chap3 - Boundary Value Problems in Electrostatics II...

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Unformatted text preview: Boundary Value Problems in Electrostatics II Friedrich Wilhelm Bessel (1784 - 1846) December 23, 2000 Contents 1 Laplace Equation in Spherical Coordinates 2 1.1 Legendre Equation and Polynomials . . . . . . . . . . . . . . . . . . . 4 1.2 Solution of Boundary Value Problems with Azimuthal Symmetry . . 12 1.2.1 Example: A Sphere With a Specified Potential . . . . . . . . . 13 1.2.2 Example: Hemispheres of Opposite Potential . . . . . . . . . . 14 1.2.3 Example: Potential of an Isolated Charge . . . . . . . . . . . . 17 1.3 Behavior of Fields in Conical Holes and Near Sharp Points . . . . . . 19 1.4 Associated Legendre Polynomials; Spherical Harmonics . . . . . . . . 23 1.5 The Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 Expansion of the Greens Function in Spherical Harmonics . . . . . . 30 2 Laplace Equation in Cylindrical Coordinates; Bessel Functions 33 2.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 B.V.P. on Large Cylinders . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Greens Function Expansion in Cylindrical Coordinates . . . . . . . . 48 1 Although this is a new chapter, we continue to do things begun in the previous chapter. In particular, the first topic is the separation of variable method in spherical polar coordinates. 1 Laplace Equation in Spherical Coordinates The Laplacian operator in spherical coordinates is 2 = 1 r 2 r 2 r + 1 r 2 sin sin + 1 r 2 sin 2 2 2 . (1) This is also a coordinate system in which it is possible to find a solution in the form of a product of three functions of a single variable each: ( r, , ) = R ( r ) P ( ) Q ( ) = U ( r ) P ( ) Q ( ) /r. (2) Operate on with 2 , and set the result equal to zero to find PQ r d 2 U dr 2 + UQ r 2 sin d d sin dP d ! + UP r 3 sin 2 d 2 Q d 2 = 0 (3) Multiply by r 3 sin 2 /UPQ to find r 2 sin 2 U d 2 U dr 2 + sin P d d sin dP d ! + 1 Q d 2 Q d 2 = 0 . (4) The first two terms are independent of while the third depends only on this variable. Thus the third must be a constant as must the sum of the first two; the first of these conditions is 1 Q d 2 Q d 2 = C or d 2 Q d 2 = CQ, (5) 2 from which it follows that Q e C . Now, a change in by 2 corresponds to no change whatsoever in spatial position; therefore, we must have Q ( + 2 ) = Q ( ) because a function describing a measurable quantity must be a single-valued function of position. Hence we can conclude that C = im where m is an integer so that e im 2 = 1. Thus C =- m 2 , and Q ( ) Q m ( ) = e im , with m = 0 , 1 , 2 , ... . We recognize that the functions Q m can be used to construct a Fourier series and are a complete orthogonal set on the interval + 2 ....
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chap3 - Boundary Value Problems in Electrostatics II...

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