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chapter 10

# chapter 10 - Chapter 10 ELECTROMAGNETIC WAVE PROPAGATION...

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Chapter 10 ELECTROMAGNETIC WAVE PROPAGATION How far you go in life depends on your being tender with the young, compas- sionate with the aged, sympathetic with the striving, and tolerant of the weak and the strong. Because someday in life you will have been all of these. —GEORGE W. CARVER 10.1 INTRODUCTION Our first application of Maxwell's equations will be in relation to electromagnetic wave propagation. The existence of EM waves, predicted by Maxwell's equations, was first in- vestigated by Heinrich Hertz. After several calculations and experiments Hertz succeeded in generating and detecting radio waves, which are sometimes called Hertzian waves in his honor. In general, waves are means of transporting energy or information. Typical examples of EM waves include radio waves, TV signals, radar beams, and light rays. All forms of EM energy share three fundamental characteristics: they all travel at high velocity; in traveling, they assume the properties of waves; and they radiate outward from a source, without benefit of any discernible physical vehicles. The problem of radia- tion will be addressed in Chapter 13. In this chapter, our major goal is to solve Maxwell's equations and derive EM wave motion in the following media: 1. Free space (<T = 0, s = e o , JX = /x o ) 2. Lossless dielectrics (a = 0, e = e,s o , JX = jx r jx o , or a <sC aie) 3. Lossy dielectrics {a # 0, e = E,E O , fx = fx r ix o ) 4. Good conductors (a — °°, e = e o , JX = ix r fx o , a ^S> we) where w is the angular frequency of the wave. Case 3, for lossy dielectrics, is the most general case and will be considered first. Once this general case is solved, we simply derive other cases (1,2, and 4) from it as special cases by changing the values of a, e, and ix. However, before we consider wave motion in those different media, it is appropriate that we study the characteristics of waves in general. This is important for proper understand- 410

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10.2 WAVES IN GENERAL 411 ing of EM waves. The reader who is conversant with the concept of waves may skip Section 10.2. Power considerations, reflection, and transmission between two different media will be discussed later in the chapter. 10.2 WAVES IN GENERAL A clear understanding of EM wave propagation depends on a grasp of what waves are in general. A wave is a function of both space and time. Wave motion occurs when a disturbance at point A, at time t o , is related to what happens at point B, at time t > t 0 . A wave equation, as exemplified by eqs. (9.51) and (9.52), is a partial differential equation of the second order. In one dimension, a scalar wave equation takes the form of d 2 E 2 d 2 E r- - U r- = 0 dt 2 dz 2 (10.1) where u is the wave velocity. Equation (10.1) is a special case of eq. (9.51) in which the medium is source free (p v , = 0, J = 0). It can be solved by following procedure, similar to that in Example 6.5. Its solutions are of the form or E =f(z~ ut) E + = g(z + ut) E=f(z- ut) + g(z + ut) (10.2a) (10.2b) (10.2c) where / and g denote any function of z — ut and z + ut, respectively. Examples of such functions include z ± ut, sin k(z ± ut), cos k(z ± ut), and e J k( - z±u '\ where k is a constant. It can easily be shown that these functions all satisfy eq. (10.1).
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chapter 10 - Chapter 10 ELECTROMAGNETIC WAVE PROPAGATION...

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