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If the position of some charge Q is known, the boundary condition is u=0 on some parabolic surface, and we know the image charge has its electric volume of Q', then how can I determine the position of the image charge?

Same questions goes for the hyperbolic curve boundary. How can I determine the position of Q'?

I think may be there is a way to transform the coordinates to make everything into an easily-handled form, but I am not sure about it. Another solution I thought about is to put this question into a general question based on basic Poisson's equation and Laplace equation, but I do not have a specific idea on how to conduct this.

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What kind of parabolic boundary? A parabolic cylinder? An elliptical paraboloid? A hyperbolic paraboloid? –  Dan Apr 6 '13 at 19:14
Also, these articles on parabolic coordinates might help: 1 2 –  Dan Apr 6 '13 at 19:22
The basic concept is that you find an image charge placement that gives you parabolic equipotential surfaces. (so, it may help to transform all equations to parabolic coordinates) –  Manishearth Apr 6 '13 at 19:26
Typical two dimension parabolic boundary like y=ax^2 is enough for me. Another boundary I deal with is x^2/a^2-y^2/b^2=1 –  Emily Apr 7 '13 at 14:49
I think, for all geometries, there is no a unique image charge for satisfying the boundary condition u=0. In fact, it may exist a distribution of image charges (not a single charge) which can satisfy the appropriate boundary condition u=0. –  Mojtaba Golshani Jul 19 '14 at 14:21

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