Direct Solution of Maxwell’s Equations

Abstract

A new solution to Maxwell's differential equations is proposed. A new approach for writing solutions to these equations under consideration uses quaternions. The equations are written as a kind of generalization of the Cauchy-Riemann equations and have a form of partial differential equation of first order. The Green's function was found for direct (without potentials) solutions of Maxwell's equations. To calculate Green's function, we use factorization of the d'Alembert operator and the fact that Green's function for the d'Alembert operator is known. Three examples of determining the electromagnetic field strength were considered. This is an example of finding the strength of the electromagnetic field created by the charge q moving with constant speed v  along the axis x₁. The example of finding the electric field strength created by a uniformly charged thin rod at a point perpendicular to the rod at a distance R from the rod. The example of finding the electric field strength created by a dipole at a point located perpendicular to the middle of the dipole at a distance R from the middle of the dipole.