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Question
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 = − B0 k (r ≤ a; a < R)
= 0 (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off?
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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"`
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