Two similar balls, each of mass m and charge q, are hung from a common point by two silk threads, each of length $l$. Prove that separation between the balls is $x={\left[\frac{q^{2}l}{2\pi {\epsilon }_{0}mg}\right]}^{1/3}$, if $\theta$ is small.
Find the rate dq/dt with which the charge should leak off each sphere if the velocity of approach varies as $v=a/\sqrt{x}$, where a is a constant

Q. Two similar balls of mass $m$ are hung from silk threads of length $\ell$ and carry similar charges $+q$ and $+q$.
(a) Prove that when the system is in equilibrium, separation between the balls is $x={\left(\frac{q^2\ell }{2\pi {\epsilon }_{0}mg}\right)}^{1/3}$, assuming the angle made by the threads with the vertical line passing through the point of suspension is small.
(b) Specify the rate $\left(dq/dt\right)$ with which the charge from each sphere leaks off if their velocity of approach varies as $v=\frac{a}{\sqrt{x}}$, where $a$ is a constant.

Q. Two small equally charged spheres, each of mass mm, are suspended from the same point by silk threads of length l. The distance between the spheres x << l. Find the rate $\frac{dq}{dt}$ with which the charge leaks off each sphere if their approach velocity varies as $v=\frac{a}{\sqrt{x}}$, where a is a constant $(k=\frac{1}{4\pi {\epsilon }_0})$.

Q. Two similar balls of mass ‘m’ are hung by a silk thread of length L and carry similar charges Q as in figure Assuming the separation to be small, the separation between the balls (denoted by x) is equal to

Q. Two identical pith balls, each of mass $m$ and carrying identical charges $q$ each, are suspended from a common point by two strings of equal length $l$. The angle between the string is $2\theta$, and initially changes at a rage $\left|\frac{d\theta }{dt}\right|={\omega }_0$, due to leakage of charge at a very slow rate. Find the rate of leakage of charge (i.e.$\left|\frac{dq}{dt}\right|$).

Q. Two identical tennis balls each having mass ${}^{\prime }{m}^{\prime }$ and charge ${}^{\prime }{q}^{\prime }$ are suspended from a fixed point by threads of length ${}^{\prime }{l}^{\prime }$. What is the equilibrium separation when each thread makes a small angle $\theta$ with the vertical ?