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

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

where the Dirichlet Green's function is written

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

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

$$+ \sum_{l=0,\infty}\sum_{m=-l,+l}\left[\alpha_{l,m}(r',\theta',\v... ...rac{\beta_{l,m}(r',\theta',\varphi')}{r^{\,l+1}}\right]Y_{l,m}(\theta,\varphi),$$ (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

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

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

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

Furthermore, it is readily demonstrated that

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

It is convenient to write

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

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

It follows from Equation (311) that

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

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

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

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

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