$A$ uniformly charged conducting sphere of diameter $14 \ cm$ has a surface charge density of $40 \ \mu C/m^2$. The total electric flux leaving the surface of the sphere is nearly (Permittivity of free space $\varepsilon_0 = 8.85 \times 10^{-12} \ C^2/N \cdot m^2$)

$A$ cylinder of radius $R$ and length $L$ is placed in a uniform electric field $E$ parallel to the cylinder axis. The total flux for the surface of the cylinder is given by-

Four closed surfaces and corresponding charge distributions are shown below. Let the respective electric fluxes through the surfaces be $\phi_1, \phi_2, \phi_3$ and $\phi_4$. Then:

An infinitely long thin non-conducting wire is parallel to the $z$-axis and carries a uniform line charge density $\lambda$. It pierces a thin non-conducting spherical shell of radius $R$ in such a way that the arc $PQ$ subtends an angle $120^{\circ}$ at the centre $O$ of the spherical shell,as shown in the figure. The permittivity of free space is $\epsilon_0$. Which of the following statements is (are) true?
$(A)$ The electric flux through the shell is $\sqrt{3} R \lambda / \epsilon_0$
$(B)$ The $z$-component of the electric field is zero at all the points on the surface of the shell
$(C)$ The electric flux through the shell is $\sqrt{2} R \lambda / \epsilon_0$
$(D)$ The electric field is normal to the surface of the shell at all points

$A$ charge $Q$ is placed at the centre of a cube of side $a$. The total flux of electric field through the six surfaces of the cube is

State Gauss's law in electrostatics and provide its mathematical expression.