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

Force of electrostatic repulsion :
`F=(q^(2))/(4piepsilon_(0)x^(2))`
For equilibrium of each sphere :
`T cos theta=mg`

`Tsintheta=F=(q^(2))/(4piepsilon_(0)x^(2))`
`implies (F)/(mg)=tantheta`
`implies (q^(2))/(4piepsilon_(0)x^(2)(mg))=tantheta~~sintheta=(x)/(2l)" "[because thetararr0]`
`implies x=((q^(2)l)/(2piepsilon_(0)mg))^(1//3)" "...(i)`
From equation (i)
`q^(2)=(2piepsilon_(0)mgx^(3))/(l)`
Differentiating both sides with respect to time :
(ii) `2q(dq)/(dt)=(2piepsilon_(0)mg)/(l)3x^(2)(dx)/(dt)`
Now, `(dx)/(dt)` = velocity of approach = v = `(a)/(sqrt(x))`
`((2piepsilon_(0)mg)/(l))^(1//2).x^(3//2)(dq)/(dt)=(3piepsilon_(0)mg)/(l)x^(2)((a)/(sqrt(x)))` using equation (i)
`implies (dq)/(dt)=(3)/(2)asqrt((2piepsilon_(0)mg)/(l))`