Statistical moments and multipole moments

[Meta Mass]

Hello,

in statistics, one can derive the moments of a distribution by using a generating function
$$<x^n> = \int dx x^n f(x) = \left( \frac {d}{dt} \int dx \exp(tx) f(x) \right)_{t=0} = \left( \frac d {dt} M(t) \right)_{t=0}$$

Is there a similar method to derive the multipole moments in electrodynamics, e.g. is there a generating function? I know that the multipole moments are derived from the expansion of
$$\frac {1}{\left|x\right|}$$
but I don't seem to get the connection to a generating function.

 

[Almsco]

Hi there,

I'm not sure if there is a similar method to deriving the multipole moments in electrodynamics using a generating function as with statistics. I think the multipole moments are derived from the expansion of \frac {1}{\left|x\right|} because this is a measure of the electric field at a given point, which is related to the multipole moments. Have you tried researching this further? Maybe someone else on the forum may have more information on this topic that can help provide more insight.

 

[charng]

Hello,

Thank you for your question. In electrodynamics, the multipole moments are derived from the expansion of the electric potential, which is given by the Coulomb potential:

V(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d^3\mathbf{r'}

The multipole moments are coefficients in the expansion of this potential in terms of spherical harmonics:

V(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \sum_{l=0}^\infty \sum_{m=-l}^l \frac{4\pi}{2l+1}q_{lm} \frac{Y_{lm}(\theta,\phi)}{r^{l+1}}

where q_{lm} are the multipole moments and Y_{lm} are the spherical harmonics.

To answer your question, there is no direct connection to a generating function in the derivation of multipole moments. However, one can use the method of moments to determine the multipole moments from the charge distribution. This involves taking the moments of the charge distribution and equating them to the multipole moments in the expansion of the potential. So in a sense, the moments of the charge distribution act as a generating function for the multipole moments.

I hope this helps clarify the connection between statistical moments and multipole moments in electrodynamics. Let me know if you have any further questions.