Page 423 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 405

11.29 Consider a left circularly polarized wave in free space that propagates in the
forward z direction. The electric field is given by the appropriate form of
Eq. (100). Determine (a) the magnetic field phasor, H s ;(b)anexpression
2
for the average power density in the wave in W/m by direct application of
Eq. (77).
11.30 In an anisotropic medium, permittivity varies with electric field direction,
and is a property seen in most crystals. Consider a uniform plane wave
propagating in the z direction in such a medium, and which enters the
material with equal field components along the x and y axes. The field
phasor will take the form:

E s (z) = E 0 (a x + a y e j βz ) e − jβz
where β = β x − β y is the difference in phase constants for waves that are
linearly polarized in the x and y directions. Find distances into the material
(in terms of β)at which the field is (a) linearly polarized and (b)
circularly polarized. (c) Assume intrinsic impedance η that is approximately
constant with field orientation and find H s and < S >.
11.31 A linearly polarized uniform plane wave, propagating in the forward z
direction, is input to a lossless anisotropic material, in which the dielectric
constant encountered by waves polarized along y( ry ) differs from that seen
by waves polarized along x( rx ). Suppose rx = 2.15, ry = 2.10, and the
wave electric field at input is polarized at 45 to the positive x and y axes.
◦
(a) Determine, in terms of the free space wavelength, λ, the shortest length
of the material, such that the wave, as it emerges from the output, is
circularly polarized. (b)Will the output wave be right or left circularly
polarized? Problem 11.30 is good background.
11.32 Suppose that the length of the medium of Problem 11.31 is made to be twice
that determined in the problem. Describe the polarization of the output
wave in this case.
11.33 Given a wave for which E s = 15e − jβz a x + 18e − jβz e jφ a y V/m in a medium
characterized by complex intrinsic impedance, η (a) find H s ;(b) determine
2
the average power density in W/m .
11.34 Given a general elliptically polarized wave as per Eq. (93):
E s = [E x0 a x + E y0 e a y ]e − jβz
jφ
(a) Show, using methods similar to those of Example 11.7, that a linearly
polarized wave results when superimposing the given field and a phase-
shifted field of the form:

e
E s = [E x0 a x + E y0 e − jφ a y ]e − jβz jδ
where δ is a constant. (b) Find δ in terms of φ such that the resultant wave is
linearly polarized along x.