Identify the correct statements:
$A$. Electrostatic field lines form closed loops.
$B$. The electric field lines point radially outward when charge is greater than zero.
$C$. The Gauss-Law is valid only for inverse-square force.
$D$. The work done in moving a charged particle in a static electric field around a closed path is zero.
$E$. The motion of a particle under Coulomb's force must take place in a plane.
Choose the correct answer from the options given below:

Which of the following physical quantities have the same dimensions?

The adjacent diagram shows a charge $+Q$ held on an insulating support $S$ and enclosed by a hollow spherical conductor. $O$ represents the centre of the spherical conductor,and $P$ is a point such that $OP = x$ and $SP = r$. The electric field at point $P$ will be

Consider a gravity-free container as shown. The system is initially at rest and the electric potential in the region is $V = (y^3 + 2) \text{ J/C}$. A ball of charge $q = -0.5 \text{ C}$ and mass $m = 2 \text{ kg}$ is released from rest from the base $(y=0)$. It starts to move up due to the electric field and collides with the shaded top face $(y=2 \text{ m})$ as shown. If its speed just after the collision is $1.5 \text{ m/s}$ and the time for which the ball is in contact with the shaded face is $0.1 \text{ s}$, find the external force required to hold the container fixed in its position during the collision, assuming the ball exerts a constant force on the wall during the entire span of the collision. (in $\text{ N}$)

The earth's surface has a negative surface charge density of $10^{-9} \; C\;m^{-2}$. The potential difference of $400 \; kV$ between the top of the atmosphere and the surface results (due to the low conductivity of the lower atmosphere) in a current of only $1800 \; A$ over the entire globe. If there were no mechanism of sustaining the atmospheric electric field,how much time (roughly) would be required to neutralize the earth's surface (in $; s$)? (This never happens in practice because there is a mechanism to replenish electric charges,namely the continual thunderstorms and lightning in different parts of the globe). (Radius of earth $= 6.37 \times 10^{6} \; m$.)

$A$ clock dial has point charges $-q, -2q, -3q, \ldots, -12q$ at the positions of the corresponding numbers on the dial respectively. The time at which the hour hand points in the direction of the net electric field at the centre of the dial is (Assume clock hands do not influence the net electric field). (in $:30$)