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A new LAB for your students

The text below can perhaps help you to develop an innovative laboratory unit
on capacitors. It is two pages long, when printed. I really tried to keep it
short. KowalskiL@alpha.montclair.edu (alpha file CAPS.TXT 2/18/97)
............................................................................
A LABORATORY PROJECT FOR YOUR STUDENTS
Introduction
-------------
Electrolytic capacitors used in this lab. activity can be purchased, as an
assorted set of 20, item 272-802 from Radio Shack, for $2.50. I suspect these
are factory rejects because nominal C often differs from what is measured.
The data presented were collected with the voltage probe (from Vernier)
connected to a computer. The probe covers the range from 0 to 5 volts, its
input resistance is very high (> 50000 megaohms). Make sure your students
understand that a charge on a capacitor can be found by discharching C
through a resistor and by measuring the current I at different times t. The
area under the I(t) curve is the charge, Q.

From the assorted set I selected two capacitors, labeling them as C1 and C2.
Nominally, C1 was 47 microF and 25 volts (it can be recognized by the purple
color and by "Ace 49"). The nominal parameters of C2 were 220 microfF and
10 volts. C2 can be recognized by its yellow color; it is a short cylinder
whose diameter (15 mm) is about as large as the height. Three D batteries in
series were used to make a source of 4.5 volts. I am assuming that capacitors
in your Radio Shack assorted set will be similar.

The C1 and C2 capacitors were selected because they differ in the ability to
hold charges. This is illustrated by the self-discharging data below.

time (min) 0 10 20 ... 30 ... 60
Volts on C1 4.5 4.30 4.15 ... 4.00 ... 3.77
Volts on C2 4.5 4.00 3.75 ... 3.55 ... 3.30

Voltage-dependent leakage resistances can be calculated from these data.

TASK 1: Finding C1 and C2
---------------------------
Instead of using a sensitive galvanometer the currents were determined from
drops of voltage, V(t), on a known resistor through which C was discharged.

Charging C1 to 4.5 volts, then discharging it through R=100 kiloohms:
t(seconds) 0 2 6 12 20 32 40
I(microA) 45 36 20 8.0 2.4 0.4 0

The area under this I(t) curve gave Q1=341 microcoulombs. The corresponding
capacitance C1=Q1/4.5=76 microfarads.

Charging C2 to 4.5 volts, then discharging it through R=100 kiloohms:
t(seconds) 0 1 5 11 19 31 47
I(microA) 301 258 105 58 19 3 0

The area under this I(t) curve gave Q2=2080 microcoulombs. The corresponding
capacitance C2=Q2/4.5=462 microfarads.

TASK 2: Distribution of charges on parallel capacitors
-------------------------------------------------------

Connect C1 and C2 in parallel and charge them from the source. Then
disconnect them and measure the charges Q1 and Q2. The result confirms the
textbook prediction that Q1/Q2=C2/C1.

TASK 3: Distribution of charges on capacitors in series
--------------------------------------------------------
Connect C1 and C2 in series with each other and with the source. Then
disconnect them and measure the charges Q1 and Q2. Textbooks tell us that Q1
and Q2 are equal but this is contradicted by experimental data below.

Discharging C1 through R=100 kiloohms:
t(seconds) 0 2 4 8 12 16 22 31 36
I(microA) 34.2 25.7 18.8 10.3 5.7 3.2 1.3 0.3 0

Discharging C2 through R=100 kiloohms:
t(seconds) 0 2 10 26 48 62 90 130 190
I(microA) 10.7 10.3 8.1 5.9 3.5 2.5 1.3 0.7 0

The charges Q1 and Q2, deduced from these data, turned out to be 232 and
436 microcoulombs, respectively. Instead of being equal the charges differ
by a large factor. The discrepancy can not be attributed to experimental
errors. Take advantage of the rare discrepancy (between what textbooks say
and what is actually observed) to promote critical thinking.

Experience shows that for some capacitors in series Q1 and Q2 are nearly the
same while for others they are very different. Another pair of capacitors,
from the Radio Shack set, was also used to demonstrate the above effect. C1
was a small green cylinder (nominally 47 micF and 25 volts) while C2 was a
large black cylinder (nominally 1000 microF and 25 volts). The actually
measured values of C1 and C2, for that set, turned out to be 57 and 1180 micF,
respectively. As for the first pair, the parallel circuit was normal (Q1/C1
and Q2/C2 were equal) while the series circuit was highly "abnormal". The
"abnormality" is not with capacitors; the textbook model is defective.

Important technical details:
----------------------------
The time to equilibrate C1 and C2 in series, (each was discharged before
being connected) turned out to be very long, about 7 hours. The idealized
theory of the equilibration process (Q1 on C1 changes through a constant
leakage resistance of C2, and vice versa) is well described in reference 2.
The real process is more complex because leakage resistances are voltage
dependent. This is a good topic for an independent study project.

Comments and references:
------------------------
References exposing the common textbook misconception about capacitors in
series are listed below. Two years ago we (phys-L-ers) had an extensive
discussion of that topic. The thread was named "A myth about capacitors in
series". You can find messages posted under this thread by linking to them
from my home page (http://www.csam.montclair.edu/~kowalski). I was wrong
in anticipating that electrolytic capacitors would not show the unequality
of charges on equilibrated capacitors in series. The main point was that
textbooks are wrong about that subject. Feel free to use my data in your
problem session. Or ask a group of students to collect data and to present
them. With your help such activities will promote critical thinking.

1) E. Noll et al., Physics Education, 31, 393, 1996
2) A. French, The Physics Teacher, 31, 156, 1993
3) L. Kowalski, The Physics Teacher, 26, 286, 1988