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Calculate a charge distribution given an electric potential.
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[QUOTE="lorentzian, post: 5927146, member: 639534"] [h2]Homework Statement[/h2] Find the distribution of charge giving rise to an electric field whose potential is $$\Phi (x,y) = 2~tan^{-1}(\frac{1+x}{y}) + 2~tan^{-1}(\frac{1-x}{y})$$where [I]x[/I] and [I]y[/I] are Cartesian coordinates. Such a distribution is called a two-dimensional one since it does not depend on the third coordinate [I]z[/I].[/B][h2]Homework Equations[/h2] Poisson's equation: $$\nabla^2 \Phi = -4 \pi \rho$$[/B][h2]The Attempt at a Solution[/h2] The first thing I noted while attempting to solve this problem is that there is a singularity when x = ±1 and y = 0. Either way, I wanted to compute the laplacian of the electric potential to see what it would result in and it reduced to zero. I thought I had made an algebraic mistake and put it in Mathematica, again reducing to zero. Then I actually took my original thought and tried to work out a solution to avoid the singularities using the Dirac delta function, the only problem is that I don't know how I could operate the Dirac delta function. Is is possible to work out the following equation? $$ \Phi (x,y) \delta(x) \delta(y) = \Phi (0,0) \delta(x) \delta(y) \\ \nabla^2 \Phi (x,y) \delta(x) \delta(y) = \nabla^2 \Phi(0,0) \delta (x) \delta(y) $$ Although I don't think my steps will lead to something useful. [/QUOTE]
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Calculate a charge distribution given an electric potential.
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