Electric Field Lines of an Arbitrarily Moving Charged Particle

Electromagnetic fields of relativistic charged particles have a broad frequency spectrum and a sophisticated spatial structure. Field lines offer a visual representation of this spatial structure. In this article, we derive a general set of equations for the field lines of any moving charged particle. The electric field lines are completely determined by the unit vector from the retarding point to the observation point. After proper transformations, the field line equations describe the rotation of this vector with an angular velocity coinciding with Thomas precession. In some cases, including all planar trajectories, the field line equations reduce to linear differential equations with constant coefficients. We present a detailed derivation of these equations and their general analytical solution. We then illustrate this method by constructing field lines for the “figure eight” motion of an electric charge moving under the influence of a plane wave, including complex field lines in three dimensions.