- #1

gop

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## Homework Statement

Consider a potential problem in the half space z>=0 with Dirichlet boundary conditions on the plane z=0.

If the potential on the plane z=0 is specified to be V inside a circle of radius a centered at the origin, and Phi=0 outside that circle, show that along the axis of the circle (rho=0) the potential is given by

[tex]\Phi=V(1-\frac{z}{\sqrt{a^2+z^2}})[/tex]

## Homework Equations

## The Attempt at a Solution

I used the Green's function (by method of images)

[tex]G(x,x')=\frac{1}{|x-x'|}-\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z+z')^2}}[/tex]

Then I converted the formula to cylindrical coordinates and computed the normal derivative -dG/dz. Then I set rho=0 and finally I got

[tex]\Phi = \frac{-z}{2\pi} \int_0^{2\pi} \int_0^a \frac{V}{(p'^2+z^2)^{3/2}} = \frac{-Va}{z \sqrt{a^2+z^2}}[/tex]

I'm somewhat certain that the computations are correct I checked them two times in a CAS so I don't really know what I did wrong.