Past exam of the optics and electrodynamics course of the University of Cambridge / 2014 / Paper 1 / 1 / c / Solution

Interpret the motion as uniform rectilinear motion: the charge is stationary in an inertial rest frame and has no radiation field. Constant speed alone is insufficient; a curved trajectory has acceleration and can radiate even in vacuum.

In a transparent medium, a charge can move faster than the light phase velocity $c/n$ without exceeding $c$. The induced polarization disturbance then adds coherently on a Cherenkov cone, with $\cos {\theta }_C=1/(n\beta )$. ThusThis is the kinematic threshold limit for the specified refractive index; appreciable emission requires exceeding it, and dispersion selects which frequencies satisfy the condition.