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:

“Anyone with a real taste for solitude who indulges that taste encounters the dangers of any other drug-taker. The habit grows. You become an addict.... Absorbed in the visions of solitude, human beings are only interruptions. What voice can equal the voices of solitude? What sights equal the movement of a single day’s tide of light across the floor boards of one room? What drama be as continuously absorbing as the interior one?”
—Jessamyn West (1902–1984)

“It is time to provide a smashing answer for those cynical men who say that a democracy cannot be honest, cannot be efficient.... We have in the darkest moments of our national trials retained our faith in our own ability to master our own destiny.”
—Franklin D. Roosevelt (1882–1945)