Question

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

Answer 1

Correct Answer - D
Let the separation between the spheres be `x`.

`tan theta=(F_(g))/(mg)=1/(4piepsilon_(0)).(q^(2))/(x^(2)mg)..(i)`
By geomatery `tan theta=(x//2)/l.....(ii)`
`(q^(2))/(4piepsilon_(0)mgx^(2))=x/(2l)`
`q^(2) prop x^(3)`
`q propx^(3//2)`
`(dq)/(dt) prop 3/2x^(1//2)(dx)/(dt)`
`prop 3/2x^(1//2)v`
`v prop x^(-1//2)`