Axial Multipole Moments - Interior Axial Multipole Moments

Interior Axial Multipole Moments

Conversely, if the radius r is smaller than the smallest for which is significant (denoted ), the electric potential may be written

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta )$

where the interior axial multipole moments are defined

$I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}}$

Special cases include the interior axial monopole moment ( the total charge)

$M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}$ ,

the interior axial dipole moment, etc. Each successive term in the expansion varies with a greater power of, e.g., the interior monopole potential varies as, the dipole potential varies as, etc. At short distances, the potential is well-approximated by the leading nonzero interior multipole term.

Read more about this topic: Axial Multipole Moments

Famous quotes containing the words interior and/or moments:

“The monk in hiding himself from the world becomes not less than himself, not less of a person, but more of a person, more truly and perfectly himself: for his personality and individuality are perfected in their true order, the spiritual, interior order, of union with God, the principle of all perfection.”
—Thomas Merton (1915–1968)

“Who among us has not, in moments of ambition, dreamt of the miracle of a form of poetic prose, musical but without rhythm and rhyme, both supple and staccato enough to adapt itself to the lyrical movements of our souls, the undulating movements of our reveries, and the convulsive movements of our consciences? This obsessive ideal springs above all from frequent contact with enormous cities, from the junction of their innumerable connections.”
—Charles Baudelaire (1821–1867)