Question

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 as shown in above figure. A uniform magnetic field extends over a circular region within the rim. It is given by,
$B=-B_{0}k(r\le a;a<R)$, or $B=0$ (otherwise).
What is the angular velocity of the wheel after the field is suddenly switched off?

Answer 1

Line charge per unit length, $\lambda =Q/2\pi r$

at distance r, the magnetic force is balanced by the centripetal force.

So, $BQv=M{v}^2/r$

also the angular velocity, $\omega =v/R=B2\pi \lambda r^2/MR$

So for $r\le a;a<R$, we get:

$\omega =-2B_o{a}^2\lambda k/MR$