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Bonewheel
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Homework Statement
A sphere of radius R carries charge Q. The distribution of the charge inside the sphere, however, is not homogeneous, but decreasing with the distance r from the center, so that ρ(r) = k/r.
1. Find k for given R and Q.
2. Using Gauss’s Law (differential or integral form), find the electric field E inside the sphere, i.e., for r < R.
Homework Equations
\int_V \rho \, dV = Q
\oint \vec E \cdot d \vec A = \frac {Q_{enclosed}} {\epsilon_0}
\vec {\nabla} \cdot \vec E = \frac {\rho} {{\epsilon}_0}
The Attempt at a Solution
1. \int_V \rho \, dV = \int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^R \frac k r r^2 \sin \theta \, dr \, d \theta \, d \phi = 2 k \pi R^2 = Q
The units check out here.
2.
Here's where I ran into a problem. I tried using both the differential and integral forms of Gauss's Law, and in both cases the r canceled out, leaving me with an expression for the electric field I know is wrong. Oddly, the units work out here as well.
\oint \vec E \cdot d \vec A = 4 \pi r^2 E = \frac {Q_{enclosed}} {\epsilon_0} = \frac {2 k \pi r^2} {{\epsilon}_0}
\vec {\nabla} \cdot \vec E = \frac 1 {r^2} \frac {\partial} {\partial r}(r^2 E_r)= \frac {\rho} {{\epsilon}_0} = \frac k {r {{\epsilon}_0}}
E = \frac k {2{{\epsilon}_0}}
Thank you so much for any help! Please let me know if you need any further information or edits for clarification.