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 work of art, just like any fragment of human life considered in its deepest meaning, seems to me devoid of value if it does not offer the hardness, the rigidity, the regularity, the luster on every interior and exterior facet, of the crystal.”
—André Breton (1896–1966)

“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 1880–1964)