1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Multipole Expansion when r<r'
Loading...