Problem 2

2) a).
First, we need to find the total charge on the sphere.
$Q_{tot} = \int \rho dV$
$Q_{tot} = 4\pi \frac{\rho_{s}}{R} \int_{0}^{R}r^{3} dr$
$Q_{tot} = \pi \rho_{s} R^{3}$

b)
We can solve for $\rho_{s}$ in part a) and find that
$rho_{s} = \frac{Q_{tot}}{\pi R^{3}}$
To find the magnitude of the electric field, we use Gauss' Law.
$E A = \frac{Q_{enclosed}}{\epsilon_{0}}$
$E (4 \pi r^{2}) = \frac{Q_{enclosed}}{\epsilon_{0}}$
From part a), the amount of charge enclosed inside a radius r is given by
$Q_{tot} = 4\pi \frac{\rho_{s}}{R} \int_{0}^{r} r^{3} dr$
$Q_{enclosed} = \pi \rho_{s} \frac{r^{4}}{R}$
So, plugging this in for Q_enclosed, we have
$E (4 \pi r^{2}) = \frac{1}{\epsilon_{0}} \pi \rho_{s} \frac{r^{4}}{R}$
$E = \frac{1}{4 \pi \epsilon_{0}} \pi \rho_{s} \frac{r^{2}}{R}$
Substituting in what we found for $\rho_{s}$, we find that
$E = k \frac{Q r^{2}}{R^{4}}$