Analytical solutions for electric field of finite rectangular sheet

[keenPenguin]

Hi,

I have been trying to find analytical solutions for a finite rectangular sheet, say, in the xy plane, with dimensions a and b. Assume it is uniformly charged.

An excellent (and short) description of the problem is here. The three integrals for Ex(x,y,z), Ey(x,y,z) and Ez(x,y,z) given on the second page are easy to derive (integrating Coulomb's law for a point charge over a plane) but hard to solve. I obtained solutions by Mathematica which qualitatively look right but quantitatively are doubtful: They have imaginary parts in the components of the E field. If there is any interest I would be happy to share the Mathematica notebook or post some vector field plots.

Maybe somebody has seen these solutions fully written out in some book or paper? I would like to use the equations for an electrodynamics simulator I am coding in my spare time, hoping to avoid doing the full nasty integration by hand.

 

[Simon Bridge]

A finite, infinitely thin, rectangular sheet of uniform charge.
You would need to look at how mathematica has performed the caculation - you may need to drop the complex part as non-physical or do some other transformation to get a real-valued field.

It may also be easier to work out the potentials instead.
But basically the edges make this very nasty.