P: 4,664 The best derivation of Cerenkov radiation by a fast charged particle I have seen is the semi-classical derivation in Schilff Quantum Mechanics (2nd Edition) pages 267-271. Schiff derives the classical E and H fields, and the resulting Poynting vector P = E X H. He then gets the number of quanta radiated per unit path length in frequency interval ω to ω+dω: $$dN=\frac{1}{137}\left(1-\frac{c^2}{n^2v^2} \right)\frac{d\omega}{c} \text{ photons per unit length}$$ which becomes for infinite index of refraction $$dN=\frac{1}{137}\frac{d\omega}{c} \text{ photons per unit length}$$ So the total number of quanta depends on wnat interval the frequency interval dω covers. Normally, the index of refraction is n ≤1 for wavelengths less than ~ 1000 Angstroms (UV).