Question

Electric charge is distributed over the disk
x^2+y^2≤2 so that the charge density at (x,y) is σ(x,y)=17+x^2+y^2 coulombs per square meter.
Find the total charge on the disk.

Answer 1

The distribution of electric charge can be described in polar coordinates as:

$x=rcos\theta, y=rsin\theta, r^2= x^2+y^2$ hence

$0\leq r\leq \sqrt{2}$ and $0 \leq \theta \leq 2\pi$

The total charge on the disk would the the double integral of the density function:

$\int_0^{2\pi}\int_0^{\sqrt{2}}(r^2+17)rdrd\theta=\int_0^{2\pi}\int_0^{\sqrt{2}}(r^3+17r)drd\theta=$

$=\int_0^{2\pi}(\frac{r^4}{4}+\frac{17r^2}{2})\mid_0^{\sqrt{2}}d\theta=\int_0^{2\pi}(\frac{4}{4}+\frac{17\cdot2}{2}-\frac{0}{4}-\frac{17\cdot0}{2})d\theta=$

$=\int_0^{2\pi}18d\theta=18\theta\mid_0^{2\pi}=18\cdot2\pi-18\cdot0=36\pi$

Answer: $36\pi$