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.

Read more about this topic:  Spherical Multipole Moments

Famous quotes containing the words special, case and/or symmetry:

    The line that I am urging as today’s conventional wisdom is not a denial of consciousness. It is often called, with more reason, a repudiation of mind. It is indeed a repudiation of mind as a second substance, over and above body. It can be described less harshly as an identification of mind with some of the faculties, states, and activities of the body. Mental states and events are a special subclass of the states and events of the human or animal body.
    Willard Van Orman Quine (b. 1908)

    The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.
    Samuel Taylor Coleridge (1772–1834)

    What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.
    George Gordon Noel Byron (1788–1824)