Sorry it may seem that my question is a homework question but it is not since I have the solution of the problem.(adsbygoogle = window.adsbygoogle || []).push({});

It is about obtaining Green function and using it to calculate the potential in space, provided the boundary conditions are satisfied.

the questions are like below (It is a problem from Jackson's book):

Consider a potential problem in the half-space defined by z≥0 with Dirichlet boundary conditions on the plane z=0(and at infinity)

(a) write down the appropriate Green's function G(x,x') .

(b) If the potential at plane z=0 is specified to be Φ=V inside a circle of radius a centered at the origin, and Φ=0 outside the circle, find an integral expression of the potential at the point P specified in terms of cylindrical coordinates(ρ,φ,z).

(c) show that along the axis of the circle (ρ=0) the potential is given by

##V=(1- \frac {z}{\sqrt{a^2+z^2}})##

(d) Show that at large distances (ρ^2 +z^2 >>a^2) the potential can be expanded in a power series (ρ^2 +z^2)^-1 and that the leading terms are

##Φ=\frac{Va^2z}{2(ρ^2+z^2)^{3/2}}[1-\frac {3a^2}{4(ρ^2 +z^2)} +\frac {5(3ρ^2a^2+a^4)}{8(ρ^2 +z^2)^2}+...]##

verify that the results of V and D are consistent with each other in their common range of validity.

I really tried to solve the first part(a) alone but I couldn't.

When there a boundary condition in the problem( such that potential or surface charge density are specified) the formal solution of potential represented by

##Φ(x)=\int_v ρ(x')G(x,x')d^3x' +(1/4π)\oint_s [G(x,x')\frac {\partial Φ}{\partial n'} -Φ(x')\frac {\partial G(x,x')}{\partial n'}]da'##

Where##G(x,x')=1/|x-x'| + F(x,x')##

In this problem the solution was like follow:

(a) The green's function

G(x,x')=1/|x-x'|+F(x,x')

Dirichlet problem Which specifies the boundary condition at the surface G(x,x')=0 and G(x,x')=0 for z<0. It is given that at z=0 and Z=∞ Φ= constant=V(Why so it is constant, the problem didn't mention this?)

If cylindrical coordinates is used, the green's function so derived should not have F(x,x')(Why so ???)

The green function is then##G(x,x')=\frac {1}{\sqrt {ρ^2 +ρ'^2-2ρρ'cosθ +(z-z')^2}}+\frac {1}{\sqrt {ρ^2 +ρ'^2-2ρρ'cosθ +(z-z')^2}}##

The second term is an contribution from the image.(did he mean that he used the method of images to obtain this result and if so where did he locate the charges?)

I'm really still not practiced enough in green functions and don't know where to use them properly in solving electrostatics problems.If somebody besides helping me understanding the problem above can give me insights about them or send me a link that explain them smoothly.

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# A Green's function in electrostatics

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