Finding Electric Field for Spherical Charge Distribution?

[jesuslovesu]

Homework Statement

There is a charge density rho that exists in a spherical region of space defined by 0 < r < a.
$$\rho (r) = Ke^{-br}$$
How do you find the electric field if a charge density varies as such?

The Attempt at a Solution

I found Q total = $$\int \int \int \rho dV$$
Now I need to find E.

My real question is can I just put Q (as a function of r) into E = kQ/r^2? Or do I need to reevaluate the integral using dq = $$\rho r^2 sin(\theta) dr d\theta d\phi$$

I get two different answers, (and I would have thought they should be the same) so which method is correct? I would have thought either would work.

 

[Mindscrape]

What do you mean? Q is the integral of the charge density over the volume. Also, I think you mean $dr = rho r^2 sin(\theta) dr d\theta d\phi$. What did you do for your integral?

 

[michalll]

Why $$sin(\theta)$$? rho depends only on r so $$dQ = 4\pi Kr^{2}e^{-br}dr$$

 

[Mindscrape]

Oh whoops, I shouldn't have had rho in there, and I missed it when you had it. You were right about the dq I was questioning. dq= rho *spherical jacobian (i.e. spherical integration differentials), which is what you had.

Yes, $$dQ = 4\pi Kr^{2}e^{-br}dr$$

This is the way you want to go. I don't really understand what other way you would have gone about it.