Question

An electric field \( \vec{E} = (2x \hat{i}) \, \text{N C}^{-1} \) exists in space. A cube of side \( 2 \, \text{m} \) is placed in the space as per the figure given below. The electric flux through the cube is __________ \( \text{N m}^2/\text{C} \).

Answer 1

Correct Answer: 16

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \begin{document} The electric flux is given by Gauss's Law: \[ \Phi = \oint \vec{E} \cdot d\vec{A}. \] The field $\vec{E} = 2x \hat{i}$ varies with $x$. For the cube, only the left ($x = 0$) and right ($x = 2$) faces contribute: \[ \Phi = E_\text{right} A - E_\text{left} A. \] Substituting $A = 4 \, \mathrm{m}^2$, $E_\text{right} = 2(2) = 4$, and $E_\text{left} = 2(0) = 0$: \[ \Phi = 4 \cdot 4 - 0 \cdot 4 = 16 \, \mathrm{Nm}^2/\mathrm{C}. \]