Two charges of equal magnitudes and at a distance $r$ exert a force $F$ on each other. If the charges are halved and distance between them is doubled, then the new force acting on each charge is

A charge ${q_1}$ exerts some force on a second charge ${q_2}$. If third charge ${q_3}$ is brought near, the force of ${q_1}$ exerted on ${q_2}$

Two identical conducing spheres having unequal positive charges $q_1$ and $q_2$ separated by distance $r$. If they are made to touch each other and then separated again to the same distance. The electrostatic force between the spheres in this case will be (neglect induction of charges)

Two pith balls carrying equal charges are suspended from a common point by strings of equal length, the equilibrium separation between them is $r.$ Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now become

Total charge $-\,Q$ is uniformly spread along length of a ring of radius $R$. A small test charge $+q$ of mass m is kept at the centre of the ring and is given a gentle push along the axis of the ring.

$(a) $ Show that the particle executes a simple harmonic oscillation.

$(b)$ Obtain its time period.

Three point charges $q,-2 q$ and $2 q$ are placed on $x$-axis at a distance $x=0, x=\frac{3}{4} R$ and $x=R$ respectively from origin as shown. If $q =2 \times 10^{-6}\,C$ and $R =2\,cm$, the magnitude of net force experienced by the charge $-2 q$ is .......... $N$