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Re: Voltaic Pile of Confusion



At 16:17 7/29/00 -0400, Tom McC asked:
>... What is the best way to view how the
>electrons all achieve a specific voltage in an arrangement where x batteries
>are hooked up in series. In other words, how is an electron produced in one
>battery affected by the presence of another battery?

At 09:23 PM 7/29/00 -0500, brian whatcott replied:

A model which can be assimilated at any level considers electrons in
metals as billiard balls which need to negotiate an array of obstacles -
lets call them nails...

Brian's balls-and-nails model is not the answer to Tom's question. It
might be the answer to some other question, perhaps a question about
currents in a resistor, but it is not even close to answering the question
about voltage in cells in series.

==========

To answer Tom's question: Consider the following diagram: three batteries
in series, connected to a load.

(C). . . . . . . . .
_.____ .
| + | .
| | .
| | .
|_-____| .
. .
(B). _.____
_.____ | |
| + | | load |
| | |______|
| | .
|_-____| .
. .
(A). .
_.____ .
| + | .
| | .
| | .
|_-____| .
. .
. . . . . . . . .
(ground ref)


First of all, one should be careful about saying that electrons are
"produced in" a battery. We believe that electrons are conserved; they
cannot be "produced" according to the usual meaning of the word. For that
matter, for present purposes it is more relevant to speak of "charge" --
and (separately from the law of conservation of electrons) we believe that
charge is conserved. To be sure, charge can flow out of one terminal of a
battery, but not because it was "produced in" the battery. Charges flow
_through_ the battery, in one terminal and out the other. Kirchhoff's law
says that charge cannot accumulate in or on the battery; this "law"
embodies the approximation that the stray capacitance between the
battery-case and other circuit elements is negligible.

Secondly, it is not true that "all" electrons have the same voltage. An
electron sitting in the wire at point (A) has a voltage (relative to the
designated ground reference) only 1/3rd as much as an electron sitting at
point (C).

So, to really answer the question: The best way to view the voltage is as
a potential. (There are no changing magnetic fields in this problem, so
the voltage _is_ a potential.) The best analogy is to altitude. The
batteries in series are like steps in a staircase. An electron that makes
it to point (C) is like a marble at the top of the staircase. It has the
same altitude (same voltage) as any other marble (electron) at the same
location. The physical and chemical processes in the battery are like
little conveyor belts that can convey an electron up one step.

In a fundamental sense, the operation of one battery is independent of the
presence or absence of other batteries. A battery cares only about the
voltage difference between its terminals, and about the current it
carries. A conveyor trying to convey a marble up the Nth step of a
staircase does not care whether N=1 or N=3 or whatever. Once a marble is
available at the input of the conveyor, nobody cares how the marble got there.

This illustrates a deep principle of physics, namely the principle of
_gauge invariance_ (which has other applications as well).