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Newmann Problem in Spherical Coordinates

According to Section 2.10, the solution to the Newmann problem, in which the charge density is specified within some volume $ V$ , and the normal derivative of the potential given on the bounding surface $ S$ , takes the form

$\displaystyle \phi({\bf r}) = -\frac{1}{\epsilon_0}\int_V G_N({\bf r},{\bf r}')...
...,dV' - \int_S G_N({\bf r}, {\bf r}')\,\frac{\partial \phi({\bf r}')}{dn'}\,dS',$ (382)

where the Newmann Green's function is written

$\displaystyle G_N({\bf r},{\bf r}') = - \frac{1}{4\pi\,\vert{\bf r}-{\bf r}'\vert} + F({\bf r},{\bf r'}).$ (383)

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

$\displaystyle \int_S G_N({\bf r},{\bf r}')\,dS = 0,$ (384)

and

$\displaystyle \frac{\partial G_N({\bf r},{\bf r}')}{\partial n} = 1\left/\int_S dS\right..$ (385)

The latter constraint holds when $ {\bf r}$ (or $ {\bf r}'$ ) lies on $ S$ . Note that we have chosen the arbitrary constant to which the potential $ \phi({\bf r})$ is undetermined such that $ \langle\phi\rangle_S = 0$ . It again follows from Sections 3.4 and 3.5 that

$\displaystyle G_N({\bf r},{\bf r}') =$ $\displaystyle -\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2\,l+1}\,\frac{r_<^{\,l}}{r_>^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi)$    
  $\displaystyle + \sum_{l=0,\infty}\sum_{m=-l,+l}\left[\alpha_{l,m}(r',\theta',\v...
...rac{\beta_{l,m}(r',\theta',\varphi')}{r^{\,l+1}}\right]Y_{l,m}(\theta,\varphi),$ (386)

where the $ \alpha_{l,m}$ and the $ \beta_{l,m}$ are chosen in such a manner that the constraints (385) and (386) are satisfied.

As a specific example, suppose that the volume $ V$ lies inside the spherical surface $ r=a$ . The physical constraint that the Green's function remain finite at $ r=0$ implies that the $ \beta_{l,m}$ are all zero. Applying the constraint (385) at $ r=a$ , we get

$\displaystyle \alpha_{0,0}(r',\theta',\varphi') = \frac{Y_{0,0}^\ast(\theta',\varphi')}{a}.$ (387)

Similarly, the constraint (386) leads to

$\displaystyle \alpha_{l,m}(r',\theta',\varphi') = - \left(\frac{l+1}{2\,l+1}\right)\frac{r'^{\,l}}{a^{\,l+2}}\,Y_{l,m}^\ast(\theta',\varphi')$ (388)

for $ l>0$ . Hence, the unique Green's function for the problem becomes

$\displaystyle G_D({\bf r},{\bf r}') =$ $\displaystyle -\left(\frac{1}{r_>}-\frac{1}{a}\right)Y_{0,0}^\ast(\theta',\varphi')\,Y_{0,0}(\theta,\varphi)$    
  $\displaystyle -\sum_{l=1,\infty}\sum_{m=-l,+l}\frac{1}{2\,l+1}\left(\frac{r_<^{...
...}{a^{\,2\,l+1}}\right) Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$ (389)

Finally, expanding $ \phi({\bf r})$ and $ \rho({\bf r})$ in the forms (378) and (379), respectively, Equations (383) and (390) yield

$\displaystyle \phi_{0,0}(r) = \int_0^r\frac{\rho_{0,0}(r')}{\epsilon_0}\left(\f...
...{\rho_{0,0}(r')}{\epsilon_0}\left(\frac{1}{r'}-\frac{1}{a}\right)r'^{\,2}\,dr',$ (390)

and

$\displaystyle \phi_{l,m}(r)$ $\displaystyle = \frac{1}{2\,l+1}\int_0^r \frac{\rho_{l,m}(r')}{\epsilon_0}\left...
...l+1}}+\frac{l+1}{l}\,\frac{r'^{\,l}\,r^{\,l}}{a^{\,2\,l+1}}\right)r'^{\,2}\,dr'$    
  $\displaystyle ~~~+ \frac{1}{2\,l+1}\int_r^a \frac{\rho_{l,m}(r')}{\epsilon_0}\l...
...\,\frac{d\phi_{l,m}}{dr}\right\vert _{r=a}\frac{1}{l}\left(\frac{r}{a}\right)^l$ (391)

for $ l>0$ .


next up previous
Next: Laplace's Equation in Cylindrical Up: Potential Theory Previous: Dirichlet Problem in Spherical
Richard Fitzpatrick 2014-06-27