Question

The electric field in a region is given by $\overrightarrow{E}=\frac{E_{0}x}{l}\overrightarrow{i}$. If the charge contained inside a cubical volume bounded by the surfaces $x=0,x=a,y=0,y=a,z=0,z=a$ is $x\times {10}^{-12}C$. Find $x$. (integer value) Take $E_0=5\times {10}^{3}N/C,l=2cm,a=1cm$

Answer 1

The electric flux will contribute only due to surfaces $x=0$ and $x=a$ because the electric filed is perpendicular to other surfaces.
Thus, $\varphi (x=0)=\frac{E_0\left(0\right)}{l}.a^2=0$ and $\varphi (x=a)=\frac{E_0\left(a\right)}{l}.a^2=\frac{E_0}{l}.a^3$
using Gauss's law, $\frac{q}{{ϵ}_0}=\varphi (x=a)=\frac{E_0}{l}.a^3$
$\Rightarrow q=\frac{E_0}{l}.a^3{ϵ}_0=\frac{5\times {10}^3\times }{2\times {10}^{-2}}\times {10}^{-6}\times 8.854\times {10}^{-12}=2.21\times {10}^{-12}\sim 2\times {10}^{-12}$