Re: induced emf again

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Sun, 5 May 2002, John S. Denker wrote:

> The retarding force must come from an E field, right?
>
> I haven't seen the Lorentz force law written in terms of
>         q E + q v ? B + retarding_forces
> I think it's just
>         q E + q v ? B

I don't buy this.

On the one hand, if I catch your drift, I guess I'd be willing to
go even farther and say that the only *real* force on an electron
is qE in the sense that mg is generally negligible and qvxB is
only a way of accounting, in the "lab" frame, for what is really
an electric force in the frame of the electron (due to the way the
Lorentz transformation turns B into E loosely speaking.)  Even the
"retarding force" due to "resistance" is just a collective effect
of collisions that are mediated by an electric force between
conduction electrons and the atomic lattice.  (I am assuming this
is what you meant by the first comment quoted above.)

On the other hand, it would be ridiculously difficult to account
for currents if we focused on the electric forces on individual
electrons in their own frame.  Indeed, neither the qE force nor
the qvxB force represent instantaneous forces on any specific
electron.  Rather, they represent average forces on a relatively
large collection of electrons in a small, but not microscopic
region.  The E that we use in qE ignores local fluctuations in E
due to the atomic lattice and the v that we use in qvxB is the
average velocity.  In the same spirit, it is useful to represent
the collective effect of collisions through a viscous like -bv
force where b is the product of the charge carrier mass and a
representative "collision frequency."

Thus, in the lab frame, we do indeed write

 F = qE + qvxB - bv

with the last term accounting for the effects of resistance.  

To be specific, in Ludwik's problem (with a 0.1 m long 1 ohm rod
moving at 10 m/s along a U-shaped frame with a total resistance of
3 ohms) we would find within the rod that 

 E     = 7.5 V/m opposite the direction of current flow, 
 vxB   = 10 V/m in the direction of current flow, 
 -bv/q = 2.5 V/m opposite the direction of current flow.  

Thus, F = 0 and as a result v is constant.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm