Page 358 - Electromagnetics

and that this is identical to the tangential magnetic field at the surface:
˜
˜ t
K(ω) =−ˆ z × H | z=0 .
If we define the surface impedance Z s (ω) of the conductor as the ratio of tangential
electric and magnetic fields at the interface, show that
1 + j
Z s (ω) = = R s (ω) + jX s (ω).
σδ
Then show that the time-average power flux entering region 2 for a monochromatic wave
of frequency ˇω is simply
1
ˇ
ˇ ∗
S av,2 = ˆ z (K · K )R s .
2
Note that the since the surface impedance is also the ratio of tangential electric field to
induced current per unit width in region 2, it is also called the internal impedance.
4.12 Consider a parallel-polarized plane wave obliquely incident from a lossless medium
onto a multi-layered material as shown in Figure 4.20. Writing the fields in each region
˜
˜ r
˜ i
n, 0 ≤ n ≤ N − 1,as H n = H n + H n where
˜ i
e
H = ˆ ya n+1 e − jk x,n x − jk z,n (z−z n+1 ) ,
n
˜ r
H =−ˆ yb n+1 e − jk x,n x + jk z,n (z−z n+1 ) ,
e
n
and the field in region N as
˜
e
H N = ˆ ya N+1 e − jk x,N x − jk z,N (z−z N ) ,
apply the boundary conditions to solve for the wave amplitudes a n+1 and b n in terms of
˜
a global reflection coefficient R n , an interfacial reflection coefficient n , and the wave
amplitude a n . Compare your results to those found for perpendicular polarization (4.313)
and (4.314).
4.13 Consider a slab of lossless material with permittivity = r 0 and permeability
µ = µ r µ 0 located in free space between the planes z = z 1 and z = z 2 . A right-hand
circularly-polarized plane wave is incident on the slab at angle θ i as shown in Figure
4.22. Determine the conditions (if any) under which the reflected wave is: (a) linearly
polarized; (b) right-hand or left-hand circularly polarized; (c) right-hand or left-hand
elliptically polarized. Repeat for the transmitted wave.

4.14 Consider a slab of lossless material with permittivity = r 0 and permeability µ 0
located in free space between the planes z = z 1 and z = z 2 . A transient, perpendicularly-
polarized plane wave is obliquely incident on the slab as shown in Figure 4.22. If the
i
temporal waveform of the incident wave is E (t), find the transient reflected field in region
⊥
0 and the transient transmitted field in region 2 in terms of an infinite superposition of
amplitude-scaled, time-shifted versions of the incident wave. Interpret each of the first
four terms in the reflected and transmitted fields in terms of multiple reflection within
the slab.

4.15 Consider a free-space gap embedded between the planes z = z 1 and z = z 2
in an infinite, lossless dielectric medium of permittivity r 0 and permeability µ 0 .A
perpendicularly-polarized plane wave is incident on the gap at angle θ i >θ c as shown

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