Re: [Phys-l] phase relationship

[John Denker <jsd@av8n.com>]

On 03/22/2010 08:10 AM, marx@phy.ilstu.edu wrote:
> Very far away from the source (such as antenna) they are in phase, 

For running waves, yes.  For standing waves, not so much.

> but nearby the source, they are not.

Agreed.


There's a simple way of explaining why this must be so.

Consider the world's simplest dipole radiator, namely 
a charge attached to a simple harmonic oscillator.
Without loss of generality assume it is located at the
origin and oscillating in the Z direction.

Now consider how things look to an observer located at
a point R in the near field.  Near field means that the 
distance |R| is small compared to the wavelength of the
radiation.

There's another name for the "near field" assumption.
It's the _electrostatics_ assumption.  If we are in
the near field, that means that we are near enough
and/or the oscillations are slow enough so the |R|/t
is small compared to the timescale t, for any 
timescale of interest ... which is electrostatics.

The quickest way to an interesting result is to consider
a point R on the Z axis.  At this point there will be
a changing electrical field.  You can calculate the
correct value using electrostatics.

Meanwhile, at this point there is *no* magnetic field.
This should be obvious by symmetry.  If the oscillation
is in the Z direction and the propagation is in the Z
direction, the system is rotationally symmetric so
there can't be any Bx or By component.  And there can't
be any Bz component because the only current is in the
Z direction, and the B field lines are always perpendicular
to the current.

So in the near field E' cannot possibly be equal to cB'.

Note that in the far field there is no E or B whatsoever
on the Z axis.  Anything you see on the Z axis must be
in the near field.

===============

To prove that things are in phase in the far field, observe 
that in the far field the spherical wave is locally 
indistinguishable from a plane wave, so it reduces to 
the problem previously solved.  E' is pointwise equal 
to +-cB'.  This is exactly true for a running plane wave 
everywhere ... and true to a good approximation for a 
running spherical wave in the far field.