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As shown in figure, a cuboid lies in a region with electric field $=2 x^{2} \hat{\imath}-4 y \hat{\jmath}+6 \hat{k} \mathrm{~N} / \mathrm{C}$ . The magnitude of charge within the cuboid is $n \in_{0} C$ .
The value of $n$ is_________ (if dimension of cuboid is $1 \times 2 \times 3 \mathrm{~m}^{3} ).$

Option: 1
12

Option: 2
-

Option: 3
-

Option: 4
-

Answers (1)

best_answer

$\overrightarrow{\mathrm{E}}=2 \mathrm{x}^2 \hat{\mathrm{i}}-4 \mathrm{y} \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$

$\\ \phi _{net}=\oint \overrightarrow{\mathrm{E}} \cdot \overrightarrow{d \mathrm{~s}} \\$

at (0,0,0) $E_x=0,E_y=0,E_z=6$

Net flux due to $E_z$ is 0 because flux incomin=flux outgoing

at x=1 , $E_x=2$

Net flux due to $E_x$ is $\phi _x=2\times 2\times 3=+12$

at y=2, $E_y=-8$

$\phi _y=-8\times 1\times 3=-24$

$\phi_{\text {net }}=+12-24$
$=-12$

$\phi_{\text {net }}=\frac{\mathrm{q}_{\text {inside }}}{\epsilon_{0}}$
$\mathrm{q}_{\text {inside }}=-12 \epsilon_{0}$

$|\mathrm{q}_{\text {inside }}|=12 \epsilon_{0}$

Ans: (12)

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Rakesh

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