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Chapter 6 ELECTROSTATIC BOUNDARY- VALUE PROBLEMS Our schools had better get on with what is their overwhelmingly most important task: teaching their charges to express themselves clearly and with precision in both speech and writing; in other words, leading them toward mastery of their own language. Failing that, all their instruction in mathematics and science is a waste of time. —JOSEPH WEIZENBAUM, M.I.T. b.1 INTRODUCTION The procedure for determining the electric field E in the preceding chapters has generally been using either Coulomb's law or Gauss's law when the charge distribution is known, or using E = — W when the potential V is known throughout the region. In most practical situations, however, neither the charge distribution nor the potential distribution is known. In this chapter, we shall consider practical electrostatic problems where only electro- static conditions (charge and potential) at some boundaries are known and it is desired to find E and V throughout the region. Such problems are usually tackled using Poisson's 1 or Laplace's 2 equation or the method of images, and they are usually referred to as boundary- value problems. The concepts of resistance and capacitance will be covered. We shall use Laplace's equation in deriving the resistance of an object and the capacitance of a capaci- tor. Example 6.5 should be given special attention because we will refer to it often in the remaining part of the text. .2 POISSON'S AND LAPLACE'S EQUATIONS Poisson's and Laplace's equations are easily derived from Gauss's law (for a linear mater- ial medium) V • D = V • eE = p v (6.1) 'After Simeon Denis Poisson (1781-1840), a French mathematical physicist. 2 After Pierre Simon de Laplace (1749-1829), a French astronomer and mathematician. 199
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200 Electrostatic Boundary-Value Problems and E = -VV Substituting eq. (6.2) into eq. (6.1) gives V-(-eVV) = p v for an inhomogeneous medium. For a homogeneous medium, eq. (6.3) becomes V 2 y = - ^ (6.2) (6.3) (6.4) This is known as Poisson's equation. A special case of this equation occurs when p v = 0 (i.e., for a charge-free region). Equation (6.4) then becomes V 2 V= 0 (6.5) which is known as Laplace's equation. Note that in taking s out of the left-hand side of eq. (6.3) to obtain eq. (6.4), we have assumed that e is constant throughout the region in which V is defined; for an inhomogeneous region, e is not constant and eq. (6.4) does not follow eq. (6.3). Equation (6.3) is Poisson's equation for an inhomogeneous medium; it becomes Laplace's equation for an inhomogeneous medium when p v = 0. Recall that the Laplacian operator V 2 was derived in Section 3.8. Thus Laplace's equa- tion in Cartesian, cylindrical, or spherical coordinates respectively is given by (6.6) (6.7) (6.8) depending on whether the potential is V(x, y, z), V(p, 4>, z), or V(r, 6, 4>). Poisson's equation in those coordinate systems may be obtained by simply replacing zero on the right-hand side of eqs. (6.6), (6.7), and (6.8) with —p v /e.
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