A line charge of length $\frac{\text { ' } a \text { ' }}{2}$ is kept at the center of an edge $B C$ of a…
A line charge of length $\frac{\text { ' } a \text { ' }}{2}$ is kept at the center of an edge $B C$ of a cube $A B C D E F G H$ 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, $\epsilon_0$ as the free space permittivity)
$\frac{\lambda \mathrm{a}}{2 \epsilon_0}$
$\frac{\lambda \mathrm{a}}{4 \epsilon_0}$
$\frac{\lambda \mathrm{a}}{16 \epsilon_0}$
$\frac{\lambda \mathrm{a}}{8 \epsilon_0}$
Solution
$\begin{aligned} & \text { Charge of the line charge }=\frac{a \lambda}{2} \\ & \text { Portion of wire inside cube }=\frac{1}{4} \\ & \therefore \quad q_{e n}=\frac{1}{4}\left(\frac{a \lambda}{2}\right)=\frac{a \lambda}{8} \\ & \qquad \phi=\frac{q_{e n}}{\varepsilon_0}=\frac{a \lambda}{8 \varepsilon_0}\end{aligned}$
