Larmor formula

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897,^[1] in the context of the wave theory of light.

When any charged particle (such as an electron, a proton, or an ion) accelerates, energy is radiated in the form of electromagnetic waves. For a particle whose velocity is small relative to the speed of light (i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula: $${\displaystyle P={\frac {2}{3}}{\frac {q^{2}}{4\pi \varepsilon _{0}c}}\left({\frac {\dot {v}}{c}}\right)^{2}={2 \over 3}{\frac {q^{2}a^{2}}{4\pi \varepsilon _{0}c^{3}}}={\frac {q^{2}a^{2}}{6\pi \varepsilon _{0}c^{3}}}=\mu _{0}{\frac {q^{2}a^{2}}{6\pi c}}{\text{ (SI units)}}}$$ $${\displaystyle P={\frac {2}{3}}{\frac {q^{2}a^{2}}{c^{3}}}{\text{ (cgs units)}}}$$ where ${\displaystyle {\dot {v}}}$ or ${\displaystyle a}$ is the proper acceleration, ${\displaystyle q}$ is the charge, and ${\displaystyle c}$ is the speed of light.^[2] A relativistic generalization is given by the Liénard–Wiechert potentials.

In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as: $${\displaystyle P={\frac {2}{3}}{\frac {m_{e}r_{e}a^{2}}{c}}}$$

One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced.