Two identical charged spheres suspended from a common point by two mass-less strings of length `l` are initially at a distance d ( `d ltlt l`) apart because of their mutual repulsion . The charge begins to leak from both the spheres at a constant rate. As a result the charge approach each other with a velocity `v`. Then as a function of distance `x` between them .

Options

A. `v prop x^(-1)`
B. `v prop x^(1//2)`
C. `v prop x`
D. `v prop x^(-1//2)`

Answer

D

Explanation

T cos `theta` =mg
`T sin theta =F_"el"=1/(4piepsilon_0)q^2/x^2`
`rArr tan theta =1/(4piepsilon_0) q^2/(mgx^2)` …(i)
In this case x is very small hence `tan theta approx sin theta =x/(2l)` where l is length of the string :
Equation (i) can be rewritten as follows :
`x/(2l)=1/(4piepsilon_0)q^2/(mgx^2)`
`rArr q=sqrt((2piepsilon_0mg)/l)x^(3//2)` ....(ii)
`rArr (dq)/(dt) =sqrt((2piepsilon_0mg)/l)d/(dx)(x^(3//2))(dx)/(dt)`
Differentiating equation (ii) we get the following :
`rArr v=(dx)/(dt)=2/3 sqrt(l/(2piepsilon0_0mg))(dq)/(dt)x^(-1//2)`