Motion of a Charge Density and the Speed of Light in Vacuum Revisited

As an extension of affirmative past work by the author on the feasibility of an infinite speed of light, this note focuses on the consequences of an infinite speed of light $c$ for a charge density in a motion that starts abruptly in time. In previous work it was proved that the scalar and vector potentials need to be non-retarded or speed of light must be infinite for such a charge density in order that Maxwell's equations be satisfied by the scalar and vector potentials. Here it is found that for this abruptly starting motion of a charge density function Maxwell's equations will fail even if the potentials are not retarded. For even though an infinite $c$ is sufficient for the Lorenz condition on the potentials to be satisfied when $t > 0$, it is not at $t=0$. Then there arises the problem of facing unsatisfied Maxwell's equations and hence the necessity of having to introduce additional source terms into these equations to render them satisfied. It is seen that with inclusion of electric charge and current density terms this objective cannot be attained, and magnetic charge and current density terms are needed as the only means. The steps for the determination of the required magnetic sources are given. The problem is seen to be reduced to that of solving an inverse source problem for Poisson's equation. The obtained electric and magnetic fields with the introduced magnetic sources are non-retarded. The obtained results apply to the case of a continuous charge distribution as well as a point charge.