A line charge A 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 = – Bo k (r ≤ a; a < R) = 0 (otherwise].What is the angular velocity of the wheel after the field is suddenly switched off?

Line charge per unit length = λ = Total Charge/Length = Q/2πr

Where, r = Distance of the point within the wheel

Mass of the wheel = M

Radius of the wheel = R

Magnetic field, B = -B₀ k

At distance r, the magnetic force is balanced by the centripetal force i.e.,

BQv = Mv²/r

Where,

v = linear velocity of the wheel

Therefore, B2πrλ = Mv/r

v =B2πλr²/M

Therefore ,Angular velocity , ω =v/R = B2πλr²/MR

For r ≤ a ≤ R, we get

ω = -(2πB₀a²λ/MR) k^