Spherical Multipole Moments - Spherical Multipole Moments of A Point Charge

Spherical Multipole Moments of A Point Charge

The electric potential due to a point charge located at is given by

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + r^{\prime 2} - 2 r^{\prime} r \cos \gamma}}.$

where is the distance between the charge position and the observation point and is the angle between the vectors and . If the radius of the observation point is greater than the radius of the charge, we may factor out 1/r and expand the square root in powers of using Legendre polynomials

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{l=0}^{\infty} \left( \frac{r^{\prime}}{r} \right)^{l} P_{l}(\cos \gamma )$

This is exactly analogous to the axial multipole expansion.

We may express in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2)

$\cos \gamma = \cos \theta \cos \theta^{\prime} + \sin \theta \sin \theta^{\prime} \cos(\phi - \phi^{\prime})$

Substituting this equation for into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical harmonic addition theorem

$P_{l}(\cos \gamma) = \frac{4\pi}{2l + 1} \sum_{m=-l}^{l} Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

where the functions are the spherical harmonics. Substitution of this formula into the potential yields

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{l=0}^{\infty} \left( \frac{r^{\prime}}{r} \right)^{l} \left( \frac{4\pi}{2l+1} \right) \sum_{m=-l}^{l} Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

which can be written as

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( \frac{Q_{lm}}{r^{l+1}} \right) \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\theta, \phi)$

where the multipole moments are defined

$Q_{lm} \ \stackrel{\mathrm{def}}{=}\ q \left( r^{\prime} \right)^{l} \sqrt{\frac{4\pi}{2l+1}} Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$ .

As with axial multipole moments, we may also consider the case when the radius of the observation point is less than the radius of the charge. In that case, we may write

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r^{\prime}} \sum_{l=0}^{\infty} \left( \frac{r}{r^{\prime}} \right)^{l} \left( \frac{4\pi}{2l+1} \right) \sum_{m=-l}^{l} Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

which can be written as

$\Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} I_{lm} r^{l} \sqrt{\frac{4\pi }{2l+1}} Y_{lm}(\theta, \phi)$

where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics

$I_{lm} \ \stackrel{\mathrm{def}}{=}\ \frac{q}{\left( r^{\prime} \right)^{l+1}} \sqrt{\frac{4\pi }{2l+1}} Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

The two cases can be subsumed in a single expression if and are defined to be the lesser and greater, respectively, of the two radii and ; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \sum_{l=0}^{\infty} \frac{r_<^{l}}{r_>^{l+1}} \left( \frac{4\pi}{2l+1} \right) \sum_{m=-l}^{l} Y_{lm}(\theta, \phi) Y_{lm}^{*}(\theta^{\prime}, \phi^{\prime})$

Read more about this topic: Spherical Multipole Moments

Famous quotes containing the words moments, point and/or charge:

“The books one reads in childhood, and perhaps most of all the bad and good bad books, create in one’s mind a sort of false map of the world, a series of fabulous countries into which one can retreat at odd moments throughout the rest of life, and which in some cases can survive a visit to the real countries which they are supposed to represent.”
—George Orwell (1903–1950)

“Competition has been shown to be useful up to a certain point and no further, but cooperation, which is the thing we must strive for today, begins where competition leaves off.”
—Franklin D. Roosevelt (1882–1945)

“Today I love myself as I love my god: who could charge me with a sin today? I know only sins against my god; but who knows my god?”
—Friedrich Nietzsche (1844–1900)