Motion of a Charged Particle in a Constant Magnetic Field and a Transverse Electromagnetic Wave Propagating along the Field

The relativistic equation of motion is examined for a charged particle in a constant magnetic field and a transverse electromagnetic wave propagating along the field. A general discussion is given of the effects at cyclotron resonance of the magnetic field of the wave and the relativistic mass increase with energy. An exact solution to the equation of motion is found for the case of a circularly polarized wave. The solution shows that when the index of refraction of the medium in which the wave propagates is not unity, the energy of the particle is a periodic function of time, the exact relationship being expressible in terms of elliptic integrals. When the index of refraction is unity, the effect of the magnetic field of the wave just compensates for the change in mass with energy, and the energy of the particle increases indefinitely at resonance. Several possible applications of this solution to classical cyclotron resonance phenomena are pointed out. As a numerical example, the case of an electron in a constant magnetic field of 1000 G initially at resonance with microwaves whose $E$ field is 0.1 esu is considered.