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chapt11 - CHAPTER 11 THE UNIFORM PLANE WAVE In this chapter...

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CHAPTER 11 THE UNIFORM PLANE WAVE In this chapter we shall apply Maxwell's equations to introduce the fundamental theory of wave motion. The uniform plane represents one of the simplest appli- cations of Maxwell's equations, and yet it is of profound importance, since it is a basic entity by which energy is propagated. We shall explore the physical pro- cesses that determine the speed of propagation and the extent to which attenua- tion may occur. We shall derive and make use of the Poynting theorem to find the power carried by a wave. Finally, we shall learn how to describe wave polarization. This chapter is the foundation for our explorations in later chapters which will include wave reflection, basic transmission line and waveguiding con- cepts, and wave generation through antennas. 11.1 WAVE PROPAGATION IN FREE SPACE As we indicated in our discussion of boundary conditions in the previous chap- ter, the solution of Maxwell's equations without the application of any boundary conditions at all represents a very special type of problem. Although we restrict our attention to a solution in rectangular coordinates, it may seem even then that we are solving several different problems as we consider various special cases in this chapter. Solutions are obtained first for free-space conditions, then for perfect dielectrics, next for lossy dielectrics, and finally for the good conductor. We do this to take advantage of the approximations that are applicable to each 348
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special case and to emphasize the special characteristics of wave propagation in these media, but it is not necessary to use a separate treatment; it is possible (and not very difficult) to solve the general problem once and for all. To consider wave motion in free space first, Maxwell's equations may be written in terms of E and H only as r ± H ± 0 @ E @ t 1 r ± E ² ² 0 @ H @ t 2 r ³ E 0 3 r ³ H 0 4 Now let us see whether wave motion can be inferred from these four equa- tions without actually solving them. The first equation states that if E is changing with time at some point, then H has curl at that point and thus can be considered as forming a small closed loop linking the changing E field. Also, if E is changing with time, then H will in general also change with time, although not necessarily in the same way. Next, we see from the second equation that this changing H produces an electric field which forms small closed loops about the H lines. We now have once more a changing electric field, our original hypothesis, but this field is present a small distance away from the point of the original disturbance. We might guess (correctly) that the velocity with which the effect moves away from the original point is the velocity of light, but this must be checked by a more quantitative examination of Maxwell's equations.
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