$(a)$ Two insulated charged copper spheres $A$ and $B$ have their centres separated by a distance of $50 \;cm$. What is the mutual force of electrostatic repulsion if the charge on each is $6.5 \times 10^{-7}\; C?$ The radii of $A$ and $B$ are negligible compared to the distance of separation.

$(b)$ What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved?

Two point charges $+8q$ and $ – 2q$ are located at $x = 0$ and $x = L$ respectively. The location of a point on the $x$-axis at which the net electric field due to these two point charges is zero is

The ratio of the forces between two small spheres with constant charge $(a)$ in air $(b)$ in a medium of dielectric constant $K$ is

Two spherical, nonconducting, and very thin shells of uniformly distributed positive charge $Q$ and radius d are located a distance $10d$ from each other. A positive point charge $q$ is placed inside one of the shells at a distance $d/2$ from the center, on the line connecting the centers of the two shells, as shown in the figure. What is the net force on the charge $q $ ?

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