A small charged ball having mass m and charge q is suspended from a rigid support by means of an inextensible thread of length l. It is made to rotate on a horizontal circular path in a uniform, time independent magnetic field of induction B which is directed upward. The time period of revolution of the ball is `T_0.` If the thread is always stretched, calculate the radius of circular path on which the ball moves.

Explanation

The situation is shown in
Fig.
`Tcos theta=mg….(i)`
and `T sin theta-qvB=((mv^2)/r)….(ii)`
From Eq. (ii),

`((mg)/(cos theta)) sin theta-qBr omega=mr omega^2`
[`:' v=r omega` and `T=(mg)/(cos theta)` from equation(i)]
From figure, `sin theta=r/l` and `cos theta=(sqrt((l^2-r^2)))/(l)`
`:. (mgr)/(sqrt(l^2-r^2))-qBr omega=mr omenga^2`
or `1/(sqrt(l^2-r^2))=(omega^2)/g+(qBomega)/(mg)`
`(l^2-r^2) =((1//omega)^2)/ [(omega//g) +(qB//mg)]^2`
or `(l^2-r^2) =(T_0//2pi)^2/ [(2pi//gT_0) +(qB//mg)]^2`
`:. r=[l^2-(T_0//2pi)^2/ {(2pi//gT_0) +(qB//mg)}^2]^(1//2)`.