Calculating Charge and Volume Density for Non-uniform Ball

[zoso335]

Homework Statement

"Charge is distributed spherically-symmetrically but nonuniformly within a ball of radius a. There is no charge outside the ball and no sheet-charge on its surface. The (radial) electric field is uniform within the ball, having the constant value E_a.
a.) Calculate the total charge.
b.) Calculate the volume charge density at any distance r<a from the center.
c.) Check that your answers to part a and part b agree.

Homework Equations

Gauss' Law. Differential form of Gauss' Law.

The Attempt at a Solution

Part a posed no problem to me, you just use Gauss' law in integral form over the sphere of radius a. This results in Q equating to E_a times the surface area of the sphere times epsilon nought.
Part b on the other hand is where I have a problem. If I use the differential form of Gauss' Law, \nabla\bullet E = \rho / \epsilon₀ then taking the gradient of a constant electric field would result in zero, and thus the volume charge density would be zero, but that physically does not make sense, so that is where I'm lost.

 

[Dick]

The electric field is only constant in magnitude, it's not constant in direction. The divergence isn't zero. I'd suggest you stick with the integral form and figure out how Q(r) varies with r.

 

[zoso335]

That's interesting though because my professor sent out and email giving hints, and for the volume charge density he suggested using Gauss' Law in differential form. So...there must be something I'm not doing right with that.

 

[Dick]

Ok, if you want to do it in differential form, you can. You'll probably want a formula for the divergence of a vector field in spherical coordinates. This always a helpful site, http://mathworld.wolfram.com/SphericalCoordinates.html