Editing Moment (physics) (section)

== Multipole moments ==
Assuming a density function that is finite and localized to a particular region, outside that region a 1/''r'' [[Scalar potential|potential]] may be expressed as a series of [[spherical harmonics]]:
:<math>
\Phi(\mathbf{r}) = 
\int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} \, d^3r' =
\sum_{\ell=0}^\infty
\sum_{m=-\ell}^\ell
\left( \frac{4\pi}{2\ell+1} \right)
q_{\ell m}\,
\frac{Y_{\ell m}(\theta, \varphi)}{r^{\ell+1}}
</math>

The coefficients <math>q_{\ell m}</math> are known as ''multipole moments'', and take the form:
:<math>
q_{\ell m} = \int 
(r')^\ell\,
\rho(\mathbf{r'})\,
Y^*_{\ell m}(\theta',\varphi')\,
d^3r'
</math>

where <math>\mathbf{r}'</math> expressed in spherical coordinates <math>\left(r', \varphi', \theta'\right)</math> is a variable of integration.  A more complete treatment may be found in pages describing [[multipole expansion]] or [[spherical multipole moments]].  (The convention in the above equations was taken from Jackson<ref>J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, Wiley, New York, (1975). p. 137</ref> – the conventions used in the referenced pages may be slightly different.)

When <math>\rho</math> represents an electric charge density, the <math>q_{lm}</math> are, in a sense, projections of the moments of electric charge: <math>q_{00}</math> is the monopole moment; the <math>q_{1m}</math> are projections of the dipole moment, the <math>q_{2m}</math> are projections of the quadrupole moment, etc.