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

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 $E_\text{left} = 2(0) = 0$:
$\Phi = 4 \cdot 4 - 0 \cdot 4 = 16 \, \mathrm{Nm}^2/\mathrm{C}.$

Answer 2

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 $E_\text{left} = 2(0) = 0$:
$\Phi = 4 \cdot 4 - 0 \cdot 4 = 16 \, \mathrm{Nm}^2/\mathrm{C}.$