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:

“Treating ‘water’ as a name of a single scattered object is not intended to enable us to dispense with general terms and plurality of reference. Scatter is in fact an inconsequential detail.”
—Willard Van Orman Quine (b. 1908)

“There are some moments in life, some feelings.... One can only point to them—and pass by.”
—Ivan Sergeevich Turgenev (1818–1883)