= Radiative spin-down of a rotating electric dipole
{title2=$\omega(t)=\omega_i[1+2K\omega_i^2t/\mathcal I]^{-1/2}$}
A permanent <electric dipole moment> of magnitude $p_0$ rotating perpendicular to a fixed axis emits $P=K\omega^4$ in the <electric-dipole approximation for radiation>, with $K=\mu_0p_0^2/(6\pi c)$. For fixed mechanical <moment of inertia> $\mathcal I$, weak radiative loss gives $\mathcal I\omega\dot\omega=-K\omega^4$ and hence the displayed spin-down law. It assumes $|\dot\omega|\ll\omega^2$: the full dipole acceleration has squared magnitude $p_0^2(\omega^4+\dot\omega^2)$, whose second term is beyond this leading adiabatic approximation.
Back to article page
