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

Answer

A

Explanation

Figure shows equilibrium positions of the two spheres. `:. T cos theta = mg`
and `T sin theta = F = (1)/(4pi epsilon_(0))(q^(2))/(d^(2))`
`:. tan theta = (1)/(4pi epsilon_(0))(q^(2))/(d^(2) mg)`
When charge begins to leak from both the spheres at a constant rate, then
`tan theta = (1)/(4pi epsilon_(0)) (q^(2))/(x^(2) mg)`
`rArr (x)/(2l) = (q^(2))/(4pi epsilon_(0)x^(2) mg) " "(because tan theta = (x)/(2l))`
or `(x)/(2l) prop (q^(2))/(x^(2))` or `q^(2) prop x^(3) rArr q prop x^(3//2)`
`(dq)/(dt) prop (3)/(2) x^(1//2)(dx)/(dt)` or `v prop x^(-1//2) " "(because (dq)/(dt) = "constant")`