Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: funny capacitor (EUREKA ?)

I wrote:
> 1) What I call the potential most certainly is work per unit charge.
> 2) It is a potential in the mathematical sense.
> 3) It is agrees completely with what Feynman calls "the electric
> potential" in section 4-3 of Volume II. He says (at the end of the
> section) that it is measured relative to "some reference point".

I wish that in item (1) I had said "energy per unit charge" rather than
"work per unit charge". Many people take "work" to mean a difference in
energy. Since we are apparently discussing how to think about absolute
potential versus differences in potential, I wish I had been more careful
about absolute energy versus differences in energy.

At 06:39 AM 3/15/01 -0500, Ludwik Kowalski wrote:

How can work be defined without a reference point from which
the probe charge is delivered to a point of interest?

The short answer is that I can *define* anything I want, so long as the
laws of physics don't forbid it.

In this case, I can *define* one potential phi_a relative to a chosen
reference point, and I can *define* another potential phi_b := phi_a + k
for some gauge k. I can cobble up instruments to measure phi_b. This just
isn't a problem.

All the known laws of physics are insensitive to the choice of k, for any
function k possibly depending on time but not depending on location. If
you dispute this, please explain what laws are violated by my definition of
phi_b.

Once a reference point is declared the potential becomes a difference of
potentials between two points.

We all know how to do subtraction problems.

That is why I still think that distinguishing a
"potential per se" from a "difference of potentials" is a significant
departure from traditional physics terminology.

I don't understand this, and I don't see the connection between this
statement and the previous statements that supposedly explain it.

Feynman does not make such distinction; his reference point, in section
4-3 is the "infinity".

No fair! No fair!

1) Generally, I think it is unscientific to argue over what is
"traditional" or not. It reminds me of fundamentalist religion, not science.

2) If we are going to go down that road, when we appeal to authority we
ought to cite it accurately! What Feynman actually says on page II.4.5
just below equation 4.22 is
"For convenience, we will often take the reference point at infinity."

He says "often" not "always" and he says it is a matter of
"convenience". To claim that Feynman endorsed the idea of a preferred
reference point (at infinity or otherwise) is just false.

>I use delta_V to represent potential differences. Some of the
> things I say about potential differences are not true about
> absolute potentials (which I represent by V).

That what confused me. The standard notation is symbol V
(or the corresponding Greek symbol used by Feynman) for
a difference of potential. I would probably not be confused
if another symbol, such as A or B, was used for the "absolute
potential".

That Greek symbol is phi. It is very commonly used to designate the
absolute potential.

Whereas in the literature people use V for a lot of things (including
absolute potential, difference of potential, various non-potential
voltages, et cetera), phi generally designates absolute potential.

Any speck of dust can be a reference to which you would
attach the black lead of a voltmeter;

BTW note we are assuming a somewhat idealized voltmeter. That's fine. I
don't want to get into wrangles about voltmeter technology.

I am more comfortable
with a grounded chassis or with "infinity".

But human comfort, tradition, and habits are not laws of physics.

The "infinity"
is not the best term, I prefer to say "a conductive enclosure."
The far-away enclosure has an advantage that its shape and
size are not significant. That is what the word infinity stand
for, in my mind.

The notion of far-away enclosure is good. I cringe a little when people
use the word "infinity" without explaining what they mean by
it. Specifying that it refers to a counterelectrode that is far enough
away that its size and shape don't matter (yet near enough that we can
exchange charge with it) is a useful clarification.

> >A traditional model, on the other hand, does not
> >allow small objects to be references.
>
> Says who?

For example, a technical assistant.

*) See previous remark about imperfections in real-world voltmeters. It
may be technically difficult to use a small object as a reference.

*) Let me point out that it is *certainly* technically difficult to use a
reference at infinity!

*) A large object not at infinity poses technical difficulties of its own.

Therefore appealing to the technical assistant is a lose/lose argument.

I suggest we nail down the conceptual issues before we start wrangling
about what is technically convenient.

> >(A traditional reference must be very very large to keep its
> >potential constant when its net charge is changing.
>
> Hogwash. Anything you choose as a reference will be constant
> by construction, by exercise of gauge freedom, no matter what
> its size or location.
>
> Remember this started with numerical Laplace-equation solvers.
> The program is perfectly happy to find Q as a function of the _
> four _absolute_potentials V (not delta_V). ...

That why, in my opinion, a potential per se belongs to mathematics
while a potential with respect to a reference object belongs to physics.

There's a germ of truth in that, but let's not exaggerate the
distinction. There are lots of things that I consider part of physics that
are not directly observable.
-- The quantum wavefunction.
-- Absolute electric potential (as opposed to difference in potential).
-- Similar gauges for various other elementary fields.
-- Absolute velocity (as opposed to velocity relative to some frame).
-- Absolute position
-- Et cetera.

Are we going to say that velocity per se belongs to mathematics, while
differences in velocity belong to physics? That seems to be going a bit
overboard.

Traditional reference objects (earth or infinity) are very large and
that is why we say V=const, no matter how large Q they receive or
loose.

No no no no no!

Read my lips: Any reference will have V=const by construction, by fiat, by
exercise of gauge freedom. That's what we mean by reference.

It's like saying that the velocity of frame A is zero relative to frame
A. It's a tautology. It tells us nothing about the size or location of
frame A.

In my opinion this is very different from saying that V=const
"by construction". I am trying to explain my confusion, not to argue
about a powerful new concept of gauge.

I don't know how to respond to this.
-- If "opinion" means personal feelings about what we wish the laws of
physics to be, there's no point in arguing about opinions and feelings.
-- If "opinion" means best guess as to what the laws of physics really
say, then we should concentrate on the laws of physics, not the opinions
per se.

The laws of physics say the potential of the reference-point can be made
zero by fiat, independent of the nature of the object (if any) at that point.

I would like to know what
other teachers think about that concept. I was not familiar with it;
would this concept be useful in an introductory physics course?

I would like to hear what others have to say about this.

My contributions include:
a) It is not necessary to spend a lot of time on it. In an introductory
course, it is not appropriate to spend a lot of time on it.

b) If you let students use spreadsheets to solve Laplace's equation, some
of them are going to populate the model-universe with objects all at
nonzero potentials, and they are going to ask what it means. Saying
"that's not allowed" would be a despicable answer. A better answer would be:
"Only differences in potential matter;
shifting the overall background potential doesn't matter.
Try adding 13V to every node in the universe and see what happens.
This is a deep principle of physics."

That takes only a few seconds to say.

c) The variable that naturally appears in the laws of physics, such as
Laplace's equation, is the absolute potential, not differences in
potential. Similarly absolute potential naturally useful in the
software. Rewriting the laws of physics (and the software) to conceal the
gauge invariance is probably possible, but at best it would be extremely
inconvenient and not worth the trouble.

d) There is an interesting pedagogical question: how to help a student
who has difficulty coming to terms with the idea that the phi(Q) problem
has multiple solutions, i.e. a solution set containing more than one element.

Standard pedagogy theory says that "learning proceeds from the known to the
unknown". So before delving into electrostatics, you have to teach that
the equation
x^2 - 1 = 0
has more than one element in the solution set. Choosing the positive root
is *not* always the right answer.

As a next step, you can teach that
sin(x) = 0.1
has a solution set with a countably infinite number of elements.

And then two planes intersect in a line, so that the system of equations
x + y + z = 1
x + 2y + 3z = 1
has a solution set with an uncountably infinite number of
elements. Pretending that one of these elements is somehow preferable to
the others is *not* the right answer.


When faced with a multiplicity of solutions:
1) Sometimes one solution *is* preferable -- in which case you need to
find some *other* equation that guides the choice. Gauges are never in
this category.

2) Sometimes it doesn't matter which one you choose, so you should just
choose one arbitrarily. Otherwise you wind up like the proverbial donkey
who starved to death midway between two piles of hay, unable to decide
which to choose. Numerical calculations with gauges are usually in this
category.

3) Sometimes you don't even need to choose, because no matter what you
choose, it will drop out in a later stage of the calculation. Symbolic
calculations with gauges are usually in this category.

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

Tangential remark: At some point this turns into a philosophical
discussion about the nature of freedom itself.

Once upon a time there was a college that gave students a lot of
freedom. Students could come and go as they pleased. Students could date
whomever they pleased. Students could take whatever courses they
pleased; nominally there were rules about course prerequisites and
graduation requirements, but there was no penalty for violating the rules
as long as what you were doing was halfway reasonable.

In contrast, down the street was another school that was completely
different. There were strict curfews. Nobody dated anybody without a
chaperone. People were told exactly what courses to take. Nobody ever cut
classes.

Now the interesting thing was that the folks at school "A" liked the system
they had. And the folks at school "B" liked the system _they_ had!

When I saw this, it was quite a revelation. I decided that some people
liked freedom, while other people liked being told what the right answer was.

The highest form of liberty is a system that allows both schools to exist.

In this spirit:
a) Gauge invariance is a freedom. You can choose any gauge you want.

b) If you are hoping to find some authoritative scripture that says
"this gauge is endorsed by Tradition and all other gauges are the work of
the devil" you are not going to find it. I recognize that such lack of
certainty makes some people uncomfortable.

c) I would like to make all people comfortable, to the extent that it is
possible -- but in this case it is not possible. No matter how strongly
you wish to establish the One True Gauge, you won't be allowed to do it,
because that would take away too much freedom from other people.