Question

In a region of space, the electric field is in the x-direction and proportional to x, i.e., $\overset{\rightarrow}{E} = E_{0}x\widehat{i}$. Consider an imaginary cubical volume of edge a, with its edges parallel to the axes of coordinates. The charge inside this cube is

• A. Zero
• B. $\varepsilon_{0}E_{0}a^{3}$
• C. $\frac{1}{\varepsilon_{0}}E_{0}a^{3}$
• D. $\frac{1}{6}\varepsilon_{0}E_{0}a^{2}$

Answer 1

Answer: (B)

$\varepsilon_{0}E_{0}a^{3}$

Answer 2

The field at the face ABCD = $E_{0}x_{0}\widehat{i}.$

∴ Flux over the face ABCD = – (E₀x₀)a²

The negative sign arises as the field is directed into the cube.

The field at the face EFGH = $E_{0}(x_{0} + a)\widehat{i}.$

∴ Flux over the face EFGH = $E_{0}(x_{0} + a)a^{2}$

The flux over the other four faces is zero as the field is parallel to the surfaces.

∴Total flux over the cube $= E_{0}a^{2} = \frac{1}{2}q$

where q is the total charge inside the cube. ∴ $q = \varepsilon_{0}E_{0}a^{3}.$