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Multipole expansion - small problem

  1. Oct 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Jackson 4.7

    Given a localized charge distribution:

    [tex]
    \rho(r)=\frac{1}{64\pi}r^{2} e^{-r} sin^{2}\theta
    [/tex]

    make the multipole expansion of the potential due to this charge distribution and determine all nonvanishing moments. Write down the potential at large distances as a finite expansion in Legendre polynomials.

    2. Relevant equations
    [tex]
    \frac{1}{x-x'}=4\pi\sum^{\inf}_{l=0}\sum^{l}_{m=-l}\frac{1}{2l+1}\frac{r^{l}_{<}}{r^{l+1}_{>}}Y^{*}_{l,m}(\theta',\phi')Y_{l,m}(\theta,\phi)
    [/tex]

    3. The attempt at a solution
    My main problem is with the [tex]\frac{r^{l}_{<}}{r^{l+1}_{>}}[/tex] Term as I don't know what I should set the r values to in this case, my original idea was to use the r< term as some constant say R then proceed with the multipole expansion but I think the solution does not have this term in it.
     
  2. jcsd
  3. Oct 15, 2014 #2

    vela

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    ##r_< = \min\{r,r'\}## and ##r_> = \max\{r,r'\}##
     
  4. Oct 15, 2014 #3
    Hey, I had another problem I input the multipole expansion into the integral

    [tex]
    \frac{1}{\epsilon_o}\int^{r}_{0}\int^{2\pi}_{0}\int^{\pi}_{0}\frac{r'^2 e^{-r'}}{64\pi}4\pi\sin^2{\theta'}\sum^{\inf}_{l=0}\sum^{l}_{m=-l}\frac{1}{2l+1}\frac{r'^{l}}{r^{l+1}}Y^{*}_{l,m}(\theta',\phi')Y_{l,m}(\theta,\phi)r'^2\sin\theta'd\theta'd\phi'dr'
    [/tex]

    Now I have to integrate but have a problem on the [tex]sin\theta'[/tex] term, I know I can express the [tex] \sin^2\theta'=1-\cos^2\theta'[/tex] which can then be used as a spherical harmonic but I do not know what to do with the sin(theta) term
     
  5. Oct 16, 2014 #4

    vela

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    What ##\sin\theta## term?
     
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