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

Let the balls be deciated by an angle `theta`, from the vertical when separtion between tehm equals `x`.
Appliying Newton's seconf law of motion for any one of the sphere, we get ,
`T cos theta = mg` (1)
and `T sin theta = F_(s)` (2)
From teh Eqs. (1) and (2)
`tan theta = (F_(e))/(mg)` (3)
But from the figure
`tan theta = (x)/(2 sqrt( l^(2) - ((x)/(2))^(2))) = (x)/(2l)` as `x lt lt l` (4)
From Eqs. (3) and (4)
`F_(e) = (mg x)/(2l)` or `(q^(2))/(4pi epsilon_(0) x^(2)) = (mg x)/(2l)`
Thus `q^(2) = (2pi epsilon _(0) x^(3)) = (l)` (5)
Differentiating Eqs. (5) with respect to time
`2q (dq)/(dt) = (2pi in_(0) mg)/(l) 3x^(2) (dx)/(dt)`
According to the problem `(dx)/(dt) = v = a//sqrt(x)` (approch velocity is `(dx)/(dt)`)
so, `((2pi epsilon_(0) mg)/(l) x^(3))^(1//2) (dq)/(dt) = (3pi epsilon_(0) mg)/(l) x^(3) (a)/(sqrt(x))`
Hence, `(dq)/(dt) = (3)/(2)a sqrt((2pi epsilon_(0) mg)/(l))`