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Q6.17   A line charge \lambda 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=\: -B_{0}K\: \: \: \: (r\leq a;\: a< R) =0\: \: \: \: \: \: \: \: \: \: (otherwise) 
What is the angular velocity of the wheel after the field is suddenly switched off?

            

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best_answer

Given:

The radius of the wheel =R 

The mass of the wheel = M

Line charge per unit length when the total charge is Q

\lambda=\frac{Q}{2\pi r}

Magnetic field :

B=\: -B_{0}K\: \: \: \: (r\leq a;\: a< R) =0

Magnetic force is balanced by centrifugal force when v is the speed of the wheel that is,

BQv=\frac{mv^2}{r}

B2\pi r\lambda=\frac{Mv}{r}

v=\frac{B2\pi \lambda r^2}{M}

The angular velocity of the wheel is given by 

w=\frac{v}{r}=\frac{B2\pi \lambda r^2}{MR}

when (r\leq a;\: a< R) 

w=-\frac{B2\pi \lambda a^2}{MR}

Hence, It is the angular velocity of the wheel when the field is suddenly shut off.

Posted by

Pankaj Sanodiya

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