Re: funny capacitor (EUREKA ?)

["John S. Denker" <jsd@MONMOUTH.COM>]

A moment ago I wrote
> OK, Folks.... Let's evaluate our progress.  Two items:
>
>   1) Consider the following proposition:  "A good way to stay out of
> trouble, especially in an introductory class, is to measure any voltage V
> relative to some Traditional reference, such as a large conducting
> enclosure tied to earth/ground."  Raise your hand if you _disagree_ with
> this proposition.
>
>
>   2) Consider another proposition:  "For describing an ordinary
> capacitor, the expression
>         Q = C V
> is simpler and therefore preferable to something like
>         Q = C delta_V
> especially in an introductory class."  Raise your hand if you _disagree_
> with this proposition.


As promised, here are my preferred answers:

  *) First of all, if you didn't object to either proposition, you've been
snookered.   To paraphrase Uncle Albert:  physics formulas should be as
simple as possible, but not simpler.  You *cannot* get away with both
simplifications suggested above.  Here is an easy counterexample:  Consider
a slowly-changing voltage applied to three capacitors in series.  Starting
from V=0 everywhere, ramp up the voltage-source to 10 volts.  Tell me the
charge on each capacitor.

                    ___________________
                   |                   |
                   |                   | V3
                   |                   |
                   |                ======== C3 = 25 microfarad
                   |                   |
                   |                   | V2
                ___|__                 |
               | ramp |             ======== C2 = 15 microfarad
               |______|                |
                   |                   | V1
                   |                   |
                   |                ======== C1 = 10 microfarad
                   |                   |
                   |                   |
                   |___________________|__  V0 = ground
reference


If you teach people that Q = C V, and that each V is to be measured
relative to the Traditional ground reference, they will get the wrong
answer every time.  The charge displaced in capacitor #3 is *not* given by
        Q3 = C3 V3              (equation ?1)
and there is no way of choosing a reference point so that any formula of
this form simultaneously applies to all three capacitors.  The sensible
solution to write something like
        Q3 = C3 V3 - C3 V2      (equation 2)

<digression>
  The following equation is not necessary in an introductory course,
  but it may help to make contact with the rest of this thread.
  Capacitor #3 has two capacitor plates:  top and bottom.
  The charges on these are:
        Q3top =  C3 V3 - C3 V2          (equations 3)
        Q3bot = -C3 V3 + C3 V2

  which has the familiar form of a *full* capacitance matrix, a 2x2
  matrix giving all the charges in terms of all the voltages.  We can
  view the over-simplified equation (?1) as involving a *diminished*
  capacitance matrix, namely the 1x1 matrix C.  Equation (2) involves
  a 1x2 matrix giving one charge in terms of the two relevant voltages.
</digression>

So what went wrong?  The answer is that gauge invariance is an option that
can only be exercised once.  Those who advocate proposition 1 are
simplifying *some* things by choosing a particular gauge.  Those who
advocate proposition 2 are simplifying some *other* things (such as
equation ?1) by using a different gauge.  These two choices are
incompatible (unless you live in a world where every capacitor has one of
its leads attached directly to ground).

I would be disappointed if students came out of a high-school level
introductory course unable to analyze the circuit diagrammed above.  I
would be very disappointed if this happened at the college level.  This
circuit makes clear the distinction:
 -- Rote application of a formula Q = C V, versus
 -- Understanding the real behavior.


I don't think my expectations are unreasonable.  I got 21,000 hits for
  http://www.google.com/search?q=high+school+physics+curriculum+OR+syllabus
http://www.google.com/search?q=high+school+physics+curriculum+OR+syllabus+OR
+objectives+circuits
which can be compared with 12,000 hits for
  http://www.google.com/search?q=high+school+physics+curriculum+OR+syllabus
http://www.google.com/search?q=high+school+physics+curriculum+OR+syllabus+OR
+objectives+circuits+series
which suggests I'm not the only one who thinks that this is "in bounds".


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

I was motivated to come up with this circuit, in order to refute the recent
suggestion that this thread had not exposed any significant misconceptions,
and to refute the suggestion that the misconceptions could be eliminated by
simply always measuring V relative to "infinity".

==========

And if you want me to be completely explicit:  I would have objected to
*both* of the propositions above.  I consider
        Q = C delta_V
or
        Q3 = C3 V3 - C3 V2
to be representations of the correct physics, and anything like
        Q = C V
to be sloppy shorthand and a trap for the unwary.  This sort of shorthand
is the *last* thing an introductory student needs.

As for the other proposition:  If you write Q = C delta_V then you won't
get into trouble, and you won't need a Traditional reference to get you out
of trouble.  Bottom line: the Traditional reference is neither necessary
nor sufficient for keeping you out of trouble.