Page 309 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 291

and from (34),
(38)
H t1 = H t2
The surface integrals produce the boundary conditions on the normal components,

D N1 − D N2 = ρ S (39)
and

B N1 = B N2 (40)
Itisoftendesirabletoidealizeaphysicalproblembyassumingaperfectconductor
for which σ is infinite but J is finite. From Ohm’s law, then, in a perfect conductor,

E = 0
and it follows from the point form of Faraday’s law that
H = 0

for time-varying fields. The point form of Amp`ere’s circuital law then shows that the
finite value of J is

J = 0
and current must be carried on the conductor surface as a surface current K. Thus, if
region 2 is a perfect conductor, (37) to (40) become, respectively,
E t1 = 0 (41)

H t1 = K (H t1 = K × a N ) (42)

D N1 = ρ s (43)
B N1 = 0 (44)
where a N is an outward normal at the conductor surface.
Note that surface charge density is considered a physical possibility for either di-
electrics, perfect conductors, or imperfect conductors, but that surface current density
is assumed only in conjunction with perfect conductors.
The preceding boundary conditions are a very necessary part of Maxwell’s
equations. All real physical problems have boundaries and require the solution of
Maxwell’s equations in two or more regions and the matching of these solutions at
the boundaries. In the case of perfect conductors, the solution of the equations within
the conductor is trivial (all time-varying fields are zero), but the application of the
boundary conditions (41) to (44) may be very difficult.
Certain fundamental properties of wave propagation are evident when Maxwell’s
equations are solved for an unbounded region. This problem is treated in Chapter 11.
It represents the simplest application of Maxwell’s equations because it is the only
problem which does not require the application of any boundary conditions.