Representation of the Electromagnetic Field of an Arbitrarily Moving Charged Particle by Electric Field Lines

Abstract

The paper discusses the representation of the electromagnetic field of an arbitrarily moving charged particle by means of electric field lines. Expressions for the field line equations are derived on the basis of exact Lienar-Wichert field formulas. Parameterization of field lines by means of light signals (dots) emitted at delayed moments of time allows us to avoid the problem of solving the retardation equation. The resulting nonlinear equations are linearized using the Lorentz transformation applied to the emission rate of these light dots in the particle’s rest frame. These linear equations coincide with the Thomas precession equation, which allows us to state that field lines can be thought of as comprised of light dots that were emitted isotropically in the particle’s rest frame at speed \(c\). The exact solution of the equations is found in the case when the ratio of the trajectory torsion to the product of the trajectory curvature by the Lorentz factor of the particle is a constant value for the trajectory. The class of such fields in particular includes all flat trajectories. Illustrations of field lines are given for two applications of practical interest – the motion of a charged particle in the field of a plane monochromatic linearly polarized wave and for a helical undulator. In addition, it is shown that the developed mathematical apparatus admits consideration of the superluminal motion of the charge. Exact solutions and illustrations of lines for the superluminal motion of a particle along a circle (superluminal synchrotron radiation) are given.