Outside the bounding sphere of the source,

(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
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.