Consider two point charges of equal magnitude and opposite sign separated by a certain distance. The neutral point due to them

The electric field due to a uniformly charged sphere of radius $R$ as a function of the distance from its center is represented by which of the following graphs?

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 $\text{ N}$)

Four charges $Q_1, Q_2, Q_3$, and $Q_4$ of the same magnitude are fixed along the $x$-axis at $x = -2a, -a, +a$, and $+2a$, respectively. A positive charge $q$ is placed on the positive $y$-axis at a distance $b > 0$. Four options regarding the signs of these charges are given in List-$I$. The direction of the net force on the charge $q$ is given in List-$II$. Match List-$I$ with List-$II$ and select the correct answer using the codes given below the lists.

List-$I$	List-$II$
$P. Q_1, Q_2, Q_3, Q_4$ all positive	$1. +x$
$Q. Q_1, Q_2$ positive; $Q_3, Q_4$ negative	$2. -x$
$R. Q_1, Q_4$ positive; $Q_2, Q_3$ negative	$3. +y$
$S. Q_1, Q_3$ positive; $Q_2, Q_4$ negative	$4. -y$

$A$ thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V_0$. $A$ hole with a small area $\alpha 4 \pi R^2$ (where $\alpha \ll 1$) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Two particles each of mass $m$ and charge $q$ are separated by distance $r_1$ and the system is left free to move at $t = 0$. At time $t$,both the particles are found to be separated by distance $r_2$. The speed of each particle is