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As we have now seen repeatedly from Chapter J6 on, in a source free
region of space, harmonic electromagnetic fields are divergenceless and
have curls given by:
By massaging these a little bit (recall
and
for
we can easily show that both
and
must be
divergenceless solutions to the HHE:
|
(12.25) |
If we know a solution to this equation for
we can obtain
from its curl from the equation above:
|
(12.26) |
and vice versa. However, this is annoying to treat directly, because of the
vector charactor of
and
which complicate the description (as
we have seen - transverse electric fields are related to magnetic
multipoles and vice versa). Let's eliminate it.
By considering the action of the Laplacian on the scalar product of
with a well-behaved vector field ,
|
(12.27) |
and using the divergenceless of
and
, we see that the scalars
and
also satisfy
the HHE:
|
|
|
(12.28) |
|
|
|
(12.29) |
We already know how to write a general solution to either of these equations
in terms of the spherical bessel, neumann, and hankel functions times
spherical harmonics.
Recall, that when we played around with multipole fields, I kept emphasizing
that electric n-pole fields were transverse magnetic and vice versa? Well,
transverse electric fields have
by definition,
right? So now we define a magnetic multipole field of order L by
Similarly, a electric multipole field of order L (which must be
transverse magnetic) is any solution such that
In these two definitions, and are arbitrary linear
combinations of spherical bessel functions12.1, two at a time. Jackson uses
the two hankel functions in (J9.113)k, but this is not necessary.
Now, a little trickery. Using the curl equation for
we get:
|
(12.34) |
so that
is a scalar solution to the HHE for
magnetic multipolar fields. Ditto for
in the
case of electric multipolar fields. Thus,
|
(12.35) |
etc. for
.
Now we get really clever. Remember that
.
Also,
. We have arranged things just so that if
we write:
we exactly reconstruct the solutions above. Neato! This gives us a
completely general TE, MM EMF. A TM, EM EMF follows similarly with
and
(and a minus sign in the second
equation).
This is good news and bad news. The good news is that this is a hell of a lot
simpler than screwing around with symmetric and antisymmetric vector
decompositions and integrations by parts ad nauseam. The radial part of
the solutions is straightforward, and the angular part is written in a concise
notation. The bad news is we have never seen that notation, good or
bad, ever before. We have two choices. Either we can laboriously crank out
the operator products and curls for each problem as we need to (which is
really just as bad as what we have been doing) or we have to work out the
algebra of these new objects once and for all so we can plug and chug out the
most difficult of answers with comparative ease.
Guess which one we're about to do.
Next: Vector Spherical Harmonics and
Up: Vector Multipoles
Previous: Angular momentum and spherical
Contents
Robert G. Brown
2007-12-28