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 ball is `x=[(q^(2)l)/(2 pi epsilon_(0)mg)]^(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.

Explanation

For momentary equilibrium of the spheres,
`T cos theta=mg and T sin theta = F`
where T is tension in the string
`rArr tan theta =(F)/(mg)`
For `x lt lt L`,

we can take `tan theta =(x)/(2L)`
So `F=(mg x)/(2L)`
Here `F=(1)/(4pi in_(0)) (q^(2))/(x^(2)) and q^(2)=(2pi in_(0) mg x^(3))/(L)`
given `(dx)/(dt) =(a) /(sqrt(x))` where a is a constant
`rArr (dq)/(dt)=(3q)/(2) sqrt((2pi in_(0) mg)/(L))`