Solve Tricky Double Integral for Charge Density Distribution in Symmetric Disc

[TheMan112]

Homework Statement

I'm trying to model the potential field in and around a symmetrically charged disc where the charge density drops exponentially from the center.

Homework Equations

This can be done by solving the double integral:

$$\int ^{2 \pi} _{0} \int ^{\infty} _{0} \frac{r e^{-r/b} (a-r sin \theta)}{(r^2 + a^2 - 2 r a cos \theta)^{3/2}} dr d \theta$$

a is the observation radius, b is the scalar length of the charge density distribution, r and theta are polar coordinates. So essentially I'm looking for a result of the form V(a).

The Attempt at a Solution

I've been trying to attack this using substitution of variables without much result. It could be done numerically but I would greatly prefer a symbolic solution.

 

[gabbagabbahey]

I'm trying to model the potential field in and around a symmetrically charged disc where the charge density drops exponentially from the center.

Huh? Are you modeling the electrostatic potential, or the electric field? Does your disk extend out to infinity? Are you only looking the model the potential (or field) at point on the disk, or also at points not on the disk?

$$\int ^{2 \pi} _{0} \int ^{\infty} _{0} \frac{r e^{-r/b} (a-r sin \theta)}{(r^2 + a^2 - 2 r a cos \theta)^{3/2}} dr d \theta$$

Even if you are just looking for the electric field at points on the disk, this integral doesn't look quite right to me... how did you get $a-r\sin\theta$ in your numerator?