- (0 pts) Carefully study section 9.1 in Jackson.
- (2 pts) Provide the details leading to (9.3)-(9.5).
- (2 pts) Argue that in the far zone () the spatial derivatives of a function that contains an exponential factor like the one in (9.3) are dominated by the derivative of that factor (small parameter ).
- (1 pts) Prove (9.7).
- (2 pts) Provide the details leading to (9.8), including a clear argument as to why is replaced by in the denominator. [Note there are two parameters: small and large in the far zone.]
- (3 pts) Note the asymptotic dependence of on in (9.8), remembering that . Use the argument from step #3 to show that in the far zone and . [You need to reconstruct the function in time domain from its frequency amplitudes. Assume we are in the far zone for all relevant frequencies.]
- (0 pts) Carefully study section 9.2 in Jackson.
- (2 pts) Show how one gets from the left-hand side of (9.14) to its right-hand side.
- (2 pts) Show the details leading from (9.16) to (9.19).
- (2 pts) Show the details leading from (9.21) to (9.22).
- (1 pts) Explain what it means for all components of to have the same phase [see above (9.23)] and how it leads to (9.23). Make a sketch showing the meaning of the angle .
- (4 pts) Eq. (9.16) shows the amplitude of at a given frequency. Integrate them to show , where the subscript “ret” means that the time argument of is retarded, .
- (3 pts) Use the results of steps #6 and #12 to show that in the far zone .
- (2 pts) Suggest two examples of a dipole radiator that radiates at a frequency with the dipole moment oriented along the axis: (1) one involving a moving charge, and (2) one involving a piece of wire carrying a current.
- (5 pts) Consider a radiator that has a dipole moment lying in the plane and rotating around the axis with an angular velocity : . Using the results of steps #13 and #6, calculate the angular distribution of the radiated power in the far zone by explicitly averaging the Poynting vector over time.
- (1 pts) For the radiator introduced in step #15, express the time-dependent dipole moment using a complex amplitude, as in . Note that the frequency is equal to the angular velocity in this case.
- (5 pts) Use (9.22) and the result of step #16 to find the angular distribution of the radiated power in the far zone. Compare with step #15. If the answers disagree, reconcile the solutions.
- (3 pts) Find the total radiated power for the rotating dipole considered in steps #14-17 by integrating the angular distribution found in steps #15 and #17.
- (2 pts) Find the total radiated power using (9.24) instead. Reconcile any disagreements with step #18.
- (3 pts) Consider a closed system of non-relativistic charged particles with the same charge-to-mass ratio. Show that for this system is proportional to the velocity of its center of mass.
- (2 pts) Explain why the result of step #20 implies that such a system of particles does not generate electric dipole radiation.
Total: 47 points.