chap2 - Boundary-value Problems in Electrostatics I Karl...

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Unformatted text preview: Boundary-value Problems in Electrostatics I Karl Friedrich Gauss (1777 - 1855) December 23, 2000 Contents 1 Method of Images 1 1.1 Point Charge Above a Conducting Plane . . . . . . . . . 2 1.2 Point Charge Between Multiple Conducting Planes . . . 4 1.3 Point Charge in a Spherical Cavity . . . . . . . . . . . . 5 1.4 Conducting Sphere in a Uniform Applied Field . . . . . . 13 2 Greens Function Method for the Sphere 16 3 Orthogonal Functions and Expansions; Separation of Variables 19 3.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Separation of Variables . . . . . . . . . . . . . . . . . . . 24 3.3 Rectangular Coordinates . . . . . . . . . . . . . . . . . . 25 3.4 Fields and Potentials on Edges . . . . . . . . . . . . . . . 29 1 4 Examples 34 4.1 Two-dimensional box with Neumann boundaries . . . . . 34 4.2 Numerical Solution of Laplaces Equation . . . . . . . . . 37 4.3 Derivation of Eq. 35: A Mathematica Session . . . . . . 40 In this chapter we shall solve a variety of boundary value problems using techniques which can be described as commonplace. 1 Method of Images This method is useful given sufficiently simple geometries. It is closely related to the Greens function method and can be used to find Greens functions for these same simple geometries. We shall consider here only conducting (equipotential) bounding surfaces which means the bound- ary conditions take the form of ( x ) = constant on each electrically isolated conducting surface. The idea behind this method is that the solution for the potential in a finite domain V with specified charge density and potentials on its surface S can be the same within V as the solution for the potential given the same charge density inside of V but a quite different charge density elsewhere. Thus we consider two dis- tinct electrostatics problems. The first is the real problem in which we are given a charge density ( x ) in V and some boundary conditions on the surface S. The second is a fictitious problem in which the charge density inside of V is the same as for the real problem and in 2 which there is some undetermined charge distribution elsewhere; this is to be chosen such that the solution to the second problem satisfies the boundary conditions specified in the first problem. Then the solution to the second problem is also the solution to the first problem inside of V (but not outside of V). If one has found the initially undetermined exterior charge in the second problem, called image charge , then the potential is found simply from Coulombs Law, ( x ) = Z d 3 x 2 ( x ) | x- x | ; (1) 2 is the total charge density of the second problem. 1.1 Point Charge Above a Conducting Plane This may sound confusing, but it is made quite clear by a simple ex- ample. Suppose that we have a point charge q located at a point x = (0 , , a ) in Cartesian coordinates. Further, a grounded conductor occupies the half-space z < 0, which means that we have the Dirichlet...
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chap2 - Boundary-value Problems in Electrostatics I Karl...

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