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:

“No man will ever bring out of that office the reputation which carries him into it. The honeymoon would be as short in that case as in any other, and its moments of ecstasy would be ransomed by years of torment and hatred.”
—Thomas Jefferson (1743–1826)

“The whole point about the true unconscious is that it is all the time moving forward, beyond the range of its own fixed laws or habits. It is no good trying to superimpose an ideal nature upon the unconscious.”
—D.H. (David Herbert)

“Martial, the things for to attain
The happy life be these, I find:
The riches left, not got with pain;
The fruitful ground, the quiet mind;
The equal friend; no grudge nor strife;
No charge of rule nor governance;”
—Martial (Marcus Valerius Martialis)