# Mechanical momentum of the EM field in a continuum?

Given an uncharged continuum the EM field produces no force on it. Yet in a discrete system we know that atoms are not uncharged locally and momentum can be exchanged due to the electric field(Which Maxwell's equations are suppose to govern).

viz., The transfer of momentum between atoms is due to the electric field(although it's not as simple as the 1/r^2 version). Maxwell's equations should describe the motion of the atoms exactly when one includes the Lorentz force and Newton's second law.

Yet when one moves up to the continuum these microscopic effects no longer exist and the electric field cannot influence the mechanical motion of density if it is not charged.

How can we include this in Maxwell's equations(or probably instead in the Lorentz force)?

I'm thinking that we simply use and it's conservation law(on the same page)

http://en.wikipedia.org/wiki/Electromagnetic_stress-energy_tensor

but there doesn't seem to be a way of controlling how well the EM energy is coupled to the material(Some materials will experience a larger motion due to the EM field it experiences).

or even

http://en.wikipedia.org/wiki/Maxwell_stress_tensor

in both cases though q = 0 for the continuum.

I think was thinking that possibly one could temporarily treat the continuum as being charged even though it isn't which will allow the momentum due to the field to propagate and one could compute forces and such. With proper scaling one might get the same effect as if modeling a discrete material. Effectively one is breaking up the material into positive and negative charge, computing the momentum then subtracting... in a continuum this is 0 but for the discrete case it cannot be zero unless there are no particles.

What is a better way? How can one model the microscopic collision transfer of momentum that can be applied to the continuous case?

-

A continuum distribution of matter, even if it is, in the bulk, electrically neutral, does not need to be electrically inert. In particularly, it is still able to obtain a Polarization density that is dependent on an external electric and magnetic field. This enables such materials to interact with electromagnetic waves.

And of course, in principle, there's nothing that is inherently banned by the laws of physics in having light just pass directly through matter without interacting with it.

-
Um, Yes, that is the point. When one moves from the discrete to the continuous charge density becomes a bi-valued function. Electrons and protons never exist at the same point(at least for all practical purposes) but when you take the limit they do and the net charge at that point is 0. This would happen for every point in the material for all bound charge. But bound charges still can experience forces in the discrete case... just not the continuous case. – Stretto Mar 15 '11 at 18:48
So when we do move to the continuum how can we compensate for the disappearance of such forces and still get an accurate model? – Stretto Mar 15 '11 at 18:49
Think of a simple crystalline lattice of atoms. When an EM field passes through the material each atom will experience forces. The net displacement of the atoms maybe zero but a polarization should take place(at least temporarily in time). Now suppose you think of the material as being continuous. In this case there is no net charge, no polarization, etc... (Note the media is not polarized in either case or at least not polarized in the average sense... the EM wave may induce a temporary polarization). – Stretto Mar 15 '11 at 18:52
The question is how can we move to the continuous case. Discrete is well defined but we seem to lose information in the continuous case. We could, for example, write p = p+ + p- but again, p = 0 for non-charged non-polar materials. If the EM induces a polarization effect then we might be able to modify p to include that so that p is not locally 0 for non-charged non-polar materials as the EM field passes which is basically what I was talking about using as an approximation in the first place. Just seems like a hack more than anything and I have no idea how to find the representation of it. – Stretto Mar 15 '11 at 18:56
@Stretto: You give the distribution a dipole moment but no monopole moment--this means that its net charge is zero, but it still interacts with matter. Even better, you can give it electric and magnetic suspectibilities so that it has a dynamic dipole moment density that can interact with EM waves. None of this needs an atomic-level description. You just describe matter with zero charge density but nonzero dipole moment. – Jerry Schirmer Mar 15 '11 at 21:30

Given an uncharged continuum the EM field produces no force on it.

This seems to be a root assumption of your question, so let's examine it a little bit. What do you mean by an 'uncharged continuum'? Would you consider a small glass bead to be an uncharged continuum? If so, this statement is disproved by optical tweezers.

If you mean something else, perhaps you could clarify your question.

EDIT:
Chapter 6.6 of Jackson's Electrodynamics has a pretty nice derivation of how you go from microscopic to macroscopic. Pay particular attention to polarizability.

-
Huh? Real materials are not continuous... do you know what continuum mechanics is? en.wikipedia.org/wiki/Continuum_mechanics – Stretto Mar 15 '11 at 18:44
Sure I know about continuum mechanics! That's why I asked 'what do you mean by an uncharged continuum?'. If by 'uncharged continuum' you mean, something which is still polarizable, then your statement that the EM field produces no force on it is false. If by 'uncharged continuum' you mean something which is not polarizable, then I would say that no such material exists. If you want to speculate on the physics of this fantasy material, perhaps the question could be clarified. – Andrew Mar 15 '11 at 19:00
Well, the continuous polarization density must be evolved along with the system for it to make any sense because most materials are not normally polarized when in equilibrium when considered as a continuum. – Stretto Mar 15 '11 at 22:21
Take a material in the continuum that is not charged. The charge density p = 0. Since the Lorentz force has p as a factor then it is 0. p = 0 at all points for such materials precisely because they are treated as a continuum. Sure polarizations and such may exist but when you move to the continuum they don't make much sense(at least to me). – Stretto Mar 15 '11 at 22:24
Suppose you look at a dipole moment as Jerry suggests. As you take the limiting effect to get to the continuum it is not at all clear that such a macroscopic effect will not be 0(since the dipole moment falls off very quickly with distance). I don't mind assuming that these things still hole in the continuum but I haven't seen any equations that use them in describing mechanical evolution of the system. Even so one must also evolve them because as the system's changes these properties can change to not just due to the EM field but due to mechanical changes. – Stretto Mar 15 '11 at 22:27