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:
<br /> \frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}+\frac{\partial^{2}V}{\partial z^{2}}=0<br />
in the region \mathbb{R}^{2}\times [\eta ,\infty ]. So basically it is the upper half z-plane where the boundary is some fixed surface \eta, I also know that on this surface:
<br /> \frac{\partial V}{\partial x}=\frac{\partial\eta}{\partial x}<br />

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 z &gt; \eta for some fixed \eta, or is \eta a function?

 

[hunt_mat]

\eta is a function but with further thought the region could be set to z\geqslant 0 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 V=\eta due to other considerations too.

 

[hunt_mat]

So I think I have solved this problem, I took V and \eta to be of the same size but small and reduced the complexity of the problem somewhat and the domain is now: \mathbb{R}^{2}\times [ 0,\infty ), using Green's second formula, I can write the solution as an integral over the boundary:
<br /> V(x,y,z)=\int_{\mathbb{R}^{2}}g\partial_{z}V-V\partial_{z}g\Big|_{z&#039;=0}d\Sigma<br />
Where g is the Green's function for Laplaces's equation for the half space given by:
<br /> g(x,y,x|x&#039;,y&#039;,z&#039;)=\frac{1}{4\pi\sqrt{(x-x&#039;)^{2}+(y-y&#039;)^{2}+(z-z&#039;)^{2}}}-\frac{1}{4\pi\sqrt{(x-x&#039;)^{2}+(y-y&#039;)^{2}+(z+z&#039;)^{2}}}<br />
Then using the boundary condition V=\eta, then the solution becomes:
<br /> u=\int_{\mathbb{R}^{2}}u\frac{\partial g}{\partial z}d\Sigma<br />