= 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 when
$$
\boxed{\beta n(\omega)>1,\qquad\cos\theta_C=\frac{1}{\beta n(\omega)}}.
$$
A 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.
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