A line charge `lambda` per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by,
`B=B_(0)k" "(ra le a,a lt R)`
=0 `" "` (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off?

Explanation

Line charge per unit length `=lambda=("Total charg e")/("Leng th")=(Q)/(2pir)`
Where,
r, Distance of the point within the wheel
Mass of the wheel = M
Radius of the wheel = R
Magnetic field, ` bar(B)=-B_(0) hat k`
At distance r,themagnetic force is balanced by the centripetal force i.e.,
`BQv=(Mv^(2))/(r)`
Where,
v= linear velocity of the wheel
`therefore B2pir lambda=(Mv)/(r)`
`v=(B2pilambdar^(2))/(M)`
`therefore` Angular velocity , `omega=(v)/(R)=(B2pilambdar^(2))/(MR)`
For `r le` and `a lt R`, we get
`omega=-(2B_(0)a^(2)lambda)/(MR)hatk`