the Larmor formula
/ LAR-mor /
Why does an accelerating charge glow? The Larmor formula gives the bottom line: any charge that accelerates radiates electromagnetic energy, and it tells you exactly how much. It is the single most important number in classical radiation theory.
For a nonrelativistic point charge q with acceleration a, the total power radiated in all directions is P = mu_0 q^2 a^2 / (6 pi c), equivalently P = q^2 a^2 / (6 pi epsilon_0 c^3). Two features matter: the power goes as the SQUARE of the acceleration (double the acceleration, quadruple the radiated power), and in this limit it does not depend on the velocity at all. The radiation is not emitted uniformly — its angular distribution is dP/dOmega = (mu_0 q^2 a^2 / 16 pi^2 c) sin^2(theta), a donut-shaped pattern with theta measured from the acceleration vector, so nothing is radiated straight along the direction of a.
The Larmor formula quietly doomed the classical atom: an orbiting electron accelerates, so it should spiral into the nucleus in about 10^-11 s — one of the crises that forced quantum mechanics. The formula as written assumes v << c; the exact relativistic version is the Liénard result, P = (mu_0 q^2 gamma^6 / 6 pi c)(a^2 - |beta x a|^2), which reduces to Larmor when gamma tends to 1.
An electron in a magnetic field, whipped around in a tight circle, undergoes a large centripetal acceleration; plugging q and a into P = q^2 a^2/(6 pi epsilon_0 c^3) gives the power it bleeds away as light — the reason electron storage rings must continuously pump energy back in.
Radiated power scales as acceleration squared; the pattern is a sin^2(theta) donut around the acceleration.
The Larmor formula is the low-speed (v << c) limit; for relativistic charges use the Liénard generalization, where the gamma^6 factor makes the losses explode. It also assumes a genuine point charge — for extended sources use the multipole expansion.