On the ambiguity of solutions of the system of the Maxwell equations

Abstract: This work, that is devoted to the memory of Dr. Andrew Chubykalo and his legacy, is the improved version of the paper published in Annales de la Fondation Louis de Broglie journal. In this article, methods for solving the Maxwell equations are analyzed. First, a method based on the direct differentiation of Maxwell's equations and their reduction to the wave equation for an electric field is considered in detail. It is shown that this method cannot give a closed-form solution that does not contain integrals. The other method, which is most often used by scientists, is based on the introduction of auxiliary quantities such as electromagnetic potentials. But rewriting the Maxwell equations via potentials requires the introduction of an additional constraint - the gauge condition. Some analysis of the expressions for the electric field obtained in two gauges, Coulomb and Lorenz gauges, shows that, contrary to the generally accepted opinion that expressions for the electromagnetic field should be identical in any gauge, the final expressions for the electric field differ in different gauges. Consequently, the uniqueness of the solution of the Maxwell equations is absent.