Spherical Charge Distribution

In summary, the conversation discusses finding the electric field when a charge density varies as \rho(r) = Ke^{-br}, where 0 < r < a in a spherical region of space. The attempt at a solution involves using the integral Q = \int \int \int \rho dV to find the total charge, and then using E = kQ/r^2 to find the electric field. An alternative approach using dq = \rho r^2 sin(\theta) dr d\theta d\phi is also mentioned, but it is determined that the first approach is correct.
  • #1

Homework Statement

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

The Attempt at a Solution

I found Q total = [tex]\int \int \int \rho dV[/tex]
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 = [tex]\rho r^2 sin(\theta) dr d\theta d\phi[/tex]

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.
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  • #2
What do you mean? Q is the integral of the charge density over the volume. Also, I think you mean [itex]dr = rho r^2 sin(\theta) dr d\theta d\phi[/itex]. What did you do for your integral?
  • #3
Why [tex]sin(\theta)[/tex]? rho depends only on r so [tex]dQ = 4\pi Kr^{2}e^{-br}dr[/tex]
  • #4
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, [tex]dQ = 4\pi Kr^{2}e^{-br}dr[/tex]

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

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