Electric field lines of relativistically moving point charges

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.