$A$ 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 linear charge density is $\lambda \; C/m$, then the total electric flux through all the faces of the cube will be . . . . . . . (Take $\varepsilon_0$ as the free space permittivity)

The figure shows the electric field lines of three charges with charges $+1, +1$,and $-1$. The Gaussian surface in the figure is a sphere containing two of the charges. The total electric flux through the spherical Gaussian surface is

An electric line of force in the $X, Y-$ plane is given by the equation $x^2 + y^2 = 1$. $A$ particle with unit positive charge is initially at rest at the point $(1, 0)$ in the $X, Y-$ plane. The particle:

The electric field in a region is given by $\overrightarrow{E} = (\frac{3}{5} E_{0} \hat{i} + \frac{4}{5} E_{0} \hat{j}) \, N/C$. The ratio of the flux of this field through a rectangular surface of area $0.2 \, m^{2}$ (parallel to the $y-z$ plane) to that through a surface of area $0.3 \, m^{2}$ (parallel to the $x-z$ plane) is $a : b$,where $a = \dots$ [Here $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the $x, y$ and $z$-axes respectively].

$A$ cubical region of side $a$ has its centre at the origin. It encloses three fixed point charges,$-q$ at $(0, -a/4, 0)$,$+3q$ at $(0, 0, 0)$,and $-q$ at $(0, +a/4, 0)$. Choose the correct option$(s)$.
$(A)$ The net electric flux crossing the plane $x = +a/2$ is equal to the net electric flux crossing the plane $x = -a/2$.
$(B)$ The net electric flux crossing the plane $y = +a/2$ is more than the net electric flux crossing the plane $y = -a/2$.
$(C)$ The net electric flux crossing the entire region is $\frac{q}{\varepsilon_0}$.
$(D)$ The net electric flux crossing the plane $z = +a/2$ is equal to the net electric flux crossing the plane $x = +a/2$.

Five point charges $\frac{1}{\pi}, \frac{2}{\pi}, \frac{3}{\pi}, \frac{4}{\pi}$ and $\frac{-5}{\pi} \ nC$ are located inside a pyramid. The total electric flux through the surface of the pyramid is