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Re: [Phys-L] circular definition of "success" .... was: standard DC circuits

As is usual in this forum, we are having four different conversations,
discussing stuff that is appropriate for ...
-- the introductory high-school physics course;
-- the introductory college electrical-engineering course;
-- the second year of the calculus-based college physics course;
-- ourselves and other who are truly deeply madly in love with
the microscopic physics details.

All that is 100% fine, so long as we keep track of which conversation
is which. In a recent message I was not as clear about this as I
should have been.


On 12/03/2013 01:26 PM, Jeffrey Schnick wrote:
http://www.av8n.com/physics/img48/battery-operating-curve.png

What charge is plotted here? The charge on the positive terminal of
the battery? How is the charge changed? How is the charge
measured?

Good question. Let me clarify that I had in mind the /engineering/
notion of charge. Imagine a world where Kirchhoff's laws apply.
The only possible relevant charge is

Q = ∫ I dt [1]

where I(t) is the current flowing through the battery.

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

Before moving on, let's clear up some side-issues:

First of all, I don't want to dwell on it, but it appears necessary to
dispel some misconceptions. A "conveyor belt" such as we find in a
van de Graaff generator is a terrible model for a battery. An ideal
battery is a constant-voltage source, whereas a van de Graaff is a
constant current source, to an excellent approximation. On an I/V
plot, one is vertical while the other is horizontal. If you are
using a van de Graaff as you mental model of a so-called battery,
you will never understand the key properties of a battery.

Secondly, sometimes people make analogies between a battery and a
capacitor. This is possible, but tricky. Among other things, we
must distinguish between the large-signal capacitance

upper-case C = Q / V [2]
Q = C V

and the small-signal capacitance

lower-case c = dQ / dV [3]

It is possible but very tricky to model the large-signal capacitance
of a battery. It involves infinite charge on an infinite capacitor.
Let's not get into it.

It is much less problematic to talk about the small-signal capacitance
of the battery. It is infinite. That corresponds to zero small-signal
impedance. If you want to avoid infinities altogether, you can work in
terms of the small-signal inverse capacitance,

ↄ = dV / dQ [4]

Note that in my diagram, the slope of the red line is dV / dQ ...
and it is zero.
http://www.av8n.com/physics/img48/battery-operating-curve.png

Assuming finite currents, the analysis for a battery goes like this:

dV dV dQ
---- = ---- ----
dt dQ dt

= ↄ I

= 0

which implies constant voltage, which is the right answer.

In engineering terms, the battery is a black box. A current flows
in one terminal and out the other, so we have nothing to lose by
saying the current flows "through" the battery. The fact that the
electron that flows out of one terminal is probably not the same
electron that flowed in to the other terminal is not our concern.
Electrons are fungible.

Similar words apply to a capacitor, in which case the result is
even more interesting, because at the microscopic level, the
electron that flows out of one terminal is /definitely/ not the
same electron that flowed into the other terminal. Still, at the
black-box level, this is not our concern. Electrons are fungible.

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

Now let us move on to the microscopic physics discussion.

We must immediately throw away Kirchhoff's so-called laws. We must also
clean up the terminology. We rewrite equation [1] as

G = ∫ I dt [5]

where G is the /gorge/ in the battery. We speak of gorging and disgorging
the battery. In contrast, we reserve the symbol Q for the real honest
physics charge.

A gorge G on a capacitor means there is a charge Q1 = +G on one plate, and
a charge Q2 = -G on the other plate.

It is also possible to put a real physics charge (Q) on a capacitor, for
instance by shorting the two leads of the capacitor and setting the whole
thing on top of a van de Graaff generator. The general case involves some
charge and some gorge, both at the same time. For details, see
http://www.av8n.com/physics/electrophorus.htm#sec-gorge

In the _Matter and Interactions_ book we find the two-word idiomatic
expression "charge separation". I am not 100% sure, but that seems mostly
(perhaps entirely) to correspond to the concept of gorge.

So ... to answer the question that was asked: The abscissa in my plot
http://www.av8n.com/physics/img48/battery-operating-curve.png
is really the gorge. See equation [5].

Yesterday, when I made the diagram, I was tempted to use the word "gorge"
to label the axis, but I resisted the temptation. I decided (for once)
to avoid over-explaining things. I decided to give the answer in terms
appropriate to engineering and to high-school physics, which is where the
discussion of "standard DC circuits" began.

Today, we are having a different conversation, a maven-to-maven conversation.
Both conversations are worth having; we just need to keep track of which is
which.

We can use our new precise terminology to restate yesterday's main point:
-- gorge is not a synonym for voltage
-- volume is not a synonym for pressure
-- entropy is not a synonym for temperature
-- horizontal is not a synonym for vertical.