Spherical Multipole Moments - General Spherical Multipole Moments

General Spherical Multipole Moments

It is straightforward to generalize these formulae by replacing the point charge with an infinitesimal charge element and integrating. The functional form of the expansion is the same

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( \frac{Q_{lm}}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)$

where the general multipole moments are defined

$Q_{lm} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime} \rho(\mathbf{r}^{\prime}) \left( r^{\prime} \right)^{l} \sqrt{\frac{4\pi}{2l+1}} Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

Read more about this topic: Spherical Multipole Moments

Famous quotes containing the words general and/or moments:

“Never alone
Did the King sigh, but with a general groan.”
—William Shakespeare (1564–1616)

“There are thoughts which are prayers. There are moments when, whatever the posture of the body, the soul is on its knees.”
—Victor Hugo (1802–1885)