 Solution

Cherenkov radiation is coherent electromagnetic emission by a charged particle moving through a dielectric faster than the phase velocity of light in that medium. The particle polarizes the material along its path. Wavelets interfere constructively on a cone whenA simple wavefront construction compares a particle displacement $vt$ with the light-wave displacement $ct/n$, giving the cone relation. The particle still travels slower than the vacuum speed of light; no violation of special relativity occurs. Dispersion makes the angle and threshold wavelength-dependent. The radiation has a continuous spectrum rather than the discrete line spectrum of an atomic transition.