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Multipole Expansion when r<r'

  1. Feb 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the multipole expansion for the case when r<r', where r is the distance from the origin to the observation point and r' is the distance from the origin to the source point.

    2. Relevant equations


    3. The attempt at a solution

    To solve this I considered the case where [tex]\cos{\theta}[/tex] is equal to one and obtained,


    The Bl terms must be zero since otherwise the sum would blow up as r goes to zero.

    Now I have Taylor expanded the 1/(r-r') to get,


    Which means,


    Which means,


    Now this is the right answer except for the minus sign. Please could someone help me out in finding where I have gone wrong? I am geussing it is in my Taylor expansion, which is as follows,


    My thanks for any help. Also Sorry about the Latex, it won't generate my first equation, even though I can detect no error in the code.
    Last edited by a moderator: Feb 27, 2009
  2. jcsd
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