Question 646: Electrostatics & Capacitance

Question

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 $1 	imes 2 	imes 3 \mathrm{~m}^{3}$ )

Accepted Answer

Flux through planes parallel to $y-z=2(1)^{2} 	imes$ Area
$$
\begin{aligned}
& =2(1)^{2} 	imes 2 	imes 3 \\
& =12 \mathrm{Nm}^{2} / \mathrm{C}
\end{aligned}
$$
Flux through planes parallel to $x-z=-4(2) 	imes$ Area
$$
\begin{aligned}
& =-4(2) 	imes 1 	imes 3 \\
& =-24 \mathrm{Nm}^{2} / \mathrm{C}
\end{aligned}
$$
Flux through planes parallel to $x-y=0$
$\Rightarrow \phi$ Total $=12-24=-12$
$\Rightarrow-12=\frac{q_{\mathrm{enc}}}{\varepsilon_{0}} \Rightarrow\left|q_{\mathrm{enc}}ight|=12 \varepsilon_{0}$
$\Rightarrow n=12$

Electrostatics & Capacitance

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} )

Integer / Numeric Answer Type