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Chapter 22
Current and Resistance
Recommended class days: 2
Background Information
This chapter shares the same difficulties for students as the previous two chapters. The material is,
to students, very abstract. They can’t see a current, they can’t feel resistance.
The language of this chapter can be a barrier as well—particularly the most familiar words.
Students all know the word “current,” but most don’t really know what a current is. There are
several misconceptions about current that most students share:
• Many students do not think of current as a “flow” of charged particles.
• Most students do not understand the conditions under which a current exists.
• A large majority of students think that batteries are a source of constant current, delivering the
same current to any circuit.
• Many students think that current is “used up” when passing through devices such as light bulbs.
In this instance, they are confusing current with energy transfer.
• Students know that current has something to do with electrons, but most students think that
electrons are the charge carriers for any and all kinds of currents.
To help address these misconceptions, much of this chapter is focused on developing a concrete
model of current flow.
Student Learning Objectives
In covering the material of this chapter, students will learn to
• Understand how charge moves through a conductor.
• Understand and apply a basic model for current, including conservation of current.
• Understand how a battery creates a current in a circuit.
• Use the concepts of resistance and resistivity to predict currents in circuits.
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• Understand the energy transfers as charge moves through simple circuits.
Pedagogical Approach
An important issue in Chapter 20 was the evidence by which we know about charges and their
properties. This chapter starts with a similar tack, explaining experiments that allow us to deduce
the presence of a current and its properties, before defining what is flowing in a current. The
discharge of a capacitor is a good starting point because it can be understood on the basis of the
charge model of Chapter 20. Nearly all students will now understand that you can charge a capacitor
by moving charge from one plate to the other. They have also seen evidence that charge can move
through a conductor. If you connect two capacitor plates together with a wire, the forces between
charges cause excess charge to leave the plates and travel through the wire until the capacitor is
discharged.
You’ll need to emphasize that a current-carrying wire is not in electrostatic equilibrium. An
electric field in the wire is not only allowed, it is essential. A more detailed look at how current
flows reveals that the internal electric field is created by a nonuniform distribution of surface
charges on the wire. This is an interesting topic because it establishes the mechanism by which a
battery sets up an electric field in a wire that curves and bends. Nonetheless, these are rather subtle
ideas and probably not worth a large investment of time.
An important new idea in this chapter is that a potential difference is created when positive and
negative charges are separated. This is what happens when a capacitor is charged, but the charge
separation (and potential difference) of a capacitor can’t be maintained if the capacitor is “used.” If
the capacitor plates are connected by a wire, charges from the positive capacitor plate “fall
downhill” through the wire to the negative plate, discharging the capacitor. If we want to maintain
the current and the potential difference, we need some means to “lift” the charges back to the
positive plate, giving them a new supply of potential energy.
Batteries are introduced as a “charge escalator” that moves charge from the negative terminal to
the positive terminal. This is a non-electric means of creating a charge separation. Whether the
charge escalator functions due to chemical reactions (batteries) or due to mechanical forces on the
charges (generators), they do work to “lift” the charge back to the positive plate. Hence the
batteries’ emf can be defined as the work per unit charge to create a charge separation and thus
cause a potential difference.
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Students find it easy to visualize the charge-escalator model. It can be used to establish three
major ideas about batteries:
• Batteries are a source of potential difference because they have an internal energy supply that is
used to separate charge, thus transferring potential energy to the charges.
• The potential difference is a fixed quantity because the escalator lifts all charges the same height.
• The battery can supply varying amounts of current by running the escalator faster or slower, thus
lifting more or less charge per second. Regardless of how much charge is lifted, it all gets lifted
the same height. Thus the potential difference doesn’t change as the current varies.
This is particularly important for countering the strong student tendency to think that batteries are
sources of constant current.
Ohm’s law is introduced in this chapter. As Arons (1990) notes, it’s the idea that is important,
not the name. The law is stated in the form I  V /R because this (like a  F /m) helps to convey
cause and effect. Note that this text consistently uses V to represent potential differences and V to
represent the potential at a point. The common statement of Ohm’s law as V  IR (or I  V /R) often
reinforces students’ misconceptions about potential.
It’s important to emphasize the cause-and-effect sequence that leads to current in a simple
circuit that consists of a battery and a wire:
1. The battery is a source of emf.
2. The battery’s emf causes a potential difference Vwire between the ends of the wire.
3. The potential difference causes an electric field E in the wire.
4. The electric field establishes a current I in the wire.
5. The current is I  V /R. It depends on both the battery (V) and the wire (R).
The focus of this chapter is understanding current and resistance. At the end of it, students will
have a very good understanding of the workings of one particular circuit. This knowledge will serve
them well in Chapter 23, where more complex circuits will appear. (This chapter does introduce
some parallel and series combinations, but only for simple problems dealing with the conservation
of current.)
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Suggested Lecture Outlines
The material of this chapter is closely tied to that of Chapter 23. There might be a temptation to
jump right into the material of Chapter 23, but students need the careful introduction to the concepts
of this chapter.
DAY 1: Basic Concepts. An important issue in Chapter 20 was the evidence by which we know
about charges and their properties. The emphasis on evidence continues here with a similar set of
experiments about current. You might start with a demonstration that mirrors the material of the
chapter.
Demonstration: Leyden jar capacitor. You can build a very nice capacitor by taking a twoliter bottle, covering it with aluminum foil, filling it with water, and putting a nail through the cap to
touch the water.
This is similar to a Leyden jar, but the thin insulator of the bottle gives it a much higher capacitance
than the typical Leyden jar you might have in your demonstration collection. If you charge such a
capacitor with a Van de Graaf or a Wimshurst machine, you can use the discharge from the bottle to
light a fluorescent tube. (Safety note: Be very careful with the charged bottle. It can deliver enough
current to light a bulb, and so it can give you a painful shock.) This demonstration convincingly
shows the connection between the “static electricity” of Chapters 20 and 21 and the “current
electricity” of Chapters 22 and 23, and it nicely sets up the questions posed at the start of Chapter
22.
This, or some similar demonstration, makes some very key points about charges and currents.
Without some motivation, many students see no connection at all between the current in a circuit
and the charges they studied in the last three chapters. “Current” simply becomes a term, and
students form no mental image of what a current is or how charges flow through a wire. Instead, or
in addition to, the above demonstration, you could use a high-capacitance capacitor for a series of
experiments that are less dramatic but just as enlightening. The following sequence would also
make a nice exercise for the laboratory.
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Demonstration: Capacitor discharge. You can get very low internal resistance capacitors
with C 1 F at many scientific supply houses these days. These remarkable devices are excellent
tools for demonstrations or laboratory experiments. If you charge such a high-capacitance capacitor
with a few volts, it will run a light bulb or a small motor for several seconds as it discharges.
Discharging the capacitor through a coil of wire will deflect a compass needle. If you then use a
battery, you can again light a bulb, run a motor, and deflect a compass needle, only this time the
process is continuous, rather than running down after a few seconds. You can then develop some of
the basic ideas of charges and current, using the following logic:
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• The wire that discharges a capacitor can light a bulb, run a motor, and deflect a compass.
• The wire attached to a battery can do the same.
• Thus there is evidence that the same physical process is happening inside the wire in both cases.
• The wire that discharges the capacitor has charge moving through it—a current.
• Therefore, the wires that make up a “circuit” have charge flowing through them, and this is what
we mean when we say that “there is a current in the wire.”
These demonstrations provide evidence that current is a flow of charge, but they give no hint
about the nature of the charge carriers. Of course, this was the situation throughout the nineteenth
century, when current was defined as the flow of positive charge. Although J. J. Thomson’s
discovery of the electron was suggestive of electrons as charge carriers, the first direct evidence
about the charge carriers didn’t appear until the Tolman-Stewart experiment of 1916. This leaves
us with the unfortunate situation that the most common charge carriers are electrons, but
“conventional” current is the flow of positive charge, something worth noting to the students. In the
context of the above demonstrations, you could explain that there is no good way to tell, through
experiments such as these, what the sign of the charge carriers really is.
After this introduction, you can continue to develop a physical model of current in a conductor.
We have had some success doing this with a motion exercise that students take part in.
Demonstration: Model of current. Choose several students to play the role of atoms in a
lattice. These students will stay put. Then, choose several other students to play the role of charge
carriers. These students will move, according to your instructions. When you introduce an electric
field, the charge carriers will move through the lattice—colliding with the atoms of the lattice as
they go. You can illustrate a few key points, including:
• You can illustrate the difference between insulators and conductors by changing the
properties of the “material”. Having the charge carriers stay fixed to a spot in the lattice will
mean that there is no current. Allowing the charge carriers to move will allow a current to
develop. Adding more charge carriers will increase the current.
• As they students move through the lattice, have them make collisions with the lattice. (If the
atoms of the lattice vibrate, this makes for more frequent collisions.) Collisions cause the
charge carriers to experience “friction” as they move through the lattice. They need a steady
force from an electric field in order to move at constant average speed through the metal.
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• The drift velocity is proportional to the field strength. There is no current if E  0. A larger
field causes a larger drift velocity and thus a larger current.
This exercise also makes a convenient tool for introducing the idea of the conservation of current. If
you make a junction, the charge carriers must go one way or the other—they won’t just disappear!
The conservation of current is such a key idea that it is worth stressing with another, physical
demonstration.
Demonstration: Conservation of current. Have three light bulbs hooked up in a row, but with
the switch open so that the bulbs are off. First examine a separate battery and light bulb, noting that
the bulb has two terminals and that the current passes through the bulb to make it light.
(A surprising number of students don’t know this!) Don’t say anything about the current being
conserved, just note that it goes through. Also note that the brightness of a bulb depends on the
amount of current, and perhaps demonstrate this. Now turn to the three bulbs. First, show the wiring
and draw a picture so that students clearly understand how the bulbs are connected. Then point out
that, when the switch is closed, bulb A will be connected to the positive side of the battery and bulb
C to the negative side. Ask students to predict the brightness of the bulbs, from most bright to least
bright. Some students, who think that current is “used up” by the bulbs, will predict A  B C. A
few, now that they “know” that current is really the flow of negative electrons, will predict
C B A. Only 50% or 60% of students in a typical class correctly predict A  B  C, although
the fraction will be higher if you let them discuss it with their neighbors before making a prediction.
After this demonstration, discuss the conservation of current idea in some detail. After this, it’s
time to test understanding with some more complex circuits.
Clicker Question: Rank the bulbs in the following circuit according to their brightness, from
brightest to dimmest.
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Clicker Question: The wires below carry currents as noted. Rate the currents I A , I B and I C :
A. I A  I B  I C
B. I B  I A  I C
C. I C  I A  I B
D. I A  I C  I B
E. I C  I B  I A
Now, it’s time to talk about batteries, and how they produce a continuous current. One of the
most difficult to erase of student misconceptions about circuits is the idea that a battery is a source
of constant current—that is, the battery always supplies the same current, no matter what the circuit
is.
This misconception will be attacked several times in the next few chapters, but an important starting
point is the model of a battery as a charge escalator. This model will reappear in Chapter 29 as the
basis for defining the emf of a battery, but for now it’s simply a model of how the battery sustains a
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current. It’s particularly important to emphasize that an escalator can run at different speeds. The
faster it runs, the more rapidly it lifts the charge and thus the larger the current in the wire.
Demonstration: Simple circuit. It’s worth setting up a simple battery-wire circuit as above, in
order to demonstrate the energy transformations. Hook a battery to a length of nichrome wire; it will
quickly heat. Discuss the energy transformations taking place. It is important for students to
recognize that it is energy that is being “used up,” not the current. You could try short and long
wires, thick and thin—the different heating will show that changing the properties of the wire affect
the current.
This demonstration is a good jumping off point to define the idea of resistance—what
determines the current in the wire? Introduce the equation for resistance at this point. Before you
go on, you might want to take more time to introduce the basic ideas. A good way to do this, in a
demonstration or as a laboratory experiment, is with resistors made from Play-Doh as described by
Jones (1993).
Demonstration: Play-Doh circuits. Play-Doh (or similar modeling compounds) has a
resistivity of a modest 0.2 m; Play-Doh resistors about a centimeter in diameter and several
centimeters long have resistances that are easy to measure. If you apply a voltage of about 10 V to
such a resistor, the current will be in the range of several tens of mA. You can probe the potential at
different points and show how it varies, which makes a nice connection with the material of the
chapter. You can also reshape the Play-Doh to produce different resistances; students can easily
predict which shapes will have high and low resistance. This is a great way to make a physical
model that reinforces the concepts.
You can finish off the day with a clicker question that tests the concepts of the simple circuits
seen so far.
Clicker Question: A battery is connected to a wire, and makes a current in the wire. Which of the
following changes would increase the current? Which would decrease the current? Which would
cause no change?
A. Increasing the length of the wire.
B. Keeping the wire the same length, but making it thicker.
C. Using a battery with a higher rated voltage.
D. Making the wire into a coil, but keeping its dimensions the same.
E. Increasing the temperature of the wire.
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DAY 2: Further Development of Concepts, Applications. You can begin the day with a
discussion of Ohm’s law. This looks very much like the definition of resistance, so some discussion
is in order. After this, you should talk about practical examples of resistors—heating wires, the
resistors one sees in circuits, variable resistors, photoresistors, and so on. This is a good place for
some “show and tell.”
Demonstration: Resistors. Show some different types of resistors, perhaps in practical circuits.
You can follow this with a question that could be combined with a demonstration.
Clicker Question: A battery is connected to a photoresistor. When light shines on this resistor, it
increases the number of charge carriers that are free to move in the resistor. Now, the room lights
are turned off, so less light falls on the photoresistor. How does this affect the current in the circuit?
A. The current increases.
B. The current decreases.
C. The current is not affected.
So far, the idea that different materials are better conductors than others has only been
introduced qualitatively. It’s time for a quantitative definition through the concept of resistivity.
Students have difficulty with the definition of resistivity, but, at this point, they have seen several
such material properties that you can refer back to. You could explain, for example, that resistivity
is to resistance as density is to mass.
The biological examples of the simple circuits in this chapter often use the concept of resistivity,
as the following example shows.
Example: The resistivity of the body is a good measure of its overall composition. A measure of
the resistance of the upper arm is a good way to estimate the percent fat in a person’s body. Let’s
model a person’s upper arm as a cylinder of diameter 8.0 cm and length 20 cm. We can model the
composition of the arm by assuming that the muscle, far, and non-conductive portions (the bone)
form simple regions. This simple model actually works quite well. For a typical adult, the bone has
a cross-sectional area of 1.0 cm2; to a good approximation, the balance of the arm is fatty tissue or
muscle.
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a) Assume that the arm has 30% fat and 70% muscle. What is the resistance of the arm?
b) Now, assume that the arm has only 10% muscle. What is the resistance now?
Example: The measurement of the resistance of the arm is made by applying a voltage and
measuring a current. Too much current can be uncomfortable and, as we will see, can be dangerous.
Suppose we wish to limit the current to 1.0 mA. For each of the above cases, what is the maximum
voltage that could be employed?
Finish off the day with a quantitative treatment of the power in simple circuits. Introduce the
basic equation, then go on to do some calculations.
Clicker Question: A resistor is connected to a 3.0 V battery; the power dissipated in the resistor
is 1.0 W. The battery is now traded for a 6.0 V battery. The power dissipated by the resistor is now
A. 1.0 W
B. 2.0 W
C. 3.0 W
D. 4.0 W
Example: An electric blanket has a wire that runs through the interior. A current causes energy to
be dissipated in the wire, warming the blanket. A new, low-voltage electric blanket is rated to be
used at 18 V. It dissipates a power of 82 W. What is the resistance of the wire that runs through the
blanket?
Conceptual Example: For the electric blanket of the above example, as the temperature of the
wire increases, what happens to the resistance of the wire? How does this affect the current in the
wire? The dissipated power?
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Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some other
suggestions for demonstrations, examples, questions, and additional topics that you could weave
into your class time.
Suggested Demonstrations
There are some other nice demonstrations that you could use to help your students develop
familiarity with some of the basic concepts. Here are two of our favorites.
Energy ball. This is sold as a magic trick. A plastic ball the size and shape of a Ping-Pong ball has
two electrodes on the back.
When a current path is connected between these two electrodes, the ball emits light and sound. You
can use this ball to make a circuit through a dozen people; breaking the circuit at the opposite side
silences the ball. As soon as the ball is reconnected, the sound resumes. This is a fun and dramatic
tool for teaching basic ideas about current and circuits.
Static sensor. If you hold one lead of a small neon indicator lamp in your hand and touch the
other lead to a grounded conductor (such as the chalk rail on your chalk board), scuffing your feet
on the carpet will cause the bulb to light.
This is a great analog to the basic circuit of the chapter, and makes another good connection to the
ideas of static electricity. It’s a good way to show a very different kind of “charge escalator.”
Sample Reading Quiz Questions
1. The charge carriers in metals are
A. electrons
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B. positrons
C. protons
D. a mix of protons and electrons
2. A battery is connected to a resistor. Increasing the resistance of the resistor will
A. increase the current in the circuit.
B. decrease the current in the circuit.
C. not affect the current in the circuit.
3. A battery is connected to a resistor. As charge flows, the chemical energy of the battery is
dissipated as
A. current
B. voltage
C. charge
D. thermal energy
Additional Student Response System (“Clicker”) Questions
1.
Several light bulbs, different rated voltages, powers. Which one has highest resistance? The
lowest?
2.
Bulb
Rated voltage
Rated power
A
10 V
1W
B
8V
1W
C
12 V
2W
D
6V
2W
E
3V
3W
A battery makes a circuit with a lightbulb as shown. Two compasses are near the wires before
and after the bulb in the circuit. Which compass experiences a larger deflection?
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A. Compass 1 experiences a larger deflection.
B. Compass 2 experiences a larger deflection.
C. Both compasses experience the same deflection.
3.
In Trial 1, a battery is connected to a single lightbulb, and the brightness noted. Now, in
Trial 2, a second, identical, lightbulb is added. How does the brightness of these two bulbs
compare to the brightness of the single bulb in Trial 1?
A. The brightness is greater.
B. The brightness is the same.
C. The brightness is less.
Additional Examples
1.
How much time does it take for 1.0 C to flow through each of the following circuit elements?
a) A 60 W reading light connected to 120 V
b) A 60 W automobile headlamp connected to 12 V
One Step Beyond: Resistance and Resistivity Measurements on the
Body
The variability of the resistivity of body tissues does provide a means for simple, non-invasive
diagnostic measurements. Your students might be aware of the measurements of resistance that
form part of a polygraph test. But there are other, more clinically interesting examples as well:
• Measuring body fat. One model for computing body fat percentage is as noted above. The
actual procedure uses an AC signal, but the details are essentially as outlined above: A
measurement of resistance allows a determination of the fraction of the body composed of
different types of tissue. Measuring the upper arm or the thigh works well; in these tissues,
the fat and muscle tends to come in layers, meaning the simple model above works quite
well. There are consumer-grade bathroom scales that measure the resistance of a current path
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between the two feet that give less accurate results, but are capable of showing long-term
trends in an individual.
• Impedance cardiography. The resistivity of blood and muscle are quite different, so the
resistance of a current path that includes the heart changes dramatically as the heart beats. It
is possible to do a simple resistance measurement that allows a physician to determine the
change in the blood volume of the heart as it beats—a surprisingly simple means to
determine this clinically interesting data.
• Blood resistivity. The resistivity of the blood changes dramatically with hematocrit, the
fraction of the blood taken up by cells—the cells are much less conductive than the fluid
they ride around in. Resistivity also varies with other blood parameters. A simple resistance
measurement can, again, give valuable insight as to the composition of the blood.
All of these make excellent examples for teaching. In this era when MRI and other such diagnostic
techniques are becoming commonplace, it’s interesting to see some lower-tech techniques like those
above that are based on such simple physical models. It’s remarkable to think that one could use the
equivalent of a battery and an ammeter to determine the volume of blood flow through the heart!
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