Electric dipole radiation

For a localized source whose size is small compared with its radiation wavelength, the radiation fields of its electric dipole moment $p$ areTheir total instantaneous power is

For a source of characteristic size $a$ and angular frequency $\omega$, the electric-dipole approximation requiresso the source is much smaller than its radiation wavelength. Observing the pure $1/R$ radiation field additionally requires the radiation zone, $R\gg c/\omega$ and $R\gg a$.

If $\theta$ is the angle between the observation direction and $\ddot{p}$, thenIntegrating over solid angle uses $\int {\sin }^2\theta d\Omega =8\pi /3$ and gives the total electric dipole radiation power.

A charge $q$ undergoing simple harmonic motion of angular frequency $\omega$ and slowly varying amplitude $A$ radiates the averaged powerIts mechanical energy $E=m{A}^2{\omega }^2/2$ therefore has exponential decay rate $\gamma ={\mu }_0{q}^2{\omega }^2/(6\pi mc)$ and half-life