Spherical Multipole Moments - Special Case of Axial Symmetry

Special Case of Axial Symmetry

The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle ). By carrying out the integrations that define and, it can be shown the multipole moments are all zero except when . Using the mathematical identity

$P_{l}(\cos \theta) \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2l+1}} Y_{l0}(\theta, \phi)$

the exterior multipole expansion becomes

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \left( \frac{Q_{l}}{r^{l+1}} \right) P_{l}(\cos \theta)$

where the axially symmetric multipole moments are defined

$Q_{l} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime} \rho(\mathbf{r}^{\prime}) \left( r^{\prime} \right)^{l} P_{l}(\cos \theta^{\prime})$

In the limit that the charge is confined to the -axis, we recover the exterior axial multipole moments.

Similarly the interior multipole expansion becomes

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} I_{l} r^{l} P_{l}(\cos \theta)$

where the axially symmetric interior multipole moments are defined

$I_{l} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime} \frac{\rho(\mathbf{r}^{\prime})}{\left( r^{\prime} \right)^{l+1}} P_{l}(\cos \theta^{\prime})$

In the limit that the charge is confined to the -axis, we recover the interior axial multipole moments.

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