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Poisson's Equation in Spherical Coordinates
Consider the general solution to Poisson's equation,
![$\displaystyle \nabla^{\,2}\phi = -\frac{\rho}{\epsilon_0},$](img395.png) |
(328) |
in spherical coordinates.
According to Section 2.3, the general three-dimensional Green's function for Poisson's equation is
![$\displaystyle G({\bf r},{\bf r}') = - \frac{1}{4\pi\,\vert{\bf r}-{\bf r}'\vert}.$](img367.png) |
(329) |
When expressed in terms of spherical coordinates, this becomes
![$\displaystyle G({\bf r},{\bf r}') = -\frac{1}{4\pi\,(r^{\,2}-2\,r\,r'\,\cos\gamma+r'^{\,2})^{1/2}},$](img744.png) |
(330) |
where
![$\displaystyle \cos\gamma = \cos\theta\,\cos\theta'+\sin\theta\,\sin\theta'\,\cos(\varphi-\varphi').$](img621.png) |
(331) |
is the angle subtended between
and
. According to Equation (298), we can write
![$\displaystyle G({\bf r},{\bf r}') = -\frac{1}{4\pi\,r}\sum_{l=0,\infty}\left(\frac{r'}{r}\right)^lP_l(\cos\gamma)$](img745.png) |
(332) |
for
, and
![$\displaystyle G({\bf r},{\bf r}') = -\frac{1}{4\pi\,r'}\sum_{l=0,\infty}\left(\frac{r}{r'}\right)^{l}P_l(\cos\gamma)$](img747.png) |
(333) |
for
. Thus, it follows from the spherical harmonic addition theorem, (322), that
![$\displaystyle G({\bf r},{\bf r}') = -\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2\...
...{>}^{\,l+1}}\right)Y^{\,\ast}_{l,m}(\theta',\varphi')\,Y_{l,m}(\theta,\varphi),$](img749.png) |
(334) |
where
represents the lesser of
and
, whereas
represents the greater of
and
.
According to Section 2.3, the general solution to Poisson's equation, (329), is
![$\displaystyle \phi({\bf r})=-\frac{1}{\epsilon_0}\int G({\bf r},{\bf r}')\,\rho({\bf r}')\,dV'.$](img752.png) |
(335) |
Thus, Equation (335) yields
![$\displaystyle \phi({\bf r}) = \frac{1}{\epsilon_0}\sum_{l=0,\infty}\sum_{m=-l,+...
...,\ast}(r)+ \frac{q_{l,m}^{\,\ast}(r)}{r^{\,l+1}}\right]Y_{l,m}(\theta,\varphi),$](img753.png) |
(336) |
where
Next: Multipole Expansion
Up: Potential Theory
Previous: Laplace's Equation in Spherical
Richard Fitzpatrick
2014-06-27