Boundary Value Problems: Numerical (Approximate) Methods
In Chapter 5, we discussed analytic methods of solution for electrostatic problems. The most outstanding feature of these methods was that the solution was exact and in the form of a mathematical relation. On the other hand, only certain classes of problems could be solved. In the case of the method of images, the question was one of finding the correct system of images, a requirement that meant, almost always, that a constant potential surface exists or can be stipulated. If this conducting surface is very complex, the condition of zero potential on the surface may not be as easy to satisfy. Similarly, separation of variable, while certainly valid in general, is often difficult to apply because of the need to satisfy complex boundary conditions. If the boundaries of the problem are not parallel to coordinates, it is next to impossible to find the constants required for solution. Even for the simple planar geometries discussed in Chapter 5, the solution required considerable skill. Furthermore, precious little was said about solution of Poisson’s equation. For another example of the difficulty in analytic solutions, consider Figure 6.1, which shows a simple parallel plate capacitor. Suppose we need to calculate its capacitance. One method we used before is to assume that the plates are very close to each other, neglect fringing, and use the formula for the parallel plate capacitor. In Figure 6.1a, the distance between the plates is very small, and the capacitance may be approximated as C = εA/d. Figure 6.1b shows the same two plates, but now the plates are much farther apart, perhaps because our design requires that this capacitor withstand high voltages. Here, we cannot neglect fringing and, therefore, the use of the formula for parallel plate capacitors is incorrect. How can we solve this problem? It seems that none of the methods of the previous chapters applies here. Yet, this type of problem is quite common.
KeywordsCharge Density Shape Function Finite Difference Method Electric Field Intensity Solution Domain
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