Re: [Phys-L] transverse waves

[John Denker <jsd@av8n.com>]

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

> Question:  What's wrong with the following diagram:
>   http://www.kshitij-iitjee.com/Study/Physics/Part6/Chapter34/70.jpg
> from
>   http://www.kshitij-iitjee.com/production-of-electromagnetic-waves-by-an-antenna
>
> and similarly
>   https://en.wikipedia.org/wiki/Dipole_antenna#/media/File:Felder_um_Dipol.svg
> and
>   https://en.wikipedia.org/wiki/Dipole_antenna#/media/File:Dipole_xmting_antenna_animation_4_408x318x150ms.gif
> from
>   https://en.wikipedia.org/wiki/Dipole_antenna
> and reused here
>   http://physics.stackexchange.com/questions/20331/understanding-the-diagrams-of-electromagnetic-waves
>
> Given the amount of qualitatively wrong information out there, it's
> a miracle that students ever understand anything.

Answer:  Those diagrams purport to show the radiation field, but
they're not transverse.

BTW there's also this:
  https://www.wired.com/wp-content/uploads/2016/01/magnetic-waves-472170802-1024x788.jpg
from
  http://www.wired.com/2016/01/use-code-to-create-sweet-3-d-visualizations-of-electric-fields/
I have no idea what it's trying to represent.



  Today I am discussing the far field, i.e. the radiative field.
  (This is different from the near field, i.e. the reactive field.)

It's not super-obvious how to draw the E-field lines correctly for
a wave in polar coordinates, such as the outgoing wave from a
small dipole at the origin.  It's clear what the E-field is doing
near the equator of the radiation pattern.  The aforementioned
loopy diagrams might even be correct near the equator.  Also it's
clear that the E-field is zero at the poles.  So the we are left
with the question, what happens in between?

Presumably you were told that the field lines can't just end in
empty space.  People with a not-very-deep understanding assume
this means the field lines have to bend over somehow and join to
form loops, as in the aforementioned diagrams.  That might seem 
almost plausible for a periodic wavefunction, but what if it's not
periodic?  What if the dipole moment just ramps up from 0 to 1 and
never comes back down?  How far does the field line have to to in
the longitudinal direction before it finds something to join up
with?

You cannot use the Coulomb field to join up the loops, because
the radiation field goes like 1/r whereas the Coulomb field
goes like 1/r^2.

Also, it turns out that the radiation field really is transverse,
even in polar coordinates, so the idea of looping back via some
longitudinal hookup is dead on arrival.  You know it has to be
transverse;  every term in the equation is r cross something.
  https://en.wikipedia.org/wiki/Electromagnetic_wave_equation#Multipole_expansion

The right answer is shown in this diagram:
  https://www.av8n.com/physics/img48/transverse-spherical-field.png

The center of the sphere is shown by a black dot. Imagine a tiny
dipole antenna there.

The B-field lines are shown in green.  I show only two octants of
these lines (a different two octants).  I show 10 semicircles out
of a possible 19 circles.  The lowest visible semicircle follows
the equator.  Note that the density of B field lines is largest
near the equator and goes to zero at the poles.

The E-field lines are shown in red. I show only two octants of
the lines. That is, I show only 5 out of 16 equally-spaced lines.
Otherwise the diagram would be too cluttered.

There are two half-right things you can imagine:
  -- The field lines can't just peter out.  Field lines, if they
   end at all, have to end on a charge.  Yet we know the field
   strength goes to zero at the poles.  So in some sense the
   field lines must peter out.
  -- The field lines get closer together as they approach the
   poles.  Other things being equal, this would mean that the
   field was getting stronger.

If you put those two ideas together, they partially cancel and
partially don't, in just the right way so that the field goes
to zero at the poles ... and ∇·E is zero everywhere on the
sphere (for r>0).

The strength of the E-field is indicated by the width of the
red line.  It is also indicated by the density of B-field lines.
In a traveling wave, E is always locally proportional to B.
For a simple dipole radiation pattern as shown here, both are
proportional to the cosine of the latitude.  It's ridiculously
easy to verify all these assertions.  Just write the divergence
operator in spherical polar coordinates.
  http://isites.harvard.edu/fs/docs/icb.topic970148.files/Spherical_coord.pdf

This is one of those things that is easier to verify than to
figure out ab_initio.  To most people, it's not obvious that 
the stripy-melon field pattern
   https://www.av8n.com/physics/img48/transverse-spherical-field.png
has ∇·E everywhere.  It's not obvious that it's even possible
to construct a field that is everywhere transverse and everywhere
divergence-free.  But once you see it, it makes sense.

Another amusing observation:  The area bounded by a pair of
B-field lines and a pair of E-field lines is the same everywhere
on the sphere.  As the E-field lines get closer together the
B-field lines get farther apart.  As the wave propagates, this
area grows in proportion to r^2.  Hint: Poynting vector.

It's also amusing that the B-field lines are equally spaced
in the z-direction (not equally spaced in latitude).

If you download the following program you can rotate the 
diagram in 3D:
  https://www.av8n.com/physics/spherical-field-lines.py
which requires:
  https://www.av8n.com/physics/spherical-field.png

It's probably straightforward to make the program run in
a browser window, platform-independently, but that's requires
more fussing than I feel like doing right now.

Again:  Given how much qualitatively wrong information there
is out there, it's a miracle that students ever understand
anything.  You could go nuts trying to understand the loopy
diagrams cited above.  The right answer is so very much 
simpler.