Re: non-potential voltage

[John Denker <jsd@MONMOUTH.COM>]

At 07:53 AM 4/13/00 -0500, Lemmerhirt, Fred wrote:

> Could you elaborate (briefly) on this objection?  I don't have a
> particularly negative reaction to the idea of a varying potential
> difference, and from your comment and John Denker's, it appears that
> probably I should.

It's not the time-varying _potential_ that is the problem.  It's just that
in electromagnetism, the time-varying situation has _nonpotential_ voltages.

Let's talk about "potential" in general.  Height on a landscape is an
example of a potential;  the height-value at a given point is independent
of how you got to that point.  Even if the potential is time-varying, the
height Z as a function of X, Y, and T is independent of the path you took
to get to that (X,Y,T) point.  This is the defining property of potential.

We define a _conservative_ force field F to be one that has the property
that the force can be written as the gradient of some potential U:  F = -
del U.  This implies that a conservative force field is necessarily
curl-free:  del × F = 0.

The _potential_ energy you gain or lose moving from point A to point B on a
landscape is independent of the path you take.  But the _total_ energy is
not, because the world is full of nonconservative forces.  If you swim from
point A  to point B the long way through a viscous swamp, you expend much
more energy than if you walked the short way along the shore.

Now, becoming less general, let's talk about the Maxwell equations.  The
electrostatic potential is a potential.

** [electrostatic case] -- In the absence of changing magnetic fields, the
force on a test charge is a conservative force field,
        F = -q E = - q del V
where in this case the voltage V is a potential.


** [electromagnetic nonstatic case] -- In general, the force on a test
charge is
        F = q * (- del phi - d/dt A + v × del × A)

where phi denotes the electrostatic potential (in horrible but
not-easily-avoidable conflict with the notation [phi=flux] of my notes
earlier this morning).

We see that the last term (v × del × A) is harmless;  it is always
perpendicular to the velocity so it cannot affect the energy.  The middle
term (d/dt A) is what causes the force field to be NONconservative.  That
term is NOT curl-free;  del × d/dt A is not generally zero.  (This is
unlike del × del phi which is always zero;  phi is a potential).  There is
*NO* potential that could generate the full electromagnetic force on the
test particle, just as there is no potential that could describe the force
on the swimmer in the swamp.

Tell your students: get used to it!  Not all forces are conservative!

Given a changing flux, if you take your test charge around a loop you pick
up energy.  If you take it around the loop again, you pick up more
energy.  Your energy is not a function of position.  It's like one of those
Escher staircases.  It's not a potential.  Talking about the potential in
an inductor is really a bad idea.

Clear enough?