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:

“Parenting forces us to get to know ourselves better than we ever might have imagined we could—and in many new ways. . . . We’ll discover talents we never dreamed we had and fervently wish for others at moments we feel we desperately need them. As time goes on, we’ll probably discover that we have more to give and can give more than we ever imagined. But we’ll also find that there are limits to our giving, and that may be hard for us to accept.”
—Fred Rogers (20th century)

“In my state, on the basis of the separate but equal doctrine, we have made enormous strides over the years in the education of both races. Personally, I think it would have been sounder judgment to allow that progress to continue through the process of natural evolution. However, there is no point crying about spilt milk.”
—Lyndon Baines Johnson (1908–1973)

“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)