A charge $q$ is placed at the centre of the open end of cylindrical vessel. The flux of the electric field through the surface of the vessel is

Given below are two statement: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

Assertion $A:$ If an electric dipole of dipole moment $30 \times 10^{-5}\,Cm$ is enclosed by a closed surface, the net flux coming out of the surface will be zero.

Reason $R$ : Electric dipole consists of two equal and opposite charges.

In the light of above, statements, choose the correct answer from the options given below:

How does the no. of electric field lines passing through unit area depend on distance ?

A circular disc of radius $R$ carries surface charge density $\sigma(r)=\sigma_0\left(1-\frac{r}{R}\right)$, where $\sigma_0$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $\phi_0$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\phi$. Then the ratio $\frac{\phi_0}{\phi}$ is. . . . . .

In $1959$ Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density $N$, which is maintained a constant. Let the charge on the proton be : 

${e_p}{\rm{ }} =  - {\rm{ }}\left( {1{\rm{ }} + {\rm{ }}y} \right)e$ where $\mathrm{e}$ is the electronic charge.

$(a)$ Find the critical value of $y$ such that expansion may start.

$(b)$ Show that the velocity of expansion is proportional to the distance from the centre.

The flat base of a hemisphere of radius $a$ with no charge inside it lies in a horizontal plane. A uniform electric field $\vec E$ is applied at an angle $\frac {\pi }{4}$ with the vertical direction. The electric flux through the curved surface of the hemisphere is