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 exterior must be joined to the interior to obtain anything from God, that is to say, we must kneel, pray with the lips, and so on, in order that proud man, who would not submit himself to God, may be now subject to the creature.”
—Blaise Pascal (1623–1662)

“There are some moments in life, some feelings.... One can only point to them—and pass by.”
—Ivan Sergeevich Turgenev (1818–1883)