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Clarence Bennett wrote (directly to me):
Ludwik, you confuse me. Are you serioues charging two Caps for seven hours
and then measuring separate charges? This will only indicate something about
the leakage. Charge them in series for only a second, and then measure the
charges. I think they then will measure nearly equal.
I have never seen an introductory textbook which tells students that the
charging time should be short to confirm the equality of charges on two
capacitors in series? On the contrary, the implied assumption is that we
must wait till the transient conditions are over. A very good mathematical
description of the transitory process was presented in TPT by French (2).
He actually gives a formula for V(t) which shows how long it takes for the
equilibrium to be established.
Yes, leakage resistances play an essential role. But this does not mean
that the equilibrated Q1 and Q2 become more and more equal when leakage
resistance become larger and larger. Compare this with the effect of air
resistance on trajectories of projectiles (where we can approach the ideal
by lowering pressure) or with the simple lens formula (which is highly
reliable for paraxial rays traversing thin lenses). The situation with
capacitors in series is different. No matter how large the leakage
resistances, their effect on final charges is always essential. We should
be careful in the use of idealizations. That is what I mean by "critical
thinking" in physics.
The main point is to use the described activity in teaching and to show
students that even textbooks can be wrong. My suggestion is to take a
textbook problem (how many volts on each capacitor), modify it to match
your lab capacitors C1, C2, and go to the laboratory for an experimental
verification. Then have a post-lab debate. Can the observed difference
(between what we predicted and what we measured) be due to experimental
errors? Note that the parallel capacitors prediction was confirmed with
the apparatus. How should this conflict be resolved? etc., etc. It can be
very exciting.
A Pasco electrometer, if available, can be used in the experiment, as
described by Noll (1). It will allow you to go beyond the 5 volts limit
of an ULI probe. You can also add a voltage divider, for example, five
resistors of 22 Mega-ohms, in series, to expand the voltage range of an
ULI probe. The input impedence of 110 mega-ohms is high enough for quick
measurements of voltages on electrolytic capacitors.
PLEASE SHARE YOUR PEDAGOGICAL OBSERVATIONS. Ludwik Kowalski