A point charge of charge -q and mass m is released with negligible speed from a distance `sqrt(3) R` on the axis of a fixed uniformly charged ring of charge Q and radius R. Find out its velocity when it reaches it reaches at the centre of the ring.

Explanation

As potential due to uniformly charged ring at its axis (at x distance)
`V = (kQ)/(sqrt(R^(2) + x^(2))` ,
So potential at point A due to ring
`V_(1) = (kQ)/(sqrt(R^(2) + 3R^(2))) = (kQ)/(2R)`
So potential energy of charge -q at point A
`P.E._(1) = (-kQ q)/(2R)`
and potential at point B `V_(2) = (kQ)/(R )`
So potential energy of charge -q at point B
`P.E._(2) = (-kQq)/(R )`
Now by energy conservation
`P.E._(1) + K.E._(1) = P.E._(2) + K.E_(2)`
`(-kQq)/(2R) + 0 = (-kQq)/(R ) + (1)/(2) mv^(2)`
`rArr v^(2) = (kQ q)/(mR)`
So velocity of charge - q at point B v
`= sqrt((kQ q)/(mR))`