Re: Series, Parallel, and Resistivity Equations

["John S. Denker" <jsd@MONMOUTH.COM>]

Ludwik Kowalski wrote:
> > One of the peculiarity of the drifting process of
> > electrons (in a conductor at common temperatures) is
> > the very high ratio between the rms velocity and the
> > drifting velocity. The flow of water through a porous
> > medium is a very different kind of "nearly random"
> > walk.

It's true that they're different.  But I'm not sure I'd
say "very different".  The same equations have the same
solutions.

Bernard Cleyet wrote:
>  just like diffusion of perfume in air.

I wouldn't have said that.

The diffusion of perfume through air is not similar
to either of the phenomena Ludwik mentioned.

In particular, it is important that water is virtually
incompressible.  That leads to an approximate "gauge
invariance" : the only thing that matters is the
pressure drop across the tube, assuming the pressure
is nowhere unreasonably low.  Adding a few PSI to
both ends won't change the flow rate.  The behavior
of electrons is the same, but for different reasons.

The behavior of perfume is not at all the same.  The
perfume molecules form a highly compressible gas,
diffusing through the air which we can consider a
porous medium.  Diffusion of _gas_ molecules through
a tube loosely packed with fluffy cotton balls would
be the same, and permits a more-controlled experiment.
If you increase the pressure at both ends, you compress
the gas.  Roughly the same number of liters of gas
will flow, but the number of _moles_ of gas will
be markedly increased.  And remember that the moles
are what's conserved (for perfume, water, and/or
electrons) so that's where the analogy must succeed
or fail.  And gasses fail to correspond with liquids
or electricity.

========================

General philosophical and pedagogical remarks:

1) It is important for students to understand the
importance of the saying:  The same equations have
the same solutions.

2) There is no reason to expect that different
equations have the same solutions, except by
rare accident.  Even seemingly similar equations
commonly have radically different solutions.

3a) The following are mutually strongly analogous:
 -- Electrostatic field lines extending through
    vacuum.
 -- Electrostatic field lines extending through a
    homogeneous dielectric.
 -- Magnetostatics.
 -- The steady-state flow of current through a
    homogenous resistive medium.
 -- The steady-state flow of heat through a homogeneous
    medium.

This category is characterized by:
        Second spatial derivative, no time derivative.

3b) The following are similar to each other but
    different from any of the above:

 -- Non-steady heat flow.
 -- Non-steady electron flow in a resistive medium,
    if we neglect magnetic and radiative effects.
 -- Non-steady diffusion of liquids.
 -- Random walk.

The diffusion equation is characterized by:
        Second spatial derivative, first time derivative

3c) The following is different from any of the above:
 -- Diffusion of compressible gasses.

Characterized by:  Nonlinear!

3d) The following are mutually similar but different
from any of the above:
 -- Electromagnetic dynamics.
 -- Sound, under ordinary conditions.
 -- Transverse waves on a flexible string.

Characterized by:
        Second derivatives in space AND time.

3e) The following is different from any of the above:
 -- Schrödinger equation.

Characterized by:
        Second spatial derivative,
        imaginary first time derivative.
That factor of i makes this more like case 3d
(waves) and not at all like case 3c (diffusion).

3f) The following is different from any of the above:
 -- Transverse waves on a springy bar.

Characterized by:
        Fourth spatial derivative.

3g) The following is different from any of the above:
 -- Fluid dyamics.

Characterized by:  Nonlinearities and every other
type of complexity you can imagine.  Fluid dynamics
is really, really tricky.  People deal with fluids
every day, and they think the physics ought to be
simple, but it's not.