Page 421 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 403

in the liquid at a known frequency. What restrictions apply? Could this
method be used to find the conductivity as well?
11.13 Let jk = 0.2 + j1.5m −1 and η = 450 + j60
for a uniform plane
propagating in the a z direction. If ω = 300 Mrad/s, find µ, , and for the

medium.
11.14 A certain nonmagnetic material has the material constants = 2 and

r
/ = 4 × 10 −4 at ω = 1.5 Grad/s. Find the distance a uniform plane

wave can propagate through the material before (a)itis attenuated by 1 Np;
(b) the power level is reduced by one-half; (c) the phase shifts 360 .
◦
11.15 A10 GHz radar signal may be represented as a uniform plane wave in a
sufficiently small region. Calculate the wavelength in centimeters and the
attenuation in nepers per meter if the wave is propagating in a nonmagnetic

material for which (a) = 1 and = 0; (b) = 1.04 and = 9.00 ×
r
r
r
r
−4
10 ;(c) = 2.5 and = 7.2.

r
r
11.16 Consider the power dissipation term, E · Jdv,inPoynting’s theorem (Eq.
(70)). This gives the power lost to heat within a volume into which
electromagnetic waves enter. The term p d = E · J is thus the power
3
dissipation per unit volume in W/m .Following the same reasoning that
resulted in Eq. (77), the time-average power dissipation per volume will be
< p d >= (1/2)Re E s · J . (a) Show that in a conducting medium,

∗
s
through which a uniform plane wave of amplitude E 0 propagates in the
2 −2αz
forward z direction, < p d >= (σ/2)|E 0 | e . (b) Confirm this result for
the special case of a good conductor by using the left hand side of Eq. (70),
and consider a very small volume.
11.17 Let η = 250 + j30
and jk = 0.2 + j2m −1 for a uniform plane wave
propagating in the a z direction in a dielectric having some finite
conductivity. If |E s |= 400 V/m at z = 0, find (a) S at z = 0 and z = 60
cm; (b) the average ohmic power dissipation in watts per cubic meter at
z = 60 cm.
11.18 Given a 100-MHz uniform plane wave in a medium known to be a good
dielectric, the phasor electric field is E s = 4e −0.5z − j20z a x V/m. Determine
e
(a) ;(b) ;(c) η;(d) H s ;(e) S ;( f ) the power in watts that is incident

on a rectangular surface measuring 20 m × 30mat z = 10 m.
11.19 Perfectly conducting cylinders with radii of 8 mm and 20 mm are coaxial.
The region between the cylinders is filled with a perfect dielectric for which
−9
= 10 /4π F/m and µ r = 1. If E in this region is (500/ρ) cos(ωt − 4z)a ρ
V/m, find (a) ω, with the help of Maxwell’s equations in cylindrical
coordinates; (b) H(ρ, z, t); (c) S(ρ, z, t) ;(d) the average power passing
through every cross section 8 <ρ < 20 mm, 0 <φ < 2π.
11.20 Voltage breakdown in air at standard temperature and pressure occurs at an
6
electric field strength of approximately 3 × 10 V/m. This becomes an issue