From: johncobb@ut-emx.cc.utexas.edu (John W. Cobb)
Newsgroups: sci.physics
Subject: Generalized Ohm's Law (was: Defining Photons)
Date: 31 Jul 92 18:17:28 GMT
Reply-To: johncobb@ut-emx.cc.utexas.edu (John W. Cobb)
Organization: The University of Texas at Austin

In article <FRANL.92Jul28173419@draco.centerline.com>,
franl@centerline.com (Fran Litterio) writes:
|>snarfy@cruzio.santa-cruz.ca.us writes:
|>
|>> ``I'M JUST TRYING TO POINT OUT THAT OHM'S LAW DOESN'T  INCLUDE  A
|>> TIME   OR   DISTANCE   FACTOR''  Screams  Snarfy.``  That  allows
|>> communication of information about changes  the  rate  of  energy
|>> transfer  back  to  the  E  transmitter from the E receiver to be
|>> instantaneous once a connection is made, by means of lowering the
|>> impedance of the receiving object! ''
|>
|>I'm not an expert on the subject, but I could swear that I've heard
|>that Ohm's Law was never an exact description of reality in the first
|>place (even before QM and relativity).  If so, the obvious question
|>is: Does whatever passes for the modern replacement for Ohm's Law (if
|>such exists) also lack a time parameter?
|>--
No, not at all, but it can get really complicated fast. Ohm's law
(or its generalizations) are not fundamental laws. They are derived from
other laws in specific situations. Usually Ohm's law is derived as
J = \sigma E jeuristically when one talks of charge carriers moving in
a solid with some scatterers. [Note the usual V=IR statement of Ohm's law
is derived from the J= \sigma E by integrating across a cross-section of
the current flow and then along the direction of flow.]

So if you want a generalized ohm's law you need to frist specify what
type of problem you are looking at. Are you looking at conduction in a
metal lattice? a semi-conductor? a Quantum fluid? a fluid plasma? a
kinetic plasms? etc. In each case knoweldge of Maxwell's equations as
well as knowledge of the microscopic dynamics of the system (classical
or quantum) are combined to try to find a relationship between the
loca lelectric field and the local current density. Except for the 
simplist cases, this relationship involves other entities like the
magnetic field or frequency of oscilation, etc.


Let me just give one example. The reference here is Krall and Trivielpeice's
plasma textbook, <Principles of Plasms Physics>. In a fluid derivation
from considering two fluids, electrons and ions. Taking the moment
equations for fluid momentum and subtracting one gets equation 3.5.9
on p. 91 which reads (orally) as:

partial J wrt to t + divergence of {sum of three dyadics[VJ + JV + 
VV*charge density]} = E * [(density*charge^2/mass)_electron + 
(density*charge^2/mass)_ion]
+{e*e[(1/Me)+(1/Mi)]/c}*mass_density* (V X B) / (Me + Mi) -
(e/c)[Mi/Me - Me/Mi]* (J X B) /(Me + Mi) - 
(e/Me)* Divergence {(Me/Mi)*(Center of Mass Ion Pressure Tensor) -
Center of mass electron Pressure tensor} + sum ov species a {Integral
over velocity space [charge_a * average_density_a * V (partial derivative
of the one-particle distribution function with respect to time due to
collisions)]}

where:
J = total curent
V = combined Velocity field (velocity of center of mass of loacl fluid)
E = Electric Field
B = Magnetic Field (Magnetic Induction for nitpickers out there)
Me = electron mass
Mi = ion mass
e = |electron charge|

This applies to free charge carriers that act as a fluid. That is it is
derivable from a Boltzman equation (first order BBGKY heirarchy). So it
excludes highly ordered systems like degenerate quantum systems or close
packed hard sphere systems. however, otherwise it is very general.

Now take the limit where B=0, V=0 and steady state (i.e.  Partial J /
partial t = 0).

The collision integral can be approximated as a collision frequency
times the current. That is the integrand is density*velocity*charge.
This is summed over species so it looks like a current if we approximate
the collisional time derivative as a collision frequency \nu.
Define the conductivity as \sigma= 1/(\eta) = ne^2/(\nu M_e).

Then the above horrible expression reduces to: E=\sigma J, and
one has derived the conductivity explicitly.

As a personal aside. Approximating the collision integral by a collision
frequencyt is valid for systems near a Maxwellian, but exotic non-maxwellain
cases exist, so there is a possibility for some really weird situaions far
from equilibrium. Just another example where systems far from equilibrium
can behave very differently.

Well, I'm afraid that what I wrote above may be difficult to decipher.
Perhaps you can write it in pencil at your deskside, or better yet go
look at Krall and Trivielpiece. I hope this is at least a little
helpful.

BTW, one of the Authors, Trivielpiece is now head of Oak Ridge. He used to
be head of DOE's OER. I don't know if he gets a chance to do much physics
anymore.

john w. cobb
jwc@fusion.ph.utexas.edu