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:

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.

At a certain distance from a point charge,the electric field intensity is $500 \ V/m$ and the electric potential is $3000 \ V$. What is this distance in $m$?

Two identical charged particles,each having a mass $10 \, g$ and charge $2.0 \times 10^{-7} \, C$,are placed on a horizontal table with a separation of $L$ between them such that they stay in limiting equilibrium. If the coefficient of friction between each particle and the table is $0.25$,find the value of $L$ in $cm$. [Use $g = 10 \, m/s^2$]

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

$A$ toy car with charge $q$ moves on a frictionless horizontal plane surface under the influence of a uniform electric field $\vec E$. Due to the force $q\vec E$,its velocity increases from $0$ to $6\, m s^{-1}$ in one second. At that instant,the direction of the field is reversed. The car continues to move for two more seconds under the influence of this field. The average velocity and the average speed of the toy car between $0$ to $3$ seconds are respectively: