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.