Re: funny capacitor

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

At 03:24 PM 3/10/01 -0500, Bob Sciamanda wrote:

> > . . .  Gauge invariance allows us to make the transformation
> >          (V1, V2, V3) --> (V1+k, V2+k, V3+k)
> > any time we want.  This applies to *any* system,
> > no matter what the size /
> > shape / location of the objects.
> > As a special case, we can choose k=-V3,
> >which is tantamount to forcing V3=0
> > and making all voltage measurements relative thereto.
> >
>
> The point is that your k = V3 is not a "constant " - it changes value
> (relative to infinity) as the system goes into different states.

Ahhh, now we are making progress!!!!!

We are free to choose a time-varying gauge!!!!!

In particular:
        del^2(phi) = del^2(phi + k(t))
for any k(t) depending on time but not depending on location.


I feel like a total oaf for not realizing this was bothering people.


> > Why is not object 3 a meaningful reference point,
> >regardless of its size / > shape / location?
>
> It is a fine reference point if you are only interested in differences of
> potential in a given, single, system state.  But how do you examine the
> behavior of the reference point V3 as the system changes from state to
> state (ie, the Q's and V's all change - except of course your V3)?

I just turn the crank.  Because of the aforementioned gauge(t) freedom, I
can let the system do whatever it wants, THEN set V3 back to zero.

> Another way of saying this is that if one were to refer your calculated
> potentials to infinity, each system state would need a different k (its
> own gauge).

A true observation -- but not a problem.  So be it!  Let each state have
its own k.

> To compare different states one must somewhere refer to a point whose
> potential is unaffected by changing system states (eg: infinity).

There is nothing special about the "point" at "infinity".  In particular,
Laplace's equation doesn't know (or need to know) whether there is any such
place as "infinity".  Laplace's equation, including gauge(t) invariance,
applies just fine in a universe with periodic boundary conditions, in which
case there obviously is no such place as "infinity".

Here's a generally useful conceptual and pedagogical technique:  given a
tricky physics problem, especially one involving "infinity", do the problem
in a finite universe with periodic boundary conditions, THEN let the size
become very large.  The periodic boundary conditions allow you to have
something that is finite yet doesn't have any nasty edges to complicate
things.

If "infinity" means anything at all, it must refer to a convergent limiting
process.  By definition of convergence, the value of the limit must be
independent of the path taken to reach the limit, so the path that goes via
a large, periodic universe must be as good as any.

Note that my spreadsheets for solving Laplace's equations
  http://www.monmouth.com/~jsd/physics/laplace.xls
and
  http://www.monmouth.com/~jsd/physics/laplace-adv.xls
are set up to implement periodic boundary conditions as described at
  http://www.monmouth.com/~jsd/physics/laplace.html

To repeat:  There is no God-given grounded shell at "infinity" with respect
to which we should measure voltages.  There can be an arbitrary gauge(t) at
infinity, just like anywhere else.