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

Answer

B

Explanation

As is clear from Fig

`tan theta = (F)/(mg) = (kq^(2))/(x^(2) mg)` ….(i)
Also, `tan theta = (x//2)/(sqrt(l^(2) -x^(2)//4)) ~= (x)/(2l)`
From(i) and (ii), `(kq^(2))/(x^(2) mg) = (x)/(2l)`
`q^(2) = (mg x^(3))/(2 kl)`
`:. q prop c^(3//2)`
`(dq)/(dr) prop (d)/(dx) (x^(3//2) (dx)/(dt) prop (3)/(2) x^(1//2).v`
As `(dq)/(dr)` is constant, therefore, `v prop x^(-1//2)`