Multipole expansion - small problem

[Eats Dirt]

Homework Statement

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.

Homework 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]

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.

 

[vela]

##r_< = \min\{r,r'\}## and ##r_> = \max\{r,r'\}##

 

[Eats Dirt]

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

 

[vela]

What ##\sin\theta## term?

 

[paranormal]

I understand your confusion with the r< and r> terms in the multipole expansion. In this case, the r< term represents the smaller of the two distances between the point of interest and the charge distribution, and the r> term represents the larger of the two distances. In this problem, the charge distribution is localized, meaning that it is confined to a specific region in space. Therefore, the r< term will always be the distance from the point of interest to the center of the charge distribution, and the r> term will be the distance from the point of interest to the boundary of the charge distribution.

To solve this problem, you can use the multipole expansion formula provided in the Homework Equations section. First, you will need to expand the term \frac{1}{x-x'} using the binomial theorem. Then, you can plug in the given charge distribution and use the fact that the Legendre polynomials are orthogonal to determine the nonvanishing moments. Finally, you can write the potential at large distances as a finite expansion in Legendre polynomials by using the Legendre expansion of the spherical harmonics.

I hope this helps clarify the multipole expansion for you. Keep up the good work in your studies of electromagnetism!