$\mathrm{C}_1$ and $\mathrm{C}_2$ are two hollow concentric cubes enclosing charges $2 Q$ and $3 Q$ respectively as shown in figure. The ratio of electric flux passing through $\mathrm{C}_1$ and $\mathrm{C}_2$ is :

A cubical region of side a has its centre at the origin. It encloses three fixed point charges, $-q$ at $(0,-a / 4,0),+$ $3 q$ 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 $x=+a / 2$.

A charge is kept at the central point $P$ of a cylindrical region. The two edges subtend a half-angle $\theta$ at $P$, as shown in the figure. When $\theta=30^{\circ}$, then the electric flux through the curved surface of the cylinder is $\Phi$ If $\theta=60^{\circ}$, then the electric flux through the curved surface becomes $\Phi / \sqrt{n}$, where the value of $n$ is. . . . . . .

For a closed surface $\oint {\overrightarrow {E \cdot } } \,\overrightarrow {ds} \,\, = \,\,0$, then 

A charged particle $q$ is placed at the centre $O$ of cube of length $L$ $(A\,B\,C\,D\,E\,F\,G\,H)$. Another same charge $q$ is placed at a distance $L$ from $O$.Then the electric flux through $BGFC$ is

A long cylindrical shell carries positive surface charge $\sigma$ in the upper half and negative surface charge $-\sigma$ in the lower half. The electric field lines around the cylinder will look like figure given in : (figures are schematic and not drawn to scale)