A line charge λ per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R.

Solution:

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

$\omega=-\frac{2 \boldsymbol{A}_{0} \quad a^{2} \lambda}{M R} \hat{k}$