Question 81 of 311

A cubical volume is bounded by the surfaces $x=0,x=a,y=0,y=a,z=0,z=a$. The electric field in the region is given by $\vec{E}={E}_{0}x\hat{i}$. Where ${E}_{0}=4\times {10}^{4}{NC}^{-1}{m}^{-1}$. If $a=2cm$, the charge contained in the cubical volume is $Q\times {10}^{–14}C$. The value of $Q$ is ______.
(Take ${ϵ}_{0}=9\times {10}^{-12}{C}^{2}{N}^{-1}{m}^{-2}$)

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Answer: 288

Given here, electric field, $\vec{E}={E}_{0}x\hat{i}$
Here, the flux passes mainly through surface areas, $ABCDandEFGH$. As the surfaces $AEFBandCGHD$ are parallel to the electric field, so flux for these surfaces are zero. Again in $EFGH$, $a=0$, thus, electric field is zero.
Hence, the flux only passes through the surface are $ABCD$.
The net flux is ${\phi }_{net}={\phi }_{ABCD}={E}_{0}a\cdot {a}^{2}$
Using Gauss's law, $\dfrac{{q}_{en}}{{ϵ}_{0}}={E}_{0}{a}^{3}$
$\Rightarrow {q}_{en}={E}_{0}{\epsilon }_{0}{a}^{3}$
$=4\times {10}^{4}\times 9\times {10}^{-12}\times 8\times {10}^{-6}$
$=288\times {10}^{-14}C$
Hence, the value of $Q=288$.