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:

“Though the railroad and the telegraph have been established on the shores of Maine, the Indian still looks out from her interior mountains over all these to the sea.”
—Henry David Thoreau (1817–1862)

“Parenting is not logical. If it were, we would never have to read a book, never need a family therapist, and never feel the urge to call a close friend late at night for support after a particularly trying bedtime scene. . . . We have moments of logic, but life is run by a much larger force. Life is filled with disagreement, opposition, illusion, irrational thinking, miracle, meaning, surprise, and wonder.”
—Jeanne Elium (20th century)