Vavilov–Cherenkov radiation for electric and magnetic multipoles

The energy of Vavilov–Cherenkov radiation (VCR) from arbitrary linear multipoles is found for a transparent medium with $\varepsilon$ and $\mu$ differing from unity. For electric multipoles, two methods are used. The force, which for $\beta n>1$ retards the motion of a system of rigidly bound charges, and the force perpendicular to the direction of motion are determined (Sec. 2). The second method consists of determining the emitted energy from the superposition of the VCR fields created by the individual charges in the moving system (Sec. 3). It is shown that both methods give identical results. The Vavilov–Cherenkov radiation from moving elementary electric and magnetic dipoles is examined in Sec. 4. The magnetic moment induced by a moving electric dipole and the electric moment induced by a moving magnetic dipole are taken into account. The formula for the VCR of an electric dipole coincides with the formula obtained in this particular case from the analysis of Secs. 2 and 3 and thus justifies the transformations relating the electric and magnetic moments. For the magnetic dipole (an elementary current loop), there is no simple analogy to the VCR of an electric dipole, especially in the case when the magnetic dipole is oriented perpendicular to its velocity. The formulas are an elementary generalization (to the case $\mu\ne1$) of the formulas obtained by the author in 1942. The Vavilov-Cherenkov radiation of hypothetical magnetic multipoles consisting of magnetic charges is examined in Sec. 5. The results are based on the analogy between the usual Maxwell equations and the equations for magnetic charges and currents. For a medium with $\mu=1$ and dipoles oriented parallel to the velocity, there is a deep analogy between the properties of the field of the usual magnetic dipole (current loop) and a hypothetical dipole consisting of magnetic charges. There is no such analogy in the general case. All formulas for magnetic charges and systems consisting of magnetic charges, including also the analog of the Lorentz force, are obtained, as expected, by interchanging $\mu$ and $\varepsilon$, $\mathbf{E}$ and $\mathbf{H}$, and $\mathbf{H}$ and – $\mathbf{E}$. The Maxwell's equations for magnetic charges and currents in a medium with e and not equal to unity are examined in the concluding Sec. 6.