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:

“There are moments when you feel free, moments when you have energy, moments when you have hope, but you can’t rely on any of these things to see you through. Circumstances do that.”
—Anita Brookner (b. 1938)

“At this age [9–12], in contrast to adolescence, girls still want to know their parents and hear what they think. You are the influential ones if you want to be. Girls, now, want to hear your point of view and find out how you got to be what you are and what you are doing. They like their fathers and mothers to be interested in what they’re doing and planning. They like to know what you think of their thoughts.”
—Stella Chess (20th century)

“Well done, good and trustworthy slave; you have been trustworthy in a few things, I will put you in charge of many things; enter into the joy of your master.”
—Bible: New Testament, Matthew 25:21.