# Multipole expansion - small problem

1. Oct 14, 2014

### Eats Dirt

1. The problem statement, all variables and given/known data
Jackson 4.7

Given a localized charge distribution:

$$\rho(r)=\frac{1}{64\pi}r^{2} e^{-r} sin^{2}\theta$$

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
$$\frac{1}{x-x'}=4\pi\sum^{\inf}_{l=0}\sum^{l}_{m=-l}\frac{1}{2l+1}\frac{r^{l}_{<}}{r^{l+1}_{>}}Y^{*}_{l,m}(\theta',\phi')Y_{l,m}(\theta,\phi)$$

3. The attempt at a solution
My main problem is with the $$\frac{r^{l}_{<}}{r^{l+1}_{>}}$$ 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. Oct 15, 2014

### vela

Staff Emeritus
$r_< = \min\{r,r'\}$ and $r_> = \max\{r,r'\}$

3. Oct 15, 2014

### Eats Dirt

Hey, I had another problem I input the multipole expansion into the integral

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

Now I have to integrate but have a problem on the $$sin\theta'$$ term, I know I can express the $$\sin^2\theta'=1-\cos^2\theta'$$ which can then be used as a spherical harmonic but I do not know what to do with the sin(theta) term

4. Oct 16, 2014

### vela

Staff Emeritus
What $\sin\theta$ term?