[Phys-L] spherical waves, near and far --> term by term

[John Denker <jsd@av8n.com>]

On 04/26/2016 02:57 PM, I wrote:
> Today I am discussing the far field, i.e. the radiative field. (This
> is different from the near field, i.e. the reactive field.)

The concepts and terminology of "near-field zone" and "far-field zone"
are conventional and very widely used, but IMHO not optimal.  Instead,
it is better to think of things on a term-by-term basis, rather than
a zone-by-zone basis.  Let me explain:

Start with the equation:
    E	=	   (1/4πє0) k^3	 (n×p)×n    	e^ikr (1/kr)	  (1a)
 	 	−i (1/4πє0) k^3	[3n(n·p) − p]	e^ikr (1/kr)^2	  (1b)
 	 	+  (1/4πє0) k^3	[3n(n·p) − p]	e^ikr (1/kr)^3	  (1c)

where:
	p =	 electric dipole moment    
	 	 including the e^iωt time dependence      
	r =	 radial coordinate                
	n =	 unit vector in the dr direction

Reference: Jackson equation 9.18.

Discussion:

Jackson derives the equation in Cartesian coordinates, but it can
equally well be interpreted in terms of spherical polar coordinates.

The first line of equation 1 is purely transverse. The other lines
are not.

In the limit of small r or small k, only the last line in equation 1
survives. This is the familiar electrostatic dipole.

In the limit of large r or large k (but assuming the source is still
tiny compared to 1/k), only the first line of equation 1 survives.
This is the familiar “far field” radiation pattern.

There are two limits (near and far), but there are three terms, so 
you cannot think of the overall field as simply the near-field term
plus the far-field term.
In particular, the electrostatic dipole term falls off like (1/kr)^3
but the middle term falls off more slowly, namely (1/kr)^2. Let’s be
clear: In the not-very-far-field zone, the dominant correction term
goes like (1/kr)^2, not (1/kr)^3.

Perhaps more importantly, it’s better to think of it on a
term-by-term basis — rather than a zone-by-zone basis. There is a
rigorous distinction between the three terms in equation 1, but there
is no sharp boundary between the near-field "zone" and the
far-field "zone".

FWIW Jackson defines three zones: The near-field zone, the
intermediate zone, and the far-field zone. Sometimes he refers to the
intermediate zone as the "induction" zone, whatever that means.
Still, though, it is better to focus on the terms (rather than the
"zones") — the (1/kr) term, the (1/kr)^2 term, and the (1/kr)^3 term.

The various terms differ only by algebraic factors of kr, not by any
exponentially-large factors. It could be argued that there is no
“zone” where you have purely transverse spherical waves, since by 
the time you get far enough away that the wave is purely transverse,
it’s locally indistinguishable from a plane wave, for all practical
purposes.

Retraction:

On 04/26/2016 01:11 PM, I wrote:

>  -- Redefining what is meant by radiation makes no sense.

>   The reactive field is real and has practical consequences and
>   is not transverse ... but it's not radiation.

I'm not so sure of that anymore.  The literature is inconsistent on
this point.  On the one hand, Jackson, among others, clearly treats
the far-field zone and the radiation zone as the same thing.

However, logic suggests that the near field deserves more respect
than it sometimes gets.  For example, near-field optical microscopy
(NSOM) puts the near-field terms to good use.

As an even more familiar example, if you play music in a small
room, the near-field terms for the bass notes are significant
everywhere in the room. If you are listening to music via headphones,
the near-field terms are significant at all frequencies.

Even if you think the near-field waves are not "radiation" they
are still waves.  Nobody thinks that low-frequency sound waves
are not waves.

  On the other hand, in the context of astronomy (where the recent
  discussions began) the incoming fields are in the far-field limit
  ... the far-far-far-far-field limit.

In any case, a hand-wavy model that captures only the near-field
behavior is a terrible model of radiation in general.  There is
AFAICT no reasonable way of using the (1/kr)^3 term as a basis for
understanding the (1/kr)^2 term, let alone the markedly different
(1/kr) term.
  -- the polarization is different
  -- the dependence on r is different
  -- the dependence on frequency is different