🚀 go-pugleaf

Paul Camp <pjcamp@coastl.edu> writes:

  >  An ideal capacitor is a break in the circuit and there is no current
  >  through it.

This conceptual model is sufficient, for example, to figure out what effect
a capacitor will have on the distribution of constant potentials on (R1+R2)
in series with a battery (when a parallel C is added). But the model does
not account for a capacitor's ability to store electric energy. And it leads
to wrong conclusions about constant voltages on capacitors in series.
(see The Physics Teacher, May 1988, p 286).

  > I don't immediately see how this conclusion is altered when one models
  > a real capacitor as an ideal capacitor in parallel with a resistor.

You are right about this. An excellent mathematical description (by A. French
from MIT) of what happens when two capacitors in series are connected to a
battery can be found in The Physics Teacher, March 1993, p 156. The article
is not limited to what happens in equilibrium; the entire charging process is
described. The initial curent (at time zero) is larger than the final
current. The final current is established at t=infinity [ practically at
t= 3*tau or so, where tau is given by (C1+C2)*R1*R2/(R1+R2) ]. Only final
current is needed in our simplified description of what happens after the
equilibrium is established.

Here is what French writes about tau. "This is therefore the time constant
that would be obtained by putting C1,C2 in parallel and R1,R2 in parallel.
Perhaps, if one has good intuition for circuits, this could have been
predicted. The present author is not blessed with such intuition."

By the way, the basic formula (#1 in his article) can easilly be verified
in a laboratory. Our students are asked to test the exponential decrease of
current when a single capacitor is charged through a 10 Mohm resistor and a
microampmeter. I am developing an additional component for that laboratory
based on the article. I use two electrolytic capacitors (C1=C2=270 microF,
from Radio Shack), two resistors (R1=970 and R2=57 kohms to simulate different
leakage resistances), a six-volt battery and a digital voltmeter (also from
Radio Shack, sale price=$10). The voltage across R2 is measured every 15
seconds for about 2 minutes and data are compared with the theoretical
formula of French. The agreement is good, as shown below

       t(s)  0    15    30    45    60    75    90    105   120  135
  theory(V) 3.0  1.925 1.283 0.900 0.672 0.535 0.454 0.405 0.376 0.359
   data (V) ***  1.90  1.27  0.91  0.68  0.54  0.46  0.41  0.38  0.36

More accurate results are obtained when larger R1, C1 and C2 are used to slow
down the process. Very large R2 is not desirable because R2 must be negligible
in comparison with the input resistance of the voltmeter (unless you want to
ask students to account for the input resistance in parallel with R2).

        HOW IS THIS ACTIVITY RELATED TO THE TOPIC OF OUR DISCUSSION?

The theory presented by French is confirmed. It tells us what happens to
capacitors in series when C1*R1 and C2*R2 are not identical. For large t
the conclusion is exactly the same as that based on our "constnat current"
model. His general theory may be too advanced for some introductory courses
but the simplified model is always appropriate, provided Ohm's law and
Q=C*V topics were covered.

                       Ludwik Kowalski   kowalsk@apollo.montclair.edu
 P.S.
       Most of us would agree with John Cooper who wrote:

> I would guess that the best that high-school faculty can do is encourage
> an aggressive curiosity in their students with a willingness to discard
> received 'truths' and dogma when the observed facts necessitate such.
>
>                                               jcooper@bucknell.edu

Yes, but there aren't too many 'dogmas' to be discarded in traditional
physics at high school level. The equality of charges on capacitors in series
is the only case I know of in which textbooks can possibly be wrong.
SEIZE THE OPPORTUNITY FOR PRACTICING SCIENTIFIC METHODOLOGY, PROMOTE
LABORATORY INVESTIGATIONS, PROMOTE CRITICAL THINKING.

            PHYSICS IS AN EXPERIMENTAL SCIENCE.