Radiation of a magnetic dipole moving more than the speed of light in a medium

Abstract

A simple asymptotic expression of the spectral density of the electric field of the Vavilov-Cherenkov radiation of a magnetic dipole with a constant magnetic moment moving uniformly at superluminal velocity (u>c/n) in a non-dispersed medium with a refractive index n is given. The spectral density was calculated directly using the time-inverse Fourier transform of the expression for electric field of an arbitrarily moving magnetic dipole using the relativistic vector magnetic potential, which we obtained earlier in a more general form. The integration was performed by the asymptotic method of saddle-point. The angular size of the radiation cone is found, where the wave vector of the emitted waves makes an angle θ with velocity, cos θ=1/β (if β=n u/c>1). It is shown that the waves of the Vavilov-Cherenkov radiation, written as the asymptotics of the cylindrical Bessel function, propagate at an acute angle θ to the direction of motion of the dipole, and the spectral density of the radiation is directly proportional to its frequency to the index of three second. The expressions obtained can be useful in further studies of the spectral analysis of the Vavilov-Cherenkov dipole radiation since the technique proposed by I.M. Frank in his works, in fact, is based on the interference of the radiation of dipole charges, which creates additional conditions and some difficulties for solving the problem in general.

Key words: cherenkov radiation, electromagnetic waves, fourier transforms, light cone.