Discussion:
Fjolsvit
2013-03-15 19:37:02 UTC
Einstein attributed the idea that light has momentum to Maxwell. IIRC, Feynman gives an example of how radiation will push an electron in the direction of the wave. I will look for that when I get a chance.

Is there a decent online discussion of how radiation pressure can be derived from Maxwell's equations? The discussions I've found rely on the QM explanation.

I know it goes something like this. The electric field drives a charged particle in a direction perpendicular to the Poynting vector, and parallel to the electric field, The moving charged particle experiences a Lorentz force due to the magnetic field which is perpendicular to both the Poynting vector and the electric field.

The particle is therefore accelerated in the direction of the pointing vector. That's an heuristic explanation, but it isn't quantitative. It also assumes the Lorentz force law, which I don't believe Maxwell relied on.
Fjolsvit
2013-03-15 20:05:28 UTC
Post by Fjolsvit
Einstein attributed the idea that light has momentum to Maxwell. IIRC, Feynman gives an example of how radiation will push an electron in the direction of the wave. I will look for that when I get a chance.
Is there a decent online discussion of how radiation pressure can be derived from Maxwell's equations? The discussions I've found rely on the QM explanation.
I know it goes something like this. The electric field drives a charged particle in a direction perpendicular to the Poynting vector, and parallel to the electric field, The moving charged particle experiences a Lorentz force due to the magnetic field which is perpendicular to both the Poynting vector and the electric field.
The particle is therefore accelerated in the direction of the pointing vector. That's an heuristic explanation, but it isn't quantitative. It also assumes the Lorentz force law, which I don't believe Maxwell relied on.
I can used the definition of the Poynting vector as power per unit aria. Take the time derivative of Newtonian kinetic energy

d/dt(p/2m) = p/m*f,

which is force times velocity. That is also power.

so <S>=vf.

Now, if I replace v with c, I get the expression shown here:

That argument seems contrived.
Timo
2013-03-15 22:59:28 UTC
Post by Fjolsvit
Is there a decent online discussion of how radiation pressure can be derived from Maxwell's equations? The discussions I've found rely on the QM explanation.
There are plenty of good classical explanations. I don't know of a good online discussion, as in a web page that covers it.

If you want to do it from the Maxwell equations, there's the Heaviside-Poynting method, where you consider the Lorentz force acting on the charge and current density. Since this force is the rate of transfer of momentum from field to charge, you get the usual expression for momentum density of an electromagnetic field (i.e., S/c^2 = ExH/c^2 for free space). Apply this to a plane electromagnetic wave, and you get the radiation pressure.

You can also consider the reflection or absorption of a plane wave by something. From the Lorentz force on induced currents (conduction current or polarisation current), you get the force density. Integrate in the direction of propagation, and you get the radiation pressure. This is easiest for a weak absorber, with negligible reflection.

Other classical methods are:

Maxwell's. In his Treatise. The main part is in vol 2, but uses stuff done in vol 1.

Umov's. Via classical thermodynamics, moving energy has momentum flux E/v, where v is the speed the energy is moved at. Apply this to an electromagnetic wave, and you have EM radiation pressure.

Bartoli's. Consider the Doppler shift on a moving mirror.

You can also take Stefan's law as a starting point, and get the ratio between energy density and pressure of thermal radiation from the exponent of 4.
Hetware
2013-03-16 04:56:39 UTC
Post by Timo
Post by Fjolsvit
Is there a decent online discussion of how radiation pressure can
be derived from Maxwell's equations? The discussions I've found
rely on the QM explanation.
There are plenty of good classical explanations. I don't know of a
good online discussion, as in a web page that covers it.
If you want to do it from the Maxwell equations, there's the
Heaviside-Poynting method, where you consider the Lorentz force
acting on the charge and current density. Since this force is the
rate of transfer of momentum from field to charge, you get the usual
expression for momentum density of an electromagnetic field (i.e.,
electromagnetic wave, and you get the radiation pressure.
You can also consider the reflection or absorption of a plane wave by
something. From the Lorentz force on induced currents (conduction
current or polarisation current), you get the force density.
Integrate in the direction of propagation, and you get the radiation
pressure. This is easiest for a weak absorber, with negligible
reflection.
Maxwell's. In his Treatise. The main part is in vol 2, but uses stuff done in vol 1.
Umov's. Via classical thermodynamics, moving energy has momentum flux
E/v, where v is the speed the energy is moved at. Apply this to an
electromagnetic wave, and you have EM radiation pressure.
Bartoli's. Consider the Doppler shift on a moving mirror.
You can also take Stefan's law as a starting point, and get the ratio
between energy density and pressure of thermal radiation from the
exponent of 4.
I don't understand why this is not treated better in so many sources. I
just checked Menzel's
http://www.amazon.com/Mathematical-Physics-Dover-Books/dp/0486600564 and
his development is opaque to me.

Pauli will require considerable attention due to his
non-(American)standard formulation.

Joos's treatment is way back in Stat Mech.

Ohanian (shockingly) doesn't seem to understand that this is an
important aspect of classical E&M.

Lewin
http://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/energy-and-momentum-in-electromagnetic-waves
defers to rotten photon tomatoes.

The energy and momentum transport of electromagnetic radiation is
clearly an essential aspect of classical E&M.

Timo
2013-03-16 06:10:27 UTC
Post by Hetware
Post by Timo
Post by Fjolsvit
Is there a decent online discussion of how radiation pressure can
be derived from Maxwell's equations? The discussions I've found
rely on the QM explanation.
There are plenty of good classical explanations. I don't know of a
good online discussion, as in a web page that covers it.
I don't understand why this is not treated better in so many sources.
Because engineers don't care? Because it isn't in the textbooks that textbook writers copied their content from?

The books that do Lagrangian field theory typically do electromagnetic momentum, via the usual Lagrangian + Noether = conserved quantities. This includes the many QED/QFT books that do this in their introductory chapter(s) on classical field theory.

Jackson gives the Heaviside-Poynting derivation. Stratton gives a clear and very simple version of it. A lot of other books either ignore it completely, or just give the result (e.g., Griffiths).

Energy, OTOH, tends to be covered in detail much more often.
Hetware
2013-03-16 06:39:17 UTC
Post by Timo
Post by Hetware
Post by Timo
Post by Fjolsvit
Is there a decent online discussion of how radiation pressure can
be derived from Maxwell's equations? The discussions I've found
rely on the QM explanation.
There are plenty of good classical explanations. I don't know of a
good online discussion, as in a web page that covers it.
I don't understand why this is not treated better in so many sources.
Because engineers don't care? Because it isn't in the textbooks that textbook writers copied their content from?
The books that do Lagrangian field theory typically do electromagnetic momentum, via the usual Lagrangian + Noether = conserved quantities. This includes the many QED/QFT books that do this in their introductory chapter(s) on classical field theory.
Jackson gives the Heaviside-Poynting derivation. Stratton gives a clear and very simple version of it. A lot of other books either ignore it completely, or just give the result (e.g., Griffiths).
Energy, OTOH, tends to be covered in detail much more often.
This is why Einstein, Schrodinger, Heisenberg, and even Bohr(, not to
mention Feynman) were uneasy with, or flat-out rejected, the
"conclusions" of QM.

A real treatment of E&M should begin with the electrostatic field;
progress to special relativity (perhaps with a bit of hand-waving about
Maxwell's equations and Galilean Relativity,) and expose the magnetic
field as a consequence of electrostatics and SR. Ohanian is 90% there.
Timo
2013-03-16 21:17:23 UTC
Post by Hetware
Post by Timo
Post by Hetware
Post by Timo
Post by Fjolsvit
Is there a decent online discussion of how radiation pressure can
be derived from Maxwell's equations? The discussions I've found
rely on the QM explanation.
There are plenty of good classical explanations. I don't know of a
good online discussion, as in a web page that covers it.
I don't understand why this is not treated better in so many sources.
Because engineers don't care? Because it isn't in the textbooks that textbook writers copied their content from?
The books that do Lagrangian field theory typically do electromagnetic momentum, via the usual Lagrangian + Noether = conserved quantities. This includes the many QED/QFT books that do this in their introductory chapter(s) on classical field theory.
Jackson gives the Heaviside-Poynting derivation. Stratton gives a clear and very simple version of it. A lot of other books either ignore it completely, or just give the result (e.g., Griffiths).
Energy, OTOH, tends to be covered in detail much more often.
This is why Einstein, Schrodinger, Heisenberg, and even Bohr(, not to
mention Feynman) were uneasy with, or flat-out rejected, the
"conclusions" of QM.
They rejected such because electromagnetic energy gets more coverage in the textbooks than electromagnetic momentum?
Post by Hetware
A real treatment of E&M should begin with the electrostatic field;
progress to special relativity (perhaps with a bit of hand-waving about
Maxwell's equations and Galilean Relativity,) and expose the magnetic
field as a consequence of electrostatics and SR. Ohanian is 90% there.
Why begin with electrostatics? Why not just start from the 4-potential (in the Lorenz gauge) as the 4-divergence of the 4-current? That's simple, and 1/2 of the Maxwell equations (the other half being a tensor identity).
Hetware
2013-03-17 04:18:03 UTC
Post by Timo
Post by Hetware
Post by Timo
Post by Hetware
Post by Timo
Post by Fjolsvit
Is there a decent online discussion of how radiation
pressure can be derived from Maxwell's equations? The
discussions I've found rely on the QM explanation.
There are plenty of good classical explanations. I don't know
of a good online discussion, as in a web page that covers
it.
I don't understand why this is not treated better in so many sources.
Because engineers don't care? Because it isn't in the textbooks
that textbook writers copied their content from?
The books that do Lagrangian field theory typically do
electromagnetic momentum, via the usual Lagrangian + Noether =
conserved quantities. This includes the many QED/QFT books that
do this in their introductory chapter(s) on classical field
theory.
Jackson gives the Heaviside-Poynting derivation. Stratton gives a
clear and very simple version of it. A lot of other books either
ignore it completely, or just give the result (e.g., Griffiths).
Energy, OTOH, tends to be covered in detail much more often.
This is why Einstein, Schrodinger, Heisenberg, and even Bohr(, not
to mention Feynman) were uneasy with, or flat-out rejected, the
"conclusions" of QM.
They rejected such because electromagnetic energy gets more coverage
in the textbooks than electromagnetic momentum?
In a sense, yes. That is to say, they still believed that classical
physics, which includes relativity, holds the proper foundational
understanding. They didn't want QM to be accepted as foundational.

It's analogous to thermodynamics. Feynman did a spectacular job in
qualitatively exposing the second law of thermodynamics as a
manifestation of the more fundamental laws of mechanics.

I have the sense that most physicist (if not all) don't really
understand the origins of QM. In particular, its connection to
classical phase space.
Post by Timo
Post by Hetware
A real treatment of E&M should begin with the electrostatic field;
progress to special relativity (perhaps with a bit of hand-waving
about Maxwell's equations and Galilean Relativity,) and expose the
magnetic field as a consequence of electrostatics and SR. Ohanian
is 90% there.
Why begin with electrostatics? Why not just start from the
4-potential (in the Lorenz gauge) as the 4-divergence of the
4-current? That's simple, and 1/2 of the Maxwell equations (the other
half being a tensor identity).
down an abstract mathematical formalism as a starting point. What is a
4-current? How will you define it a-priori?
Hetware
2013-03-17 04:39:45 UTC
Post by Fjolsvit
Einstein attributed the idea that light has momentum to Maxwell.
IIRC, Feynman gives an example of how radiation will push an electron
in the direction of the wave. I will look for that when I get a
chance.
Is there a decent online discussion of how radiation pressure can be
derived from Maxwell's equations? The discussions I've found rely on
the QM explanation.
I know it goes something like this. The electric field drives a
charged particle in a direction perpendicular to the Poynting vector,
and parallel to the electric field, The moving charged particle
experiences a Lorentz force due to the magnetic field which is
perpendicular to both the Poynting vector and the electric field.
The particle is therefore accelerated in the direction of the
pointing vector. That's an heuristic explanation, but it isn't
quantitative. It also assumes the Lorentz force law, which I don't
believe Maxwell relied on.
I guess for a truly modern approach I will need to read Sommerfeld.
Hetware
2013-03-17 17:08:09 UTC
Post by Fjolsvit
Einstein attributed the idea that light has momentum to Maxwell.
IIRC, Feynman gives an example of how radiation will push an electron
in the direction of the wave. I will look for that when I get a
chance.
Is there a decent online discussion of how radiation pressure can be
derived from Maxwell's equations? The discussions I've found rely on
the QM explanation.
I know it goes something like this. The electric field drives a
charged particle in a direction perpendicular to the Poynting vector,
and parallel to the electric field, The moving charged particle
experiences a Lorentz force due to the magnetic field which is
perpendicular to both the Poynting vector and the electric field.
The particle is therefore accelerated in the direction of the
pointing vector. That's an heuristic explanation, but it isn't
quantitative. It also assumes the Lorentz force law, which I don't
believe Maxwell relied on.
This gives about the same treatment as Feynman. Feynman's treatment is
more quantitative than I remembered.

http://www.people.fas.harvard.edu/~djmorin/waves/electromagnetic.pdf

It's still somewhat unsatisfying because a charged "point" mass cannot
absorb all of the momentum without either changing its rest mass (which
electrons, for example, don't do) or violating the law of conservation
of momentum.

See Exercise 2.15

http://www.pma.caltech.edu/Courses/ph136/yr2012/1202.1.K.pdf

What I'm really interested in is the momentum and energy of a pulse of
radiation treated classically. In particular how these quantities
behave under a Lorentz transformation.

My expectation and goal is to show that they give the same results as
are given by quantum physics. The difficulty is that classical E&M
doesn't give a simple relationship between frequency and 4-momentum.

I'm pretty sure what I need to do is consider a 3-space volume selected
as follows: consider a plane wave moving in the x-direction. Select a
unit square in the y-z plane. Mark one point on the x-axis at the
leading edge of the pulse. Mark the other end at the trailing edge.

Integrate over the volume to get the total momentum and energy therein.
Then I need to do something like transform this volume. The problem
is, I no longer have a volume representing an instant in time, in terms
of the new reference frame.

So I should probably describe a similar spacelike slice in the new
reference frame. But then I'm not sure who to compare the 4-momenta of
the two samples.

Another approach which doesn't seem as satisfying is to transform the
change in 4-momentum of our above mentioned test particle.
Timo
2013-03-17 19:18:41 UTC
Post by Hetware
What I'm really interested in is the momentum and energy of a pulse of
radiation treated classically. In particular how these quantities
behave under a Lorentz transformation.
My expectation and goal is to show that they give the same results as
are given by quantum physics. The difficulty is that classical E&M
doesn't give a simple relationship between frequency and 4-momentum.
It only gives the "standard quantum result" (E=hbar*omega, p=hbar*k) in a special case, since the quantum result is for a special case: the infinite plane wave. Of course, you won't get quantisation (E=hbar*omega) classically.

The infinite plane wave + any of the classical derivations gives p=E/c, which is the same as the quantum ratio. If you want to see how they behave under Lorentz transformation, you can use the densities, since the total energy and momentum are infinite.