Radiation Assignment

1. Derive the integral expression for spherical bessel functions in terms of plane waves at the same wavenumber.
2. The addition theorems:

   $$N_0({\bf r} - {\bf r'}) = n_0(k \mid {\bf r} - {\bf r'}) \frac{1}{\sqrt{4 \pi}} = \sqrt{4 \pi} \sum_{L} N_L({\bf r}_>) J_L({\bf r}_<)^\ast$$ (13.89)

   
   
   and

   $$H^\pm_0({\bf r} - {\bf r'}) = h_0^\pm (k \mid {\bf r} - {\bf r'})... ...{\sqrt{4 \pi}} = \sqrt{4 \pi} \sum_{L} H^\pm_L({\bf r}_>) J_L({\bf r}_<)^\ast .$$ (13.90)

   
   
   are derived someplace, for both this special case and for the general case. Find at least one such place (for $L = 0,0$ ), copy the derivation (with acknowledgement), and hand it in. If you work in a group, see how many places you can find it and compare. LEARN what you can from the process, that is, read the text accompanying the derivation(s) you find and try to understand it. Work it out. For extra credit, find in the literature the original paper that derives the general addition theorem. Hints: JMP, Danos and Maximon. Study it.
3. Derive the Green's function for the Helmholtz equation in free space (zero boundary conditions at infinity). Do not use the addition theorem, since you do not (in principle) know its form yet and so do not know that it is a Neumann or Hankel function. Naturally, you can follow Wyld or Jackson or Arfken, but acknowledge your source and show the entire derivation.
4. Make a neat sheet with Everything You Never Wanted To Know About Spherical Bessel/Neumann/Hankel Functions but were Afraid Not To Ask on it. Don't hand it in, this will be your guide through life (for at least a few weeks). Do NOT simply photocopy my notes. Do it by hand. Pool your sheets with those of your friends -- put together everything to make a ``best'' sheet and then photocopy it. I use the term ``sheet'' loosely. I expect it will fill several (it did in my notes).
5. Using the addition theorem derived above (in the form of the Green's function) and the asymptotic relations on your worksheet, derive the static result for the vector potential A we previously obtained for the near field zone (my equation 66). Find the lowest order correction to this expression. This will, of course, involve finding more out about spherical waves than I have so far told you! item Using the same addition theorem and the other asymptotic relations, derive an expression for the v.p. A in the far zone. Is there a correspondance of some sort with our previous result (Jackson 9.9)?
6. Show that
   $${\bf A}(\vx) = i k h_1^+(kr) \sum_{m = -1}^1 Y_{1,m}(\hat{r}) \int \vJ({\bf x'}) j_1(kr') Y_{1,m}(\hat{r'})^\ast d^3x'$$
   is equivalent to
   $${\bf A}(\vx) = \frac{e^{ikr}}{4\pi r} \left ( \frac{1}{r} - ik \right ) \int \vJ({\bf x'}) ({\bf n} \cdot {\bf x'}) d^3x'$$
   for $kd << 1$ .

7. Any vector quantity can be decomposed in a symmetric and an antisymmetric piece. Prove that, in the case of the $\ell = 1$ term derived above, the current term can be decomposed into
   $$\vJ({\bf n} \cdot {\bf x'}) = \frac{1}{2} [({\bf n} \cdot {\bf x'... ...f n} \cdot \vJ) {\bf x'}] + \frac{1}{2} ({\bf x'} \times \vJ) \times {\bf n}$$

8. Evaluate the antisymmetric piece. Show (from the magnetic dipole vector potential) that
   $$\vB = \frac{\mu_0}{4\pi} \left\{k^2 ({\bf n} \times {\bf m}) \tim... ... m}) - {\bf m}] \left ( \frac{1}{r^3} - \frac{ik}{r^2} \right ) e^{ikr}\right\}$$
   and
   $$\vE = -\frac{1}{4\pi} \sqrt{\frac{\mu_0}{\epsilon_0}} k^2 ({\bf n} \times {\bf m}) \frac{e^{ikr}}{r} \left ( 1 - \frac{1}{ikr} \right ).$$

   Remark upon the similarities and differences between this result and the electric dipole result.

9. Next start to evaluate the integral of the symmetric piece. Show that you get:
   $$\frac{1}{2} \int [({\bf n} \cdot {\bf x'}) \vJ + ({\bf n} \cdot {... ...\frac{i\omega}{2} \int {\bf x'}({\bf n} \cdot {\bf x'}) \rho({\bf x'}) d^3x'$$
   The steps involved are:
   1. integrate by parts (working to obtain divergences of $\vJ$ ).
   2. changing $\deldot \vJ$ into a $\rho$ times whatever from the continuity equation (for a harmonic source).
   3. rearranging and recombining.
   Don't forget the boundary condition at infinity!

10. Homemade tables, part II. What you did for spherical bessel functions, do for spherical harmonics. In particular, derive the commutation rules for the raising and lowering operators from the cartesian commutation relations for L. From the commutation rules and $L_z \Ylm = m\Ylm$ derive the (normalized) action of $L_\pm$ on $Y_{\ell,m}$ .

11. Jackson, problems 9.2, 9.3, 9.4