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Old Feb 24th 2010, 07:18 PM   #1
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Total charge of sphere with varying charge density?

"A nonconducting solid sphere of radius R has a volume charge density that is proportional to the distance from the center. That is, ρ = Ar for r R, where A is a constant. (Use pi for π, epsilon_0 for ε0, A for A, r for r, and R for R, as necessary.)
(a) Find the total charge on the sphere."

When I read this I thought of there being an infinite number of infinitely small, hollow spheres ("dr") each with its own charge density. I tried taking the integral of A*r from 0 to R and I end up with A*R^2/2. I take this number to be p, and then multiplied it by V (=4/3*pi*R^3) and I get [4/3*pi*R^3*A*R/2] which then simplifies to [2/3*pi*R^4*A] which the website we enter our answers into says is incorrect.

I know how simple this is if the charge density was uniform, but I don't understand how I am supposed to take into account the variations in the charge density... any help would be much appreciated!
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Old Feb 24th 2010, 07:59 PM   #2
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Originally Posted by dspaulding09 View Post
"A nonconducting solid sphere of radius R has a volume charge density that is proportional to the distance from the center. That is, ρ = Ar for r R, where A is a constant. (Use pi for π, epsilon_0 for ε0, A for A, r for r, and R for R, as necessary.)
(a) Find the total charge on the sphere."

When I read this I thought of there being an infinite number of infinitely small, hollow spheres ("dr") each with its own charge density. I tried taking the integral of A*r from 0 to R and I end up with A*R^2/2. I take this number to be p, and then multiplied it by V (=4/3*pi*R^3) and I get [4/3*pi*R^3*A*R/2] which then simplifies to [2/3*pi*R^4*A] which the website we enter our answers into says is incorrect.

I know how simple this is if the charge density was uniform, but I don't understand how I am supposed to take into account the variations in the charge density... any help would be much appreciated!
Welcome to PHF!
I'm not really sure about the integral you've made.
Have you taken calculus 3 yet? In other words, do you know how to deal with triple integrals?
You can't multiply this "p" by the volume. If p was the charge density of the sphere it would be ok as long as the charge density is constant. Since it is not constant, you'll have to set up a triple integral. More precisely, set up the triple integral in spherical coordinates and integrate the volume charge density. Don't forget the dV element while integrating.
If you have any difficulty on doing this, I'm willing to assist you with Latex.
Good luck.
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