Finding potential using Greens function

[sayebms]

Homework Statement

A potential $\phi(\rho, \phi ,z)$ satisfies $\nabla^2 \phi=0$ in the volume $V={z\geqslant a}$ with boundary condition $\partial \phi / \partial n =F_{s}(\rho, \phi)$ on the surface $S={z=0}$.
a) write the Neumann Green's function $G_N (x,x')$ within V in cylindrical coordinates $\rho, \phi, z (and \rho', \phi', z')$. Evaluate G and its normal derivative $\partial G/ \partial n'$ for x' on S.
b) For zero charge density and with boudnary condition $F_S=E_0$ (constant) within the circle $\rho < a$ and zero outside, find the potential on the z-axis. compare the limit z-->0 of your solution with the given boundary condition.
c) Find the first two nonvanishing terms in the potential for $r=\sqrt(\rho ^2 +z^2)>> a$. Compare with (b) where the two overlap. what is the charge inferred from the large-r potential?

Homework Equations

$\phi=<\phi>_S +\frac{1}{4 \pi \epsilon}\int \rho G_N d^3x' +\frac{1}{4 \pi} \int\frac{\partial \phi}{\partial \phi} G_N d^2a'$[/B]

The Attempt at a Solution

a) the green function is $G_N = \frac{1}{|\vec x -\vec x'|}$
$|\vec x -\vec x'|^2 = \vec x.\vec x + \vec x'.\vec x' -2\vec x.\vec x'$
thus in cyllindrical coordinates:
$G_N = \frac{1}{\sqrt (\rho^2 + z^2 +\rho'^2 + z'^2 -2\sqrt((\rho^2 + z^2)(\rho'^2 + z'^2)cos\gamma)}$
so its normal derivative for x' on S will vanish and its value for x' on S is (z'=0):
$G_N = \frac{1}{ (\rho^2 + z^2 +\rho'^2 -2\rho' \sqrt((\rho^2 + z^2)cos\gamma)^{1/2}}$

b) on the z axis $\rho =0$ (this is not the charge density $\rho$ from $\nabla ^2 \phi=0$ we know that charge density is zero that's why the volume term vanishes )and $\theta=0 --> cos\gamma =cos\theta'$ and on the sruface z'=0 $cos\gamma=0$
$\phi=<\phi>_S +\frac{1}{4 \pi} \int E_0 \frac{1}{ (z^2 +\rho'^2 )^{1/2}} d^2a'$
as $d^2a' = \rho' d\rho' d\phi'$ we have the following

$\phi=<\phi>_S +\frac{E_0}{2} \int_{0}^{a} \frac{\rho' d\rho'}{ (z^2 +\rho'^2 )^{1/2}}$
I would like to ask if till this point I have done anything wrong or if I have missed anything? and also how should I proceed with the surface term: $<\phi>_S$, since I don't know the value of $\phi$. I appreciate any help. thank you for your time.

 

[Fred Wright]

You must expand your Green's function in terms of a sum over Bessel functions and modified Bessel functions. For a good explanation of this procedure see http://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap3/chap3.pdf

 

[sayebms]

You must expand your Green's function in terms of a sum over Bessel functions and modified Bessel functions. For a good explanation of this procedure see http://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap3/chap3.pdf

I have actually found out a way to do it, its not through bessels functions though. but thank you for the help