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Dirichlet Problem in Spherical Coordinates

We saw in Section 2.10 that the solution to the Dirichlet problem, in which the charge density is specified within some volume $ V$ , and the potential given on the bounding surface $ S$ , takes the form

$\displaystyle \phi({\bf r}) = -\frac{1}{\epsilon_0}\int_V G_D({\bf r},{\bf r}')...
...\,dV' + \int_S \phi({\bf r}')\,\frac{\partial G_D({\bf r},{\bf r}')}{dn'}\,dS',$ (371)

where the Dirichlet Green's function is written

$\displaystyle G_D({\bf r},{\bf r}') = - \frac{1}{4\pi\,\vert{\bf r}-{\bf r}'\vert} + F({\bf r},{\bf r'}).$ (372)

Here, $ {\bf F}({\bf r},{\bf r}')$ is solution of Laplace's equation (i.e., $ \nabla^{\,2} F=0$ ) which is chosen so as to ensure that $ G_D({\bf r},{\bf r}')=0$ when $ {\bf r}$ (or $ {\bf r}'$ ) lies on $ S$ . Thus, it follows from Sections 3.4 and 3.5 that

$\displaystyle G_D({\bf r},{\bf r}') =$ $\displaystyle -\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2\,l+1}\,\frac{r_<^{\,l}}{r_>^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi)$    
  $\displaystyle + \sum_{l=0,\infty}\sum_{m=-l,+l}\left[\alpha_{l,m}(r',\theta',\v...
...rac{\beta_{l,m}(r',\theta',\varphi')}{r^{\,l+1}}\right]Y_{l,m}(\theta,\varphi),$ (373)

where the $ \alpha_{l,m}$ and the $ \beta_{l,m}$ are chosen in such a manner that the Green's function is zero when $ {\bf r}$ lies on $ S$ .

As a specific example, suppose that the volume $ V$ lies between the two spherical surfaces $ r=a$ and $ r=\infty$ . The constraint that $ G_D({\bf r},{\bf r}')\rightarrow 0$ as $ r\rightarrow \infty$ implies that the $ \alpha_{l,m}$ are all zero. On the other hand, the constraint $ G_D({\bf r},{\bf r}')=0$ when $ r=a$ yields

$\displaystyle \beta_{l,m}=\frac{1}{2\,l+1}\,\frac{a^{\,2\,l+1}}{r'^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi').$ (374)

Hence, the unique Green's function for the problem becomes

$\displaystyle G_D({\bf r},{\bf r}') =-\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2...
...1}\,r_>^{\,l+1}}\right)Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$ (375)

Furthermore, it is readily demonstrated that

$\displaystyle \left.\frac{\partial G_D}{\partial r'}\right\vert _{r'=a} = -\sum...
...a^{\,l-1}}{r^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$ (376)

It is convenient to write

$\displaystyle \phi(r,\theta,\varphi)$ $\displaystyle =\sum_{l=0,\infty}\sum_{m=-l,+l}\phi_{l,m}(r)\,Y_{l,m}(\theta,\varphi),$ (377)
$\displaystyle \rho(r,\theta,\varphi)$ $\displaystyle = \sum_{l=0,\infty}\sum_{m=-l,+l}\rho_{l,m}(r)\,Y_{l,m}(\theta,\varphi).$ (378)

It follows from Equation (311) that

$\displaystyle \phi_{l,m}(r)$ $\displaystyle =\oint \phi(r,\theta,\varphi)\,Y_{l,m}^\ast(\theta,\varphi)\,d{\mit\Omega},$ (379)
$\displaystyle \rho_{l,m}(r)$ $\displaystyle = \oint \rho(r,\theta,\varphi)\,Y_{l,m}^\ast(\theta,\varphi)\,d{\mit\Omega}.$ (380)

Thus, Equations (372), (376) and (377) yield

$\displaystyle \phi_{l,m}(r)$ $\displaystyle =\frac{1}{2\,l+1}\int_a^r \frac{\rho_{l,m}(r')}{\epsilon_0}\left(...
...\,l}}{r^{\,l+1}}-\frac{a^{\,2\,l+1}}{r'^{\,l+1}\,r^{\,l+1}}\right)r'^{\,2}\,dr'$    
  $\displaystyle ~~~+\frac{1}{2\,l+1}\int_r^\infty \frac{\rho_{l,m}(r')}{\epsilon_...
...}\,r^{\,l+1}}\right)r'^{\,2}\,dr' +\phi_{l,m}(a)\left(\frac{a}{r}\right)^{l+1}.$ (381)


next up previous
Next: Newmann Problem in Spherical Up: Potential Theory Previous: Axisymmetric Charge Distributions
Richard Fitzpatrick 2014-06-27