Wednesday, March 5, 2014

Rewriting Maxwell's equations

When I was in my fourth year studying physics, I felt a bit overwhelmed by the E&M courses.  Every thing we learned in these courses was a corollary, ultimately, of Maxwell's equations.  Yet if I had gone into the final having only memorized them or written them on my cheat-sheet, I was sure to fail.

Fast forward several (many) years and I'm working on atmospheric science stuff.  To do atmospheric science successfully requires an intersection of knowledge from many different branches of physics: classical and fluid dynamics, thermodynamics, electricity and magnetism.  We even use some quantum mechanics.  Here again, it's mostly fairly narrow theorems and rarely delving into fundamentals.  In the area of quantum mechanics, for instance, I've never touched a wave-function, just used spectral lines in radiative transfer calculations.

Some of the most interesting work I've done (although I didn't think so at the time) has been simulating surface emissivities, specifically sea water and sea ice.  It's a very fundamental problem: how electromagnetic radiation interacts with matter, but you wouldn't think so from the way it's actually done in the field.  Here again I was using corollaries of more fundamental theories, specifically the Fresnel equations which can be derived from Maxwell's equations.

Here's where it gets interesting: I started off with a radiative transfer equation that was derived fairly directly from Maxwell's equations.  But this equation had at least two errors so I re-derived it much more simply using the Fresnel equations as a basis.  This and a couple of other things got me thinking about Maxwell's equations.  The model requires a series of dielectric constants for the media being simulated.  These constants are quite difficult to calculate, even for homogeneous media such as salt water, and most of the models for them are at least semi-empirical.  For sea ice, it's almost impossible and most of the models handed down to me have only a passing connection to reality.

So it really got me thinking about how to simplify the laws to make them more tractable and also wondering, are they really complete (or perhaps I should say complete enough, since I'm not sure any physics law can ever be complete, efforts to find a "final theory" not withstanding) if they need these finicky and difficult to calculate constants?  Physics laws like Maxwell's equations have deep "roots" and people would sooner question their own measurements or application of the laws rather than the laws themselves.  I've experienced this myself: you keep beating away at the solution until you get the "right" answer, that is the one that agrees with your measurements.

The first thing I came up with was the plane-wave solution: why do we always throw away the imaginary part?  Why not keep it: this is the other field.  If your solution is for the E-field, then the imaginary part is the B-field and vice-versa.  The vacuum equations become:

div(R)     =  p

curl(R)   =    i    d R
                   -----  ------
                  c^2  d t

where R=E+Bi and p is charge.  In light of the equivalence of the electric and magnetic forces--an electric field turns into a magnetic field under a Lorentz transformation--merging the two fields would seem to make sense.  It works for the plane wave solution and as I understand it, every other solution can be decomposed into a series of plane wave solutions.

This is about as far as I got with it.  As you can tell from my ham-fisted attempts to analyse my "Vernier" clock, I'm not that good at theory, although I'd like to do more of it.  To be fair my earlier mistake was more because I couldn't be bothered to take the time, sit down and work things out properly.  I'd like to write more about that later.

There are still problems with this: you can't really merge the electromagnetic permittivity and magnetic permeability into one constant because there will be cross terms.  So obviously I haven't made any progress towards understanding the electromagnetic properties of sea ice--unless that's the answer: there really are cross terms and we haven't accounted for them!

I'm thinking you might be able to take it further merging the reduced set of equations and so further reduce them to only one by generalizing the curl and incorporating time into a four-space.  We did something similar in fourth year--I dug it out recently but all the details escape me.

Two other things bother me about the Maxwell equations.  You can solve the equations using a complex permittivity.  In this case, the imaginary part is the conductivity (or some part of it--I'm pretty sure the two are not precisely equivalent).  But conductivity isn't part of the Maxwell equations.  Why is it directly proportional to the electric field, instead of related through the first derivative as in Newtonian physics?  This is obviously an approximation--an empirical hack.

Of course real physicists don't bother with the classical E & M equations since we've long since moved beyond them, although I'm not sure how much use quantum electrodynamics is for real world problems such as the one above.  I still have a bunch of really dumb questions related to E & M and light, such as: What is the quantum formulation for a single photon? (Can you write out the Schroedinger equation for a photon?  How come we never did that in undergrad?) and how does it relate to the classical formulation for light?  How much information does a single photon transmit?

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