Radiation outside the source

Outside the bounding sphere of the source,

$$\mbox{\boldmath$A$}(\mbox{\boldmath$r$}) = ik \sum_L H^+_L(... ...x{\boldmath$r$}') J_L(\mbox{\boldmath$r$}')^{(\ast)} d^3r' .$$ (11.117)

At last we have made it to Jackson's equation 9.11, but look how elegant our approach was. Instead of a form that is only valid in the far zone, we can now see that this is a limiting form of a convergent solution that works in all zones, including inside the source itself! The integrals that go into the $C_L(r)$ and $S_L(r)$ may well be daunting to a person armed with pen and paper (depending on how nasty $\mbox{\boldmath$J$}(\mbox{\boldmath$x$}')$ is) but they are very definitely computable with a computer!

Now, we must use several interesting observations. First of all, $J_L(\mbox{\boldmath$r$})$ gets small rapidly inside $d$ as $\ell$ increases (beyond $kd$). This is the angular momentum cut-off in disguise and you should remember it. This means that if $\mbox{\boldmath$J$}({\bf r})$ is sensibly bounded, the integral on the right (which is cut off at $r' = d$) will get small for ``large'' $\ell$. In most cases of physical interest, $kd << 1$ by hypothesis and we need only keep the first few terms (!). In practically all of these cases, the lowest order term ($\ell = 0$) will yield an excellent approximation. This term produces the electric dipole radiation field.