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Chapter 23
Circuits
Recommended class days: 3
Background Information
There have been many detailed studies that have given us good insight into students’ understanding
of circuits. From the research, we learn that:
• Students do not differentiate between the concepts of current, voltage, energy, and power. To
them, it’s all just “electricity.” The situation is analogous to students’ use of the term “motion”
to describe either velocity or acceleration.
• Students think almost exclusively about current, rarely or never about potential difference. Thus
the majority of students, even after seeing experimental evidence that suggests otherwise,
continue to believe that batteries are constant-current sources.
• Students cannot reason with the concept of potential difference, and they rarely invoke potential
difference spontaneously. Their understanding of potential difference is not so much a
misconception as no conception.
• Students reason locally, not globally. In most circumstances, they do not see that changing one
circuit component will affect the voltage and current at other points in the circuit.
• Students have no micro/macro understanding of circuits. They do not see any connection
between macroscopic quantities, such as current or resistance, and their previous study of
charges, forces, and fields. To students, circuits are a subject entirely independent of
electrostatics.
These conclusions are perhaps not surprising, because students lack familiarity with circuits.
Surveys at the University of Washington have found that less than 20% of students report even
rudimentary experience with battery-and-bulb types of circuits—even though such experiments
should be a part of every child’s K-12 science instruction. What is surprising is that students show
essentially no improvement in their conceptual understanding of circuits following conventional
instruction.
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After conventional instruction, many students can successfully analyze simple circuits. Following
such procedures as reducing parallel and series combinations apparently does not imply that students
understand the physical concepts or that they can reason about circuits. Indeed, several studies have
found that students, when asked a question such as “What happens if I remove this light bulb,”
immediately begin trying to apply formulas rather than reasoning—they want to “plug and chug.”
In addition, other research has found:
• Students generally have no trouble with series and parallel resistance when just two resistors
are involved. However, unusual circuit arrangements can cause confusion. For example, some
students fail to recognize that R1 and R2 in the circuit below are in parallel. Apparently they
consider the definition to refer to a geometric relationship rather than an electric relationship.
More serious difficulties appear when there are more than two resistors. For example, many
students consider R3 and R4 to be in series.
• Students interpret schematic diagrams literally. They find it difficult to relate a schematic diagram
to the layout of an actual circuit. Even simple rearrangements of the diagram can cause confusion.
For example, many students perceive the two diagrams below to represent different circuits.
• For the most part, students do not understand how meters function, what their properties are, or
how they’re used in a circuit.
• Most students do not know what it means for a light bulb to be a “100 watt bulb.” The majority
interpret this to mean that the bulb always dissipates 100 W of power.
• Even though they use the term freely, almost no students know what grounding a circuit means
or why it is done.
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We take some pains to address all of these issues in the text. All of these difficulties and
misconceptions are explicitly noted and addressed, but you will need to reinforce these lessons in
class, as noted below.
Student Learning Objectives
In covering the material of this chapter, students will learn to
• Draw and interpret basic circuit diagrams.
• Understand and analyze circuits containing resistors in series and parallel.
• Understand the physical basis for Kirchhoff’s laws and use these laws to analyze circuits.
• Understand the time-varying voltage and current in RC circuits.
• Develop an electrical model for signal propagation in the nervous system.
Pedagogical Approach
The basic goal of this chapter is to establish the physics of circuits. Compared to many texts, there is
a reduced emphasis on the detailed analysis of complex circuits. This is something that will be of
very limited utility to these students in the future careers. More importantly, without a decent
physical understanding, the analysis of complex circuits is a formal exercise that will have very little
meaning for students. As the research results reveal, their most urgent need is a conceptual model
for thinking about basic circuit properties.
It is difficult to overcome students’ lack of familiarity with simple circuits and to convey the
basic ideas of a conceptual model with a classroom presentation. Hands-on experience at a studentcontrolled pace is essential. Although most schools have one or two “circuits laboratories,” research
has found that conventional measurement-oriented labs provide essentially no improvement in a
student’s conceptual understanding of circuits. Standard laboratories, despite how simple they
appear to the instructor, are mismatched to the conceptual level where most students are starting.
Instead, instructors are urged to devote one or two lab periods to hands-on experiences and
questions of the type described in Evans (1978) or Shaffer and McDermott (1992), preferably
starting before circuits are introduced in lecture. These are “guided discovery” experiments are
designed to help students acquire a mental model of basic circuit processes. We have excellent
success with such an approach at our institutions. At Colorado State University, the first circuits
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laboratory the students do is entirely qualitative and requires students to reason using basic concepts
such as the conservation of current. This initial exposure greatly aids students in understanding
material presented in lecture, and ultimately improves their ability to solve quantitative problems
when the time comes.
If such lab experiences are not possible, an alternative is to spend an entire class period working
through similar demonstrations in an interactive mode, asking students to predict, explain, and
reason. Such an approach would add an extra day to your treatment of this material. Whether it is
hands-on experience or interactive demonstrations, it is vital that students have an opportunity to see
and to reason about circuits qualitatively, building conceptual understanding prior to beginning any
quantitative analysis.
To compensate for students’ lack of experience and understanding, the textbook presentation
starts at a more basic level than usual. For instance, the chapter begins with a detailed description of
the translation between a real, physical circuit and an abstract circuit diagram. Many such basic
concepts are not treated in conventional texts, because they seem too “obvious.” As we have seen,
they may be obvious to us, but are not obvious to our students!
RC circuits are treated briefly, because of the importance of exponential decays, and because
they will be important in the final section of the chapter. Such time-dependent circuits are quite
challenging for students; plan a significant amount of time to explain what goes on in an RC circuit
before giving any exercises.
The final section of the chapter deals with the electrophysics of conduction of signals in the
nervous system. This is a wonderful “capstone” topic. Chapters 20–23 form a natural sequence of
topics on electric forces and fields and the practical use of such forces and fields to move charges
around circuits. At the end of this sequence, before moving on to magnetism, it makes great sense to
pause to review the connected set of concepts just completed. A study of electricity in the nervous
system is an ideal way to do this. A full treatment of the topic involves an understanding of the
motion of charges, forces on charges, field, potential, capacitance, resistance, and basic circuits. In
addition, the context is one that will be very interesting to students in the bio sciences. Many of our
students will go on to study this material in other courses, but they will be much better prepared to
do so if they have seen how it connects to the basic physics.
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Suggested Lecture Outlines
The main suggestion we have for instructors teaching this material is this: Make the subject
concrete. Students have a difficult time understanding electricity in general, and their
misconceptions concerning circuits are well-documented. Students use electrical circuits on a daily
basis, but the operation of the circuits is hidden from view. For them to understand the physical
basis of circuits, we need to make things visible. Students need to see real circuits in order to
understand the meaning of the diagrams that represent them. You should introduce or develop
concepts with real, practical circuits whenever possible. We have many suggestions for such
demonstrations and activities below. Ideally, this material should be supplemented with one or two
days of experimentation in the laboratory.
DAY 1: Kirchhoff’s Laws, Series and Parallel Circuits. Students have seen simple circuits
in the previous chapter, but the semi-realistic drawings we have been making won’t be practical
now, as the circuits rapidly grow more complex. It’s worth spending some time explaining to your
students how we use symbols to represent circuits. This translation between circuit elements and
wires to abstract symbols is quite challenging for students.
Clicker Question: The following circuit has a battery, two capacitors, and a resistor.
Which of the following circuit diagrams is the best representation of the above circuit?
Chapter 22 introduced Kirchhoff’s junction law. Chapter 23 follows with a discussion of the
loop law, which is much more difficult for students to understand, given their difficulties with the
concept of potential difference. Consistent with our suggestion to make topics concrete wherever
possible, we suggest introducing the loop law with a physical demonstration on a circuit, using a
voltmeter to measure potential differences.
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Demonstration: Kirchhoff’s loop law. Construct a
circuit with three resistors of similar but different values
connected to a (nominal) 9 V battery. Measure the potential
differences of the elements, being certain to note the sign,
and show that they sum to zero. Now replace one the
resistors with one of a different value and perform the same
Typical values for resistors and actual
battery voltage for the demonstration.
exercise. The potential differences will all change (a good
way to illustrate the global change resulting from a local alteration, something the students have
great difficulties with) but the sum will still be zero. It’s good to use ½ W resistors for this exercise,
as they are larger and easier to see.
You can follow the demonstration with a formal discussion of both of Kirchhoff’s laws, after
which you can test student understanding.
Clicker Question: The diagram below shows a segment of a circuit. What is the current in the
200  resistor?
A. 0.5 A
B. 1.0 A
C. 1.5 A
D. 2.0 A
E. There is not enough information to decide.
Clicker Question: The diagram below shows a circuit with two batteries and three resistors. What
is the potential difference across the 200  resistor?
A. 2.0 V
B. 3.0 V
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C. 4.5 V
D. 7.5 V
E. There is not enough information to decide.
Once students have a handle on these basic circuit analysis principles, you can delve into series
and parallel combinations of resistors. Again, we suggest starting with a demonstration.
Demonstration: Series lightbulbs. Connect one light bulb to a battery so that it glows quite
brightly. Now, add another bulb in series, and note the brightness of both bulbs. Now add a third.
Ask students to explain the change in brightness.
Demonstration: Windspeed indicator. A change in one element of a series circuit will affect
the circuit as a whole, and so will affect the current through each element—something students have
difficulty understanding. This principle is the basis for a very nice demonstration that connects
nicely to the material of the previous chapter. Take a 100 W lightbulb, and carefully break the glass
envelope so that the bulb’s filament stays intact. Use this filament as the resistor in a circuit with a
battery and a bulb as shown below.
Now, blow hard on the exposed filament. The filament isn’t hot enough to glow, but it will be
warm; blowing on it will cool it and thus decrease its resistance. The current in the circuit will
increase, and the 3.0 V bulb will noticeably brighten. This is a very nice demonstration that can be a
springboard for an enlightening class discussion.
The idea of replacing two or more resistors with an “equivalent resistance” is quite mysterious to
students. A simple demonstration can make this concept clear, and is a good choice before you
begin doing calculations using this concept.
Demonstration: Secret series circuit. Connect several resistors in series, and place the
combination in a box with two external connections. Now, use a multimeter to measure the
resistance of the contents of the box. The meter will have no trouble “deciding” on a resistance
value. You can switch the ends of the box and get the same result. After showing this, open the box
to reveal the contents—the actual resistors whose equivalent resistance was just shown.
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After such an introduction, you can give the formula for the equivalent resistance of a series
combination, then give students a chance to apply it.
Example: There is a current of 1.0 A in the circuit below. What is the resistance of the unknown
circuit element?
After series circuits, we suggest a similar development for parallel combinations of resistors.
Demonstration: Parallel lightbulbs. Connect one light bulb to a battery so that it glows quite
brightly. Now, add another bulb in parallel, and note the brightness of the two bulbs. Next, add a
third bulb in parallel. Ask students to explain their observations.
Demonstration: Secret parallel circuit. Connect several resistors in parallel, and place the
combination in a box with two external connections. Now, use a multimeter to measure the
resistance of the contents of the box. The meter will have no trouble “deciding” on a resistance
value. You can switch the ends of the box and get the same result. After showing this, open the box
to reveal the contents—the actual resistors whose equivalent resistance was just shown.
Finish the day by introducing the formula for computing the equivalent resistance of resistors
connected in parallel, and an example that uses this formula.
Example: What is the current supplied by the battery in the following circuit?
DAY 2: More Complex Circuits. The full analysis of circuits with series and parallel elements
together is quite challenging. The text gives a few examples, but you should work through more
examples in full detail in class if you expect students to be able to solve such circuits. This is a timeconsuming exercise, but it is useful because it brings together many of the ideas of circuits explored
to this point.
Example: What is the equivalent resistance of the following circuit?
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Example: Find the current in and the potential difference across each element in the following
circuit.
A discussion of household electricity is certainly in order—it’s the one element of the chapter
we can be certain our students will use in the future. There is a thorough discussion of this topic in
Chapter 26, so you might want to wait until Chapter 26 to cover this topic, but there is some
argument for doing it here, in concert with parallel circuits.
Example: The following devices are plugged into outlets on the same 120 V circuit in a house.
This circuit is protected with a 15 A circuit breaker.
Device
Power
Computer
250 W
Heater
900 W
Lamp
100 W
Stereo
120 W
Is there too much current in the circuit—that is, does the circuit breaker blow?
The chapter covers capacitors in series and parallel. If you need to trim topics in this chapter,
this is one that you could perhaps skip, but it can be dealt with quite quickly; once students have the
idea of an equivalent resistance, an equivalent capacitance isn’t too much of a stretch.
Clicker Question: Which of the following combinations of capacitors has the highest
capacitance? The lowest?
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DAY 3: Extensions and Applications. Start Day 3 with RC circuits—a concept that will be
important for the final topic of the day, electricity in the nervous system. We suggest starting with a
demonstration or two, to give students a feeling for what these circuits do. With the large-value
capacitors with low internal resistance now available you can do some wonderful classroom
demonstrations that will make the operation of these circuits clear to your students before you dive
into the mathematics.
Demonstration: RC circuit, part I. Charge a 1.0 F capacitor to 5.0 V and attach it to a single
lightbulb. Watch the bulb’s brightness fade until it is no longer visible with the room lights on, then
turn off the room lights to see that it still glows. This is a nice way to dramatize the long tail of the
exponential decay.
Demonstration: RC circuit, part II. Charge a 1.0 F capacitor to 5.0 V and attach it to a single
lightbulb. Measure the time for the bulb’s brightness to decrease to some level. Next, do the same
for two bulbs in parallel, then three. Finally, switch the 1.0 F capacitor for a 0.47 F capacitor, and do
a similar series of experiments. Ask your students to explain what they see. You can easily lead
them to hypothesize that the time for the decay depends on the resistance and capacitance.
The above demonstration can be used to “prove” the formula for the time constant of the decay,
which you can then present with very little additional fanfare. Do make a connection with the
previous example of an exponential decay in Part IV, then give students a chance to apply the
formula.
Clicker Question: The following circuits contain capacitors that are charged to 5.0 V. All of the
switches are closed at the same time. After 1 second has passed, which capacitor is charged to the
highest voltage? The lowest?
After this, you can move into a discussion of electricity in the nervous system. As noted above,
this is a great capstone piece for Chapters 20–23. Usually, we advocate doing different examples
than those in the book, but this topic is different. The numbers in the examples are based on human
physiology, and the relevant examples are all treated in the flow of the chapter, so you can simply
review and reinforce the examples treated in the text.
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There aren’t many good demonstrations available for this material, but there is one good
simulation of saltatory conduction that also makes a point about nerve conduction.
Demonstration: Salatory conduction, part I. Have a series of students (at least 10, but more
is nice) stand in a line, with each student placing his or her right hand on the shoulder of the student
to the right. The student at the right end of the line holds a ball in his or her hand. Now, tap the right
shoulder of the student at the left end of the line. Once he or she feels the tap, he or she taps the
shoulder of the student to the right—and so on down the line. Once the last student in the line feels
the tap, he or she drops the ball. The whole process will take several seconds; students in the class
can see the “signal” jump from student to student down the line. This is a good simulation of the
concept of saltatory conduction: Individual students represent insulated segments of axon, the
breaks between students the nodes between segments.
Demonstration: Salatory conduction, part II. Repeat the above example, but with one twist:
the signal isn’t communicated with a tap on the shoulder. Instead, students will use the right foot to
tap on the left foot of the student to the right. The longer distance the real nerve signal must travel
(from foot to brain to foot, rather than shoulder to brain to hand) means a much longer delay; the
total time for travel of the “signal” is much greater.
Something similar to this delay happens in certain diseases.
Conceptual Example: There are some diseases that result in a thinning of the myelin sheath that
surrounds peripheral neurons—those that carry signals between the spinal cord and the limbs. How
will this thinning affect nerve conduction speed? Explain this using the model for nerve conduction
developed in the chapter.
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
ideas for demonstrations, examples, questions, and additional topics that you could weave into your
class time.
Suggested Demonstrations
Which is brighter? Have a 60 W and 100 W light bulb and a circuit board where you can place
the two either in series or parallel with standard 120 V line voltage. Place the bulbs in parallel, and
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show that they have the predicted brightness—the 100 W bulb is brighter. Now, draw a diagram
showing how you’re going to place them in series. Ask students to predict which will be brighter.
Now, turn the circuit on; your students will be surprised to see that the 100 W bulb barely glows
at all. This demonstration points out that most students are unsure what it means for bulbs to be
labeled as “60 W” or “100 W.” Many think that the bulb will always dissipate this much power, and
essentially none will recognize that this has anything to do with the bulb’s resistance. After the
demonstration has captured their attention, explain these points, and show how students could do a
correct analysis of this system.
Minilight circuits. If you take one bulb from a string of minilights, a single battery can make it
light. Clearly, the operating voltage of individual bulbs is low, because there are so many in series.
If you remove one bulb, you break the circuit, and the whole string goes dark. But if you crush one
bulb, breaking the filament, and then reinsert it, the string will light—a shunt in the bulb base
carries the current, as described in the chapter. (Normally, the shunt is a very good insulator. But
when the bulb with the broken filament is inserted in the circuit, the entire 120 V appears across the
bulb. At this high voltage, the shunt changes its character, and becomes a reasonable conductor. Its
resistance is less than that of the filament, of course (otherwise, it would get very hot) so the current
in the string of lights increases slightly—though too little to notice. But if you break several bulbs
the way, the individual bulbs will noticeably brighten.
Sample Reading Quiz Questions
1. The bulbs in the circuit below are connected __________.
A. in series
B. in parallel
2. Which terminal of the battery has a higher potential?
A. the top terminal
B. the bottom terminal
3. When three resistors are combined in series the total resistance of the combination is
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A. greater than any of the individual resistance values.
B. less than any of the individual resistance values.
C. the average of the individual resistance values.
Additional Student Response System (“Clicker”) Questions
1.
In the circuit below, the switch is initially open and bulbs A and B are of equal brightness.
When the switch is closed, what happens to the brightness of the two bulbs?
A. The brightness of the bulbs is not affected.
B. Bulb A becomes brighter, bulb B dimmer.
C. Bulb B becomes brighter, bulb A dimmer.
D. Both bulbs become brighter.
2.
In the circuit shown below, the switch is initially closed and the bulb glows brightly. When the
switch is opened, what happens to the brightness of the bulb?
A. The brightness of the bulb is not affected.
B. The bulb gets dimmer.
C. The bulb gets brighter.
D. The bulb initially brightens, then dims.
E. The bulb initially dims, then brightens.
Additional Examples
1.
In the circuit shown below:
a. Rank in order, from most to least bright, the brightness of bulbs A–D. Explain.
b. Describe what, if anything, happens to the brightness of bulbs A, B, and D if bulb C is
removed from its socket. Explain.
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In the circuit shown below, rank in order, from most to least bright, the brightness of bulbs A–
E. Explain.
3.
In the circuit shown below:
a. How much power is dissipated by the 12 V resistor?
b. What is the value of the potential at points a, b, c, and d?
One Step Beyond: Strongly Electric Fish
“Wonderful as are the laws and phenomena of electricity when made evident to us in inorganic or dead
matter, their interest can bear scarcely any comparison with that which attaches to the same force when
connected with the nervous system and with life.”
Michael Faraday, in a 1839 article detailing his studies of Gymnotus, the electric eel
Instructors may be tempted to skip the final section of the chapter on electricity in the nervous
system, because it may seem a topic more appropriate for a biology course. But the early
investigators of electricity spent significant time considering this “animal electricity.” Michael
Faraday did a series of studies on the electric eel that influenced not only other scientists developing
a theory of electricity but those developing an understanding of neuroanatomy. Faraday might have
done much more work on this topic than he did but for a shipment of electric eels that did not arrive.
A model for electricity in the human nervous system is described in detail in the chapter. As
noted above, most of the examples one might envision are already part of the chapter. You could ask
for, say, the capacitance of a thicker cell membrane, but the value given in the example in the
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chapter is the appropriate value for the human nervous system. There isn’t a natural source of
alternate systems; most animals have nervous systems quite similar to ours.
There is a type of animal electricity, though, for which many different models exist—the
electricity of strongly electric fish, whose dramatic discharges so intrigued Faraday and many other
early investigators. These fish use series circuits composed of electrocytes, modified nerve or
muscle cells that produce a net voltage across each cell of a bit more than 100 mV. The series
combination of thousands of these cells can make significant voltages. The over 200,000 series
electrocytes in the electric organ of a full-grown electric eel can produce a voltage of over 600 V,
and a peak current of over 1 A. The fish uses its electric abilities to hunt, paralyzing prey that it
subsequently swallows. (Fish are quite susceptible to electric shock; researchers frequently survey
fish populations by passing a current through a section of a river or stream, paralyzing fish present
so that they can be assessed and analyzed. Most fish recover from this treatment with no ill effects.)
The pulse is quite short, so all of the hundreds of thousands of electrocytes must be simulated at the
same time—a remarkable degree of synchronization. This synchronization is achieved by having
different conduction speeds for the nerves that enervate electrocytes at different distances: different
nerves have myelin sheaths of different thicknesses. Close electrocytes are connected with nerves
having thin sheaths, and thus modest conduction speeds; distant electrocytes require the faster signal
speeds of nerves with thicker sheaths.
As noted in the chapter, the electric eel isn’t really an eel; it’s a fish. Why is it so long and
skinny? The electric organs that produce the electric eel’s remarkable pulse take up most of the
length of its body. The electric eel’s length is crucial—the electrocytes have a certain size, and they
must be connected in series. One might speculate that this fish developed its shape due to the needs
of its electrical system; a shorter, fatter eel wouldn’t produce as high a voltage. A shorter, fatter eel
would have more electrocytes in parallel, and so could produce a larger current—but the current is
primarily limited by the high resistivity of the freshwater environment in which this fish lives, not
by the output of the electrocytes.
We can see the connection between length and voltage more directly: A longer fish makes a
higher voltage. As electric eels grow, they add electrocytes to their electric organs, increasing the
output voltage. Young eels of 10 cm length can produce 100 V; mature eels measuring over 1 meter
may put out over 600 V. This increase in voltage with length is found in other strongly electric fish
as well. Electric catfish steadily increase in length and output voltage as they age.
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Saltwater electric fish such as the torpedo ray are put together quite differently. (The ray’s name
may seem odd, but “torpedo” comes from the same root as “torpor,” the Latin verb torpere, be
numb. The ancients were well aware of this fish’s ability to deaden the limbs of the body with its
powerful shocks; in fact, physicians used the shocks of the torpedo ray to treat certain ailments.) The
fish is nearly as wide as it is long, suiting its electric organs, which are composed of many parallel
stacks, each stack containing fewer electrocytes than the organs of the electric eel. Torpedo rays put
out lower voltages than electric eels, but are capable of higher currents, suiting their more
conductive saltwater environment. These rays aren’t shy, and will approach—and shock—divers
who get too close. The 45 V pulse of a typical ray isn’t enough to kill a diver, but it will be a painful
reminder to treat future encounters with more care.
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