Page 393 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 375

Figure 11.1 (a) Arrows represent the instantaneous values of E x0 cos[ω(t − z/c)] at
t = 0 along the z axis, along an arbitrary line in the x = 0 plane parallel to the z axis, and
along an arbitrary line in the y = 0 plane parallel to the z axis. (b) Corresponding values
of H y are indicated. Note that E x and H y are in phase at any point in time.

Although we have considered only a wave varying sinusoidally in time and
space, a suitable combination of solutions to the wave equation may be made to
achieve a wave of any desired form, but which satisfies (14). The summation of
an infinite number of harmonics through the use of a Fourier series can produce a
periodic wave of square or triangular shape in both space and time. Nonperiodic
waves may be obtained from our basic solution by Fourier integral methods. These
topics are among those considered in the more advanced books on electromagnetic
theory.

D11.1. The electric field amplitude of a uniform plane wave propagating in
the a z direction is 250 V/m. If E = E x a x and ω = 1.00 Mrad/s, find: (a) the
frequency; (b) the wavelength; (c) the period; (d) the amplitude of H.

Ans. 159 kHz; 1.88 km; 6.28 µs; 0.663 A/m

D11.2. Let H s = (2 −40 a x − 3 20 a y )e − j0.07z A/m for a uniform plane
◦
◦
wave traveling in free space. Find: (a) ω;(b) H x at P(1, 2, 3) at t = 31 ns; (c)
|H| at t = 0at the origin.
Ans. 21.0 Mrad/s; 1.934 A/m; 3.22 A/m

11.2 WAVE PROPAGATION IN DIELECTRICS
We now extend our analytical treatment of the uniform plane wave to propagation
in a dielectric of permittivity and permeability µ. The medium is assumed to be
homogeneous (having constant µ and with position) and isotropic (in which µ and