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Re: funny capacitor



John,
I disdain ad hominem remarks, but your post, below, is akin to flag-waving
and vague heresy implications without specifications. This is a (not
unfamiliar) debating tactic which is effective politically but is only
distracting in a scientific context. At the same time, you ignored my last
exchange about your reading of "partial derivative".

A law of the universe, such as the conservation of charge, does not
preclude me from making valid and useful statements about only a part of
the universe in which that constraint is relaxed. Ex: Gauss' law is
valid for any closed region of space. The flux out of that region is
proportional to the net charge inside the region. The law is still valid
(with the same proportionality constant) if I change the net charge
enclosed (either gedankenly in calculation, or by physically transferring
charge from outside the region). Ex: Laplace's equation is solvable for
a finite (charge-free) region where the potential (to some gauge) is
defined everywhere on the bounding surfaces. Once the solution of a
particular geometry is acquired, containing the boundary V's as
parameters, the solution is valid for varied values of the V's - which may
involve an exchange of charge with the "rest of the universe", either
gedankenly or physically. (You recently did this with Excel.)

I strongly advise that you go to the derivation of Qi = SUM Cij Vj in a
reputable text (eg:,Smythe) and see that no heresy is committed in proving
that:
Given an isolated (far removed from all else) system of N conductors with
a fixed geometry, a single set of geometrical numbers (Cij) exist for
which Qi = SUM Cij Vj describes the relations among the N
charges/potentials of the N conductors for all possible states of the
system - preserving only the geometry and the reference point for V (most
authors simply assume V is measured from infinity - a natural choice for
any real, localized system). No constraint of a fixed, total system
charge is ever imposed; some derivations explicitly speak of "bringing a
charge in from infinity" in order to change system states. Caution: some
elementary texts employ dubious hand waving in expediting the derivation -
but rigorous treatments exist, based heavily on uniqueness theorems.

I apologize for my ad hominem remarks.

Bob Sciamanda
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "John S. Denker" <jsd@MONMOUTH.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Wednesday, March 07, 2001 07:00 AM
Subject: Re: funny capacitor


Hi Folks --

By way of analogy, consider analyzing the motion of the following
dynamical
cart setup:
P post
________ P
| |$$$$$$$$$$$$$$$$$P
| cart | spring P
|________| P
===================== track P


Now suppose some student tried to analyze the motion of the cart,
ignoring
the fact that the spring is connected to the lamp-post, saying "I only
care
about the cart; the post is not part of the system under
consideration".

Such an analysis would be greatly at risk of getting the wrong answer.
The
post is sufficiently part of "the system" that it can exchange momentum
with the cart; therefore its effects really ought to be considered in
the
analysis.

Otherwise the analysis will probably violate of conservation of momentum
and all sorts of nonsense will ensue.

At 07:17 PM 3/6/01 -0500, Bob Sciamanda wrote:
The "object at infinity" is not a part of the one-conductor system
under
consideration.

This was in the context of:

At 05:50 PM 3/6/01 -0500, Bob Sciamanda wrote:
>
> >Consider our old friend the single, isolated conductor and its
> >description:
> >Q = C V, where C = C11, the only Cij of this one-conductor system.
> >Let's make it a sphere of radius a. Then if V is referred to
infinity
> >C=4*PI*epsilon*a.
> >OTOH if V is referred to a space-point located a distance 2a from
the
> >sphere's center, then C=8*PI*epsilon*a.


1) There is no God-given "object at infinity". What we really have,
sometimes, are physical objects that are
a) distant enough that their shape and location don't matter much,
yet
b) close enough that we can transfer charge to and from them.

2) Physically there is no such thing as a one-terminal capacitor. Such
a
thing would violate gauge invariance. There must be a second object
somewhere. It must be part of the physical setup, because you need to
transfer charge from it in order to charge the sphere of
interest. Afterwards it remains part of the physical setup because we
can
follow the "lines of force" [or, more formally, the electric flux = E .
d(area)] connecting it to the sphere.

Sometimes the second object's shape and exact location don't matter, but
its charge and electric flux always matter. Electrostatics is a "long
range interaction".

There are short-range arguments also, because conservation of charge and
conservation of electric flux are *local* conservation laws, just like
conservation of momentum. In the case of the cart, even if you only
care
about the cart, you need to notice that momentum is flowing through the
boundary of the "system under consideration" via the spring. Similarly,
even if you only care about the sphere, you need to notice that flux is
flowing through the boundary of the "system under consideration". If
the
voltage of the second object is changed, it will transfer a changed
amount
of flux through this boundary. This changed flux is part of the physics
of
capacitance, and it must be considered.

To calculate the capacitance of the sphere, excluding from consideration
the second object, is like calculating the kinematics of the cart,
excluding from consideration the lamp-post. Almost any law of physics
can
be rendered meaningless if such exclusions are allowed.