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 (Figure). A uniform magnetic field extends over a circular region within the rim. It is given by,

B = − B_{0} 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 = λ = `"Total charge"/"Length"` = `"Q"/(2π"r")`

Where,

r = Distance of the point within the wheel

Mass of the wheel = M

Radius of the wheel = R

Magnetic field, `vec"B" = -"B"_0hat"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

∴ `"B"2pi"r"lambda = ("Mv")/"r"`

`"v" = ("B"2pilambda"r"^2)/"M"`

∴ Angular velocity, `omega = "v"/"r" = ("B"2pilambda"r"^2)/("MR")`

For r ≤ and a < R, we get:

`omega = -(2pi"B""aa"^2lambda)/("MR")hat"k"`