Green's function representation of electric potential

hunt_mat

Hi,

I have the following problem, I have an electric field (which no charge) which satisfies the usual Laplace equation:
[tex]
\frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}+\frac{\partial^{2}V}{\partial z^{2}}=0
[/tex]
in the region [itex]\mathbb{R}^{2}\times [\eta ,\infty ][/itex]. So basically it is the upper half z-plane where the boundary is some fixed surface [itex]\eta[/itex], I also know that on this surface:
[tex]
\frac{\partial V}{\partial x}=\frac{\partial\eta}{\partial x}
[/tex]

I can do this in 2D by the use of the Hilbert transform. Any suggestions?

Muphrid

Let me make sure I understand: this is a region for [itex]z > \eta[/itex] for some fixed [itex]\eta[/itex], or is [itex]\eta[/itex] a function?

hunt_mat

[itex]\eta[/itex] is a function but with further thought the region could be set to [itex]z\geqslant 0[/itex] and I think that this will make the problem easier.

Muphrid

And if I read what you said correctly, you only know the value of the x-component of the electric field on this surface?

hunt_mat

I think that you can also say that [itex]V=\eta[/itex] due to other considerations too.

hunt_mat

So I think I have solved this problem, I took [itex]V[/itex] and [itex]\eta[/itex] to be of the same size but small and reduced the complexity of the problem somewhat and the domain is now: [itex]\mathbb{R}^{2}\times [ 0,\infty )[/itex], using Green's second formula, I can write the solution as an integral over the boundary:
[tex]
V(x,y,z)=\int_{\mathbb{R}^{2}}g\partial_{z}V-V\partial_{z}g\Big|_{z'=0}d\Sigma
[/tex]
Where [itex]g[/itex] is the Green's function for Laplaces's equation for the half space given by:
[tex]
g(x,y,x|x',y',z')=\frac{1}{4\pi\sqrt{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}}}-\frac{1}{4\pi\sqrt{(x-x')^{2}+(y-y')^{2}+(z+z')^{2}}}
[/tex]
Then using the boundary condition [itex]V=\eta[/itex], then the solution becomes:
[tex]
u=\int_{\mathbb{R}^{2}}u\frac{\partial g}{\partial z}d\Sigma
[/tex]