chapt07 - CHAPTER 7 POISSON'S AND LAPLACE'S EQUATIONS A...

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CHAPTER 7 POISSON'S AND LAPLACE'S EQUATIONS A study of the previous chapter shows that several of the analogies used to obtain experimental field maps involved demonstrating that the analogous quan- tity satisfies Laplace's equation. This is true for small deflections of an elastic membrane, and we might have proved the current analogy by showing that the direct-current density in a conducting medium also satisfies Laplace's equation. It appears that this is a fundamental equation in more than one field of science, and, perhaps without knowing it, we have spent the last chapter obtaining solu- tions for Laplace's equation by experimental, graphical, and numerical methods. Now we are ready to obtain this equation formally and discuss several methods by which it may be solved analytically. It may seem that this material properly belongs before that of the previous chapter; as long as we are solving one equation by so many methods, would it not be fitting to see the equation first? The disadvantage of this more logical order lies in the fact that solving Laplace's equation is an exercise in mathe- matics, and unless we have the physical problem well in mind, we may easily miss the physical significance of what we are doing. A rough curvilinear map can tell us much about a field and then may be used later to check our mathematical solutions for gross errors or to indicate certain peculiar regions in the field which require special treatment. With this explanation let us finally obtain the equations of Laplace and Poisson. 195
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7.1 POISSON'S AND LAPLACE'S EQUATIONS Obtaining Poisson's equation is exceedingly simple, for from the point form of Gauss's law, r Á D ± ± v ² 1 ³ the definition of D , D ± ² E ² 2 ³ and the gradient relationship, E ±´r V ² 3 ³ by substitution we have r Á D ±r Á ² ² E ³±´r Á ² ² r V ³± ± v or rµr V ±´ ± v ² ² 4 ³ for a homogeneous region in which ² is constant. Equation (4) is Poisson's equation , but the ``double r '' operation must be interpreted and expanded, at least in cartesian coordinates, before the equation can be useful. In cartesian coordinates, A ± @ A x @ x @ A y @ y @ A z @ z r V ± @ V @ x a x @ V @ y a y @ V @ z a z and therefore V ± @ @ x @ V @ x ±² @ @ y @ V @ y @ @ z @ V @ z ± @ 2 V @ x 2 @ 2 V @ y 2 @ 2 V @ z 2 ² 5 ³ Usually the operation r Á r is abbreviated r 2 (and pronounced ``del squared''), a good reminder of the second-order partial derivatives appearing in (5), and we have 196 ENGINEERING ELECTROMAGNETICS
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r 2 V ± @ 2 V @ x 2 ² @ 2 V @ y 2 ² @ 2 V @ z 2 ±³ ± v ² ´ 6 µ in cartesian coordinates. If ± v ± 0, indicating zero volume charge density, but allowing point charges, line charge, and surface charge density to exist at singular locations as sources of the field, then r 2 V ± 0 ´ 7 µ which is Laplace's equation . The r 2 operation is called the Laplacian of V .
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This note was uploaded on 03/30/2010 for the course EE 2317 taught by Professor Wilton during the Spring '10 term at University of Houston.

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chapt07 - CHAPTER 7 POISSON'S AND LAPLACE'S EQUATIONS A...

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