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

Answer

A

Explanation

From figure, `T cos theta = mg` …(i)
`T sin theta = (kq^(2))/(x^(2))`
From eqns. (i) and (ii), `tan theta = (kq^(2))/(x^(2) mg)`
Since `theta` is small, `:. tan theta ~~ sin theta = (x)/(2l)`
`:. (x)/(2l) = (kq^(2))/(x^(2) mg) rArr q^(2) = x^(3)(mg)/(2lk)` or `q prop x^(3//2)`
`rArr (dq)/(dt) prop (3)/(2) sqrt(x)(dx)/(dt) = (3)/(2)sqrt(xv)`.
Since, `(dq)/(dt)` = constant
`:. v prop (1)/(sqrt(x))`