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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 Green’s 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 Green’s 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 θ ∂ ∂θ sin θ ∂ ∂θ + 1 r 2 sin 2 θ ∂ 2 ∂φ 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: Φ( r, θ, φ ) = R ( r ) P ( θ ) Q ( φ ) = U ( r ) P ( θ ) Q ( φ ) /r. (2) Operate on Φ with ∇ 2 , and set the result equal to zero to find PQ r d 2 U dr 2 + UQ r 2 sin θ d dθ ˆ sin θ dP dθ ! + UP r 3 sin 2 θ d 2 Q dφ 2 = 0 (3) Multiply by r 3 sin 2 θ/UPQ to find r 2 sin 2 θ U d 2 U dr 2 + sin θ P d dθ ˆ sin θ dP dθ ! + 1 Q d 2 Q dφ 2 = 0 . (4) The first two terms are independent of φ 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 dφ 2 = C or d 2 Q dφ 2 = CQ, (5) 2 from which it follows that Q ∼ e √ Cφ . Now, a change in φ by 2 π corresponds to no change whatsoever in spatial position; therefore, we must have Q ( φ + 2 π ) = Q ( φ ) because a function describing a measurable quantity must be a singlevalued function of position. Hence we can conclude that √ C = im where m is an integer so that e im 2 π = 1. Thus C = m 2 , and Q ( φ ) → Q m ( φ ) = e imφ , 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 φ ≤ φ ≤ φ + 2 π ....
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This note was uploaded on 01/08/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.
 Fall '11
 Electrodynamics
 Electrostatics

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