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Homework Help: 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

    [tex]\frac{1}{\sqrt{r^{2}+r'^{2}-2rr'\cos(\theta)}}=\sum_{l=0}^{\infty}(A_{l}r^l+B_{l}r^{-l-1})[/tex]

    3. The attempt at a solution

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

    [tex]\frac{1}{r-r'}=\sum_{l=o}^{\infty}A_{l}r^{l}[/tex]

    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,

    [tex]\frac{1}{r-r'}=\frac{1}{r'}\frac{1}{\frac{r}{r'}-1}=\frac{-1}{r'}*(1+\frac{r}{r'}+\frac{r^{2}}{r'^{2}}+...)[/tex]

    Which means,

    [tex]\frac{1}{r-r'}=-\sum_{l=0}^{\infty}\frac{r^{l}}{r'^{l+1}}[/tex]

    Which means,

    [tex]-A_{l}=\frac{1}{r'^{l+1}}[/tex]

    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,

    [tex]\frac{1}{(-1)+x}=-[1+x+x^{2}+x^{3}+...][/tex]

    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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