FWLec22 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #22 Normal Reflection from Conducting Half Spaces Objectives: This note considers the problem of uniform plane waves reflected from perfectly conducting boundaries, when the wave is incident from the normal direction. Maxwell’s equations and the boundary conditions at the surface of a perfect conductor are used to obtain a solution. The concept of a standing wave is introduced. Reflection from conducting half spaces Figure 1 shows the interface at z = 0 between a lossless region (Region 1) and a half space of perfect conductor (Region 2). Suppose that a wave is incident from Region 1 toward the boundary. Such a wave might have a phasor expression of the form E z x E e i i j z ( ) ˆ = - 0 1 b (22.1) H z y E e i i j z ( ) ˆ = - 0 1 1 h b (22.2) where the amplitude E i 0 is determined by a source located beyond the region under consideration (an infinite current sheet as explained in Note #18). The fields in (22.1)–(22.2) are known to be a valid solution of Maxwell’s equations in an infinite region, but in the problem posed in Figure 1 the fields must vanish in Region 2. Thus, the expressions in (22.1) and (22.2) are not complete solutions in this situation. The full solution must also satisfy the boundary conditions at the interface, which are given in the general case by ˆ ( ) n E E ¥ - = 1 2 0 interface (22.3) ˆ ( ) n E E s - = e e r 1 1 2 2 interface (22.4) ˆ ( ) n H H J s ¥ - = 1 2 interface (22.5) ˆ ( ) n H H - = m m 1 1 2 2 0 interface (22.6) where ˆ n is the normal vector pointing into Region 1 (in Figure 1, ˆ ˆ n z = - ). Since a time- varying field vanishes in a perfect conductor, E 2 0 = and H 2 0 = in this case. Therefore, the boundary conditions at the perfect conductor simplify to ˆ n E ¥ = 1 0 interface (22.7) ˆ n E s = e r 1 1 interface (22.8)
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 ˆ n H J s ¥ = 1 interface (22.9) ˆ n H = m 1 1 0 interface (22.10) Clearly, the first of these four conditions is not satisfied by (22.1), since ˆ ( ˆ) ( ) ˆ n E z E y E i i ¥ = - ¥ = - π 1 0 0 0 interface (22.11) This implies that the simple plane wave solution in (22.1)–(22.2) is not sufficient.
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