Finding potential using Greens function

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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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Answers and Replies

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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