Larmor formula

Remarks

Proof idea: differentiate the Liénard–Wiechert potentials and keep only the part of the fields that decays like $1/R$ at distance $R$. For small speed this part is $E = \frac{q}{4 \pi \varepsilon_0 c^2 R} \, n \times (n \times a)$ and $B = n \times E / c$, where $n$ is the unit vector toward the field point. The Poynting vector then has size $\frac{q^2 \abs{a}^2 \sin^2 \theta}{16 \pi^2 \varepsilon_0 c^3 R^2}$, which gives $dP / d\Omega$. Integrating over the sphere with $\int \sin^2 \theta \, d\Omega = 8\pi/3$ gives $P$.

Example: a charge that oscillates as $x(t) = x_0 \cos(\omega t)$ along a line has $\abs{a}^2 = x_0^2 \omega^4 \cos^2(\omega t)$. Its average radiated power over a period is $\frac{q^2 x_0^2 \omega^4}{12 \pi \varepsilon_0 c^3}$, the power of an oscillating dipole. The hypothesis of small speed is needed: for arbitrary speed, Liénard's generalization reads $P = \frac{q^2 \gamma^6}{6 \pi \varepsilon_0 c} \left( \abs{\dot\beta}^2 - \abs{\beta \times \dot\beta}^2 \right)$ with $\beta = v / c$ and $\gamma = (1 - \abs{\beta}^2)^{-1/2}$.

Going deeper

For a student

The formula is the basic estimate of radiation losses in classical electrodynamics: antennas, synchrotron radiation and bremsstrahlung all start from it or from its relativistic form. It shows the failure of the classical model of the atom: an electron that circles a nucleus accelerates, so it would lose energy and fall in. The energy carried away must be balanced by a force on the charge itself, which leads to the Abraham–Lorentz force.