Concept

Classical electron radius — where it appears

The size at which an electron's electrostatic energy would equal its rest energy, e²/(4πε₀mc²), about 2.82 femtometres. It is not the electron's size, which no experiment has found, but it sets the Thomson cross-section and marks where classical electrodynamics stops being self-consistent.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as self-energy — the same set of essays touches all of them, so they are one junction rather than several.

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.

astrophysics · Radiating charge $Where a moving charge keeps its field's energy and its momentum. Left: the density of field energy and of field momentum around a charged shell moving slowly to the right, by direction, each drawn as the mass per unit volume it contributes — energy density over c², momentum density over velocity. The energy density is the same in every direction; the momentum density goes as sin²θ, zero ahead and behind and largest broadside, because it is ε₀E × B and the magnetic field of a moving charge vanishes along its line of motion. Broadside it is twice the energy's; averaged over all directions sin²θ is 0.6667, and twice that is the 4/3. Right: the fraction of the field's energy, and of its momentum, lying within a distance r of the centre of a shell of radius a. Both are 1 − a/r, because both densities fall as the fourth power of distance: half of each lies within 2a and nine-tenths within 10a.$

The mass a charge's field gets wrong by a third

Give a charged sphere's field its energy U and divide by c², and the field has a mass. Move the sphere and ask the field's momentum what the mass is, and it says four-thirds of that. The discrepancy took from 1881 to the 1960s to understand, and the answer is not a better calculation of the field. It is that a charge cannot hold itself together, and whatever does has energy too.

electromagnetism · Field energy

Named alongside it

The objects these essays reach for when they reach for this one.

Self-energy Acceleration Causality Conservation of energy Dipole radiation Electromagnetic mass Energy flux Field energy Field momentum Four-momentum Idealisation Larmor formula

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