Outside the bounding sphere of the source,
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 and may well be daunting to a person armed with pen and paper (depending on how nasty is) but they are very definitely computable with a computer!
Now, we must use several interesting observations. First of all, gets small rapidly inside as increases (beyond ). This is the angular momentum cut-off in disguise and you should remember it. This means that if is sensibly bounded, the integral on the right (which is cut off at ) will get small for ``large'' . In most cases of physical interest, by hypothesis and we need only keep the first few terms (!). In practically all of these cases, the lowest order term () will yield an excellent approximation. This term produces the electric dipole radiation field.