Green's function representation of electric potential

In summary, the electric field satisfies the usual Laplace equation in the region bounded by a fixed surface, and the solution can be written as an integral over the boundary.
  • #1
hunt_mat
Homework Helper
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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?
 
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  • #2
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?
 
  • #3
[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.
 
  • #4
And if I read what you said correctly, you only know the value of the x-component of the electric field on this surface?
 
  • #5
I think that you can also say that [itex]V=\eta[/itex] due to other considerations too.
 
  • #6
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]
 
Last edited:

FAQ: Green's function representation of electric potential

1. What is the Green's function representation of electric potential?

The Green's function representation of electric potential is a mathematical tool used to calculate the electric potential at a point in space due to a distribution of charges. It involves solving an integral equation, known as the Green's function, which takes into account the effects of all charges in the system.

2. How is the Green's function related to the electric potential?

The Green's function is a solution to the Poisson's equation, which relates the electric potential to the charge distribution in a system. By using the Green's function representation, the electric potential can be calculated at any point in space, taking into account the contributions from all charges.

3. What are the advantages of using the Green's function representation?

One advantage of using the Green's function representation is that it allows for the calculation of the electric potential in complex systems with multiple charges, which may not be easily solved using other methods. Additionally, the Green's function approach can provide a more intuitive understanding of how different charges contribute to the overall potential.

4. Are there any limitations to the Green's function representation?

One limitation of the Green's function representation is that it may not be applicable to all systems, as it relies on the assumption of linearity and isotropy. This means that the charges must be small and well-separated, and the medium in which they are located must have uniform properties.

5. How is the Green's function representation used in practical applications?

The Green's function representation is commonly used in the field of electrostatics, as it provides a powerful tool for calculating electric potentials in complex systems. It is also used in other areas of physics, such as quantum mechanics and electromagnetism, where similar integral equations can be solved using similar methods.

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