Ncert Exercises & Solutions

Q 17.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=−B₀k(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 $= λ = \frac{Total charge}{Length} = \frac{Q}{2 πr}$

Where,

r = Distance of the point within the wheel

Mass of the wheel = M

The radius of the wheel = R

The magnetic field, null

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

Where,

v = linear velocity of the wheel

\begin{array}{l} ∴ B 2 πrλ = \frac{Mv}{r} \\ v = \frac{B 2 {πλr}^{2}}{M} \end{array}

\begin{array}{l} ∴ Angular velocity, ω = \frac{v}{R} = \frac{B 2 {πλr}^{2}}{MR} \\ For r \leq a \leq R, we get \\ ω = - \frac{2 {πB}_{0} a^{2} λ}{MR} \hat{k} \end{array}

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