Past exam of the optics and electrodynamics course of the University of Cambridge / 2016 / Paper 1 / 1 / c

Write $C=e^2/(4\pi {ϵ}_0)=m{c}^2{r}_0$. For a slowly evolving circular Coulomb orbit, force balance and the mechanical energy giveThe supplied Larmor formula therefore yieldsSet $dE/dt=-P$. Since $dE/dt=C\dot{r}/(2r^2)$,Thus the classical radiation collapse of a Coulomb orbit obeysThe decreasing mechanical energy means a more tightly bound orbit, not a smaller speed. The formal collapse time is $a_0^3/(4r_0^{2}c)$. The nonrelativistic, small-loss-per-orbit approximation eventually fails; the result demonstrates the classical instability rather than describing an actual hydrogen atom.