Theory of equations by James Victor Uspensky

By James Victor Uspensky

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8b) In these equations, σ stands for the free surface charge density at any given point on Σ, and K for the free surface current density, stands for the unit normal on Σ at the point under consideration, directed from the region ‘1’ into region ‘2’, while the suffix ‘t’ is used to indicate the tangential component (along the surface Σ) of the respective vectors. Expressed in words, the above equations tell us that the normal component of the magnetic intensity and the tangential component of the electric intensity are continuous across the surface, while the normal component of the electric displacement vector and the tangential component of the magnetic field strength may possess discontinuities, the change in these quantities across the surface being related to the free surface charge density and the free surface current density respectively.

In this book, however, my approach will be to trace the origins of the working rules of these approximation schemes to the principles of electromagnetic theory. In working out solutions to the Maxwell equations, it is often found convenient to look at regions of space where there are no free charge or current sources as distinct from those containing the sources. These sources are commonly situated in some finite region of space and the field they create satisfies the inhomogeneous wave equation in these regions.

The complex representation has been introduced here for a real time dependent (and possibly space dependent) vector A since the electromagnetic field variables are vectorial quantities. Evidently, an analogous complex representation can be introduced for spaceand time dependent scalar functions as well. The complex representation for the harmonically varying electric field described by eq. 32b) The complex amplitude is often expressed in brief as Ẽ (or even simply as E, by dropping the tilde), keeping its space dependence implied.

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