Electromagnetism PHY 712
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This 3 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 712 at Wake Forest University taught by Natalie Holzwarth in Fall. Since its upload, it has received 17 views. For similar materials see /class/230730/phy-712-wake-forest-university in Physics 2 at Wake Forest University.
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Date Created: 10/28/15
January 27 2009 Notes for Lecture 5 We are concerned with nding solutions to the Poisson equation v2ltIgtltrgt e lt1 80 and the Laplace equation v2qgtr 0 2 In fact the Laplace equation is the homogeneous77 version of the Poisson equation The Greens function allows us to determine the electrostatic potential from volume and surface integrals lt1gt1 V dgrprGrr i S Grr v ltigtr 7 ltIgtrVGI39r f d2r 3 7 F80 This general form can be used in 1 2 or 3 dimensions In general the Greens function must be constructed to satisfy the appropriate Dirichlet or Neumann boundary conditions For some special cases we can use the results of the method of images to construct Dirichlet Green7s functions as described in Section 26 of your text In some cases it may be dif cult or inconvenient to nd a Green7s function that generates a solution with the correct boundary conditions In these situations we can still use Eq 3 to obtain a solution to the Poisson equation 1 and then add the appropriate linear combinations of solutions to the Laplace equation 2 to adjust the boundary values In lecture notes 4 we discussed one method of constructing Green7s functions that works for one dimensional systems Below we discuss another method that is generalizable for higher dimensional systems Orthogonal function expansions and Greens functions Suppose we have a complete set of orthogonal functions de ned in the interval 1 3 x 3 2 such that A ugcumz dab 6W 4 We can show that the completeness of this functions implies that unzunz 595 7 ya 5 This relation allows us to use these functions to represent a Green7s function for our system For the 1 dimensional Poisson equation the Greens function satis es 62 G 747T6 7 x 6 Therefore if d2 where also satisfy the appropriate boundary conditions then we can write the Greens functions as Gwan 47f 8 TL an For example consider the interval 0 S x S a 1 0 and x2 a A set of orthogonal functions de ned in this interval which vanish at both end points is given by i2a sinn7Txa so that we can construct the Greens function for this case as am 2 a n I This form of the one dimensional Green7s function only allows us to nd a solution to the Poisson equation within the interval 0 S x S a from the integral 9 i 1 T 4713980 z Oa dx Ga may 10 since by assumption the boundary values vanish ltIgt0 a 0 In fact from our previous analysis of the one dimensional Poisson equation we know that this boundary condition is not appropriate The boundary corrected full solution within the interval 0 S x S a would include additional terms from the solutions of the Laplace equation 1x 1 and 2x x qorrectedz A 1 The orthogonal function expansion method can easily be extended to two and three dimen sions For example if NAM and denote the complete functions in the s y and 2 directions respectively then the three dimensional Green7s function can be written G 7 7 7 7272 471 ulul1myimywnzwnz7 12 where d2 d2 d2 wu x ialu z i mvmy and 774042 13 See Eq 3167 in Jackson for an example As discussed in Lecture Notes 4 an alternative method of nding Green7s functions for second order ordinary differential equations is based on a product of two independent solu tions of the homogeneous equation u1x and u2 which satisfy the boundary conditions at 1 and 1 respectively 47139 Gxx Ku1zltu2xgt where K E W 14 1 d1 with zlt meaning the smaller of z and x and zgt meaning the larger of z and x For example we have previously discussed the example of the one dimensional Poisson equation with the boundary condition lt1gt0 0 and 33 0 to have the form Gxx 47Tzlt 15 For the two and three dimensional cases we can use this technique in one of the dimensions in order to reduce the number of summation terms These ideas are discussed in Section 311 of Jackson For the two dimensional case for example we can assume that the Greens function can be written in the form Gx77y7y Zunungny7y39 If the functions satisfy Eq 7 then we must require that G satisfy the equation 32 WGZWWWNAWWMWF WWWW an The yidependence of this equation will have the required behavior if we choose 6 which in turn can be expressed in terms of the two independent solutions Um and vn2y of the homogeneous equation 62 7EWwimmiw m d2 dig 7 an m 07 19 and a constant related to the Wronskian 47139 Kn E dun 7 Z dim2 39 20 dy 2 1 dy If these functions also satisfy the appropriate boundary conditions we can then construct the 2 dimensional Green7s function from GWJCyw ZUnUn Knvm 98 11gt 21 For example a Green7s function for a two dimensional rectangular system with 0 S x S a and 0 S y S b which vanishes on each of the boundaries can be expanded 00 sin m sin n sinh MK sinh Mb 7 y Glt 77y7y8Z a a n gt a gtgt n1 n sinh T 22 g m m V As an example we can use this result to solve the 2 dimensional Laplace equation in the square region 0 S x S 1 and 0 S y S 1 with the boundary condition lt1gt0 lt1gt0y lt1gt1y 0 and lt1gt 1 Vb In this case in determining lt1gtxy using Eq 3 there is no volume contribution since the charge is zero and the surface77 integral becomes a line integral 0 S x S 1 for y 1 Using the form from Eq 22 with a b 1 it can be shown that the result takes the form 271 17Txsinh2n 17Ty MaiPi 4V0 ml 271 17Tsinh2n 17T 23
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