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
$ρ = \frac{Q}{π R^{4}} r$
where $r$ is the radial distance from that center, and notice $Q / (π R^{4})$ is a constant.
(a) Show that the total charge of the sphere is $Q \cdot [6 pts]$

Answer

Expert–verified

Hide Steps

Answer

$Q = \frac{Q}{π R^{4}} \cdot \frac{2 π R^{4}}{4} \cdot 2 π$

Steps

Step 1 ： $Q = \int_{0}^{R} \int_{0}^{π} \int_{0}^{2 π} \frac{Q}{π R^{4}} r \cdot r^{2} \sin ϕ d r d θ d ϕ$

Step 2 ： $Q = \frac{Q}{π R^{4}} \int_{0}^{R} \int_{0}^{π} \int_{0}^{2 π} r^{3} \sin ϕ d r d θ d ϕ$

Step 3 ： $Q = \frac{Q}{π R^{4}} \cdot \frac{2 π R^{4}}{4} \cdot 2 π$