Two similar conducting balls of mass m are hung from silk threads of length L and carry simillar charges q as shown in the figure. Assuming 0 to be small, the distance x between the ball is :

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.

In the given figure two tiny conducting balls of identical mass m and identical charge q hang from non-conducting threads of equal length L. Assume that $\theta$ is so small that $tan\theta \approx sin\theta$, then for equilibrium x is equal to

Q. 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 ball $A$ and $B$ of same mass $\left(M\right)$ and charges $Q,-Q$ are suspended by two strings of same length from two different suspension points $S_1$ and $S_2$ if $S_1{S}_2=3x$ and $AB=x$ then show that-
tension in the string is $T=\frac{Q^{2}L}{4\pi {\in }_0{x}^3}$
$\tan \theta =\frac{Q^2}{4\pi {\in }_0{x}^{2}Mg}$

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.