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EMLecture7 - Lecture 7 Notes 95.657 Electromagnetic Theory...

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Lecture 7 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. Expansion of Green Functions in Spherical Coordinates - Consider the problem of a spherical boundary with radius a , the potential is known on the boundary, there is charge present, and we wish to find the the potential anywhere external to the sphere. - We have already solved this problem using Green's functions:  x = 1 4  0  x ' G D d 3 x ' 1 4 d G D d n ' da ' External to a spherical boundary we have already found the Green's function, it being the potential created by a unit charge and its image: G x , x ' = 1 x x ' 1 x ' a x a x ' x ' - When solving problems with both charges and boundary surfaces, the mathematics is simplified if the Green's function is expanded in spherical harmonics. - The addition theorem was used to find the spherical harmonics expansion of a unit charge: 1 r r 0 = 4 l = 0 m =− l l 1 2 l 1 r l r 0 l 1 Y l m *  ' , ' Y lm  ,  if r r 0 and 1 r r 0 = 4 l = 0 m =− l l 1 2 l 1 r 0 l r l 1 Y lm *  ' , ' Y lm  ,  if r r 0 - Both the point charge and its image can be expanded in this way to give the spherical Green's
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