Sphere with non-uniform charge density

[Bonewheel]

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.

 

[Bonewheel]

I just realized that I made a very stupid mistake and never found k... Marking as solved.