Uniformly-charged disk electric field at ANY point (i.e., off-axis)?

[TTTTTTrent]

Homework Statement

I need to find the equation for the electric field intensity $$\vec{E}(\vec{R})$$ of a uniformly-charged disk lying in the xy-plane with radius $$\rho_{0}$$ and surface charge density $$\rho_{s}'$$ at any point.

Homework Equations

I know that a charged disk doesn't have the symmetry needed for Gauss's Law to be useful, and finding the electric field intensity on the z axis is easily found using Coulomb's Law, so I'm using that again for this more general problem.

The Attempt at a Solution

I set up the position vectors for the observation point and the source point, with

$$\vec{R} = \rho\hat{\rho}+z\hat{z}$$ for the observation point
$$\vec{R}' = \rho'\hat{\rho}'$$ for the source point (prime designating source quantities throughout this problem).

Using cylindrical coordinates I found dA to be $$\rho'd\phi'd\rho'$$. When plugging the position vectors into the integral, though, I had to convert $$\hat{\rho}$$ and $$\hat{\rho}'$$ to x and y components because they didn't correspond to the same unit vectors and also varied with $$\phi '$$. Taking the constant surface charge density out of the equation and simplifying as best I could with the magnitude in the denominator, I was left with the integral

$$\vec{E}(\vec{R}) = \frac{\rho_{s}'}{4\pi\epsilon_{0}}\int^{\rho_{0}}_{\rho '=0}\rho'd\rho'\int^{2\pi}_{\phi '=0}\frac{(\rho cos \phi - \rho'cos \phi')\hat{x} + (\rho sin \phi - \rho'sin \phi')\hat{y} + z\hat{z}}{[\rho^{2}-2\rho\rho'cos(\phi - \phi') + (\rho')^{2} + z^{2}]^{3/2}}d\phi'$$

As you can see, this integral seems like it would take some monumental calculus skills to evaluate. I've tried breaking the integral up into fractions with numerators being each term of the original numerator, and a common denominator for all. Even with the $$z\hat{z}$$ term I can't see a way to evaluate that integral. There seem to be too many nested functions within one another. I've looked around online and there are thousands of pages solving an on-axis charged disk problem, but I've only found 2 or 3 sites that even discuss solving an off-axis one. They've referred to elliptic integrals and legendre polynomials...both of which I know nothing about and which haven't been spoken of in my course ever. Am I missing something? I've never been the strongest in integration techniques, but this is just ridiculous! Help please?!

 

[weejee]

I suggest evaluating the potential first and then obtain the field by taking a derivative.
Even so, it is very unlikely that you will be able to get the solution in a closed form. It seems you should expand the integrand in terms of Legendre polynomials. Consult with Jackson's EM book or hopefully, Wiki.

 

[TTTTTTrent]

Thanks for the quick reply and good suggestions, weejee. My professor likes to watch us squirm, so I'm not surprised that the solution to this problem will probably require knowledge outside of our course. I'll definitely try the potential to field method first, though. Good thinking.