Illustrated Physics
physics, drawn
Series

Retardation — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The shell of news, and the kink inside it. A charge that was moving at 0.6c to the right, stopped over a short interval, and has been at rest ever since. Outside the sphere of radius ct nothing has heard: the field there still points at the position the charge would have reached had it carried on, marked ahead of it. Inside, the field is that of a charge at rest. In between is a shell one deceleration-time thick, and across it the field line has to bend, because a field line cannot simply stop in empty space — the two ends are joined here to 8.5e-14 pixels. That bend is transverse to the radius, and it is the radiation. It is not an extra thing the charge emitted; it is the join between two static fields that do not line up, and it travels outward at c because that is where the news front is.

    The field that points where the charge is now

    The field here was set by what the charge was doing a distance over c ago, so it ought to point at where the charge used to be. For a charge moving steadily it points at where the charge is — not approximately, exactly — and nothing has outrun light. What breaks the arrangement is a change of motion, and the break is the whole of radiation.

    part 1 · electromagnetism
  2. Three terms, and one distance. The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of.

    The distance where a field changes its mind

    An oscillating source has three fields around it, falling as the inverse cube, the inverse square and the inverse first power of distance. They are all equal at one radius, and that radius is the wavelength over 2π — a number containing nothing about the source at all. Inside it a source mostly stores energy; outside it, mostly loses it, and the two behaviours are different technologies rather than different strengths.

    part 2 · electromagnetism
  3. What a row of sources does that one cannot. The pattern of 8 identical sources in a row, spaced 0.5 wavelengths apart, all driven in phase. Each source alone radiates the same in every direction drawn here; together they radiate almost entirely along one. The sum being performed is the sum over path differences across the row, which is the sum a diffraction grating performs over its slits — the same function with the same first null, at sin θ = 1/Nd, measured here off the curve. What has been exploited is retardation: the contributions arrive at different times, and the pattern is a map of where they arrive in step.

    When the source is not heard all at once

    The dipole approximation is not a statement that a source is small. It is a statement that every part of it is heard at the same retarded time, and dropping that assumption turns one source into a sum over a source. The sum is the same one a diffraction grating performs over its slits, with the same first null and the same extra orders — so a phased array and a grating are one piece of arithmetic met twice.

    part 3 · electromagnetism
  4. Two solutions, and nothing in the equations to choose between them. A spherical pulse leaving a point and a spherical pulse arriving at one, each drawn at three times 0.35, 0.6, 0.85 in units where the speed is one. Both are exact solutions of the same wave equation, which is checked here by differencing the drawn samples twice in space and twice in time and requiring the residual to vanish for each. The one on the left is what is always used; the one on the right is discarded, and the equations do not do the discarding. The incoming pulse grows as it converges for the same reason the outgoing one decays as it spreads — the same energy through a smaller sphere — and it is as consistent with conservation as its mirror image is.

    The solution that is thrown away

    Maxwell's equations admit a field that converges on a charge exactly as readily as one that leaves it, and nothing in them prefers either. Retardation is a boundary condition rather than a law. Which boundary condition is right has been argued about for a century, one of the answers makes the arrow of time a property of there being absorbers, and the laboratory version of the question — whether an atom emits at all — has a measured answer that depends on what is listening.

    part 4 · electromagnetism
  5. London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm.

    The attraction that weakens when light is too slow

    Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

    part 5 · electromagnetism
  6. The Casimir pressure between warm plates, from vacuum to heat. The attractive pressure between two parallel, perfectly reflecting plates at 300 K against their separation, on logarithmic axes, from the Lifshitz sum over Matsubara frequencies. Dashed: the zero-temperature law π²ħc/240a⁴. Dotted: the classical law ζ(3)kT/4πa³, which contains no Planck constant. At 1 μm the full pressure is 0.0013 Pa, within 0.2 per cent of the vacuum value; at 10 μm it is 3.96·10⁻⁷ Pa, 3.05 times the vacuum value and within 0.002 per cent of the classical law. The two laws cross at 3.28 μm; the thermal length ħc/kT is 7.63 μm. The second solid curve removes one term — the static magnetic fluctuation, which a metal of finite conductivity does not reflect — and sits 15 per cent lower at 1 μm and a factor of two lower far out. The plates are otherwise ideal mirrors: a model, chosen to isolate that one term.

    The force that forgets Planck's constant

    Two mirrors in a vacuum attract, with a pressure that contains Planck's constant and nothing else about the world but the speed of light. Warm the mirrors and pull them apart, and past a few micrometres at room temperature the Planck constant drops out of the answer. The force is still there and is now larger than the vacuum alone would give, but it has become classical: a count of thermal fluctuations of the static field, with kT of energy each. Half of that classical force comes from one fluctuation — a static magnetic field — and whether a real metal mirror reflects it is a question that has divided the people who measure the force for twenty years.

    part 6 · electromagnetism
  7. Two descriptions of gold that agree everywhere but at zero. How strongly gold answers a slowly varying field, (ε(iξ) − 1)ξ²/ωₚ² in the two forms of its permittivity at imaginary frequency ξ, against ξ on a logarithmic axis: the plasma model (dashed), which ignores the electrons' collisions and gives exactly 1, and the Drude model (solid), which includes them at the measured rate ħγ = 35 meV and falls to zero below about γ = 5.3·10¹³ per second. The Lifshitz force samples this curve only at the Matsubara frequencies 2πlkT/ħ (dots, 300 K): the first, 2.47·10¹⁴ per second, lies above γ, and there the two descriptions differ by 18 per cent, at the next by 10, and less beyond. At l = 0, ξ = 0, they part completely: 1 against 0. Everything the two models disagree about is in that one point.

    The static field a metal may or may not reflect

    The Casimir force between two gold plates at room temperature has been computed two ways for twenty-five years, and the two answers differ by forty per cent at three micrometres. Both use the same measured optical properties of gold. They part company over a single number: whether gold reflects a magnetic field that does not change in time. Common sense and every electrical engineer say it does not, since a static field passes straight through a copper sheet. The most precise force measurements say it does. The disagreement is not about the Casimir effect at all; it is about what a metal's electrons do with a fluctuation slower than their collisions.

    part 7 · electromagnetism

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