Page 405 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 387

The second-harmonic component of the integrand in (75) integrates to zero, leaving
only the contribution from the dc component. The result is

1 E 2 x0 −2αz
S z = e cos θ η (76)
2 |η|
Note that the power density attenuates as e −2αz , whereas E x and H y fall off as e −αz .
We may finally observe that the preceding expression can be obtained very easily
by using the phasor forms of the electric and magnetic fields. In vector form, this is

1
∗
S = Re(E s × H ) W/m 2 (77)
s
2
In the present case
E s = E x0 e − jβz a x
and
E x0 + jβz E x0 jθ + jβz
∗ e e e
H = a y = a y
s
η ∗ |η|
where E x0 has been assumed real. Eq. (77) applies to any sinusoidal electromagnetic
wave and gives both the magnitude and direction of the time-average power density.
D11.6. At frequencies of 1, 100, and 3000 MHz, the dielectric constant of
ice made from pure water has values of 4.15, 3.45, and 3.20, respectively, while
the loss tangent is 0.12, 0.035, and 0.0009, also respectively. If a uniform plane
wave with an amplitude of 100 V/m at z = 0is propagating through such ice,
find the time-average power density at z = 0 and z = 10 m for each frequency.

2
2
Ans. 27.1 and 25.7 W/m ; 24.7 and 6.31 W/m ; 23.7 and 8.63 W/m 2
11.4 PROPAGATION IN GOOD
CONDUCTORS: SKIN EFFECT

As an additional study of propagation with loss, we will investigate the behavior of a
good conductor when a uniform plane wave is established in it. Such a material sat-
isfies the general high-loss criterion, in which the loss tangent, / 1. Applying

this to a good conductor leads to the more specific criterion, σ/(ω ) 1. As before,

we have an interest in losses that occur on wave transmission into a good conductor,
and we will find new approximations for the phase constant, attenuation coefficient,
and intrinsic impedance. New to us, however, is a modification of the basic problem,
appropriate for good conductors. This concerns waves associated with electromag-
netic fields existing in an external dielectric that adjoins the conductor surface; in
this case, the waves propagate along the surface. That portion of the overall field that
exists within the conductor will suffer dissipative loss arising from the conduction
currents it generates. The overall field therefore attenuates with increasing distance