Re: non-potential voltage

[John Denker <jsd@MONMOUTH.COM>]

At 01:27 PM 4/17/00 -0400, Eugene P. Mosca wrote:

> However, Kirchhoff's loop rule -- that the sum of the
> potential differences around a loop equals zero -- is a statement about
> potential differences, not emfs/voltages.  It seems to me this is
> equivalent to stating that an irrotational (curl free) electric field
> exists in the region.

Kirchhoff's laws are tantamount to making two simplifying approximations:
  a) A capacitor is a two-terminal black box, and there are no significant
capacitances outside of capacitors;
  b) An inductor is a two-terminal black box, and there are no significant
inductances outside of inductors.

As soon as you make those approximations, you are _assuming_ that at
every  point (excluding inaccessible points inside the black boxes), the
electric field has no significant curl.

>  For an inductor, the net electric field has both
> an irrotational (curl free)** part and a rotational part.

Right, or to be more specific, _inside_ an inductor, there is some curl.

> Emf is
> determined only by the rotational part whereas potential difference is
> determined only by the irrotational part.

I've never heard that distinction made before -- and it seems
unhelpful.  Can you cite a reference and/or argue why such a distinction is
useful?

>                  V = E - Ir = -LdI/dt - Ir.

I've never before seen an expression of that form -- and it seems
unhelpful.  Can you cite a reference and/or argue why distinguishing E
(EMF) from V (voltage) is useful?

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

No matter whether you speak in terms of voltage or EMF or both or neither,
the fact remains that (in the presence of a changing magnetic field) the
energy of a test charge is _not_ simply a function of position.

It's OK to talk about the scalar potential and vector potential (phi and A)
at a point.  But it doesn't make much sense to talk about one without the
other, nor to try to find a simple scalar field (with units of voltage or
otherwise) that summarizes both contributions.

Pedagogical suggestion:  Part of the problem here is that the students'
physics is getting ahead of their mathematics.  They tend to think that all
equations are about _functions_ but in fact the world is full of relations
that are not functions.  The set of all points (x,y) for which x^2 + y^2 =
1 is an example;  y is not a function of x, and x is not a function of
y.  Get used to it.

In this case, an even more apropos example is z = arctangent(x/y).  This is
easily modelled using popsicle sticks;  drill a hole in the middle of each,
stack them on a string or rod, and then arrange them in a nice multi-turn
helix.  There will be innumerably many z values for each (x,y)
position.  An ant crawling around on the model can follow a path which when
viewed from above projects onto a closed loop in the (x,y) plane;  this is
analogous to close-loop motion of a test charge in the presence of a
changing magnetic field.  In so doing, the ant can climb up the
helix;  this is analogous to the increase in the test particle's
energy.  Emphasize that x and y represent real space, while z abstractly
represents energy value, which is a nonfunction of x and y.

(Fanatics note: This isn't my favorite representation of arctangent;  I
much prefer the two-argument form z = atan2(x,y) which involves drilling a
hole at the _end_ of each stick... slope is the same but the z-axis period
is twice as large... alas this is harder to build.)