Question 285 of 311

As shown in figure, a cuboid lies in a region with electric field $E=2{x}^{2}\hat{i}-4y\hat{j}+6\hat{k}N{C}^{-1}$. The magnitude of charge within the cuboid is $n{\epsilon }_{0}C$.
The value of $n$ is ______ (if dimension of cuboid is $1\times 2\times 3{m}^{3}$)

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Answer: 12

Given:
$\vec{E}=2{x}^{2}\hat{i}-4y\hat{j}+6\hat{k}$

For top and bottom faces, as value of x=0&y=0, electric field becomes constant and therefore net flux through these two faces will be zero(${\phi }_{z}=0$).
Now, ${E}_{x}=2{x}^{2}$, ${E}_{y}=-4y$ and ${E}_{z}=6$.
Change in net flux through face at x=1 and at x=0 parallel to y-z plane can written as, ${\phi }_{x}=\left(2\times {1}^{2}\right)\times \left(2\times 3\right)-0=12$.
Similarly, net flux through face at y=2 and y=0 parallel to x-z plane can be written as, ${\phi }_{y}=-\left(4\times 2\right)\times \left(1\times 3\right)-0=-24$
${\phi }_{net}=12-24=-12$
Using Gauss's law,
${\phi }_{net}=\dfrac{q}{{\epsilon }_{0}}$
$\Rightarrow \left|q\right|=12{\epsilon }_{0}$