1. (0 pts) Carefully study section 9.3 in Jackson.

2. (2 pts) Note that the far-zone limit $$kr\gg 1$$ eliminates the term $$1/r$$ in the brackets in (9.30) and the second term in the brackets in (9.33) and in (9.38). Rewrite (9.35) and (9.36) in this limit.

3. (3 pts) An example of a magnetic dipole radiator is a circular loop of wire of radius carrying AC current $$I(t)=I_0\cos\omega t$$. Use a reference frame with the $$z$$ axis perpendicular to the plane of the loop and passing through its center. Calculate the complex amplitude of the magnetic dipole moment. At what frequency does the magnetic moment oscillate?

4. (3 pts) Use (9.35), (9.36) and step #2 to find the fields in the far zone for the wire loop from step #3.

5. (3 pts) Find the angular distribution of the radiated power for the wire loop from steps #3.

6. (1 pts) Compare the result of step #5 with the angular distribution of radiated power for an oscillating electric dipole oriented along the z axis.

7. (2 pts) Describe the polarization of the fields in the far zone for the wire loop from step #3 and compare it with the electric dipole radiator from step #6. [Make sure you understand the first paragraph on page 414.]

8. (2 pts) A circular loop of wire of radius $$a$$ carries a constant current $$I$$ and rotates with angular velocity $$\omega$$ around an axis containing its diameter. Align the $$z$$ axis with the axis of rotation. Introduce the complex amplitude for the magnetic dipole moment and find it.

9. (3 pts) Find the angular distribution of the time-averaged power radiated by the wire loop from step #8.

10. (3 pts) Show that the magnetic moment of a closed system of particles with the same charge-to-mass ratio is proportional to the angular momentum of this system.

11. (3 pts) Show that the magnetic moment of a closed system of any two (different) particles is proportional to the angular momentum of this system.

12. (2 pts) Explain why the results of steps #10 and #11 imply that such systems of particles do not emit magnetic dipole radiation.

13. (2 pts) We now turn to the electric quadrupole term. Prove (9.37) via integration by parts.

14. (3 pts) Prove (9.37) by switching from an integral to a sum over charged particles, recognizing the full time derivative, switching back to an integral, introducing a complex amplitude, and replacing the time derivative by $$-i\omega$$.

15. (3 pts) Verify (9.42), (9.44), and (9.45).

16. (5 pts) Use (9.16), (9.33), (9.38), (9.42) to reassemble the real-time vector-potential in the far zone: $\mathbf{A}(\mathbf{r})=\frac{\mu_0}{4\pi r}\left\{\dot{\mathbf{p}}_{ret}+\frac{1}{c}\dot{\mathbf{m}}_{ret}\times\mathbf{n}+\frac{1}{6c}\ddot{\mathbf{Q}}_{ret}\right\} ,$ where $$\mathbf{Q}$$ is defined in (9.43) and “ret” indicates the regarded time argument.

17. (4 pts) Use $$\mathbf{H}=\frac{\dot{\mathbf{p}}\times\mathbf{n}}{\mu_0 c}$$ in the radiation zone and the corresponding $$\mathbf{E}$$ field to derive the expressions for the time-dependent radiated power given in Jackson problem 9.7.

18. (2 pts) Consider an electric quadrupole radiator for which the quadrupole tensor in the given reference frame is diagonal, oscillates at frequency $$\omega$$, and has $$Q_{xx}=Q_{yy}=-\frac12 Q_0$$ and $$Q_{zz}=Q_0$$. [Note the quadrupole tensor is always traceless.] Provide an explicit example of such a radiator.

19. (5 pts) Use (9.45) to find the angular distribution of the radiated power for the quadrupole radiator from step #18. The angular-dependent factor should come out as $$\sin^2 2\theta$$.

20. (3 pts) A square of side $$a$$, which has charges $$\pm q$$ of alternating signs at its corners, rotates with angular velocity $$\omega$$ about an axis normal to its plane and passing through its center. Work out the time-dependent quadrupole moment of this system, assuming that positive charges are on the $$x$$ axis at $$t=0$$.

21. (3 pts) Rewrite the quadrupole tensor from step #20 in terms of its complex amplitude. At what frequency does this tensor oscillate?

22. (5 pts) Use (9.45) to find the angular distribution of the radiated power for the quadrupole radiator from step #20. The angular-dependent factor should come out as .

23. (8 pts) Three equal point charges are glued to the corners of an equilateral triangle that lies in the $$xy$$ plane and rotates with an angular velocity $$\omega$$ around the $$z$$ axis passing through its center. Show that this system does not emit electric dipole, magnetic dipole, or electric quadrupole radiation.

24. (5 pts) Calculate the total power of the cyclotron radiation emitted by an electron (charge $$e$$, mass $$m$$) moving with a non-relativistic velocity perpendicular to the magnetic field $$B$$.

25. (2 pts) Consider the center-fed linear antenna from Jackson’s section 9.4. Show that this antenna carries an oscillating electric dipole moment, and, therefore, its radiation is dominated by the electric dipole term at $$kd\ll1$$.

26. (3 pts) Follow the calculation of the radiated power (9.56) that is done without the multipole expansion. Expand (9.56) to the leading order in $$kd$$ and show that the resulting angular dependence is that of an electric dipole radiator with the correct orientation in space.