Boundary Value Problems: Analytic Methods of Solution

Abstract

The relations and methods introduced in Chaps. 3and 4dealt primarily with point and distributed charges and the electric fields they produce. If the charges were known, the electric field and potential could be determined. However, many practical situations exist in which the charges are either unknown or are distributed in a complex fashion. The use of the simple formulas for the calculation of fields in these geometries is not always possible. In still other geometries, we have no knowledge of charges but only of fields and potentials. For example, in an overhead transmission line, we may know the potential but not the charge on the line. How can we then calculate the electric field intensity everywhere in space? Similarly, when designing an electric instrument, such as an electrostatic filter, the engineer is not going to calculate “how much charge must be present on the electrode.” This information, while important in itself, is not normally a design parameter simply because we do not usually use “charge supply sources” and we are ill equipped to measure charge or charge density. The more common problem in design would be to calculate the required potential on the electrodes of the device to produce the needed effect. This information is important because with it, the power supply required can be designed. Although the principles in Chaps. 3and 4and the formulas developed for calculation of fields, potentials, and energy are applicable to these types of problems as well, the difficulty is in applying them.