A ring of mass $m$ and radius $r$ is rotated in uniform magnetic field $B$ which is perpendicular to the plane of the loop with constant angular velocity ${\omega }_0$. Find the net ampere force on the ring and the tension developed in the ring if there is a current i in the ring. Current and rotation both are clockwise.

A long non-magnetic cylindrical conductor with inner radius a and outer radius $\mathrm{b}$ carries a current $\mathrm{I}$. The current density in the conductor is uniform. Assume end effects can be neglected.

(a) Prove by the Biot-Savart law that the magnetic field, if any, at a point is always tangential to the circle passing through the point, with the centre of the circles on the axis of the cylinder.

(b) Find the magnetic field due to the current as a function of radius.

(i) inside the hollow space $(\mathrm{r} < \mathrm{a})$ (ii) within the conductor $(\mathrm{a} < \mathrm{r} < \mathrm{b})$(iii) outside the conductor $(\mathrm{r} > \mathrm{b})$

(c) A beam of particles, each with positive charge $\mathrm{q}$ and mass $\mathrm{m}$ travels with initial velocity $\mathrm{v}$ anti-parallel to the direction of the current. Assume that the length of cylinder is $\mathrm{L}$. Find the deflection of a particle as a function of its initial distance r from the axis if $\mathrm{r} > \mathrm{b}$ when it goes from one end to other end. You can assume that the velocity is high enough such that the velocity is constant and the deflection is small.