In a region of space, the electric field is in the x-direction and proportional to x, i.e. $\vec{E}=E_o x \hat{i}$. Consider an imaginary cubical volume of edge a, with its edges parallel to the axes of coordinates. The charge inside this volume is
Answer & explanation
Correct answer: option 2
The field at the face ABCD $=E_0 x_0 \hat{i}$
∴ flux over the face ABCD = $-\left(E_0 x_0\right) a^2$
The negative sign arises as the field is directed into the cube. The field at the face EFGH $=E_0\left(x_0+a\right) \hat{i}$
∴ flux over the face EFGH $=E_0\left(x_0+a\right) a^2$
The flux over the other four faces is zero as the field is parallel to the surfaces.
∴ total flux over the cube = $E_0 a^3=\frac{1}{\varepsilon_0} q$
where q is the total charge inside the cube.
∴ $q=\varepsilon_0 E_0 a^3$
∴ (b)