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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.