An oil drop carries six electronic charges,has a mass of $1.6 \times 10^{-12} \text{ g}$ and falls with a terminal velocity in air. The magnitude of the vertical electric field required to make the drop move upward with the same speed as it was formerly moving is ........$kN/C$.

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 $\epsilon_0 = 9 \times 10^{-12} F / m$):

How much work is done in moving a charge $+q$ once around a circle of radius $r$ in the presence of a central charge $+Q$?

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.

Ten charges are placed on the circumference of a circle of radius $R$ with constant angular separation between successive charges. Alternate charges $1, 3, 5, 7, 9$ have charge $(+q)$ each, while $2, 4, 6, 8, 10$ have charge $(-q)$ each. The potential $V$ and the electric field $E$ at the centre of the circle are respectively (Take $V = 0$ at infinity).

The following figures show regular hexagons with charges placed at their vertices. In which of the following cases is the electric field at the center zero?