Radiative spin-down of a rotating electric dipole ($\omega (t)={\omega }_i[1+2K{\omega }_i^{2}t/\mathcal{I}{]}^{-1/2}$)

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 $K={\mu }_0{p}_0^2/(6\pi c)$. 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.