NEC-LIST: A last comment on radiation

Hello,

I'll give one last response to the issues that Dr. Dreher has raised.
If anyone wants to discuss anything further off line, I'd be glad to
do so. Rather than fully including each of his statements, I'll just
use the first line or two.

> the discussion about diploe radiation and charge oscillations
> is very interesting and challenging, but I guess it could be
> continued endlessly.

I would generally agree with Dr. Dreher's observations, but also note
that there is some physical truth and reality that our models, if
correct, should properly represent.

> 1.) O.k., Maxwell's equations do not deal directly with physical electrons.

By limiting oneself to the frequency domain, one possibly also limits
understanding made possible by separating cause and effect due to
causality as is possible in the time domain. And why do electrons
need to be brought into the picture? They're not required in a theory
limited to perfect conductors.

> 2.) The conductivity in a metal is given by: conductivity =
> Q*mobility*(electron density).

In discussing mobility, I had in mind the use of equivalent current
and charge. I should probably not have used a term associated with
physical charge. Of course, in using equivalent sources, there seems
to be no reason not to assume the mobility can become infinite in
discussing a PEC; whatever is needed for the conductivity to become
infinite in that case seems to be appropriate, whether infinite
mobility or density. It is important to note that charge and mass
must be properly included when dealing with a medium such as a plasma,
where an electron plasma frequency and electron cyclotron frequency do
have physical meaning and can be measured in various ways. However, a
PEC in the sense normally meant, as I understand it, is not a real
physical medium, but rather something assumed for simplification in
solving boundary-value problems.

> 3.) I strongly disagree: Propagating waves are mathematical
> solutions of Maxwell's equations, but the same holds for
> eigensolutions or modes, if suitable boundary conditions apply. If
> you measure the field of a standing wave, the sensor

The sensor measures the superposion of the various component
quantities to which it is sensitive, but I'm not sure what that
proves. The physical reality, it seems to me, is that a more complex
phenomenon, a standing wave, can be decomposed into two propagating
waves moving at constant speed. Perhaps I'm missing something, but I
don't see how adding two such waves that have charges moving at
constant speed somehow then produces charges that oscillate
physically.

> 4.) As already mentioned, the charges do not oscillate with equal
> amplitude along the wire and are related to the currents just by the
> continuity relation. There is no contradiction to radiated power by
> these equivalent currents.

It may be worthwhile here to explain why I've come to view the problem
of a sinusoidal current filament (SCF) in the way previously
described, i.e., as two counter-propagating Q/I waves moving at
constant speed. When I first began to pose the question about why,
where and how much radiation comes from a physical PEC object, or
perhaps more simply, a SCF, the explanation most-often offered was
"oscillating charge." This observation at first thought seemed a
reasonable, though not very quantitative one. But when looking at the
SCF more closely, I became aware of the fact that the total power
radiated by such a source of length L, longer than a wavelength or
two, oscillated between two limits that themselves grow as Log(kL).
By comparison, it's worth noting that a spatially uniform,
time-harmonic current of length L radiates a total power that grows in
proportion to L. Why this difference between the two current
distributions?

Let's ask what is implied by an assumption that the charges do
actually oscillate along the SCF. Upon applying the continuity
equation to a sinsusoidal current filament (SCF) where the current is
given by sin[k(L/2 - |z|)] one obtains a similar, but spatially
shifted, charge wave. Since the maximum current aplitude along the
filament is unity and repeats periodically, the charge density will
also exhibit a similar spatial behavior, repeating periodically. If
there is some amount of "oscillation" per wavelength that produces
some amount of radiation, then over several wavelengths one should
expect that the total radiated power would be several times the amount
that comes from one wavelength since it would appear that one
wavelength of Q/I is no different from any other. Thus, it would seem
that the total power radiated by a SCF L wavelengths long should be
L/x times that from one x wavelengths long. But it can be shown
analytically and in other ways that the amount of power radiated by
the above current filament grows in proportion to log(kL) rather than
in proportion to L. How can this log(kL) behavior be explained?
Apparently, even if one accepts that constant-speed,
counter-propagating Q/I waves do produce oscillating charges, not all
oscillation is equally effective in producing radiated power.

As a somewhat different approach to the problem I would suggest than
anyone interested look at the article "On Radiation from Antennas" by
S. A. Schelkunoff and C. B. Feldman, in the Proceedings of the IRE,
November 1942, pp. 511-516. In this article the authors develop an
expression for a position-dependent radiation resistance of a
filamentary current which they then specialize to a SCF given by
Isin[k(L/2 - |x|)].

For the special case where L is an odd integer number of half
wavelengths, their expression for the radiation resistance per unit
length simplifies to

    R(x) = 60h/[(L/2)^2 - x^2].

When this expression is multiplied by I(x)^2/2 (something they didn't
do in the article), and examined as a function of x, the results are
essentially identical to those obtained from FARS. Rather
interesting. Furthermore, it's possible to generalize their
expression for the radiation resistance to arbitrary lengths, results
from which also match FARS for the distributed power. This will be
discussed in my June "PCs for AP" column. One might infer that
[I(x)^2]R(x)/2 represents the power radiated per unit length at point
x. Of course, this is not uniform along the filament, varying due to
both I(x) and R(x).

However, I readily admit to not having a fully self-consistent
understanding of even the simple SCF, especially how it compares with
a wire modeled as a boundary-value problem.

Best wishes,

Ed

--
Dr. Edmund K. Miller
3225 Calle Celestial
Santa Fe, NM 87501-9613
505-820-7371 (Voice & FAX)
e.miller_at_ieee.org