Outside the bounding sphere of the source,

(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 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.