The time period of revolution of a charge `q_(1)` and of mass m moving in a circular path of radius r due to Coulomb force of attraction with another charge `q_(2)` at its centre is

Options

A. `sqrt ((16 pi epsi _(0) mr ^(3))/( q _(1) q _(2)))`
B. `sqrt (( 8pi ^(2) epsi _(0) mr ^(3))/( q _(1) q _(2)))`
C. `sqrt (( epsi _(0) mr ^(3))/( 1 6 q _(1) q _(2)))`
D. `sqrt (( 16pi ^(3) epsi _(0) mr ^(3))/( q _(1) q _(2)))`

Answer

D

Explanation

Centripetal force `F = ( mv ^(2))/( r)" "...(i)`
and electrostatic force
`F = K. (q _(1).q _(2))/(r ^(2)) " "...(ii)`
Equating both equation
`(mv ^(2))/( r ) = K . (q _(1) q _(2))/( r ^(2))`
`v = sqrt (( k )/(m ) . ( q _(1) q _(2))/( r ))`
The time period of revolution
`T = (2pi r )/( v)`
`= 2pi r sqrt (( m.r )/( k. q _(1) q _(2)))`
`= sqrt (( 4pi ^(2) r ^(2) mr )/( K . q _(1) q _(2) )) " "(because K = (1)/( 4pi epsi _(0)))`
`T = sqrt (( 4pi ^(2) r ^(2) mr. 4pi epsi _(0))/( q _(1 )q _(2)))`
`= sqrt (( 16pi ^(2) epsi _(0) mr ^(3))/( q _(1) q _(2)))`