(a) Derive the expression for the torque on a rectangular current carrying loop suspended in a uniform magnetic field.
(b) A proton and a deuteron having equal momenta enter in a region of uniform magnetic field at right angle to the direction of the field. Depict their trajectories in the field.

OR

(a) A small compass needle of magnetic moment ‘m’ is free to turn about an axis perpendicular to the direction of uniform magnetic field ‘B’. The moment of inertia of the needle about the axis is ‘I’. The needle is slightly disturbed from its stable position and then released. Prove that it executes simple harmonic motion. Hence deduce the expression for its time period.

(b) A compass needle, free to turn in a vertical plane orients itself with its axis vertical at a certain place on the earth. Find out the values of (i) horizontal component of earth’s magnetic field and (ii) angle of dip at the place.

A rectangular loop ABCD of dimensions l and b, carrying a steady current is placed in uniform magnetic field as shown in fig; such that normal of the plane is at angle  with the magnetic field lines.

The force F_BC and F_AD on arms BC and AD are equal, opposite and along the axis of the coil, so they cancel each other.

The forces F_AB and F_CD are also equal and opposite, but are not collinear, so they constitute a couple, and the magnitude of the torque can be given as, 

Since, 

Since, magnetic moment m I |A|

; which is the expression of torque. 

b) If a charge particle enters right angle to the direction of magnetic field, it follows a circular trajectory, and radius can be given as,
 

Since momentum are equal and they having equal charges.

So, r_p : r_d = 1:1

OR

If magnetic compass of dipole moment  is placed at angle q in uniform magnetic field, and released it experiences a restoring torque.

Torque is given by, 

b) If compass needle orients itself with its axis vertical at a place, then