Radiation damping of a harmonically oscillating charge
= Radiation damping of a harmonically oscillating charge
A charge $q$ undergoing <simple harmonic motion> of <angular frequency> $\omega$ and slowly varying amplitude $A$ radiates the averaged power
$$
\langle P\rangle=\frac{\mu_0q^2A^2\omega^4}{12\pi c}.
$$
Its mechanical energy $E=mA^2\omega^2/2$ therefore has <exponential decay> rate $\gamma=\mu_0q^2\omega^2/(6\pi mc)$ and <half-life>
$$
t_{1/2}=\frac{6\pi mc\log2}{\mu_0q^2\omega^2}.
$$