Motion of a Charge Density, Necessary Magnetic Sources and Solution of Maxwell's Equations including Magnetic Sources by Employing Potentials

We treat the problem of determining the magnetic sources that have been shown to come into existence when an electric charge density abruptly starts an arbitrary motion and hence an infinite speed of light becomes necessary for the satisfaction of the Lorenz condition for the scalar and vector potentials. The change that the Poynting theorem equation undergoes when an infinite speed of light constraint is enforced is the basis for the development and also use is made of the fact that the required magnetic and electric charge density functions share the same support and the same velocity in their motion. Reference to Green's functions in the source region is made to obtain the solution for the magnetic sources. The magnetic sources are computed to be in the form $m=Z\rho$ and $\vec{M}=Z\vec{J}$ where $Z$ is the wave impedance of the lossless, simple medium which can be vacuum. Also, the solution of Maxwell's equations incorporating the magnetic charge and current densities is given in terms of potentials with an infinite speed of propagation for the waves emitted by the charge functions. The magnetic scalar potential is a summation in the form of an integral of monopole moments. For $t > 0$ even though the resulting differential equations involve biharmonic operators, particular solutions of these equations are essentially solutions of Poisson's equations. When $t > 0$ the same Lorenz condition for the case with no magnetic sources appears also in this case with magnetic sources. The solutions for the potentials at $t=0$ where the Lorenz condition fails are obtained using the Helmholtz theorem.