- #1

lorentzian

- 2

- 0

## Homework Statement

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

*x*and

*y*are Cartesian coordinates. Such a distribution is called a two-dimensional one since it does not depend on the third coordinate

*z*.[/B]

## Homework Equations

Poisson's equation: $$\nabla^2 \Phi = -4 \pi \rho$$[/B]

## The Attempt at a Solution

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.