ABSTRACT
Classical electromagnetics is summed up in the Maxwell-Lorentz equations [56, 57, 133], and in the absence of charges the electric and magnetic fields, E(r, t) and B(r, t), satisfy the equations
∇×E(r, t) = − ∂ ∂t B(r, t), (2.1)
∇×B(r, t) = c−2 ∂ ∂t E(r, t), (2.2)
∇ · E(r, t) = 0, (2.3) ∇ ·B(r, t) = 0, (2.4)
in space (r)-time (t). Eqs. (2.3) and (2.4) specify that both E and B are divergence-free (solenoidal) fields in matter-free regions of space. The magnetic field remains divergence-free in matter-filled domains, and this is so because our present theory is based on the fact that there is no experimental evidence for the existence of magnetic charges or monopoles. Since electric charges do exist, the electric field will not be divergence-free in matter-filled regions, and Eq. (2.3) thus must be modified in such regions. Whether a region can be characterized as matter-filled in the context of classical electromagnetics requires some remarks. In the macroscopic Maxwell theory matter is conceived as a continuum and the characterization complies with this. In the microscopic Maxwell-Lorentz theory all relevant charged particles (electrons, protons, ions) are treated as point-like entities. In consequence matter is present only in discrete points, and in these the charge density is infinite. In the covering theory of classical electrodynamics, named semiclassical electrodynamics [206], the dynamics of the charged elementary particles (electrons, etc.) is treated on the basis of quantum mechanics. Although we think of these particles as point-like entities, quantum theory does not allow one to determine (at a given time) a particle’s position precisely. The probabilistic nature of quantum mechanics in a way leads us back to a continuum view of matter, yet in a quantum statistical sense to be described later on.
