1. Two point charges, q each, are attached to the ends of a rigid rod of length . The rod is rotated with an angular speed  about an axis through the centre of the rod and perpendicular to it. Calculate the electric and the magnetic dipole moment  of the system taking the centre of the rod as the origin. What is the total power radiated by this system?

The electric dipole moment about the centre is zero as the position vector of the two (equal) charges are always directed oppositely.  Magnetic dipole moment can be calculated by observing that charge 2q is rotating with a time period , which is equivalent to a current. The area of the current loop is . Thus the magnetic moment is given by . Both the electric and the magnetic dipoles are constant in time and hence there is no radiation emitted by the system.

2. An electric dipole of strength  initially lies along the x-axis centred at origin. The dipole rotates in the x-y plane about the z-axis at an angular frequency  Calculate the radiation field and the radiated power as seen by an observer in the x-z plane at an angle  to the z-axis.

The dipole vector can be written as

Since the observer is in x-z plane, y-direction is perpendicular to the direction of observation . We can write . Note that


Using these, we can write the complex magnetic field as

The corresponding electric field is given by

The average Poynting vector is given by



The power flowing out per unit solid angle is given by



The total power radiated is given by integrating the above over the solid angle. Since there is no azimuthal dependence, we have,

3. Neutron stars are a class of very dense stars consisting primarily of neutrons.  The neutron star rotates with an angular frequency  about an axis which makes an angle  with the direction of the magnetic moment.  Assume the radius of the star to be 10 km and the angular frequency of rotation to be 100 rad/s about the z-axis which makes angle of  with the magnetic moment vector of the neutron star.  Take the strength of the magnetic field at the magnetic equator to be T. Calculate the total power radiated.

The magnetic field of a dipole at  is given by the expression

where is the angle  between   and . At the equator,  so that the strength of the magnetic field is


Substituting the given values, the magnitude of the magnetic field is  Am².



As the star rotates about an axis which does not coincide with the magnetic moment vector, the direction of the magnetic moment vector  precesses about the z-axis. This accounts for the radiation. It was pointed out that the magnetic dipole radiation power is proportional to the square of the second derivative of the magnetic moment. We can write

The magnetic moment vector as



It is seen that . The power radiated is given by



Substituting numerical values, one gets the power radiated to be  W.