Finding Green's Function From Known Potential

  • #1

Homework Statement


My question comes from problem 2 of this homework set, but is dependent on problem 1 of this same homework set. In problem 1 I used the method of images to find the potential everywhere in two dimensions due to an infinite uniform line charge located some distance from a dielectric slab. Problem 2 asks to use the results of this problem to solve for the potential between two infinite conducting planes. One plane contains a microstrip with some charge distribution on it, and the other one is grounded. The goal is to replace the charge distribution with an infinite line charge and find the green's function in between the planes due to the line charge. Knowing the potential on the grounded plane, we can then use the green's function and the known potential to solve for the charge distribution on the microstrip.


The Attempt at a Solution


My problem is that I don't really know how to use my results from problem 1 to solve for the potential using the green's function method. If I already have the solution to the potential in the region between the conducting plates, using the same method of images from problem 1, am I supposed to be able to solve for the green's function from this known potential? If so, I'm not really sure how to do this. I know that the potential I have must satisfy the equation

[itex]\nabla^{2}[/itex][itex]\phi[/itex]=[itex]\rho[/itex][itex]\delta(x^{'}-d)[/itex][itex]\delta(y^{'}-0)[/itex]

The green's function will then satisfy the homogeneous equation
[itex]\nabla^{2}[/itex][itex]\phi[/itex]=0

If I had the green's function, I would solve for the potential by multiplying the source function with the green's function and integrating over the source. Not really sure how to do this process backwards to solve for the green's function.

I was also thinking that I could solve directly for the Green's function
[itex]\nabla^{2}[/itex][itex]\phi[/itex]=0
using the boundary condition that V=0 at the conducting plane. But if I did that, I wouldn't really be using the results of problem 1 which is what the professor suggest I do.

SO I guess my question is how do I find the Green's function to problem 2 if I have the potential already.

THanks
 

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