Next: Newmann Problem in Spherical
Up: Potential Theory
Previous: Axisymmetric Charge Distributions
We saw in Section 2.10 that the solution to the Dirichlet problem, in which the charge density is specified
within some volume
, and the potential given on the bounding surface
,
takes the form
![$\displaystyle \phi({\bf r}) = -\frac{1}{\epsilon_0}\int_V G_D({\bf r},{\bf r}')...
...\,dV' + \int_S \phi({\bf r}')\,\frac{\partial G_D({\bf r},{\bf r}')}{dn'}\,dS',$](img828.png) |
(371) |
where the Dirichlet Green's function is
written
![$\displaystyle G_D({\bf r},{\bf r}') = - \frac{1}{4\pi\,\vert{\bf r}-{\bf r}'\vert} + F({\bf r},{\bf r'}).$](img829.png) |
(372) |
Here,
is solution of Laplace's equation (i.e.,
) which is chosen so as to ensure that
when
(or
) lies on
. Thus, it follows from Sections 3.4 and 3.5
that
where the
and the
are chosen in such a manner that the Green's function is zero when
lies on
.
As a specific example, suppose that the volume
lies between the two spherical surfaces
and
.
The constraint that
as
implies that the
are all zero.
On the other hand, the constraint
when
yields
![$\displaystyle \beta_{l,m}=\frac{1}{2\,l+1}\,\frac{a^{\,2\,l+1}}{r'^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi').$](img836.png) |
(374) |
Hence, the unique Green's function for the problem becomes
![$\displaystyle G_D({\bf r},{\bf r}') =-\sum_{l=0,\infty}\sum_{m=-l,+l}\frac{1}{2...
...1}\,r_>^{\,l+1}}\right)Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$](img837.png) |
(375) |
Furthermore, it is readily demonstrated that
![$\displaystyle \left.\frac{\partial G_D}{\partial r'}\right\vert _{r'=a} = -\sum...
...a^{\,l-1}}{r^{\,l+1}}\,Y_{l,m}^\ast(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$](img838.png) |
(376) |
It is convenient to write
It follows from Equation (311) that
Thus, Equations (372), (376) and
(377) yield
Next: Newmann Problem in Spherical
Up: Potential Theory
Previous: Axisymmetric Charge Distributions
Richard Fitzpatrick
2014-06-27