Answer
The total charge is $\frac{2\pi}{3}$ coulombs.
Work Step by Step
Given:
- A disk $x^2+y^2\leq 1$ (or in polar coordinates $0\leq r\leq 1$)
- Electric charge density given by $\sigma(x,y)=\sqrt{x^2+y^2}$ (or in polar coordinates $\sigma(r,\theta) = \sqrt{r^2}=r$
The total charge is given by $T=\int_0^1\int_0^{2\pi} \sigma(r,\theta) \cdot rd\theta dr$:
Evaluate $T$:
$T=\int_0^1\int_0^{2\pi} r\cdot rd\theta dr$
$=\int_0^1\int_0^{2\pi}r^2d\theta dr$
$=\int_0^1r^2\theta]_0^{2\pi}dr$
$=\int_0^12\pi r^2dr$
$=\frac{2\pi r^3}{3}]_0^1$
$=\frac{2\pi}{3}$
