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-

Consider the charge configuration and spherical Gaussian surface as shown in the figure. When calculating the flux of the electric field over the spherical surface the electric field will be due to

Electric field in a region is uniform and is given by $\vec{E}=a \hat{i}+b \hat{j}+c \hat{k}$. Electric flux associated with a surface of area $\vec{A}=\pi R^2 \hat{i}$ is

Gauss’s law is true only if force due to a charge varies as

If a charge $q$ is placed at the centre of a closed hemispherical non-conducting surface, the total flux passing through the flat surface would be

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.