Re: funny capacitor (EUREKA ?)

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

At 04:56 PM 3/13/01 -0500, Ludwik Kowalski wrote:

> 1) Keep in mind that each V is a difference of
> potentials with respect to a reference.

OK.  You call it V, I call it delta_V, but I understand.

> This term reference is used for a very special object.

Not very special.

> What makes it special? IT IS NOT HOW FAR IT IS.

OK...

> It is the fact that its potential remains constant NO
> MATTER HOW much charge is received or given away.
> Our planet is big enough to nearly satisfy this
> requirement in most cases.

No no no no no!

The thing that makes it special, and for present purpose the *only* thing
that makes it special, is that Ludwik has soldered the black lead of his
voltmeter to it.

The beautiful thing about gauge invariance is that it allows you to pick
*any* node, big or small, near or far, and use that as the reference.

> The answer to a funny capacitor problem (either
> theoretical or experimental) will depend on the
> distance to the enclosure, and on the shape of
> that enclosure, unless it is very far away.

OK.

> A common silent assumption in electrostatic is that
> the reference is a very distant body but this
> is not essential.

Right.

> So why was I confused? Because I accepted John's
> potentials. They are not physical concepts defined
> in terms of work per unit charge.

I wouldn't have said that.

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".

4) The only difference is that Ludwik has insisted on choosing one of the
nodes of the circuit as the reference point.  My equations are more
general, and they _include_ (as a special case) the possibility of choosing
V4 as the reference.

> His potentials,
> and potential vectors, are set of numbers which
> can be converted to traditional potentials.

What is the evidence that my usage is less than 100% traditional?
I didn't invent gauge invariance!

> The unsolvable set of equations was created by treating
> the reference object in the same way as any other
> object in which a change of Q leads to a change
> in V.

I would rather say that I used an arbitrary reference.  In my formulation
there is no need to designate one of the nodes of the circuit as "the
reference object".

> Any conductor, no matter how small, can be
> used as a gauge in John's model of reality.

Indeed an arbitrary point in space can be used.

For that matter, a gauge can be chosen such that *no* point in space
actually has that potential.

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

Says who?

> (A traditional reference maust 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.

> John was able to find a unique solution by a trick
> of dropping a row and a column from the "full Cij
> matrix".

OK.  Indeed any row can be dropped, and any column.

> In my opinion it is not a good method of
> modeling electrostatic Q(V) and V(Q) problems.

Everyone is entitled to his own opinion.

> What
> do we gain by turning a problem into something that
> seems to be unsolvable and which can only be solved
> by a trick?

We did not choose this problem.  It found us.

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).  The program does not know that Ludwik is going
to solder the black lead of his voltmeter to node 4.  Even if Ludwik always
measure things in such a way that V4=0, it is only a matter of time before
some student sets V1=0 V2=0 V3=0 V4=1 and asks what that means.  That is a
very reasonable question, and we had better be prepared to answer.  The
program can handle this case with ease.  The equations of physics can
handle this case with ease.

There is nothing pathological or unphysical about this case.  The
Laplace-equation solver has no reason to treat node 4 differently from the
other nodes.  If humans feel a burning desire to treat this node
differently, it is only a sign of human prejudices.

The 4x4 full capacitance matrix allows us to have four _independent_
voltages, measured with respect to some _arbitrary_ reference, and allows
us to view the world the same way the program views it, the same way the
laws of physics view it.  The 16 Cij values are gauge independent, and make
gauge-independent predictions for the four Qi as a function of the four Vj.

==========

There are some statements that are true no matter what the frame of
reference.  For instance in relativistic dynamics,
        E^2 - p^2 c^2 = m^2 c^4
is true no matter what Lorentz frame is used to measure E and p.

Other statements are obviously true in only one reference frame.  For instance
        speed limit = 55 mph
cannot possibly be frame-independent.

Whenever I see a frame-dependent or gauge-dependent statement, I look
around to see what is the corresponding frame-independent or
gauge-independent statement.  Evidently this confuses some people.  That
makes me sorry, but that's not enough to shake my confidence in the
advantages of using a gauge-invariant approach.

I say again I did not go looking for a gauge-invariant approach in this
case;  the software foisted it on us.  But if it had not I probably _would_
have gone looking for it!

Gauge theories are central to modern physics.  The gauge of electrostatics
is just the teeniest tip of the iceberg.

  http://www.nobel.se/physics/laureates/1999/press.html

  http://www.nobel.se/physics/laureates/1982/wilson-autobio.html

  http://www.nobel.se/physics/laureates/1979/press.html