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A line charge , $λ$ per unit length is pasted uniformly on to the rim of a wheel of mass m and radius R. The wheel has light non-conducting spokes and is free to rotate about a vertical axis as shown in figure. A uniform magnetic field extends over a radial region of radius r given by $B = - B_{0} \overset{\land}{k} (r \leq a; a < R) = 0$ (otherwise). What is the angular velocity of the wheel when this field is suddenly switched off?

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$\frac{- 2 B_{0} π a^{2} r}{m R} \overset{\land}{k}$

$\frac{- B_{0} π a^{2} r}{3 m R} \overset{\land}{k}$

$\frac{B_{0} π a^{2} λ}{m R} \overset{\land}{k}$

$\frac{- B_{0} π a^{2} λ}{m R} \overset{\land}{k}$

answer is D.

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Detailed Solution

Induced electric field (in clockwise sense):

$E = \frac{a^{2}}{2 R} \frac{d B}{d t}$

$I ω = \int τ d t$

$\Rightarrow m R^{2} ω = \int q E R d t = \int \frac{q a^{2}}{2 R} \frac{d B}{d t} R d t = \frac{λ 2 π R}{2 R} a^{2} R \int d B$

$\Rightarrow ω = \frac{B_{0} π a^{2} λ}{m R}$

$\Rightarrow \vec{ω} = - \frac{B_{0} π a^{2} λ}{m R} \overset{\land}{k} (b e c a u s e c l o c k w i s e s e n s e)$

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