$A$ uniformly charged thin spherical shell of radius $R$ carries a uniform surface charge density of $\sigma$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to

Under the influence of the Coulomb field of charge $+Q$,a charge $-q$ is moving around it in an elliptical orbit. Find out the correct statement$(s)$.

Electric charges $Q$ are placed on the $x$-axis at $x = 1, 2, 4, 8, \dots \text{meters}$ respectively. What are the electric field and electric potential at $x = 0$?

Two charged particles,each of mass $3 \ g$ and charge $0.2 \ \mu C$,stay in (vacuum) equilibrium on a horizontal surface with a separation of $20 \ cm$. The coefficient of friction is $\left[\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \ Nm^2 C^{-2}\right]$ and $\left(g=10 \ ms^{-2}\right)$.

Six infinitely large and thin non-conducting sheets are fixed in configurations $I$ and $II$. As shown in the figure,the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is $d = 1 \mu m$. The various regions between the sheets are denoted as $1, 2, 3, 4$ and $5$. If $\sigma_0 = 9 \mu C / m^2$,then which of the following statements is/are correct: (Take permittivity of free space $\epsilon_0 = 9 \times 10^{-12} F / m$):

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 constant. Let the charge on the proton be: $e_p = -(1 + y)e$,where $e$ 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.