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:

“When General Motors has to go to the bathroom ten times a day, the whole country’s ready to let go. You heard of that market crash in ‘29? I predicted that.... I was nursing a director of General Motors. Kidney ailment, they said; nerves, I said. Then I asked myself, “What’s General Motors got to be nervous about?” “Overproduction,” I says. “Collapse.””
—John Michael Hayes (b. 1919)

“There are moments when you feel free, moments when you have energy, moments when you have hope, but you can’t rely on any of these things to see you through. Circumstances do that.”
—Anita Brookner (b. 1938)