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MIT 6 001 - ELECTRODYNAMIC FIELDS: THE BOUNDARY VALUE POINT OF VIEW

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MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms13 ELECTRODYNAMIC FIELDS: THE BOUNDARY VALUE POINT OF VIEW 13.0 INTRODUCTION In the treatment of EQS and MQS systems, we started in Chaps. 4 and 8, re-spectively, by analyzing the fields produced by sp ecified (known) sources. Then we recognized that in the presence of materials, at least some of these sources were induced by the fields themselves. Induced surface charge and surface current den-sities were determined by making the fields satisfy boundary conditions. In the volume of a given region, fields were composed of particular solutions to the gov-erning quasistatic equations (the scalar and vector Poisson equations for EQS and MQS systems, respectively) and those solutions to the homogeneous equations (the scalar and vector Laplace equation, respectively) that made the total fields satisfy appropriate boundary conditions. We now embark on a similar approach in the analysis of electrodynamic fields. Chapter 12 presented a study of the fields produced by specified sources (dipoles, line sources, and surface sources) and obeying the inhomogeneous wave equation. Just as in the case of EQS and MQS systems in Chap. 5 and the last half of Chap. 8, we shall now concentrate on solutions to the homogeneous source-free equations. These solutions then serve to obtain the fields produced by sources lying outside (maybe on the boundary) of the region within which the fields are to be found. In the region of interest, the fields generally satisfy the inhomogeneous wave equation. However in this chapter, where there are no sources in the volume of interest, they satisfy the homogeneous wave equation. It should come as no surprise that, following this systematic approach, we shall reencounter some of the previously obtained solutions. In this chapter, fields will be determined in some limited region such as the volume V of Fig. 13.0.1. The boundaries might be in part perfectly conducting in the sense that on their surfaces, E is perpendicular and the time-varying H is tangential. The surface current and charge densities implied by these conditions 12 Electrodynamic Fields: The Boundary Value Point of View Chapter 13 Fig. 13.0.1 Fields in a limited region are in part due to sources induced on boundaries by the fields themselves. are not known until after the fields have been found. If there is material within the region of interest, it is perfectly insulating and of piece-wise uniform p ermittivity � and permeability µ. 1 Sources J and ρ are specified throughout the volume and ap-pear as driving terms in the inhomogeneous wave equations, (12.6.8) and (12.6.32). Thus, the H and E fields obey the inhomogeneous wave-equations. ∂2H � 2H − µ� ∂t2 = −� × J (1) 2E − µ�∂2E = �� ρ� + µ∂J (2)� ∂t2 � ∂t As in earlier chapters, we might think of the solution to these equations as the sum of a part satisfying the inhomogeneous equations throughout V (partic-ular solution), and a part satisfying the homogeneous wave equation throughout that region. In principle, the particular solution could be obtained using the su-perposition integral approach taken in Chap. 12. For example, if an electric dipole were introduced into a region containing a uniform medium, the particular solution would be that given in Sec. 12.2 for an electric dipole. The boundary conditions are generally not met by these fields. They are then satisfied by adding an appropriate solution of the homogeneous wave equation.2 In this chapter, the source terms on the right in (1) and (2) will be set equal to zero, and so we shall be concentrating on solutions to the homogeneous wave equation. By combining the solutions of the homogeneous wave equation that satisfy boundary conditions with the source-driven fields of the preceding chapter, one can describe situations with given sources and given boundaries. In this chapter, we shall consider the propagation of waves in some axial direction along a structure that is uniform in that direction. Such waves are used to transport energy along pairs of conductors (transmission lines), and through 1 If the region is one of free space, � �o and µ µo.→ →2 As pointed out in Sec. 12.7, this is essentially what is being done in satisfying boundary conditions by the method of images.Sec. 13.1 TEM Waves 3 waveguides (metal tubes at microwave frequencies and dielectric fibers at optical frequencies). We confine ourselves to the sinusoidal steady state. Sections 13.1-13.3 study two-dimensional modes between plane parallel con-ductors. This example introduces the mode expansion of electrodynamic fields that is analogous to the expansion of the EQS field of the capacitive attenuator (in Sec. 5.5) in terms of the solutions to Laplace’s equation. The principal and higher order modes form a complete set for the representation of arbitrary b oundary conditions. The example is a model for a strip transmission line and hence serves as an intro-duction to the subject of Chap. 14. The higher-order modes manifest properties much like those found in Sec. 13.4 for hollow pipe guides. The


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MIT 6 001 - ELECTRODYNAMIC FIELDS: THE BOUNDARY VALUE POINT OF VIEW

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