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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?
Line charge per unit length $=\lambda=\frac{\text { Total charge }}{\text { Length }}=\frac{Q}{2 \pi r}$
Where,
r = Distance of the point within the wheel
Mass of the wheel = M
Radius of the wheel = R
Magnetic field, $\vec{B}=-B_{0} \hat{k}$
At distance r,themagnetic force is balanced by the centripetal force i.e.,
$B Q v=\frac{M v^{2}}{r}$
Where,
$v=$ linear velocity of the wheel
$\therefore B 2 \pi r \lambda=\frac{M v}{r}$
$v=\frac{B 2 \pi \lambda r^{2}}{M}$
$\therefore$ Angular velocity, $\omega=\frac{v}{R}=\frac{B 2 \pi \lambda r^{2}}{M R}$
For $r \leq$ and $a
$\omega=-\frac{2 \boldsymbol{A}_{0} \quad a^{2} \lambda}{M R} \hat{k}$
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