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:

“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 1880–1964)

“If you give me a short shot I will attack you. I’m not a baseliner who rallies. I try to get the point over with.”
—Venus Williams (b. 1980)

“It is hereby earnestly proposed that the USA would be much better off if that big, sprawling, incoherent, shapeless, slobbering civic idiot in the family of American communities, the City of Los Angeles, could be declared incompetent and placed in charge of a guardian like any individual mental defective.”
—Westbrook Pegler (1894–1969)