Question

An electric field in a region is $800\sqrt{x}\hat{i}$. The charge contained in a cubical volume bounded by the surfaces $x=0,x=a,y=0,y=a,z=0$ and $z=a$ is :

• A. $\frac{800\sqrt{a}}{1/2}{\epsilon }_0$
• B. $800a^{5/2}{\epsilon }_0$
• C. $800a^2{\epsilon }_0$
• D. $800\sqrt{a}{\epsilon }_0$

Answer 1

The correct option is B $800a^{5/2}{\epsilon }_0$

As $E$ is parallel to x axis so surfaces $x=0$ and $x=a$ will contribute flux. For other surfaces it will be zero because $E.dS=0$.
also flux for surface $x=0$ will be zero because $E.dS=800\sqrt{0}.a^2=0$
Thus the total flux through cube is $\varphi =E.dS=800\sqrt{a}.a^2=800.a^{5/2}$ for $x=a$
By Gauss's law, $\varphi =\frac{Q_{en}}{{ϵ}_0}$
$\therefore Q_{en}=\varphi {ϵ}_0=800.a^{5/2}{ϵ}_0$