Two similar balls, each of mass m and charge q, are hung from a common point by two slik 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.

Answer

`-(3)/(2)a sqrt((2 pi epsilon_(0)mg)/(l))`

Explanation

` tan theta=(F)/(mg)=(kq^2)/(x^(2)mg)`
` (x)/(21)=(q^(2))/(4 pi epsilon_(0)mg)`
`x=[(q^(2)l)/(2 pi epsilon_(0)mg)]^(1//3)`. Hence proved.
Now `q^(2)=(2 pi epsilon_(0)mgx^(3))/(l)`
or, `q=sqrt((2 pi epsilon_(0)mg)/(l))x^(3//2)`
or, `(dq)/(dt)=sqrt((2 pi epsilon_(0)mg)/(l)) (3)/(2) sqrt(x)(dx)/(dt)`
or, `sqrt(x)(dx)/(dt)=sqrt(x)(-upsilon)`
`= -sqrt(x)a//sqrt(x)= -a`
or, `(dq)/(dt)=-(3)/(2)a sqrt((2 pi epsilon_(0)mg)/(l))`.