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.

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