Calculating Charge Density and Total Charge Using Gauss's Law

[spaghetti3451]

Homework Statement

Suppose the electric field in some region is found to be $\vec{E} = kr^{3} \hat{r}$, in spherical coordinates (k is some constant).

(a) Find the charge density ρ.

(b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)

Homework Equations

The Attempt at a Solution

(a) Use the formula for the divergence of a vector in spherical basis to get $\nabla . \vec{E} = 5kr^{2}$ so that $\rho = 5k\epsilon_{0}r^{2}$.

(b) Q is the volume integral of $\rho$ over the volume of the sphere. So, we integrate over d$\phi$, integrate sinθ over dθ, integrate the $\rho$ times $r^{2}$ over dr and multiply the three results. The process gives us 4$\pi \epsilon_{0} R^{5}$.

Q can also be found using the integral form of Gauss's law, where the surface integral of the electric field is taken with the infinitesimal area, which is R² sinθ dθ d∅ r, where r is the unit vector in the radial direction. We take the constants out of the integral and integrate 1 over phi and sinθ over theta to obtain the same Q as above.

Please could you check if the process and the answers are correct?

 

[dikmikkel]

a) Looks good to me. I get the same.
b) Is it a hollow spherical shell. Then you can use that the electric field is constant over the surface(fixed r) and only multiply by the area of the sphere(shell) because:
$\oint \vec{E}\cdot\vec{da} = \oint E\,\text{d}a = E\oint 1\text{d}a = E 4\pi R^2 = Q/\epsilon_0 \Leftrightarrow Q = 4\pi \epsilon_0 kR^5$
The dot product $\vec{E}\cdot\vec{da}$ is just da times E cause they point in the same direction.