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Magnetic Dipole and Electric Quadrupole Radiation Fields

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

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

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



Subsections
next up previous contents
Next: Magnetic Dipole Radiation Up: Radiation Previous: Energy radiated by the   Contents
Robert G. Brown 2014-08-19