Two identical balls having like charges and placed at a certain distance apart repel each other with a certain force. They are brought in contact and then moved apart to a distance equal to half their initial separation. The force of repulsion between them increases $4.5$ times in comparison with the initial value. The ratio of the initial charges of the balls is

Two identical charged spheres suspended from a common point by two massless strings of lengths $l$ are initially at a distance $d$ $(d << l)$ apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result,the spheres approach each other with a velocity $v$. Then $v$ varies as a function of the distance $x$ between the spheres,as:

Two fixed,identical conducting plates $(\alpha)$ and $(\beta)$,each of surface area $S$,are charged to $-Q$ and $q$,respectively,where $Q > q > 0$. $A$ third identical plate $(\gamma)$,free to move,is located on the other side of the plate with charge $q$ at a distance $d$ as per the figure. The third plate is released and collides with the plate $(\beta)$. Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongst $(\beta)$ and $(\gamma)$.
$(a)$ Find the electric field acting on the plate $(\gamma)$ before collision.
$(b)$ Find the charges on $(\beta)$ and $(\gamma)$ after the collision.
$(c)$ Find the velocity of the plate $(\gamma)$ after the collision and at a distance $d$ from the plate $(\beta)$.

Consider the points lying on a straight line joining two fixed opposite charges. Between the charges,there is:

Two identical charged spheres of material density $\rho$,suspended from the same point by inextensible strings of equal length,make an angle $\theta$ between the strings. When suspended in a liquid of density $\sigma$,the angle $\theta$ remains the same. The dielectric constant $K$ of the liquid is

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?