Magnetic Dipole and Electric Quadrupole Radiation Fields

The next term in the multipolar expansion is the $\ell = 1$ term:

$${\bf A}(\mbox{\boldmath$x$}) = i k \mu_0 h_1^+(kr) \sum_{m ... ...\boldmath$J$}({\bf x'}) j_1(kr') Y_{1,m}(\hat{r'})^\ast d^3x'$$ (11.140)

When you (for homework, of course)

1. $m$-sum the product of the $Y_{\ell,m}$'s
2. use the small $kr$ expansion for $j_1(kr')$ in the integral and combine it with the explicit form for the resulting $P_1(\theta)$ to form a dot product
3. cancel the $2 \ell + 1$'s
4. explicitly write out the hankel function in exponential form

you will get equation (J9.30, for - recall - distributions with compact support):

$${\bf A}(\mbox{\boldmath$x$}) = \frac{\mu_0}{4\pi} \frac{e^... ... \mbox{\boldmath$J$}({\bf x'}) ({\bf n} \cdot {\bf x'}) d^3x'.$$ (11.141)

Of course, you can get it directly from J9.9 (to a lower approximation) as well, but that does not show you what to do if the small $kr$ approximation is not valid (in step 2 above) and it neglects part of the outgoing wave!

There are two important and independent pieces in this expression. One of the two pieces is symmetric in $\mbox{\boldmath$J$}$ and $\mbox{\boldmath$x$}'$ and the other is antisymmetric (get a minus sign when the coordinate system is inverted). Any vector quantity can be decomposed in this manner so this is a very general step:

$$\mbox{\boldmath$J$}({\bf n} \cdot {\bf x'}) = \frac{1}{2} [... ...{1}{2} ({\bf x'} \times \mbox{\boldmath$J$}) \times {\bf n}.$$ (11.142)