Re: [Phys-l] Charge Neutrality and 'infinities'

[John Denker <jsd@av8n.com>]

On 02/08/2008 09:27 AM, chuck britton wrote:
> Thoughts resurrected by this current discussion of capacitors:
>
> Is there any evidence that the universe is 'charge neutral'?
>
> 	(It doesn't seem to be 'anti-matter' neutral)
>
>
> If there were a 'very large' volume of uniform charge density - could  
> any experiment be done within that volume that would detect the  
> charge density?


> Gauss's Law would be toast (?) if the universe had a uniform charge  
> density?!     [G]

Let's start by dealing with the last sentence.

Gauss's law is a consequence of the structure of the Laplacian 
operator, which is a *local* operator.  Therefore I trust
Gauss's law, no matter what else is going on.


I understand the argument that leads to statement [G], but I 
would use it in reverse.  That is to say:  If you hypothesize
a universe with unbalanced charge and find a violation of
Gauss's law, I would keep Gauss's law and conclude that the
hypothesis is toast.

In a closed universe, that's exactly what happens, and IMHO
constitutes a proof of global charge balance.

The proof goes like this.  Imagine a sphere as a model, an
example of a closed manifold.  Draw a loop on the sphere.
This divides the universe into two parts.  We can arbitrarily 
label the two parts "A" and "B".  Now suppose there is *one*
unbalanced charge, and that it resides in part A.  Integrate
over part A and conclude that there is a field on the 
boundary.  Integrate over part B and conclude that there is
no field on the boundary.  This is impossible.

As another way of coming to the same conclusion, consider the
idea of continuity of field lines.  An unbalanced charge launches
a field line that keeps going and going ....  In a closed universe,
this field line is going to get hopelessly snarled.

=====

Obtaining the same result in an infinite universe is trickier.
The Gaussian loop divides the universe into two parts, one
of which is infinite, and somebody might try to argue that
we don't know how to do the integral over the infinite part.
Or maybe there is some bizarre boundary condition "at infinity"
such that Gauss's law is valid everywhere except "at infinity".

I don't buy any of that, and I put scare quotes around the
whole idea of anything happening "at infinity" because strictly
speaking there is no such thing.  Infinity exists only as the
limiting case of a sequence of larger and larger finite cases.

So far I've only considered the case of a /single/ unbalanced
charge.  The case of a universe with a uniform /density/ of
unbalanced charges is even more implausible.  There would be
all sorts of problems.  For starters, consider a sequence of 
successively larger regions within the universe and apply 
Gauss's law to them.  As the regions get larger and larger, 
the field strength at the surface gets larger and larger.  
Field strength is directly observable.

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

If you want to consider really weird possibilities, you could
hypothesize that "our" part of the universe is inside a Faraday 
cage, and that there is a huge unbalanced charge on the cage.

It's true that if you're inside the cage, you cannot observe
the charge on the cage.  But you can observe the cage itself!
And there's precisely no observational evidence for such a
thing.

So yeah, maybe there is an unbalanced charge "outside our cage".
I have not disproved this possibility.  All I can say is that
I'm not very interested in investigating things that have no
effect on anything we can ever observe.