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CHAPTER 7 :POISSON'S AND LAPLACE'S EQUATIONSA

Tham khảo tài liệu 'chapter 7 :poisson's and laplace's equationsa', khoa học tự nhiên, vật lý phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 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 quantity 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 solutions 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 mathematics 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 I e-Text Main Menu Textbook Table of Contents 196 ENGINEERING ELECTROMAGNETICS 7.1 POISSON S AND LAPLACE S EQUATIONS Obtaining Poisson s equation is exceedingly simple for from the point form of Gauss s law V D pv 1 the definition of D D eE 2 and the gradient relationship E -VE 3 by substitution we have V D V E -V eVỤ pv or V-VE - 4 for a homogeneous region in which 6 is constant. Equation 4 is Poisson s equation but the double V operation must be interpreted and