Electric field lines of relativistically moving point charges

TL;DR

This paper derives a simple method to visualize electric field lines of relativistically moving charges, providing analytical solutions for key cases like rectilinear, wiggler, and synchrotron motions, linking classical electrodynamics with relativity.

Contribution

It introduces an equation for constructing electric field lines of moving charges and solves it analytically for important special cases.

Findings

Derived an equation for electric field line construction.

Solved the equation analytically for rectilinear motion.

Provided solutions for wiggler and synchrotron motions.

Abstract

Generation of electromagnetic fields by moving charges is a fascinating topic where the tight connection between classical electrodynamics and special relativity becomes particularly apparent. One can gain direct insight into the fascinating structure of such fields by visualizing the electric field lines. However, the calculation of electric field lines for arbitrarily moving charges is far from trivial. Here, we derive an equation for the director that points from the retarded position of a moving charge towards a specific field line position, which allows for a simple construction of these lines. We analytically solve this equation for several special but important cases: for an arbitrary rectilinear motion, for the motion within the wiggler magnetic field of a free electron laser, and for the motion in a synchrotron.