Abstract

Maxwell's equations from the 19th century and the almost equally old Lorentz force equation provide the theoretical basis of all of electrical engineering and consequently the foundation for the majority of all modern technologies. For point charges, this system of partial differential equations reduces to the Weber-Maxwell wave equation. In this article, it is shown that this wave equation can be solved analytically for arbitrarily moving point charges, including accelerated charges, and that it is possible to present an analytical solution in the style of Coulomb's law. This finding demonstrates that classical electrodynamics, with all of its numerous wave phenomena, can also be represented without differential equations. This is surprising and unexpected. Moreover, this work enables the development of a novel class of electromagnetic field solvers that are highly superior in terms of speed and quality to existing solvers based on finite-difference time-domain methods, the method of moments, or finite element methods. In the future, this solution will make it possible to simulate any electromagnetic task in interactive form at previously unattainable quality, even on low-performance computers.