Radiation of a Point Magnetic Dipole Moving in a Medium with Superluminal Speed

Simple asymptotic expressions have been obtained for the spectral density of the field and energy losses of Cherenkov radiation from a magnetic dipole with a constant magnetic moment moving uniformly in a medium with superluminal speed. The spectral density is calculated by Fourier transforming in time an arbitrarily moving magnetic dipole previously obtained in a more general form from the relativistic vector magnetic potential. The Fourier inversion integration was performed using the asymptotic saddle-point method. The conditions and angular size of the Vavilov-Cherenkov cone of radiation are derived. It is shown that the radiation waves propagate at a sharp angle to the direction of the dipole's motion, and the spectral density of the radiation field is directly proportional to its frequency raised to the power of three halves. The results are compared with previously known ones. It is found that the expression for energy losses per unit length of the dipole's path is identical to Frank's result when the dipole moment is parallel to the velocity of motion.