🚀 go-pugleaf

On 3/22/95 Arthur B. Western, (the author of "University Physics: Models
and Applications", published by Wm. C. Brown, Dubuque, 1994,) wrote:

    > Colleagues; I'm a newcomer to the list. And I'll probably hate myself
    > in the morning for rising to the bait, but ...

Welcome to our network; you will not be the first one to feel that way.
Brave participants make this group's discussions interesting.

    > ..... Ludwik Kowalski then concludes, "Yes, the story is imaginary but
    > the problem is real. Physical reality does not have to agree with our
    > ideas but IN PHYSICS our ideas must agree with reality. Otherwise we are
    > not doing physics. Why is common knowledge of practitioners ignored by
    > the writers of physics textbooks? Can you find an author and ask for an
    > answer which can be shared with all of us?"

    > As one of the offending authors I will have to step up and take my
    > scolding.  The point which Kowalski makes regarding the importance
    > of leakage resistance of capacitors in series is well taken.  I will
    > certainly see to it that at least a couple of problems are devoted to
    > the effect in the second edition of [our book].

I think you should do more than this. Textbooks already cover too much material;
why should the electrostatic model be there (in deriving the 1/C=1/C1+1/C2
formula)? The constant current model is much more appropriate and it should
be used instead (if you think the topic is important). The equivalence
formula belongs to the alternating and transient currents; it should be
introduced when students are ready for the subject. WHY SHOULD YOU BE
OBLIGATED TO FOLLOW OLDER TEXTBOOKS?

    > Having thus agreed with the validity of Kowalski's admonition, I must
    > say I am far less exercised about the matter than he. ...
    > Should introductory students be burdened with all of this on their first
    > encounter with capacitance?  My answer is, "No."  The point of the
    > capacitors in series and parallel to allow students to practice with the
    > concepts of conservation of charge and summation of potentials. The
    > ideal capacitor, like the frictionless incline, provides a convenient
    > and relatively uncomplicated context in which to explore these concepts.

Obviously you are not aware of previous messages in which I tried to
deal with idealizations. I will summarize them for you below. Unlike
mathematicians we are not free to make arbitrary idealizations. Practicing
with the concept of conservation of charge can be done on the basis of
parallel capacitors while resistive circuits provide a lot of opportunities
to deal with summations of potentials. Nothing is gained from using a wrong
model.

    > .... So, if Professor Kowalski is open to plea bargaining, I will plead
    > *nolo contendere* to the charge of missing an opportunity to point out a
    > limitation of the ideal capacitor model when leakage resistance is
    > important.  But as for the charge of ignoring the real world, I want a
    > jury trial.  (Please vote.  We're writing the second edition right now!)
    > arthur.western@rose-hulman.edu    Rose-Hulman Institute of Technology

******************************************************************************
THE REST OF THIS MESSAGE, EXCEPT THE LAST TWO PARAGRAPHS, SIMPLY REITERATES
PREVIOUS ARGUMENTS; SKIP IT IF YOU ARE ALREADY CONVINCED THAT IDEAL CAPACITORS
ARE VERY DIFFERENT FROM FRICTIONLESS SURFACES, MASSLESS PULLEYS AND THIN LENSES.
******************************************************************************

Idealizations are essential in science; teaching and learning would be
practically impossible without them. An idealization, however, is acceptable
only when the factors ignored can be made negligible, at least in principle.
Air resistance, for example, is essential for aiming cannons but it can be
ignored in student-oriented problems. This is acceptable because a simplified
solution becomes more and more realistic when the air pressure is reduced.
The same is true for the so-called "thin lens" formula; it does not account
for numerous aberrations but becomes reliable for paraxial rays when radii
of curvature are large in comparison with the lens thickness. The ideal
theory is presented first and additional factors are considered later, as
needed. This is always acceptable when a gradual transition from an ideal
model to reality is possible. Massless cords, rigid rods, frictionless
surfaces, ideal fluids, etc. can be added to increase the number of examples.

Now consider what happens in the case of two capacitors connected to a
source of constant voltage, after the equilibrium is established. (The
problem of time needed to reach the steady state condition is usually
ignored when capacitors are introduced in a first physics course.) Here is
how I would describe the situation, at this level of learning, if I were
asked to write a textbook.

To simplify assume that capacitors have parallel plates. Consider C1 and C2
in series with a source of constant voltage, V. The leakage current is
i=V/(R1+R2), where R1 and R2 are leakage resistances. The voltage on C1 is
V1=V*R1/(R1+R2) and therefore Q1=C1*V1. Similarly, the voltage on C2 is
V2=V*R2/(R1+R2) and therefore Q2=C2*V2. Thus the ratio Q1/Q2=(C1*R1)/(C2*R2).
Express C1 and C2 in terms of the parallel plate formula, C=e*Area/dist,
where e stands for the dielectric constant. Also express R1 and R2 in terms
of the wire formula R=r*dist/Area, where r stands for the resistivity. Then
show that C1*R1=e1*r1 and C2*R2=e2*r2 (the geometrical factors, distance
and area cancel for each capacitor). Consequentely, the ratio of charges
Q1/Q2=(e1*r1)/(e2*r2) depends only on material constants, e and r. That ratio
is equal to unity when the dielectric materials are identical in two
capacitors in series. In general, however, Q1 and Q2 are expected to be
different.

Suppose, for example, that for a particular pair of capacitors R1=1*(10^10)
and R2=9*(10^10) ohms so that Q2=10*Q1. (Textbooks predict Q2=Q1, which is
wrong.) What happens when R1 and R2 are increased by several orders of
magnitude? You still have Q2=10*Q1. The voltage ratio does not come closer
to the "ideal capacitor prediction", Q2=Q1. That is why the ideal capacitor
model is not acceptable. It can not be justified by saying that a solution of
a problem dealing with the distribution of charges or voltages (for an
equilibrated system) becomes more and more reliable when R1 and R2 are very
large.

Compare this with an ideal pulley situation in which the simplified model
solution does becomes more and more reliable when the mass of the pulley is
progressively reduced. Or compare this with the approach in which the stray
fields of the capacitors are ignored; that field become less and less
important when the ratio of area/distance becomes large.

The formula 1/C=1/C1+1/C2, derived from the electrostatic model, leads to
conclusions which are contradicted by experimental data and it should not be
used in dealing with equilibrated DC circuits. (It is a very reliable formula,
however, for dealing with the nonstatic circuits.) You are correct that
capacitors in series often take a long time to be fully charged, "leaving
students plenty of time for measurements". But why should you make them
believe that such measurements are verifications of the static model of ideal
capacitors in series? The dynamic theory of capacitors in series was described
in The Physics Teacher (by A.P. French, March, 1993) and it should be used
instead of the incorrect static theory, when students are ready.

The main point is that leakage resistances, no matter how large, are always
essential for the staticly equilibrated capacitors in series. The textbook
descriptions of such capacitors can NOT be justified in the same way in which
we justify absence of friction. A theory ignoring always-essential factors can
not be called an idealization. Physics is not mathematics; our assumptions
must agree with reality. The mathematical derivation of the 1/C formula is
correct but the underlying assumption of equal charges is not acceptable.

By the way, the equal charge hypothesis was "experimentally verified", about
one hundred years ago, by J.J. Thomson. He used two Leyden jars, capacitors
made from the same material, glass. In that case Q2=Q1 because e2*r2=e1*r1,
not because capacitors are "ideal". Take two "leaky" capacitors made from the
same material (such as paper) and the equilibrated charges will still be equal.

DID I CONVINCE YOU THAT THE "RULE OF IDEALIZATION" IS NOT USED PROPERLY
      IN MOST OF OUR TEXTBOOKS WHEN CAPACITORS IN SERIES ARE DESCRIBED?
            HOW DO YOU DISTINGUISH "I AM WRONG" FROM "I AM IDEALIZING"?

You wrote "I want a jury trial"; ME, TOO. BUT NEW EXPERIMENTAL DATA MUST FIRST
BE SUPPLIED BY INDEPENDENT INVESTIGATORS. PHYSICS IS AN EXPERIMENTAL SCIENCE.
WE NEED MORE EVIDENCE BEFORE THE TRIAL; I HOPE IT WILL START COMMING SOON.

Can you, Art, collect some of the evidence in your excellent laboratories
and share the numerical results on this network? Just make sure that the two
capacitors are made from very different dielectric materials (in terms of e*r)
so that no "Thomsonian illusion" (of equal static charges) is created. The
teaching methodologies seem to shift toward situations in which students are
expected to rediscover (rather than confirm) some physics through the
laboratory work. I think that as a textbook writer you should be following
this trend. Capacitors in series are simple enough and students may be guided
to discover that old textbooks are wrong.

                     Ludwik Kowalski      kowalski@apollo.montclair.edu