Page 258 - Electromagnetics

ˇ
ˇ
ˇ
ˇ
This implies a relationship between E 0 , D 0 , B 0 , and H 0 . Since V is arbitrary we see that
ˇ
ˇ
ˇ
ˇ
one possible relationship is simply to have one of each pair (E 0 , D 0 ) and (H 0 , B 0 ) equal to
ˇ
ˇ
ˇ
ˇ
zero. Then, by (4.163) and (4.164), E 0 = 0 implies B 0 = 0, and D 0 = 0 implies H 0 = 0.
ˇ
ˇ
Thus E 1 = E 2 , etc., and the solution is unique throughout V . However, we cannot in
general rule out more complicated relationships. The number of possibilities depends on
ˇ
ˇ
ˇ
ˇ
the additional constraints on the relationship between E 0 , D 0 , B 0 , and H 0 that we must
supply to describe the material supporting the field — i.e., the constitutive relationships.
c
For a simple medium described by ˜µ(ω) and ˜ (ω), equation (4.166) becomes

ˇ
ˇ
2 c 2
c∗
∗
|E 0 | [˜ ( ˇω) − ˜ ( ˇω)] +|H 0 | [ ˜µ( ˇω) − ˜µ ( ˇω)] dV = 0
V
or

ˇ
ˇ
2
2 c

|E 0 | ˜ ( ˇω) +|H 0 | ˜µ ( ˇω) dV = 0.
V

For a lossy medium, ˜ c < 0 and ˜µ < 0 as shown in § 4.5.1. So both terms in the
integral must be negative. For the integral to be zero each term must vanish, requiring
ˇ
ˇ
E 0 = H 0 = 0, and uniqueness is guaranteed.
When establishing more complicated constitutive relations we must be careful to ensure
that they lead to a unique solution, and that the condition for uniqueness is understood.

= 0 implies that the tangential components of
In the case above, the assumption ˆ n×E 0
ˇ
S
ˇ
ˇ
E 1 and E 2 are identical over S — that is, we must give specific values of these quantities

= 0.
on S to ensure uniqueness. A similar statement holds for the condition ˆ n × H 0
ˇ
S
In summary, the conditions for the fields within a region V containing lossy isotropic
materials to be unique are as follows:
1. the sources within V must be specified;
2. the tangential component of the electric field must be specified over all or part of
the bounding surface S;
3. the tangential component of the magnetic field must be specified over the remainder
of S.
We may question the requirement of a lossy medium to demonstrate uniqueness of the
phasor fields. Does this mean that within a vacuum the specification of tangential fields
is insufficient? Experience shows that the fields in such a region are indeed properly
described by the surface fields, and it is just a case of the mathematical model being
slightly out of sync with the physics. As long as we recognize that the sinusoidal steady
state requires an initial transient period, we know that specification of the tangential
fields is sufficient. We must be careful, however, to understand the restrictions of the
mathematical model. Any attempt to describe the fields within a lossless cavity, for
instance, is fraught with difficulty if true time-harmonic fields are used to model the
actual physical fields. A helpful mathematical strategy is to think of free space as the
limit of a lossy medium as the loss recedes to zero. Of course this does not represent
the physical state of “empty” space. Although even interstellar space may have a few
particles for every cubic meter to interact with the electromagnetic field, the density of
these particles invalidates our initial macroscopic assumptions.
Another important concern is whether we can extend the uniqueness argument to all
of space. If we let S recede to infinity, must we continue to specify the fields over S,or
is it sufficient to merely specify the sources within S? Since the boundary fields provide
information to the internal region about sources that exist outside S, it is sensible to
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