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:

“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

“As the tragic writer rids us of what is petty and ignoble in our nature, so also the humorist rids us of what is cautious, calculating, and priggish—about half of our social conscience, indeed. Both of them permit us, in blessed moments of revelation, to soar above the common level of our lives.”
—Robertson Davies (b. 1913)