# Finding potential using Greens function

## 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 thats 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.

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