Re: Capacitor Charged, Right term

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Joel Rauber, being dissatisfied with the idea of charges at infinity, wrote:
> I have never been happy with the statement that the field lines on the +Q 
> charge must continue to -Q's at infinity.   Why?  Does this mean a universe 
> that consists of one positive electron is impossible??
> ...
> My opinion, is that saying that this single conductor capacitor implies 
> negative charges at infinity; is simpy a mental construct made solely to 
> allow one the mental convenience of saying all field lines orginate on 
> positive charges and terminate on negative charges; *always*.  I don't think 
> it is necesary to say this nor even correct; Because I think a universe of 
> one proton only is consistant with Maxwell's equations, and would be a 
> universe with no negative charges upon which the Field lines from the proton 
> would terminate.

Whether or not the idea of a universe with a net electric charge is a
necessity or just a convenience depends on the asymptotic topological
structure of the spatial sections of the spacetime of the universe.  If the
universe is spatially infinite then the idea of neutralizing charges at
infinity is a convenience.  If the universe is topologically compact, (i.e.
closed and bounded in extent) then the universe *must* be charge-neutral.  In
such a universe there is no 'infinity' for the excess field lines to escape
to.  A spatially finite universe *cannot* contain just 1 positively charged
particle (or have any nonzero net charge of any amount).  Consider a universe
whose spatial sections (of spacetime) have the topology of the 3-d 'surface'
of a 4-d solid sphere.  Suppose you try to place a single positive charge
some place.  The field lines radially emanating from this charge diverge
radially (like meridians away from the North pole) up until they get 1/4 of
the circumference of the universe from the charge.  Beyond this distance the 
field lines begin to reconverge and come back together (in the opposite 
hemi'sphere') to the point in universe which is antipodal to the original
positive charge.  Since all the field lines reconverge to this point this
point *must* be a negative charge of the same magnitude as the original
positive charge.  Thus this universe ends up having one positive charge and
one negative charge, and is, therefore, charge-neutral after all.

If one writes the Poisson equation for the potential on a compact space with
a net nonzero source charge (i.e. integral of the charge density over the
entire space) the equation will admit no solution.  If the whole compact
space is charge-neutral, however, the solution is unique up to an arbitrary
additive constant.

BTW, I don't mind at all saying that a capacitor is charged and discharged.
I *do* believe, however, that before ever using the word 'charge' as a verb
in connection with a capacitor, that the instructor very carefully define
the meaning of that verb in terms of a process where net positive charge
accumulates on one electrode 'plate' via conduction from a connection to an
external circuit *and* a compensating net negative charge accumulates on the
other electrode 'plate' via conduction to the other side of the external
circuit in such a way that the total net charge on both electrode plates
together remains effectively zero. 

David Bowman
dbowman@gtc.georgetown.ky.us