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DC Circuits
4. DC Circuits*
Objectives: In this unit you will explore current flow in circuits.
The learning objectives are the following:
1. To be able to explain what electric current is, what a circuit is, and how current flow
through a resistor relates to the voltage drop across it.
2. To know how to use a voltmeter to measure potential difference between points in a
circuit, and to know how to connect an ammeter to measure current flow in some branch
of a circuit.
3. To understand the hydraulic analogy—which is the analogy between water circulating
though a closed network of pipes and electric current flowing in a circuit—and to use
this analogy to predict circuit properties.
4. To understand when circuit elements are in series, or when they are in parallel.
5. To be able to use (and to explain the origin of) the rule for combining resistances in
series or in parallel.
(These ideas, except the hydraulic analogy, are presented in your textbook.)
Reading assignment: Review of basic concepts of current, voltage and circuits, series and
parallel and on measurements.
Read the following sections. (Section numbers may be slightly different depending on the edition
of your textbook: Check the section titles.)
Knight, Jones and Field : 22.1 A Model of Current , 22.2 Defining and Describing Current, 22.3
Batteries and emf , 22.4: Connecting Potential and Current, 22.5 Ohm's Law and Resistor
Circuits 17.1: Electric Current
Serway and Vuille (212): 17.2 A Microscopic View: Current and Drift Speed, 17.3 Current and
Voltage Measurements in Circuits, 17.4 Resistance, Resistivity, and Ohm's Law
Serway and Jewett (252): 27.1 Electric Current, 27.2 Resistance, 27.3 A Model for Electrical
Conduction
Pre-Lab Exercises: Answer the following based on your reading, and bring your solutions with
you to the lab. The instructor will check off that you have done them, and later you will include
these solutions in your lab report.
1. A continuous metal wire connects the two ends of a 3V battery with a rectangular loop as
shown. Let the negative terminal of the battery (point c) be chosen as the reference point
where V = 0 Volts. Let’s assume the wire has some fixed resistance per unit length.
(Caution: if the wire is made of a good conductor you should not connect this way? Why?)
b
a
3V
c
d
(a) Label and mark with small x’s the points along the wire
where the potential would be 3 Volts, 2 Volts and 1 Volt
(three different points).
(b) Estimate the value of  Vad, the voltage difference (or
potential difference) from point a to point d along the wire.
______________________________________________________________________________
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*© William A Schwalm 2012
DC Circuits
(c) As a small amount of charge moves along the wire from point b to point c, does the electric
potential increase or decrease?
(d) In moving through the battery from b to c, does the potential increase or decrease? Or does
it remain the same? If it changes, how much does it change by?
(e) In going all the way around the loop in a clockwise direction, starting at b and ending up back
at b, is the net change in potential positive, negative or zero?
3. State Ohm’s law that relates current though a resistance to the voltage drop across it.
Scenario: Your team is working on curriculum materials for teaching the basic physics of electric
circuits. The idea is to base these materials on the hydraulic analogy, which has not been
popular recently as a teaching aid. Thus you will need to help
Water (yes)
write descriptions of how certain aspects of fluid flow in a
plumbing circuit are similar to certain aspects of electric current
flow in an electric circuit. Of course it is assumed that you have
done your assigned reading, because now you will need to
apply what you have learned in that pre-lab reading. As you go
through the activities as a group, discuss how these activities
relate to the learning objectives listed above.
Hydraulic analogy: In some ways, electricity flowing in a circuit
electricity
is like water flowing in a pipe. This leads to what is called the
(no)
hydraulic analogy. Now, of course, water flowing in a pipe and
electricity flowing in a circuit are different in important ways too.
The main difference is that while water can pour out of the pipe
when you open one end, electricity cannot usually do this. The
reason is that, while the air of the room offers no resistance to
water flow, the same air acts as an insulator for the electricity.
Electric charge cannot easily flow through air (except across
high voltage differences) while water can. So air is more like
the metals walls of the pipe for electricity, not like a way out. The end of a wire in a circuit is
similar, in the hydraulic analogy, to a pipe with an end cap on it so that no water can flow out onto
the floor.
Thus somewhat unlike water in pipes, electricity can only flow in complete circuits. This is like
water or another incompressible fluid flowing around a complete or closed circuit of pipes. For
electricity, this means an electric current can only flow in complete circuits.
Electric current flowing in a circuit is like water flowing (in complete circuits or “round-trips”)
through pipes. The rate of flow of water is analogous to the electric current, and the pressure
difference between points in the plumbing is like the voltage difference in the electric circuit.
What flows in a circuit instead of water is electric charge. Here is the hydraulic analogy spelled
out in a table:
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DC Circuits
electric
corresponds to
1. current in Amps (or Coulombs )
Second
2. voltage (potential) in Volts
3. wires forming electric circuit
4. voltage source (battery, generator)
5. electrical resistor
6. volt meter measures potential
7. ammeter measures current
hydraulic
1. rate of flow ( Gallons )
Second
2. pressure difference
3. pipes forming hydraulic circuit
4. a circulating pump
5. a filter offering resistance to flow
6. pressure gauge measures pressure
7. flow meter measures rate of flow
In-class group response questions:
1. A lifeguard at a swimming pool monitors a filter. A pump pushes the water through the filter,
and then the water re-circulates back to the pool.
The rate of flow H of water, in gallons per second, must relate in some way to three things,
namely the pressure in the water before the filter, the pressure in the water after the filter, and
some constant k that denotes the resistance of the filter to the flow of water.
FILTER
P1
P2
H
After discussion in your problem-solving group, see if you can come up with a formula for the rate
of flow H in gallons per second in terms of P1, P2 and k. Write it on the white board and be ready
to report. Record it also here:
2. Recall that electric currents flow in complete circuits. Here is an electric circuit (using standard
symbols) and the analogous hydraulic circuit appears on the next page.
Ammeter
measures
current
A
A. A
Resistor
Battery
+
V
_
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Voltmeter
measures
voltage
difference
Your group should prepare a
table explaining which device
or which variable in the
electric circuit is analogous to
which item or variable in the
hydraulic circuit.
This table should be written
on the white board, and also
recorded by each group
member for submission later.
DC Circuits
Rate of Flow
Meter
Filter
Pump
Record here:
Pressure
Difference
Meter
2. A student writes to you that a potential of 4.5 Volts “flows through” the resistor in the circuit
above. Speaking grammatically, what’s wrong with this? How do I know right of the bat
something is wrong with the way the student is thinking? Explain.
Summary comment on the use of meters: By the hydraulic analogy you can see that current
meters (ammeters) and voltmeters must be placed very differently in circuits.
The ammeter has a very low internal resistance. Notice that it is placed so that all the
current you want to measure flows through it. It should offer little or no resistance to the flow.
Current flows out of the source, through the ammeter, through the rest of the circuit, and then
back into the source. Only the rest of the circuit should offer resistance to the flow, not the
ammeter. For this reason, never place the ammeter by itself across the battery, so as to give the
current a zero-resistance short cut, or “short circuit” from the + to the - terminals of the battery by
flowing only through the meter.
On the other hand, the voltmeter has a very high internal resistance. It is placed between
two points in the circuit when you need to know voltage difference, or potential difference
between the two points. Very little current therefore actually flows through the voltmeter. Thus
you should never place the voltmeter in such a way that the current would have to flow through
the voltmeter in order to make the circuit work. Because the circuit would not work. The high
resistance of the voltmeter would block the current flow.
Ammeter measures current. It is placed in the circuit by interrupting one of
the wires, so that current must flow through the ammeter. It has almost no
resistance to current flow, so
Never place an ammeter directly across a battery. 
Voltmeter measures potential difference between two points. It is placed
between these two points, one lead wire connected to each. It must have
a very high resistance to current flow. Thus, unlike the ammeter,
A voltmeter is never placed in the circuit, but always
across it, between the two points.
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DC Circuits
Problem: Use the hydraulic analogy in which an ammeter is like a rate-of-flow meter and a volt
meter is like a pressure-difference meter to explain the difference in how these two meters would
be placed in a circuit. You will be using hydraulic terms like pressure and gallons per second,
etc. in your explanation. (See the figures on page 4.)
Exploring the apparatus: The figure at the right
represents the instructional tool we want to use for
teaching our employees about voltage, current and
resistance. To understand how it works you should
play with it. You have two multi-meters for making
measurements. One type of multi- meter is shown
below.
1
2
3
10

20

30

Measuring voltage: When you measure DC voltage, set the rotary dial to one of the upper lefthand settings, depending on the highest voltage you need to measure,
as shown in the figure. The wires from the circuit to the meter plug into
to sockets at the base, as you’ll see. To measure voltage, one wire
plugs into the COM terminal and the other to the V-  terminal. You
touch the meter wires (called probes, sometimes) to the circuit you’re
measuring on. A positive voltage reading means that the point where the
OFF
V
V
V-  probe is touching is that many Volts more positive than the point
where the COM probe is touching. A negative reading means that, on
the other hand, the COM probe is at the higher voltage.
Measuring current: For DC current measurements, set the rotary dial
over to the lower right-hand sector and select the scale appropriate for
the maximum current you expect for full scale. You’re better off to over
estimate at first, so as not to damage anything. Remember that when

A
you measure current, the current must actually flow through the meter.
Thus you have to replace one of the wires of the circuit by the meter.
We say we put the meter “in” the circuit. You connect the meter via the
V  COM A 10A
A and the COM terminals. The current should go into the meter via the A
terminal (often denoted 200mA) and out of the meter via the COM
terminal. This will give a positive reading. When it flows the other way, you get a negative
reading. When you want to measure current above 200 mille amperes you should use the
special terminal 10A, rather than the A terminal.
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A
DC Circuits
There’s a fuse in the meter that can burn out, so if your meter stops working, you may have blown
a fuse. Report this situation. Finally, the batteries in the meters can go bad. Your instructor can
help you check the battery.
Completing your circuit: The power source will sometimes be the special wall receptacles near
your station. Usually the even-numbered ones are used. Red means positive, or where the
current comes out of the wall, and black is negative, or where the current goes back into the wall.
(The actual power supply is in another room.) Usually, however, there will be a separate DC
power supply instead of the wall receptacle.
Use the plug box of your experimental set up to construct a circuit such as the one shown
schematically in the figure at the right. The current flows out of the source at C, along the wire
to the resistor, then through the resistor to the ammeter, then through the ammeter and back into
the source via D. If the voltmeter has a very high resistance, then almost no current will get
around the resistor by flowing through the voltmeter. On the other hand, if the ammeter has very
little resistance, it will not impede the flow. Be sure to use at least 10 rather than a smaller
value. Make the indicated connections by plugging the banana plug wires into appropriate
banana jacks. When you have set up this experiment, record the current and voltage readings.
Take time to experiment with the meters and the circuitry so that you can figure out how
everything works. (Can you describe the route the current takes from C to D in the case shown?)
Resistance
plug box
A
V
Ohm’s law:
From the hydraulic analogy,
1. What equation do you expect to hold between the current I flowing through the resistor R and
the voltages Va and Vb ?
R
a
b
I
The resistance to the flow of electric current R is analogous to the resistance k of the filter to the
flow of water. To what extent does this theoretical prediction agree with what you measure?
(“What extent,” means you need to give numbers, percentages or something.)
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DC Circuits
Now to study this relation between voltage difference across a branch of a circuit, and current
flowing through the branch in more detail, we need a way of applying a small, well controlled
voltage to the test circuit. Consider the following device.
___________________
(If you are using a power supply instead of the rheostat voltage divider, skip this.)
Voltage divider: It is handy to apply a rather low voltage to the experimental part of the circuit.
One good way to do this is to use a potentiometer voltage divider. You may or may not want to
set up and use this kind of voltage divider. If the divider is used, Some higher voltage (only a
couple of volts in any case) is provided at the wall connections A and B, and voltage you actually
use is obtained at terminals C and D. Adjusting the slide on the variable resistor adjusts the
voltage difference between C and D. That means the effective “battery voltage” for your
experiment appears between C and D and is adjusted by using the slider on the variable resistor.
The figure on the left shows how the voltage divider looks in the lab, while the schematic diagram
on the right shows how it works. Notice that B and D are actually at the same voltage. The
potential drop from A to C plus the voltage drop from C to D must equal the whole voltage drop
from A to B provided at the wall receptacle. Thus, by moving the adjustable contact on the
resister, you can adjust the voltage between C and D. This is the voltage you use for the
experiment, if you decide to use this set-up.
A
A B
Red
(positive)
B
Black
(negative)
B. B
C
D
C
D
2. The physical unit used to measure resistance is the Ohm (  ). From the equation you figured
out in question 1, express the Ohm in terms of the volt (unit of potential) and the amp (unit of
current).
______________________
(Continue here if you have skipped the voltage divider.)
The relationship among current, voltage difference and resistance for any linear circuit element in
a DC circuit is called Ohm’s law. This is what you should have come up with by the hydraulic
analogy or through your reading. The first project is to set up to demonstrate Ohm’s law using
the equipment you have at hand, now that you know how it all works.
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DC Circuits
Problem 1
Devise a good way to use the equipment to demonstrate Ohm’s law.
Measurement plan: This plan needs to tell what kind of graphs would be needed, what data to
take and how it should be taken. Think about varying relevant quantities two at a time. Record
the plan here. (Remember the things that need to be in a measurement plan.)
Implementation: Now carry out your plan and record data here.
Resistors in series: When we say some circuit components are “in series” (with respect to a
certain current flow) we mean the whole current has to flow first through one, then through
another, then through another, and so on. Here we show you just a schematic view of the circuit
we have in mind. It’s up to you to see how to wire it up using the plug box.
Notice that the resistors in the figure are “in series.” Why? The circle with A in it represents an
ammeter, which is also in series. Record the current reading from the ammeter. (Is there
another place to put the ammeter and get the same result?)
Let’s walk through a measurement plan that would use the equipment to illustrate how resistors
add in series. After that, you are asked to do your own plan for adding resistors in parallel.
List each resistor separately, and tell how much current flows through each.
1  resistor?
P
2  resistor?
Q
1
3  resistor?
Now, figure out how to use the voltmeter V to
measure the voltage difference across each
resistor in turn. Draw a picture of the plug box
R
S T
2
A
4-8
U
3
DC Circuits
to show at least one case for an example. So, the voltage drop across the 3 resistor would be
VUT = VU - VT. You will need to show how to attach the voltmeter wires.
Using simple algebra, show that the sum of the voltage drops VUT+VSR+VQP should be equal
the voltage drop VUP all the way across the combination of three resistors. You would start with
the meaning of these voltage drops as in the previous paragraph. Explain. (Hint: What does
VUP mean in terms of VU and VP?)
Then measure the voltage drop VUP, and see if you prediction holds true. Record the
measurement.
Now suppose I ask you: What one, single resistor could replace the series combination and give
the same result? In other words, suppose the source voltage remains the same. What single
resistor would let the same amount of current flow? Put another way, what single resistance
value represents the effective resistance of these three resistors in series?
To answer this question, notice that due to Ohm’s law the effective resistance would have to obey
the relation
 VUP  I Rseries .
Thus, show from your measurements that in this case
Rseries  1  2  3  6.
The effective resistance of a series
combination of resistors is the sum of
the individual resistances.
It is now easy to see why this is so. The voltage drops across the individual resistors add up to
the total drop, while the same current flows through each resistor. So,
Rseries 
Vtotal VUT  VSR  VQP

 RUT  RSR  RQP .
I
I
In other words, the reason the resistances add for resistors in series is that the voltage drops
across the individual resistors (the numerators) add, while the currents through the individual
resistors (the denominators) are all the same.
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DC Circuits
Now you might say, “Of course! This is obvious!” But the next exercise shows that combining
resistors need not be so obvious.
Thus you have demonstrated a general result:
Problem 2
Considering the combination of resistors shown at the right. These are in parallel. (Why?) Your
team needs to make up an activity involving the plug box that will illustrate how three resistors
combine in parallel to form an equivalent resistance.
10 
20 
Measurement Plan: Work out a plan to
30 
demonstrate adding resistors in parallel.
Besides the usual components, you need to
make a drawing showing how you would wire
the plug box to make the circuit shown at the left. To accomplish this, notice that some of the
banana plugs can be stacked, or double plugged. Thus you can plug several plugs piggy-back
into the same banana jack. Show where you would put the voltmeter to determine the net
effective Rparallel. Describe what you would need to measure and how you would measure it.
Record your plan here.
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A
DC Circuits
Implementation: Put your plan in operation and record the data here. On the way out of lab, your
instructor will want to see the plan including the wiring sketch and the also the data that result
from actually making the measurements.
Analysis and conclusion: As part of your report, you need to write a procedure, including an
explanation of what’s going on, that will illustrate combining three resistors in parallel. Your writeup should make reference to the hydraulic analogy, because that’s the work assigned to your
group.
At the end of the lab period, you will need to review the objectives and comment on how your lab
activities related to these objectives.
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DC Circuits
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