Class_15new - 1 PHYS809 Class 15 Notes Green function in...

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Unformatted text preview: 1 PHYS809 Class 15 Notes Green function in spherical polar coordinates To illustrate construction of a Green function in spherical polar coordinates consider the Dirichlet problem in a region bounded by two concentric sphere of radii a and b with a < b . The Green function is the solution of ( ) ( ) ( ) ( ) 2 2 4 , , , , , cos cos , G r r r r r = ---- (15.1) with G = 0 on r = a and r = b . Since the completeness relation for the spherical harmonics is ( ) ( ) ( ) ( ) * , , cos cos , l lm lm l m l Y Y = =- =-- (15.2) we look for a solution of form ( ) ( ) ( ) ( ) * , , , , , , , , . l lm lm lm l m l G r r g r r Y Y = =- = (15.3) Remembering that ( ) , l lm r Y is a solution of Laplaces equation, substitution of (15.3) into equation (15.1) gives ( ) ( ) ( ) ( ) ( ) ( ) 2 * * 2 2 2 1 1 4 , , , , . l l lm lm lm lm lm lm l m l l m l l l r g g Y Y r r Y Y r r r r r = =- = =- + - = -- (15.4) Since the spherical harmonics are linearly independent, we must have ( ) ( ) 2 2 2 2 1 1 4 . lm lm l l r g g r r r r r r r + - = -- (15.5) The homogeneous equation has solution of form ( ) 1 , , l l lm mn mn g r r A r B r- - = + (15.6) where the coefficients will depend on . r Because of the boundary conditions at r = a and r = b , the solution is of form 2 ( ) 1 1 , , , , ....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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Class_15new - 1 PHYS809 Class 15 Notes Green function in...

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