A line charge λ 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 = − B0 k (r ≤ a; a < R)
= 0 (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off?
Solution
Line charge per unit length
Where,
r = Distance of the point within the wheel
Mass of the wheel = M
Radius of the wheel = R
Magnetic field, `vecB=-B_0hatk`
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 B2pirlambda=(Mv)/r`
`v=(B2pilambdar^2)/M`
∴Angular velocity, `omega=v/r=(B2pilambdar^2)/(MR)`
For r `<= ` and a <R, we get :
`omega=-(2piBa^2lambda)/(MR)hatk`
