Question

The electric field in a region is radially outward with magnitude E = 2r. The charge contained in a sphere of radius a = 2m centred at the origin is $4x\pi {ϵ}_0$. Find the value of x.

Answer 1

Flux is given by

${\varphi }_E=\int \overrightarrow{E}.d\overrightarrow{S}=\int \overrightarrow{E}.dS\cos 0=E\left(4\pi r^2\right)=Ar\left(4\pi r^2\right)=2r\left(4\pi r^2\right)$

According to Gauss's theorem

${\varphi }_E=\frac{q}{{\epsilon }_0}$

$\therefore q={\varphi }_E{\epsilon }_0=Ar\left(4\pi r^2\right){\epsilon }_0=2r\left(4\pi {\epsilon }_0{r}^2\right)$

Now,

$4x\pi {\epsilon }_0=2r\left(4\pi {\epsilon }_0\right)r^2$

$x=2\times r^3$

$x=2\times 2^3$

$x=16$