# Multipole Expansion when r<r'

1. Feb 26, 2009

### Vuldoraq

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

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

3. The attempt at a solution

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

$$\frac{1}{r-r'}=\sum_{l=o}^{\infty}A_{l}r^{l}$$

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,

$$\frac{1}{r-r'}=\frac{1}{r'}\frac{1}{\frac{r}{r'}-1}=\frac{-1}{r'}*(1+\frac{r}{r'}+\frac{r^{2}}{r'^{2}}+...)$$

Which means,

$$\frac{1}{r-r'}=-\sum_{l=0}^{\infty}\frac{r^{l}}{r'^{l+1}}$$

Which means,

$$-A_{l}=\frac{1}{r'^{l+1}}$$

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,

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

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