Two identical charged spheres suspended form a common point by two massless strings of lengths l, are initially at a distance `d(d gt gt l)` apart because of their mutual repulsion. The charges brgin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as

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

Given `(dq)/(dt)`= constant
Tension is string , at equilibrium state,

T sin `theta =F_e = (Kq^2)/x^2`
`T cos theta =mg`
`thereforetan theta =(Kq^2)/(x^2mg)`
`theta` is very small so, tan `theta approx sin theta =x/(2l)`
`(Kq^2)/(x^2mg) =x/(2l)`
`x^3=(2Kl)/(mg)q^2`
`x^3 prop q^2 rArr q=K_1x^(3//2)` ...(i)
Differentiate w.r.t. time
`3x^2 (dx)/(dt)=(2K_2 q(dq)/(dt))`
`3x^2 (dx)/(dt) =2K_2qI`
`v=(2K_2I)/3 qx^(-2)`
From equation (i)
`v=(2K_2K_1I)/3 x^(-1//2)`
`rArr v prop x^(-1//2)`