We formulate an existence theorem that states that given localized scalar and vector timedependent sources satisfying the continuity equation, there exist two retarded fields that satisfy a set of four field equations. If the theorem is applied to the usual electromagnetic charge and current densities, the retarded fields are identified with the electric and magnetic fields and the associated field equations with Maxwell's equations. This application of the theorem suggests that charge conservation can be considered to be the fundamental assumption underlying Maxwell's equations.

Maxwell's displacement current has been the subject of controversy for more than a century.Questions on whether the displacement current represents a true current like the conduction current and whether it produces a magnetic field have recently been discussed in the literature. Similar interpretations for the Faraday induction current have also been controversial. These basic questions are answered in this paper by considering the relation between the displacement and conduction currents as well as the relation between Faraday induction and conduction currents.It is pointed out that the displacement current contributes to the magnetic field and that the induction current contributes to the electric field. However, the displacement and induction currents cannot be considered as the conduction current because they are nonlocal. Both relations are used to implement an instantaneous formulation of Maxwell's equations with local and nonlocal sources.

The covariant formulation of Maxwell's equations can be expressed in a form independent of the usual systems of units by introducing the constants α, β and γ into these equations. Maxwell's equations involving these constants are then specialized to the most commonly used systems of units: Gaussian, SI and Heaviside-Lorentz by giving the constants α, β and γ the values appropriate to each system.

Two results support the idea that the scalar and vector potentials in the Lorenz gauge can be considered to be physical quantities: (i) they separately satisfy the properties of causality and propagation at the speed of light and not imply spurious terms and (ii) they can naturally be written in a manifestly covariant form. In this paper we introduce expressions for the Lorenzgauge potentials at the present time in terms of electric and magnetic fields at the retarded time.These expressions provide a third result in favor of a physical interpretation of the Lorenz-gauge potentials: (iii) they can be regarded as causal effects of the observed electric and magnetic fields.