Multipole Expansion

At a large distance \(d\) from a charge source, we can approximate the charge to be a point source. However, we wish to generalize for any distance away from a source. Consider a charge source well-structured in an \(n\)-pole configuration. The potential located at \(\boldsymbol r\) from the origin and \(d\) distance away from the a geometric slice of the source \(\mathrm d \mathcal V\) located at \(\boldsymbol r'\) is given by,

\[ V(\boldsymbol{r}) = \frac{1}{4\pi \epsilon_0}\int{\frac{\rho(\boldsymbol{r'})}{d} \mathrm d \mathcal V} \]

The distance here however changes as the geometric slice \(\mathrm d \mathcal V\) changes. It turns out that the inverse distance \(1/d\) are the Legendre polynomials.

\[ \frac{1}{d} = \frac{1}{r} \sum_{n=0}^{\infty}\left(\frac{r'}{r}\right)^n P_n (\cos \alpha) \]

Where \(\alpha\) is the angle between \(\boldsymbol{r}\) and \(\boldsymbol{r'}\). Thus the exact potential is,

\[ V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\sum_{n=0}^{\infty}\frac{1}{r^{n+1}}\int{(r')^n}P_n(\cos\alpha)\rho(\boldsymbol r') \mathrm d \mathcal V \]