth, three tenths and one times γωₚ, where ωₚ is the medium's plasma frequency. The scale is αħγ²/π² per unit frequency. Nothing is emitted straight ahead; the light comes out in a narrow hollow cone with its brightest ring at γθ = 0.98, 0.88, 0.68 for the three. At a tenth of γωₚ the peak is about ten times higher than at γωₚ. The charge never accelerates: it is the field's sudden change of shape at the surface, from the flattened disc it carries in vacuum to the one the medium's electrons allow, that is radiated.

    The light a charge makes by changing medium

    A charge moving at constant speed in a straight line radiates nothing in a vacuum, and nothing in glass either if it is slower than light there. Let it cross from one into the other and it radiates at the boundary, though it neither turns nor slows. The field it carries has one shape in each medium and has to be rebuilt as it crosses, and what is shed in the rebuilding is light. The energy shed grows in proportion to the particle's Lorentz factor, the one property of a fast particle that almost nothing else measures at high energy, and this is why detectors stack hundreds of plastic foils in a particle's path to collect one or two X-rays.

    part 8 · astrophysics
  9. A line carved into the X-rays of a magnetised star. A model X-ray spectrum of the kind fitted to accreting pulsars, drawn as energy per logarithmic interval on a logarithmic scale: a hard continuum that bends over above about 25 keV, with the photons near the cyclotron energy removed by resonant scattering. Dashed, the same continuum without the field. The fundamental is placed at 38 keV, where lines of this kind are seen; thermal motion of the scattering electrons along the field at kT = 8 keV broadens it to a width of about 4.8 keV. The second harmonic falls at 73.0 keV, a little below twice the fundamental. Read through ħeB/m with the relativistic level spacing and the redshift of a 1.4 solar-mass, 12 km star, z = 0.24, the fundamental says the field where the line forms is 4.24 × 10⁸ T. The depths and widths here are illustrative, not fitted to a star.

    The circling that comes in quanta

    An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

    part 9 · astrophysics
  10. Every colour at its own angle, from one charge. The wavelength radiated by a charge passing over a grating, as a multiple of the grating's period d, against the angle from the charge's direction, for speeds of 0.3, 0.776, 0.99 of light's, in the first order: λ = d(1/β − cos θ). The shaded band is visible light for a grating of period 1.67 micrometres, the one Smith and Purcell used in 1953. At 0.776c — 300 keV electrons, their speed — visible light comes out between 0° and 30° from the beam, blue-green straight ahead and red at the wider angles; straight ahead the wavelength is 0.29d, 482 nm, and straight back 2.29d. A slow charge at 0.3c radiates only at wavelengths longer than 2.33d. A fast one at 0.99c sends 1.0 per cent of a period straight ahead, the d/2γ² of an undulator.

    The rainbow a passing electron leaves on a grating

    An electron moving steadily in a straight line through empty space does not radiate, at any speed. Send it skimming over a ruled metal grating, without touching it, and it lights up — every colour at its own angle, blue-green ahead and red further out, from an electron that is neither turning nor slowing. The grating hands the electron's field the one thing it lacked: a wavenumber that lets part of it escape. It is the same mechanism as the blue glow in a reactor pool, with a ruled surface standing in for the slow light in the water.

    part 10 · astrophysics
  11. The circle of electrons a wave can talk to. Electron velocities along and across the magnetic field, in units of c. Shaded: a horseshoe of 10 keV electrons (speed 0.195c), the shape a beam takes when it is driven down into a strengthening field, with the electrons moving up along the field missing. Curves: the electrons a wave travelling across the field can exchange energy with, for wave frequencies of 0.995, 0.980, 0.960 times the cyclotron frequency. Because a faster electron is heavier and circles more slowly, at Ω/γ, each resonance is a semicircle of radius c√(1 − ω²/Ω²): 0.100, 0.199, 0.280c. A wave at 0.981Ω resonates with the horseshoe's middle.

    The radio that only heavier electrons can make

    Above every aurora the Earth broadcasts a gigawatt of radio that no one on the ground can hear. It comes out just below the frequency at which electrons circle the field there, and it is a maser — amplified by the electrons themselves. The surprise is what makes the amplifier work. Without relativity it could not: summed over every electron, a wave at the circling frequency is always absorbed. The few per cent by which a fast electron is heavier, and circles more slowly, is not a correction to the mechanism. It is the mechanism.

    part 11 · astrophysics

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