[2008EM/HMW-11]

IMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008 MM : 20 Set-XI : Tutorial-VII Induction
 
       [$\oslash$] Read the following two questions and select hints from a list at the end. Arrange the useful hints in an ordered sequence and solve the problems.

1. A wooden cylinder of mass $m=0.5$kg, radius $R=3$cm, length $\ell=10$cm, is placed on an inclined plane. It has 10 turns of wire wrapped around it longitudinally so that the plane of the wire contains the axis of the cylinder and is parallel to the inclined plane, see Figure. Assuming no friction, what is the current that will prevent the cylinder from rolling down the inclined plane in presence of a uniform magnetic field of 0.5T?. Describe what happens if the block is a rectangular instead of a cylindrical one? What will be the current that will prevent the block from moving down the plane?
2. A square wire of length $L$, mass $m$, and resistance $R$ slides without friction down parallel rails of negligible resistance, as in . The rails are connected to each other at the bottom by a resistance-less rail parallel to the wire so that the wire and rails form a closed rectangular conducting loop. The plane of the rails makes an angle $\theta$ with the horizontal, and a uniform vertical magnetic field $\vec{B}$ exists in the region.

1. Show that the wire acquires a steady state velocity of magnitude $$v= \frac{mgR\sin\theta}{B^2 L^2\cos^2\theta}$$
2. Show that the above result is consistent with conservation of energy.
3. What changes will be necessary in the above results, if the direction of magnetic field is reversed?

Food for your thought

1.  Q[1]
   1. If there is no rolling, the cylinder will not move, the required condition must be follow if the value of the component of all the forces along the inclined plane is equated to zero.
   2. We must compute the torque on the loop due to the current and equate it to the torque of the remaining forces.
   3. The cylinder will always slip, but may or may not roll depending on the strength of the magnetic field.
   4. The required condition cannot be obtained by balancing the component of all the forces along the inclined plane, but we must use something else.
   5. Find the angle between the normal to the loop and the magnetic field. This is needed to compute the magnetic force opposing the rolling.
   6. There cannot be a nonzero {magnetic force} on the cylinder/ rectangular block.
2. Q[2]
   1. The force opposing the motion can be computed by $I\vec{l}\times\vec{B}$, where the value of the current $I$ is to be figured out.
   2. Find the rate of change of flux cut by the rod as it moves down the plane. This is needed to compute the induced emf.
   3. There are no frictional forces so steady state is not possible.
   4. When the rod moves with steady velocity all the forces on the rod must balance.
   5. The angle between the velocity of the rod and the magnetic field is \par (i) $\pi/2$ (ii) $\pi/2+ \theta$ (iii) $\pi/2- \theta$ (iv)$\theta$.
   6. The angle between the induced current in rod and the magnetic field is \par (i) $\pi-\theta$ (ii) $\pi/2+ \theta$ (iii) $\pi/2- \theta$ (iv)$\pi/2$.
   7. List all the forces on the rod when it is moving.
   8. There will be an induced current which will oppose the motion.
   9. As the rod moves its gravitational potential energy decreases and conservation of energy requires that the kinetic energy will increase. Therefore, it cannot be in a steady motion with constant velocity.