Page 359 - Electromagnetics

in Figure 4.22. Here θ c is the critical angle for a plane wave incident on the single
interface between a lossless dielectric of permittivity r 0 and free space. Apply the
boundary conditions and find the fields in each of the three regions. Find the time-
average Poynting vector in region 0 at z = z 1 , in region 1 at z = z 2 , and in region 2 at
z = z 2 . Is conservation of energy obeyed?

4.16 A uniform ferrite material has scalar permittivity ˜ = and dyadic permeability
˜ ¯ µ. Assume the ferrite is magnetized along the z-direction and has losses so that its
permeability dyadic is given by (4.118). Show that the wave equation for a TEM plane
wave of the form
˜
˜
H(r,ω) = H 0 (ω)e − jk z z
is
2 ˜ 2 ˜
k H 0 = ω ˜ ¯µ · H 0
z
where k z = β − jα. Find explicit formulas for the two solutions k z± = β ± − jα ± . Show
that when the damping parameter α 1, near resonance α + α − .

4.17 A time-harmonic, TE-polarized, uniform cylindrical wave propagates in a lossy
medium. Assuming |kρ| 1, show that the power per unit length passing through a
cylinder of radius ρ is given by
e −2αρ
ˇ
2
P av /l = Re Z ∗ |H z0 | .
TE
8|k|
If the material is lossless, show that the power per unit length passing through a cylinder
is independent of the radius and is given by
ˇ 2
η|H z0 |
P av /l = .
8k
4.18 A TM-polarized plane wave is incident on a cylinder made from a perfect electric
conductor such that the current induced on the cylinder is given by (4.365). When the
cylinder radius is large compared to the wavelength of the incident wave, we may ap-
proximate the current using the principle of physical optics. This states that the induced
current is zero in the “shadow region” where the cylinder is not directly illuminated by
the incident wave. Elsewhere, in the “illuminated region,” the induced current is given
by
˜
˜ i
J s = 2ˆ n × H .
Plot the current from (4.365) for various values of k 0 a and compare to the current com-
puted from physical optics. How large must k 0 a be for the shadowing effect to be signif-
icant?

4.19 The radar cross section of a two-dimensional object illuminated by a TM-polarized
plane wave is defined by
˜ s 2
|E |
z
σ 2−D (ω, φ) = lim 2πρ .
˜ i 2
ρ→∞ |E |
z
This quantity has units of meters and is sometimes called the “scattering width” of the
object. Using the asymptotic form of the Hankel function, determine the formula for
the radar cross section of a TM-illuminated cylinder made of perfect electric conductor.

© 2001 by CRC Press LLC