A line charge of length $a/2$ is kept at the centre of an edge $BC$ of a cube $ABCDEFGH$ having edge length $a$, as shown in the figure. If the density of line charge is $\lambda\ \text{C}$ per unit length, then the total electric flux through all the faces of the cube will be (Take $\epsilon_0$ as the free space permittivity)
Answer: (D) $\dfrac{\lambda a}{8\epsilon_0}$
The charge on the line is $q = \lambda\cdot\dfrac{a}{2}$.
It lies along an edge of the cube. An edge is shared by four identical cubes that together surround the charge completely, so by symmetry this cube receives one quarter of the total flux $q/\epsilon_0$:
$$\phi = \frac{1}{4}\cdot\frac{\lambda a/2}{\epsilon_0} = \frac{\lambda a}{8\epsilon_0}$$
Solution by Sreeraj P, M.Sc Physics