JEE MAIN - Physics (2023 - 30th January Evening Shift - No. 25)
As shown in figure, a cuboid lies in a region with electric field $E=2 x^{2} \hat{i}-4 y \hat{j}+6 \hat{k} \mathrm{~N} / \mathrm{C}$. The magnitude of charge within the cuboid is $n \in_{0} C$.
The value of $n$ is _________ (if dimension of cuboid is $1 \times 2 \times 3 \mathrm{~m}^{3}$ )
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The value of $n$ is _________ (if dimension of cuboid is $1 \times 2 \times 3 \mathrm{~m}^{3}$ )
Answer
12
Explanation
Flux through planes parallel to $y-z=2(1)^{2} \times$ Area
$$ \begin{aligned} & =2(1)^{2} \times 2 \times 3 \\\\ & =12 \mathrm{Nm}^{2} / \mathrm{C} \end{aligned} $$
Flux through planes parallel to $x-z=-4(2) \times$ Area
$$ \begin{aligned} & =-4(2) \times 1 \times 3 \\\\ & =-24 \mathrm{Nm}^{2} / \mathrm{C} \end{aligned} $$
Flux through planes parallel to $x-y=0$
$\Rightarrow \phi$ Total $=12-24=-12$
$\Rightarrow-12=\frac{q_{\mathrm{enc}}}{\varepsilon_{0}} \Rightarrow\left|q_{\mathrm{enc}}\right|=12 \varepsilon_{0}$
$\Rightarrow n=12$
$$ \begin{aligned} & =2(1)^{2} \times 2 \times 3 \\\\ & =12 \mathrm{Nm}^{2} / \mathrm{C} \end{aligned} $$
Flux through planes parallel to $x-z=-4(2) \times$ Area
$$ \begin{aligned} & =-4(2) \times 1 \times 3 \\\\ & =-24 \mathrm{Nm}^{2} / \mathrm{C} \end{aligned} $$
Flux through planes parallel to $x-y=0$
$\Rightarrow \phi$ Total $=12-24=-12$
$\Rightarrow-12=\frac{q_{\mathrm{enc}}}{\varepsilon_{0}} \Rightarrow\left|q_{\mathrm{enc}}\right|=12 \varepsilon_{0}$
$\Rightarrow n=12$
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