On 2/23/2012 9:26 PM, Timo Nieminen wrote:
> On Feb 23, 8:35 pm, Jos Bergervoet<
jos.bergerv...@xs4all.nl> wrote:
>> On 2/21/2012 9:15 PM, Timo Nieminen wrote:> On Feb 21, 8:42 pm, Jos Bergervoet<
jos.bergerv...@xs4all.nl> wrote:
>>
>>> There is no "interface as a thing"; there is only a mathematical
>>> boundary that you can place, where there are no dipoles on one side.
>>
>> Let's again look at the other side where the dielectric is.
>> Everything works fine if we add an infinitesimal dissipation
>> in the dielectric. The polarization current then has a tiny
>> in-phase component with the B-field and the time-averaged
>> bulk Lorentz force gives the correct radiation pressure.
>> No extra interface force is present. It is just the very
>> weak bulk force in a very large volume (the penetration
>> depth following from this very small dissipation).
>
> I strongly suspect that this won't give the right answer. For a semi-
> infinite medium with low reflectivity, this looks like it gives
> approximately F=I/c, the force on a perfect absorber.
What else would you want for this "medium with low
reflectivity"? It then is approximately a perfect
absorber so shouldn't this be the force you expect?
Anyhow, the general case with reflectivity R gives
F = (1+R^2) I/c, i.e. all transferred momentum is
(eventually) transferred to the crystal, because no
wave is left after absorption has taken place..
But completely without absorption there would
be less force on the crystal because the Abraham
momentum remains in the wave. In that case:
F = 2 R^2 I/c [force on the crystal]
F = (1-R^2) I/c [creating Abraham momentum]
+ -------------
F = (1+R^2) I/c [total force incoming radiation]
> Given that we _know_ that the reflection isn't sudden at an interface
> for an atomic medium, why look further?
My objective is to compute the Lorentz force from
the B-field on the polarization current, wherever
it occurs! And if it occurs deeper in the medium
then I have to look there. In the case of normal
incidence there is no Coulomb force, so the Lorentz
force is all I need to find the complete momentum
transfer to the crystal.
> Especially since the continuum
> version of such a distributed reflection model works.
But the continuum description is exactly what I'm
looking at! (I prefer to avoid the microscopic
description if possible). The Lorentz force in
the bulk of the dielectric, described by the
continuum limit. That's all. (That's all that's
needed!)
>> It doesn't change the outcome for normal incidence of a
>> wave on loss-less dielectric. Once you make the transition
>> layer thin w.r.t. the wavelength the result smoothly goes
>> back to the stepped-index result.
>
> Yes, that's the point. And we can calculate, without undue difficulty,
> the force on the region over which the change in index happens. And we
> can see that the total force, integrated over this region remains
> constant as we reduce the thickness of this region.
It goes to zero! (The Lorentz force in this transition
layer.) To have a nonzero limit the force would have to
diverge if the thickness shrinks. But it cannot do so
because neither the current nor the B-field are diverging.
So I'm puzzled what remaining force you expect to see!
> So we shouldn't
> assume that it suddenly goes to zero as the thickness becomes zero; we
> should assume that the transition is smooth.
Anyhow, it goes to zero, so we could just as well
have started without the transition region, which
also gives zero. And it hass continuous fields,
so they already _are_ smooth in that case.
> So, we _can_ assume a short, but continuous transition, and get the
> correct answer.
But also without it we get the correct answer
As I said: we can compute 2+2 as a limiting
case (of 2+2+epsilon, for instance). But why?
The correct answer is: "No force in the transition
layer if its thickness goes to zero."
>> So, after a century of doubt,
>> finally the Abraham-Mikowsky controversy has been resolved.
>> (Unless one disagrees..)
>
> The controversy isn't about the total force. Both Abraham and
> Minkowski give the same total force, so calculating the force doesn't
> resolve anything.
It resolves it completely if you accept that the
force gives the momentum transfer to the crystal
and the remaining momentum per volume area is the
momentum in the wave. i think that is a reasonable
point of view. (And the result is the Abraham
momentum!)
I find it surprising why there is a "controversy"
since the whole thing is as simple as described
above!
Also interesting: this Lorentz force only is present
in regions where there either are standing waves, or
the medium is lossy, or the amplitude of the passing
wave is increasing! (The last one is crucial, since
for the crystal momentum you need the history of
forces from the time before the wavefront arrives
to the steady state. The integral gives the acquired
momentum.
> There is no physics in Abraham-Minkowski, only the
> philosophy of what part of the force to call "electromagnetic" (in the
> simple case of a medium completely characterised by a constant
> permittivity and permeability, Minkowski says "It's all
> electromagnetic!").
As I explained, I was just trying to base it all on
the Lorentz force. Obviously the momentum transfer
to the lattice by this force would mean that only
the "left over" momentum in the wave should be
called "electromagnetic". Unless of course there
are mixed terms messing it up (momentum and energy
are quadratic in the amplitude, after all..)
> An atomic medium strongly suggests that Abraham is correct (or at
> least more correct), but that's not new. Again, Gordon 1973. Or our
> newest paper on this (Journal of Optics, 13(4), 044017, 2011).
My simple "Lorentz force" view gives the same! At
least for case 5e) of the earlier list, it leaves
exactly the Abraham momentum in the wave. And my
analysis is not based on an atomic medium but on the
continuum description. Of course my usage of a force
acting on the polarization current is ultimately
justified by saying "these are physically moving
charges at the microscopic level". Anyhow, this does
not "strongly suggest" but it "simply proves"
that Abraham is correct! (More difficult would be
to prove that Minkowsky is necessarily wrong..)
> Assuming an atomic medium makes a bigger difference in the angular
> momentum controversy,
Let's please save that subject for another thread! :-)
(Actually we had that other thread already with Radi.
There I also was only interested in whether I could
understand the forces, not the philosophy.)
-jrb