Page 78 - Electromagnetics Handbook

Figure 2.6: Derivation of the electromagnetic boundary conditions across a discontinuous
source layer.

sense) across its boundary, and we can employ the Maxwell–Boffi equations to describe
the relationship between the “equivalent” sources and the electromagnetic fields.
We should note, however, that the limiting concept is not without its critics. Stokes
suggested as early as 1848 that jump conditions should never be derived from smooth
solutions [199]. Let us therefore pursue the boundary conditions for a surface of true
field discontinuity. This will also allow us to treat a material modeled as having a true
discontinuity in its material parameters (which we can always take as a mathematical
model of a more gradual transition) before we have studied in a deeper sense the physical
properties of materials. This approach, taken by many textbooks, must be done carefully.
There is a logical difficulty with this approach, lying in the application of the large-
scale forms of Maxwell’s equations. Many authors postulate Maxwell’s equations in point
form, integrate to obtain the large-scale forms, then apply the large-scale forms to regions
of discontinuity. Unfortunately, the large-scale forms thus obtained are only valid in the
same regions where their point form antecedents were valid — discontinuities must be
excluded. Schelkunoff [167] has criticized this approach, calling it a “swindle” rather
than a proof, and has suggested that the proper way to handle true discontinuities
is to postulate the large-scale forms of Maxwell’s equations, and to include as part
of the postulate the assumption that the large-scale forms are valid at points of field
discontinuity. Does this mean we must reject our postulate of the point form Maxwell
equations and reformulate everything in terms of the large-scale forms? Fortunately, no.
Tai [192] has pointed out that it is still possible to postulate the point forms, as long
as we also postulate appropriate boundary conditions that make the large-scale forms,
as derived from the point forms, valid at surfaces of discontinuity. In essence, both
approaches require an additional postulate for surfaces of discontinuity: the large scale
forms require a postulate of applicability to discontinuous surfaces, and from there the
boundary conditions can be derived; the point forms require a postulate of the boundary
conditions that result in the large-scale forms being valid on surfaces of discontinuity.
Let us examine how the latter approach works.
Consider a surface across which the constitutive relations are discontinuous, containing
electric and magnetic surface currents and charges J s , ρ s , J ms , and ρ ms (Figure 2.6).
We locate a volume region V 1 above the surface of discontinuity; this volume is bounded
by a surface S 1 and another surface S 10 which is parallel to, and a small distance δ/2
above, the surface of discontinuity. A second volume region V 2 is similarly situated below
the surface of discontinuity. Because these regions exclude the surface of discontinuity

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