2. [25 pts total] A nonconducting sphere of radius \( R \) has positive charge with a distribution which is spherically symmetric but not uniform and is given by volume density
\[
\rho=\frac{Q}{\pi R^{4}} r
\]
where \( r \) is the radial distance from that center, and notice \( Q /\left(\pi R^{4}\right) \) is a constant.
(a) Show that the total charge of the sphere is \( Q \cdot[6 \mathrm{pts}] \)
Answer
\( Q = \frac{Q}{\pi R^4} \cdot \frac{2 \pi R^4}{4} \cdot 2 \pi \)
Step 1 ：\( Q = \int_{0}^{R} \int_{0}^\pi \int_{0}^{2\pi} \frac{Q}{\pi R^{4}} r \cdot r^{2} \sin{\phi} dr d\theta d\phi \)
Step 2 ：\( Q = \frac{Q}{\pi R^4} \int_{0}^{R} \int_{0}^\pi \int_{0}^{2\pi} r^3 \sin{\phi} dr d\theta d\phi \)
Step 3 ：\( Q = \frac{Q}{\pi R^4} \cdot \frac{2 \pi R^4}{4} \cdot 2 \pi \)
