Page 357 - Electromagnetics

4.3 Show that the Debye permittivity formulas
ωτ( s − ∞ )
s − ∞

˜ (ω) − ∞ = , ˜ (ω) =− ,
2 2
2 2
1 + ω τ 1 + ω τ
obey the Kronig–Kramers relations.
4.4 The frequency-domain duality transformations for the constitutive parameters of
an anisotropic medium are given in (4.197). Determine the analogous transformations
for the constitutive parameters of a bianisotropic medium.
4.5 Establish the plane-wave identities (B.76)–(B.79) by direct differentiation in rect-
angular coordinates.

4.6 Assume that sea water has the parameters = 80 0 , µ = µ 0 , σ = 4 S/m, and that
these parameters are frequency-independent. Plot the ω–β diagram for a plane wave
propagating in this medium and compare to Figure 4.12. Describe the dispersion: is it
normal or anomalous? Also plot the phase and group velocities and compare to Figure
4.13. How does the relaxation phenomenon affect the velocity ofa wave in this medium?

4.7 Consider a uniform plane wave incident at angle θ i onto an interface separating
two lossless media (Figure 4.18). Assuming perpendicular polarization, write the explicit
forms of the total fields in each region under the condition θ i <θ c , where θ c is the critical
angle. Show that the total field in region 1 can be decomposed into a portion that is
a pure standing wave in the z-direction and a portion that is a pure traveling wave in
the z-direction. Also show that the field in region 2 is a pure traveling wave. Repeat for
parallel polarization.

4.8 Consider a uniform plane wave incident at angle θ i onto an interface separating
two lossless media (Figure 4.18). Assuming perpendicular polarization, use the total
fields from Problem 4.7 to show that under the condition θ i <θ c the normal component
of the time-average Poynting vector is continuous across the interface. Here θ c is the
critical angle. Repeat for parallel polarization.

4.9 Consider a uniform plane wave incident at angle θ i onto an interface separating
two lossless media (Figure 4.18). Assuming perpendicular polarization, write the explicit
forms of the total fields in each region under the condition θ i >θ c , where θ c is the critical
angle. Show that the field in region 1 is a pure standing wave in the z-direction and that
the field in region2 is an evanescent wave. Repeat for parallel polarization.
4.10 Consider a uniform plane wave incident at angle θ i onto an interface separating
two lossless media (Figure 4.18). Assuming perpendicular polarization, use the fields
from Problem 4.9 to show that under the condition θ i >θ c the field in region 1 carries no
time-average power in the z-direction, while the field in region 2 carries no time-average
power. Here θ c is the critical angle. Repeat for parallel polarization.

4.11 Consider a uniform plane wave incident at angle θ i from a lossless material onto
a good conductor (Figure 4.18). The conductor has permittivity 0 , permeability µ 0 ,
and conductivity σ. Show that the transmission angle is θ t ≈ 0 and thus the wave in
the conductor propagates normal to the interface. Also show that for perpendicular
polarization the current per unit width induced by the wave in region 2 is
1 − j
˜
˜
˜
K(ω) = ˆ yσ T ⊥ (ω)E ⊥ (ω)
2β 2
© 2001 by CRC Press LLC