Tutorial: Long-wave limit and electric dipole radiation

1. (0 pts) Carefully study section 9.1 in Jackson.

2. (2 pts) Provide the details leading to (9.3)-(9.5).

3. (2 pts) Argue that in the far zone ($$r\gg\lambda$$) 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 $$1/kr$$).

4. (1 pts) Prove (9.7).

5. (2 pts) Provide the details leading to (9.8), including a clear argument as to why $$|\mathbf{r}-\mathbf{r}'|$$ is replaced by $$r$$ in the denominator. [Note there are two parameters: small $$kd$$ and large $$kr$$ in the far zone.]

6. (3 pts) Note the asymptotic dependence of $$\mathbf{A}(\mathbf{r})$$ on $$\mathbf{r}$$ in (9.8), remembering that $$\mathbf{n}=\mathbf{r}/r$$. Use the argument from step #3 to show that in the far zone $$\mathbf{B}(\mathbf{r},t)=\frac{1}{c} \dot{\mathbf{A}}(\mathbf{r},t)\times\mathbf{n}$$ and $$\mathbf{E}=Z_0\mathbf{H}\times\mathbf{n}$$. [You need to reconstruct the function in time domain from its frequency amplitudes. Assume we are in the far zone for all relevant frequencies.]

7. (0 pts) Carefully study section 9.2 in Jackson.

8. (2 pts) Show how one gets from the left-hand side of (9.14) to its right-hand side.

9. (2 pts) Show the details leading from (9.16) to (9.19).

10. (2 pts) Show the details leading from (9.21) to (9.22).

11. (1 pts) Explain what it means for all components of $$\mathbf{p}$$ 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 $$\theta$$.

12. (4 pts) Eq. (9.16) shows the amplitude of $$\mathbf{A}$$ at a given frequency. Integrate them to show $$\mathbf{A}(\mathbf{r},t)=\frac{\mu_0}{4\pi r}\dot{\mathbf{p}}_{ret}$$, where the subscript “ret” means that the time argument of $$\mathbf{p}$$ is retarded, $$t-r/c$$.

13. (3 pts) Use the results of steps #6 and #12 to show that in the far zone $$\mathbf{H}(\mathbf{r},t)=\frac{\ddot{\mathbf{p}}_{ret}\times\mathbf{n}}{4\pi rc}$$.

14. (2 pts) Suggest two examples of a dipole radiator that radiates at a frequency $$\omega$$ with the dipole moment $$\mathbf{p}_\omega$$ oriented along the $$z$$ axis: (1) one involving a moving charge, and (2) one involving a piece of wire carrying a current.

15. (5 pts) Consider a radiator that has a dipole moment lying in the $$xy$$ plane and rotating around the $$z$$ axis with an angular velocity $$\omega$$: $$\mathbf{p}(t)=p_0(\hat x \cos\omega t + \hat y \sin \omega t)$$. 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.

16. (1 pts) For the radiator introduced in step #15, express the time-dependent dipole moment using a complex amplitude, as in $$A(t)=\mathrm{Re} A_\omega e^{-i\omega t}$$. Note that the frequency is equal to the angular velocity in this case.

17. (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.

18. (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.

19. (2 pts) Find the total radiated power using (9.24) instead. Reconcile any disagreements with step #18.

20. (3 pts) Consider a closed system of non-relativistic charged particles with the same charge-to-mass ratio. Show that $$\dot{\mathbf{p}}$$ for this system is proportional to the velocity of its center of mass.

21. (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.