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Suppose that we are in the near zone. Then by definition
and
This makes the integral equation into the ``static'' form already
considered in chapter 5 (cf. equation (5.32)). We see that
is just the Green's function for the good old
Poisson equation in this approximation and can be expanded in harmonic
functions just like in the good old days:
|
(11.74) |
Note Well: I will use
freely and without
warning in this course. The sum is over all . Hopefully, by
now you know what they run over. If not, read the chapter in Wyld on
spherical harmonics and review Jackson as well. This is
important!
This means that (if you like)
|
(11.75) |
We will use expressions like this (derived from the multipolar
expansion of the Green's function) frequently in what follows. For
that reason I suggest that you study it carefully and be sure you
understand it.
Since (for fixed r outside the source)
we see that this limit is reached (among other times) when
(relative to the size of the source and point of measurement)! But
then the IHE turns back into the Poisson equation (or inhomogeneous
Laplace equation, ILE) as it should, come to think about it. The near
fields oscillate harmonically in time, but are spatially
identical to the fields produced by a ``static'' current with the
given spatial distribution. That's why we also call the near zone the
``static zone''.
Next: The Far Zone
Up: Simple Radiating Systems
Previous: The Zones
Contents
Robert G. Brown
2007-12-28