Radiation outside the source

Outside the bounding sphere of the source,

$$\vA(\vr) = ik \sum_L H^+_L(\vr) \int_0^\infty \mu_0 \vJ(\vr') J_L(\vr')^{(\ast)} d^3r' .$$ (13.44)

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 $\vJ(\vx')$ is) but they are very definitely computable with a computer!

Now, we must use several interesting observations. First of all, $J_L(\vr)$ 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 $\vJ({\bf r})$ 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.