A small ball of mass m and charge q is attached to the bottom end of a piece of negligible mass thread of length l, whose top end is fixed. The system formed by the thread and ball is in vertical plane and is in uniform horizontal magnetic field B, which is perpendicular to the plane of figure and points into the paper

The ball is started with a velocity `v_(0)` from lower most point of circle in a direction perpendicular both to the magnetic induction and to direction of thread. The ball moves along a circular path such that thread remains tight during the whole motion. Neglect any loss of enery.
What is the magnitude of magnetic induction B, if the minimum initial speed at which the described motion of ball (complete vertical circular motion) occurs is `V_(0)=(1)/(2)sqrt(17gL)`

Options

A. `(3mg)/(2qsqrt(gL))`
B. `(mg)/(2sqrt(gL))`
C. `(2mg)/(3qsqrt(gL)`
D. Situation is not possible

Answer

A

Explanation

`V=sqrt(v_(0)^(2)-2gL(1-cosalpha))`
`v_(0)^(2)=17/4gLimpliesV=3/2sqrt(gL(1+8/9cosalpha))`

From FB.D of particle
`T-mgcosalpha-qVB=(mV^(2))/(L)`
`therefore=Tmg[3cosalpha+9/4+(qB)/(2mg)sqrt(qL(9+8cosalpha)]]`
Here `0 lt alpha lt 180^(@)`
`Trarr0 atcos alpha=-1impliesB=3/2(mg)/(qsqrt(gL))` And
`T=mg[3 cos alpha+9/4+3/4sqrt(9+8cosalpha)]`
For `T_(A)` put `alpha=0` and for `T_(c)` put `alpha=90`
So `(T_(A))/(T_(C))=(21+3sqrt(17))/(18)`