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

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

where the Newmann Green's function is written

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

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

and

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

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

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

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

Similarly, the constraint (386) leads to

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

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

$$G_D({\bf r},{\bf r}') = -\left(\frac{1}{r_>}-\frac{1}{a}\right)Y_{0,0}^\ast(\theta',\varphi')\,Y_{0,0}(\theta,\varphi)$$

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

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

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

and

$$\phi_{l,m}(r) = \frac{1}{2\,l+1}\int_0^r \frac{\rho_{l,m}(r')}{\epsilon_0}\left... ...l+1}}+\frac{l+1}{l}\,\frac{r'^{\,l}\,r^{\,l}}{a^{\,2\,l+1}}\right)r'^{\,2}\,dr'$$

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

for $l>0$ .