Plane-wave first approximation density

The longitudinal current density in vacuo is investigated, first in the plane-wave first approximation, by taking the real part of the potential... 

In the plane-wave first approximation, the current density is therefore... 

The vacuum charge density is also structured in general, but in the plane wave, first approximation is given by... 

In SI units, the transverse vacuum current densities are given in the plane-wave first approximation by... 

In general, the vacuum current density has a definite structure in the vacuum that is much richer than in the first plane-wave approximation a structure that has to be computed because analytical solutions to Eq. (183) are not available. 

In this section, we illustrate the self-consistent calculation of these charge current densities in the plane-wave approximation, using plane waves in the X, Y, and Z directions. In general, the solution of the field equation (459) must be found numerically, and it is emphasized that the plane-wave approximation is a first approximation only. In the internal space, there is the real vector ... 

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