A line charge of length $$\frac{'a'}{2}$$is kept at the center of an edge BC of a cube ABCDEFGH having edge length 'a' as shown in the figure. If the density of line charge is $$\lambda C$$ per unit length, then the total electric flux through

all the faces of the cube will be . (Take, $$\epsilon _o$$ as the free space permittivity)

Use Gauss law.

Length of line charge placed on edge BC:

$$\frac{a}{2}$$

Charge enclosed in cube:

$$q=\lambda\cdot\frac{a}{2}$$

Now, because charge lies along an edge, it is shared by 4 identical cubes if we imagine 4 cubes arranged around that edge.

So this cube encloses only one-fourth of that charge effectively for flux through this cube.

Effective enclosed charge:

$$q_{\text{eff}}=\frac{1}{4}\left(\lambda\frac{a}{2}\right)=\frac{\lambda a}{8}$$

By Gauss law,

$$\Phi=\frac{q_{\text{eff}}}{\epsilon_0}$$

therefore it is

$$\frac{\lambda}{8\epsilon_o} a$$