Poisson's Equation in Spherical Coordinates

Consider the general solution to Poisson's equation,

$$\nabla^{\,2}\phi = -\frac{\rho}{\epsilon_0},$$ (328)

in spherical coordinates. According to Section 2.3, the general three-dimensional Green's function for Poisson's equation is

$$G({\bf r},{\bf r}') = - \frac{1}{4\pi\,\vert{\bf r}-{\bf r}'\vert}.$$ (329)

When expressed in terms of spherical coordinates, this becomes

$$G({\bf r},{\bf r}') = -\frac{1}{4\pi\,(r^{\,2}-2\,r\,r'\,\cos\gamma+r'^{\,2})^{1/2}},$$ (330)

where

$$\cos\gamma = \cos\theta\,\cos\theta'+\sin\theta\,\sin\theta'\,\cos(\varphi-\varphi').$$ (331)

is the angle subtended between ${\bf r}$ and ${\bf r}'$ . According to Equation (298), we can write

$$G({\bf r},{\bf r}') = -\frac{1}{4\pi\,r}\sum_{l=0,\infty}\left(\frac{r'}{r}\right)^lP_l(\cos\gamma)$$ (332)

for $r'<r$ , and

$$G({\bf r},{\bf r}') = -\frac{1}{4\pi\,r'}\sum_{l=0,\infty}\left(\frac{r}{r'}\right)^{l}P_l(\cos\gamma)$$ (333)

for $r'>r$ . Thus, it follows from the spherical harmonic addition theorem, (322), that

$$G({\bf r},{\bf r}') = -\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2\... ...{>}^{\,l+1}}\right)Y^{\,\ast}_{l,m}(\theta',\varphi')\,Y_{l,m}(\theta,\varphi),$$ (334)

where $r_<$ represents the lesser of $r$ and $r'$ , whereas $r_>$ represents the greater of $r$ and $r'$ .

According to Section 2.3, the general solution to Poisson's equation, (329), is

$$\phi({\bf r})=-\frac{1}{\epsilon_0}\int G({\bf r},{\bf r}')\,\rho({\bf r}')\,dV'.$$ (335)

Thus, Equation (335) yields

$$\phi({\bf r}) = \frac{1}{\epsilon_0}\sum_{l=0,\infty}\sum_{m=-l,+... ...,\ast}(r)+ \frac{q_{l,m}^{\,\ast}(r)}{r^{\,l+1}}\right]Y_{l,m}(\theta,\varphi),$$ (336)

where

$$p_{l,m}(r) = \int_r^\infty \oint\frac{1}{r'^{\,l+1}}\,\rho(r',\theta,\varphi)\,Y_{l,m}(\theta,\varphi)\,r'^{\,2}\,d{\mit\Omega}\,dr',$$ (337)

$$q_{l,m}(r) = \int_0^r \oint r'^{\,l}\,\rho(r',\theta,\varphi)\,Y_{l,m}(\theta,\varphi)\,r'^{\,2}\,d{\mit\Omega}\,dr'.$$ (338)