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Multipole Expansion
Consider a bounded charge distribution that lies inside the sphere
. It follows that
in the region
. According to the previous three equations, the electrostatic potential in the region
takes the form
![$\displaystyle \phi({\bf r}) = \frac{1}{\epsilon_0}\sum_{l=0,\infty}\sum_{m=-l,+l} \frac{q_{l,m}^{\,\ast}}{2\,l+1}\,\frac{Y_{l,m}(\theta,\varphi)}{r^{\,l+1}},$](img759.png) |
(339) |
where the
![$\displaystyle q_{l,m}^{\,\ast}= \int r^{\,l}\,\rho(r,\theta,\varphi)\,Y_{l,m}(\theta,\varphi)\,dV$](img760.png) |
(340) |
are known as the multipole moments of the charge distribution
. Here, the integral is over all space.
Incidentally, the type of expansion specified in Equation (340) is called a multipole expansion.
The most important
are those corresponding to
,
, and
, which are known as monopole, dipole,
and quadrupole moments, respectively. For each
, the multipole moments
, for
to
, form an
th-rank tensor with
components.
However, Equation (310) implies that
![$\displaystyle q_{l,m}^{\,\ast} = (-1)^m\,q_{l,-m}.$](img768.png) |
(341) |
Hence, only
of these components are independent.
For
, there is only one monopole moment. Namely,
![$\displaystyle q_{0,0}^{\,\ast}=\int \rho({\bf r}')\,Y_{0,0}^{\,\ast}(\theta,\varphi)\,dV = \frac{1}{\sqrt{4\pi}}\int\rho({\bf r})\,dV = \frac{Q}{\sqrt{4\pi}},$](img770.png) |
(342) |
where
is the net charge contained in the distribution, and use has been made of Equation (312). It follows from Equation (340) that, at sufficiently large
, the charge
distribution acts like a point charge
situated at the origin. That is,
![$\displaystyle \phi({\bf r})\simeq \phi_0({\bf r})=\frac{q_{0,0}^{\,\ast}}{\epsilon_0}\,\frac{Y_{0,0}(\theta,\varphi)}{r} = \frac{Q}{4\pi\,\epsilon_0\,r}.$](img771.png) |
(343) |
By analogy with Equation (195), the dipole moment of the charge distribution is written
![$\displaystyle {\bf p} = \int \rho({\bf r})\,{\bf r}\,dV.$](img772.png) |
(344) |
The three Cartesian components of this vector are
On the other hand, the spherical components of the dipole moment take the form
where use has been made of Equations (313)-(315).
It can be seen that the three spherical dipole moments are independent linear combinations of the three Cartesian moments. The potential
associated with the dipole moment is
![$\displaystyle \phi_1({\bf r}) = \frac{1}{3\,\epsilon_0}\left(\frac{q_{1,-1}^{\,...
...}+q_{1,0}^{\,\ast}\,r\,Y_{1,0}+q_{1,+1}^{\,\ast}\,r\,Y_{1,+1}}{r^{\,3}}\right).$](img785.png) |
(351) |
However, from Equations (313)-(315),
Hence,
![$\displaystyle \phi_1({\bf r})= \frac{1}{4\pi\,\epsilon_0}\,\frac{p_x\,x+p_y\,y+p_z\,z}{r^{\,3}}= \frac{1}{4\pi\,\epsilon_0}\,\frac{{\bf p}\cdot{\bf r}}{r^{\,3}},$](img792.png) |
(355) |
in accordance with Equation (200). Note, finally, that if the net charge,
, contained in the distributions is non-zero then it is always possible to
choose the origin of the coordinate system in such a manner that
.
The Cartesian components of the quadrupole tensor are defined
![$\displaystyle Q_{ij} = \int \rho({\bf r})\,(3\,x_i\,x_j-r^{\,2}\,\delta_{ij})\,dV,$](img794.png) |
(356) |
for
,
,
,
. Here,
,
, and
. Incidentally, because the quadrupole tensor is symmetric (i.e.,
)
and traceless (i.e.,
), it only possesses five independent Cartesian components.
The five spherical components of the quadrupole
tensor take the form
![$\displaystyle q_{2,-2}^{\,\ast}$](img803.png) |
![$\displaystyle =\left(\frac{5}{96\pi}\right)^{1/2}\,(Q_{11}+2\,{\rm i}\,Q_{12}-Q_{22}),$](img804.png) |
(357) |
![$\displaystyle q_{2,-1}^{\,\ast}$](img805.png) |
![$\displaystyle =\left(\frac{5}{24\pi}\right)^{1/2}\,(Q_{13}+{\rm i}\,Q_{23}),$](img806.png) |
(358) |
![$\displaystyle q_{2,0}^{\,\ast}$](img807.png) |
![$\displaystyle =\left(\frac{5}{16\pi}\right)^{1/2}\,Q_{33},$](img808.png) |
(359) |
![$\displaystyle q_{2,+1}^{\,\ast}$](img809.png) |
![$\displaystyle =-\left(\frac{5}{24\pi}\right)^{1/2}\,(Q_{13}-{\rm i}\,Q_{23}),$](img810.png) |
(360) |
![$\displaystyle q_{2,+2}^{\,\ast}$](img811.png) |
![$\displaystyle =\left(\frac{5}{96\pi}\right)^{1/2}\,(Q_{11}-2\,{\rm i}\,Q_{12}-Q_{22}).$](img812.png) |
(361) |
Moreover, the potential associated with the quadrupole tensor is
![$\displaystyle \phi_2({\bf r})= \frac{1}{5\,\epsilon_0}\sum_{m=-2,+2} \frac{q_{2...
...3}}=\frac{1}{8\pi\,\epsilon_0}\sum_{i,j=1,3}\frac{Q_{ij}\,\,x_i\,x_j}{r^{\,5}}.$](img813.png) |
(362) |
It follows, from the previous analysis, that the first three terms in the multipole expansion, (340), can be written
![$\displaystyle \phi({\bf r})\simeq \phi_0({\bf r})+\phi_1({\bf r})+\phi_2({\bf r...
..._0\,r^{\,3}}+ \sum_{i,j=1,3}\frac{Q_{ij}\,x_i\,x_j}{8\pi\,\epsilon_0\,r^{\,5}}.$](img814.png) |
(363) |
Moreover, at sufficiently large
, these are always the dominant terms in the expansion.
Next: Axisymmetric Charge Distributions
Up: Potential Theory
Previous: Poisson's Equation in Spherical
Richard Fitzpatrick
2014-06-27