Page 75 - Electromagnetics Handbook

and the two Gauss’s law expressions

D · ˆ n dS = ρ dV,
S(t) V (t)

B · ˆ n dS = ρ m dV.
S(t) V (t)
Magnetic sources also allow us to develop equivalence theorems in which difficult prob-
lems involving boundaries are replaced by simpler problems involving magnetic sources.
Although these sources may not physically exist, the mathematical solutions are com-
pletely valid.

2.8 Boundary (jump) conditions
If we restrict ourselves to regions of space without spatial (jump) discontinuities in
either the sources or the constitutive relations, we can find meaningful solutions to the
Maxwell differential equations. We also know that for given sources, if the fields are
specified on a closed boundary and at an initial time the solutions are unique. The
standard approach to treating regions that do contain spatial discontinuities is to isolate
the discontinuities on surfaces. That is, we introduce surfaces that serve to separate space
into regions in which the differential equations are solvable and the fields are well defined.
To make the solutions in adjoining regions unique, we must specify the tangential fields
on each side of the adjoining surface. If we can relate the fields across the boundary, we
can propagate the solution from one region to the next; in this way, information about
the source in one region is effectively passed on to the solution in an adjacent region. For
uniqueness, only relations between the tangential components need be specified.
We shall determine the appropriate boundary conditions (BC’s) via two distinct ap-
proaches. We first model a thin source layer and consider a discontinuous surface source
layer as a limiting case of the continuous thin layer. With no true discontinuity, Maxwell’s
differential equations hold everywhere. We then consider a true spatial discontinuity be-
tween material surfaces (with possible surface sources lying along the discontinuity). We
must then isolate the region containing the discontinuity and postulate a field relationship
that is both physically meaningful and experimentally verifiable.
We shall also consider both stationary and moving boundary surfaces, and surfaces
containing magnetic as well as electric sources.

2.8.1 Boundaryconditions across a stationary, thin source layer
In § 1.3.3 we discussed how in the macroscopic sense a surface source is actually a
volume distribution concentrated near a surface S. We write the charge and current in
terms of the point r on the surface and the normal distance x from the surface at r as
ρ(r, x, t) = ρ s (r, t) f (x, ), (2.179)
J(r, x, t) = J s (r, t) f (x, ), (2.180)

where f (x, ) is the source density function obeying
∞

f (x, ) dx = 1. (2.181)
−∞

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