Free-space Maxwell equations current density

As noted elsewhere [67], Eq. (14) means that the continuity condition does not prohibit the existence of an electromagnetic current density J in free space. It is stressed that Eq. (14) is a mathematical prediction of Maxwell s equations, completely independent of any interpretation. 

These four vector relations compactly summarize the experimental laws describing all known electrical and magnetic phenomena. In these expressions, p is the electric charge density, J, the current density, E, the electric field and B, the magnetic induction. Maxwell s equations in free space (in the absence of dielectric or magnetic media) can be written... 

The spatial dependence of the electric field E(x, y, z) and the magnetic field H(x, y, z) of an optical waveguide is determined by Maxwell s equations. We assume an implicit time dependence exp( — iwt) in the field vectors, current density J and charge density a. The dielectric constant s(x, y, z) is related to the refractive index n(x, y, z) by e = n CQ, where Eq is the dielectric constant of free space. For the nonmagnetic materials which normally constitute an optical waveguide, the magnetic permeability p is very nearly equal to the free-space value Pq. Thus for convenience we assume p = Po throughout this book unless otherwise stated. Under these conditions. Maxwell s equations are expressible in the form[l]... 

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