2. Learning Objectives
Electric circuits
Current, resistance, power
Students should understand the definition of electric
current, so they can relate the magnitude and direction of
the current to the rate of flow of positive and negative
charge.
Students should understand conductivity, resistivity, and
resistance, so they can:
Relate current and voltage for a resistor.
Describe how the resistance of a resistor depends upon
its length and cross-sectional area, and apply this result
in comparing current flow in resistors of different
material or different geometry.
Apply the relationships for the rate of heat production in
a resistor.
3. Learning Objectives
Steady-state direct current circuits with batteries and
resistors only
Students should understand the behavior of series and
parallel combinations of resistors, so they can:
Identify on a circuit diagram whether resistors are in
series or in parallel.
Determine the ratio of the voltages across resistors
connected in series or the ratio of the currents through
resistors connected in parallel.
Calculate the equivalent resistance of a network of
resistors that can be broken down into series and
parallel combinations.
Calculate the voltage, current, and power dissipation for
any resistor in such a network of resistors connected to
a single power supply.
Design a simple series-parallel circuit that produces a
given current through and potential difference across
one specified component, and draw a diagram for the
circuit using conventional symbols.
4. Learning Objectives
Steady-state direct current circuits with batteries and
resistors only
Students should understand the properties of ideal and real
batteries, so they can:
Calculate the terminal voltage of a battery of specified
emf and internal resistance from which a known current
is flowing.
Students should be able to apply Ohm’s law and
Kirchhoff’s rules to direct-current circuits, in order to:
determine a single unknown current, voltage, or resistance.
Students should understand the properties of voltmeters
and ammeters, so they can:
State whether the resistance of each is high or low.
Identify or show correct methods of connecting meters
into circuits in order to measure voltage or current.
5. Learning Objectives
Capacitors in circuits
Students should understand the t = 0 and steady-state
behavior of capacitors connected in series or in parallel, so
they can:
Calculate the equivalent capacitance of a series or
parallel combination.
Describe how stored charge is divided between
capacitors connected in parallel.
Determine the ratio of voltages for capacitors connected
in series.
Calculate the voltage or stored charge, under steady-
state conditions, for a capacitor connected to a circuit
consisting of a battery and resistors.
6. Table of Contents
1. Electromotive Force and Current
2. Ohm’s Law
3. Resistance & Resistivity
4. Electric Power
5. Alternating Current (Not AP)
6. Series Wiring
7. Parallel Wiring
8. Series and Parallel Wiring
9. Internal Resistance
10. Kirchoff’s Rules
11. The Measurement of Current and Voltage
12. Capacitors in Series and Parallel Circuits
13. RC Circuits (Not AP)
14. Safety and the Physiological Effects of Current (Not AP)
8. Electric Circuits
In an electric circuit, an energy source and an energy
consuming device are connected by conducting wires through
which electric charges move.
9. Electromotive Force
Within a battery, a chemical reaction occurs that transfers
electrons from one terminal to another terminal.
The maximum potential difference across the terminals is
called the electromotive force (emf).
10. The electric current is the amount of charge per unit time
that passes through a surface that is perpendicular to the
motion of the charges.
t
Q
Iavg
∆
∆
=
One coulomb per second equals one ampere (A).
Electric Current
11. Types of Current
If the charges move around the circuit in the same direction at
all times, the current is said to be direct current (dc).
If the charges move first one way and then the opposite way,
the current is said to be alternating current (ac).
12. Example 1 A Pocket Calculator
The current in a 3.0 V battery of a pocket calculator is 0.17 mA. In one hour
of operation, (a) how much charge flows in the circuit and (b) how much energy
does the battery deliver to the calculator circuit?
(a)
(b)
( )tIQ ∆=∆
( )( )V0.3C61.0=
t
Q
Iavg
∆
∆
=
( )( )s3600A1017.0 3−
×= C61.0=
QVU = J8.1=
13. Direction of Current
Conventional current is the hypothetical flow of positive
charges that would have the same effect in the circuit as the
movement of negative charges that actually does occur.
14. 20.1.1. In which one of the following situations does a conventional
electric current flow due north?
a) Protons in a beam are moving due south.
b) A water molecule is moving due north.
c) Electrons in a beam are moving due south.
d) Electrons in a wire connected to a battery are moving from south
to north.
15. 20.1.2. The battery capacity of a lithium ion battery in a digital
music player is 750 mA-h. The manufacturer claims that the player
can operate for eight hours if the battery is initially fully charged.
Given this information, determine the number of electrons that flow
through the player as you listen to your favorite songs for three hours.
a) 6.2 × 1018
electrons
b) 1.0 × 103
electrons
c) 2.4 × 109
electrons
d) 6.3 × 1021
electrons
e) 8.1 × 1028
electrons
17. Resistance
The resistance (R) is defined as
the ratio of the voltage V applied
across a piece of material to the
current I through the material.
To the extent that a wire or an
electrical device offers resistance
to electrical flow, it is called a
resistor.
18. SI Unit of Resistance: volt/ampere (V/A) = ohm (Ω)
I∝V
Ohm’s Law
The V is proportional to I,
where V is the voltage applied
across a piece of material and
I is the current through the
material:
IRV =
19. Example 2 A Flashlight
The filament in a light bulb is a resistor in the form
of a thin piece of wire. The wire becomes hot enough
to emit light because of the current in it. The flashlight
uses two 1.5-V batteries to provide a current of
0.40 A in the filament. Determine the resistance of
the glowing filament.
I
V
R =
IRV =
A0.40
V0.3
= Ω= 5.7
20. 20.2.1. In a certain circuit containing a battery and a resistor, Ohm’s law is obeyed.
An instrument to measure the current in the circuit, an ammeter, is connected in
between one of the terminals of the battery and one end of the resistor. The ammeter
indicates that the current in the circuit is I. The battery is then removed and replaced
with another battery. This time, the ammeter indicates the current is 2I. Which one of
the following statements concerning the resistor is true?
a) When the second battery was placed in the circuit, the resistance increased to twice its
initial value.
b) When the second battery was placed in the circuit, the resistance decreased to one half its
initial value.
c) When the second battery was placed in the circuit, the resistance increased to four times
its initial value.
d) When the second battery was placed in the circuit, the resistance increased to one fourth
its initial value.
e) When the second battery was placed in the circuit, the resistance did not change.
21. 20.2.2. Consider the circuit containing a battery and a resistor shown.
For which one of the following combinations of current and
voltage does R have the smallest value?
a) V = 9 V and I = 0.002 A
b) V = 12 V and I = 0.5 A
c) V = 1.5 V and I = 0.075 A
d) V = 6 V and I = 0.1 A
e) V = 4.5 V and I = 0.009 A
22. 20.2.3. A certain circuit contains a battery and a resistor. An instrument to measure the current in
the circuit, an ammeter, is connected in between one of the terminals of the battery and one
end of the resistor. The graph shows the current in the circuit as the voltage is increased.
Which one of the following statements best describes the resistor in this circuit?
a) The resistor does not
obey Ohm’s law.
b) The resistor obeys Ohm’s
law for voltages between zero
and twenty-five volts.
c) The resistor obeys Ohm’s law
for voltages between zero and
thirty-five volts.
d) The resistor obeys Ohm’s law for
voltages between zero and forty volts.
e) The resistor obeys Ohm’s law for voltages between thirty and forty volts.
24. A
L
R
ρ
=
resistivity in units of ohm·meter
Resistance in Materials
For a wide range of materials, the resistance of a piece of
material of length L and cross-sectional area A is:
25.
26. Example 3 Longer Extension Cords
The instructions for an electric lawn mower suggest that a 20-gauge extension
cord can be used for distances up to 35 m, but a thicker 16-gauge cord should
be used for longer distances. The cross sectional area of a
20-gauge wire is 5.2x10-7
Ω·m, while that of a 16-gauge wire is 13x10-7
Ω·m.
Determine the resistance of (a) 35 m of 20-gauge copper wire and (b) 75 m of
16-gauge copper wire.
A
L
R
ρ
=
(a)
(b) ( )( )
27-
8
m1013
m75m1072.1
×
⋅Ω×
=
−
( )( )
27-
8
m105.2
m35m1072.1
×
⋅Ω×
=
−
Ω= 2.1
A
L
R
ρ
= Ω= 99.0
27. ( )[ ]oo TT −+= αρρ 1
temperature coefficient
of resistivity
( )[ ]oo TTRR −+= α1
Temperature Effects
28. 20.3.1. For which combination for the length L and radius R of a wire
will the resistance have the smallest value?
a) L = 0.50 m and R = 0.03 m
b) L = 0.25 m and R = 0.08 m
c) L = 0.40 m and R = 0.2 m
d) L = 0.80 m and R = 0.1 m
e) L = 0.10 m and R = 0.05 m
29. 20.3.2. The ends of a wire are connected to the terminals of a battery.
For which of the following changes will the resulting current in the
circuit have the largest value?
a) Replace the wire with one that has a larger resistivity.
b) Replace the wire with one that has a larger radius.
c) Replace the wire with one that has a longer length.
32. IVP =
SI Unit of Power: watt (W)
( ) RIIRIP 2
==
R
V
V
R
V
P
2
=
=
Electric Power
When there is current in a circuit as a result of a voltage, the
electric power delivered to the circuit is:
33. Example 5 The Power and Energy Used in a
Flashlight
In the flashlight, the current is 0.40A and the voltage
is 3.0 V. Find (a) the power delivered to the bulb and
(b) the energy dissipated in the bulb in 5.5 minutes
of operation.
(a)
(b)
IVP =
PtE =
( )( )V0.3A40.0= W2.1=
( )( )s330W2.1= J100.4 2
×=
34. 20.4.1. An automatic coffee maker uses a resistive heating element to boil
the 2.4 kg of water that was poured into it at 21 °C. The current
delivered to the coffee pot is 8.5 A when it is plugged into a 120 V
electrical outlet. If the specific heat capacity of water is 4186
J/kgC°, approximately how long does it take to boil all of the water?
a) 5 minutes
b) 8 minutes
c) 10 minutes
d) 13 minutes
e) 15 minutes
35. 20.4.2. The insulated wiring in a house can safely carry a maximum current
of 18 A. The electrical outlets in the house provide an alternating
voltage of 120 V. A space heater when plugged into the outlet operates
at an average power of 1500 W. How many space heaters can safely be
plugged into a single electrical outlet and turned on for an extended
period of time?
a) zero
b) one
c) two
d) three
e) four
36. 20.4.3. A portable CD player was recently introduced that has a
“special power saving technology.” The manufacturer claims
that with only two standard AA batteries (together: 3.0 V, 20 kJ
energy storage) that the player can be played for about 25 hours.
What is the approximate resistance in the CD player’s electrical
circuitry?
a) 41 Ω
b) 0.010 Ω
c) 300 Ω
d) 1.5 Ω
e) 15 Ω
37. 20.4.4. A wire is used as a heating element that has a resistance that is
fairly independent of its temperature within its operating range.
When a current I is applied to the wire, the energy delivered by the
heater each minute is E. For what amount of current will the
energy delivered by the heater each minute be 4E?
a) 2I
b) 4I
c) 0.5I
d) 0.25I
e) 8I
41. In circuits that contain only resistance, the current reverses direction
each time the polarity of the generator reverses.
( ) ( )ftIft
R
V
R
V
I o
o
ππ 2sin2sin ===
peak current
42. ( )ftVIIVP oo π2sin2
==
( )ftII o π2sin= ( )ftVV o π2sin=
45. Example 6 Electrical Power Sent to a
Loudspeaker
A stereo receiver applies a peak voltage of
34 V to a speaker. The speaker behaves
approximately as if it had a resistance of 8.0 Ω.
Determine (a) the rms voltage, (b) the rms
current, and (c) the average power for this
circuit.
47. Conceptual Example 7 Extension Cords and a Potential Fire Hazard
During the winter, many people use portable electric space heaters to keep
warm. Sometimes, however, the heater must be located far from a 120-V wall
receptacle, so an extension cord must be used. However, manufacturers often
warn against using an extension cord. If one must be used, they recommend
a certain wire gauge, or smaller. Why the warning, and why are smaller-gauge
wires better then larger-gauge wires?
48. 20.5.1. The graph shows the current as a function of time for an
electrical device plugged into a outlet with an rms voltage of 120
V. What is the resistance of the device?
a) 24 Ω
b) 21 Ω
c) 17 Ω
d) 14Ω
e) 12 Ω
49. 20.5.2. Consider the circuits shown in parts A and B in the picture. In part A, a light bulb is
plugged into a wall outlet that has an rms voltage of 120 volts. A current I passes through the
circuit and the bulb turns on. In part B, a second, identical light bulb is connected in series in
the circuit. How does the current in circuit B compare with that in circuit A?
a) The current is the same, I, as in part A.
b) The current is twice as much, 2I, as in part A.
c) The current in part B is zero amperes.
d) The current is one fourth as much, 0.25I, as in part A.
e) The current is one half as much, 0.5I, as in part A.
51. Series Wiring
There are many circuits in which more than one device is
connected to a voltage source.
Series wiring means that the devices are connected in
such a way that there is the same electric current
through each device. (One Path)
52. 21 VVV +=
+++= 321 RRRRSSeries resistors
Resistance in a series Circuit
As we will discuss later, the sum of all voltage in a circuit must
equal zero.
Voltage supplied by battery is lost by resistors
21 IRIR += ( )21 RRI += SIR=
∑=
i
iS RR
53. Example 8 Resistors in a Series Circuit
A 6.00 Ω resistor and a 3.00 Ω resistor are connected in series with a 12.0 V
battery. Assuming the battery contributes no resistance to the circuit, find
(a) the current, (b) the power dissipated in each resistor, and (c) the total
power delivered to the resistors by the battery.
(a)
(b)
(c)
Ω=Ω+Ω= 00.900.300.6SR
SR
V
I =
RIP 2
=
RIP 2
=
W31.5W6.10 +=P
Ω
=
00.9
V0.12
A33.1=
( ) ( )Ω= 00.6A33.1
2
W6.10=
( ) ( )Ω= 00.3A33.1
2
W31.5=
W9.15=
54. 20.6.1. Consider the circuit shown in the drawing. Two identical light bulbs, labeled
A and B, are connected in series with a battery and are illuminated equally.
There is a switch in the circuit that is initially open. Which one of the following
statements concerning the two bulbs is true after the switch is closed?
a) Bulbs A and B will be off.
b) Bulbs A and B will be equally
illuminated.
c) Bulb A will be brighter and bulb B
will be off.
d) Bulb A will be off and bulb B
will be brighter.
e) Both bulbs will be dimmer than before the switch was closed.
56. Parallel Wiring
Parallel wiring means that the
devices are connected in such a
way that the same voltage is applied
across each device.
Multiple paths are present.
When two resistors are connected in
parallel, each receives current from the
battery as if the other was not present.
Therefore the two resistors connected
in parallel draw more current than does
either resistor alone.
58. parallel resistors…
+++=
321
1111
RRRRP
Parallel Wiring
As we will discuss later, the total
current flowing into any point must
equal the total current flowing out.
21 III +=
21 R
V
R
V
+=
+=
21
11
RR
V
=
PR
V
1
∑=
i iP RR
11
59. Simplifying Circuits
∑=
i iP RR
11
R1 = 5 Ω
R2 = 3 Ω
21
11
RR
+=
21
21
RR
RR
R
+
=
21
121
RR
RR
R
+
=
( )( )
Ω+Ω
ΩΩ
=
35
35
Ω= 89.1R
0
5
RTotal
OR
60. Example 10 Main and Remote Stereo Speakers
Most receivers allow the user to connect to “remote” speakers in addition
to the main speakers. At the instant represented in the picture, the voltage
across the speakers is 6.00 V. Determine (a) the equivalent resistance
of the two speakers, (b) the total current supplied by the receiver, (c) the
current in each speaker, and (d) the power dissipated in each speaker.
61. (a)
Ω
+
Ω
=
00.4
1
00.8
11
PR
Ω= 67.2
(b)
PR
V
I rms
rms =
R
V
I rms
rms =(c)
R
V
I rms
rms =
(d) rmsrmsVIP =
rmsrmsVIP =
∑=
i iP RR
11
Ω
=
00.8
3
Ω
=
67.2
V00.6
A25.2=
Ω
=
00.8
V00.6
A750.0= A
00.4
V00.6
Ω
= A50.1=
( )( )V00.6A750.0= W50.4=
( )( )V00.6A50.1= W00.9=
62. Conceptual Example 11 A Three-Way Light Bulb
and Parallel Wiring
Within the bulb there are two separate filaments.
When one burns out, the bulb can produce only
one level of illumination, but not the highest.
Are the filaments connected in series or
parallel?
How can two filaments be used to produce three
different illumination levels?
63. 20.7.1. Consider the three resistors and the battery in the circuit
shown. Which resistors, if any, are connected in parallel?
a) R1 and R2
b) R1 and R3
c) R2 and R3
d) R1 and R2 and R3
e) No resistors are connected in parallel.
64. 20.7.2. Consider the circuits shown in parts A and B in the picture. In part A, a
light bulb is plugged into a wall outlet that has an rms voltage of 120 volts. A
current I passes through the circuit and the bulb turns on. In part B, a second,
identical light bulb is connected in parallel in the circuit. How does the total
current in circuit B compare with that in circuit A?
a) The current is the same, I, as in part A.
b) The current is twice as much, 2I, as
in part A.
c) The current in part B is zero amperes.
d) The current is one fourth as much, 0.25I, as in part A.
e) The current is one half as much, 0.5I, as in part A.
65. 20.7.3. Two light bulbs, one “50 W” bulb and one “100 W” bulb, are
connected in parallel with a standard 120 volt ac electrical outlet.
The brightness of a light bulb is directly related to the power it
dissipates. Therefore, the 100 W bulb appears brighter. How does
the brightness of the two bulbs compare when these same bulbs
are connected in series with the same outlet?
a) Both bulbs will be equally bright.
b) The “100 W” bulb will be brighter.
c) The “50 W” bulb will be brighter.
68. 20.8.1. Consider the three identical light bulbs shown in the circuit.
Bulbs B and C are wired in series with each other and are wired in
parallel with bulb A. When the bulbs are connected to the battery as
shown, how does the brightness of each bulb compare to the others?
a) Bulbs B and C are equally bright,
but bulb A is less bright.
b) Bulbs B and C are equally bright,
but less bright than bulb A.
c) All three bulbs are equally bright.
d) Bulbs A and B are equally bright, but bulb C is less bright.
e) Only bulb A is illuminated.
69. 20.8.2. A circuit is formed using a battery, three identical resistors,
and connecting wires as shown. How does the current passing
through R3 compare with that passing through R1?
a) I3 < I1
b) I3 = I1
c) I3 > I1
d) This cannot be determined without knowing the amount of current
passing through R2.
70. 20.8.3. What is the approximate equivalent resistance of the five
resistors shown in the circuit?
a) 21 Ω
b) 7 Ω
c) 11 Ω
d) 14 Ω
e) 19 Ω
78. Internal Resistance
Batteries and generators add some resistance to a circuit.
This resistance is called internal resistance.
The actual voltage between the terminals of a battery is
known as the terminal voltage.
79. Example 12 The Terminal Voltage of a Battery
The car battery has an emf of 12.0 V and an internal
resistance of 0.0100 Ω. What is the terminal voltage when
the current drawn from the battery is (a) 10.0 A and (b)
100.0 A?
(a) IrV =
V10.0V0.12 −
(b) IrV =
V0.1V0.12 −
( )( )Ω= 010.0A0.10 V10.0=
11.9V=
( )( )Ω= 010.0A0.100 V0.1=
11.0V=
80. 20.9.1. In physics lab, two students measured the potential difference
between the terminals of a battery and the current in a circuit connected
to the battery. The students then made a graph of the two parameters as
shown. They then drew a best fit line through the data. From their
results, determine the approximate internal resistance of the battery.
a) 0.002 Ω
b) 0.08 Ω
c) 0.1 Ω
d) 0.3 Ω
e) 0.6 Ω
82. Loop Rule
The loop rule expresses conservation of energy in terms of
the electric potential.
States that for a closed circuit loop, the total of all potential
rises is the same as the total of all potential drops.
83. Junction Rule
Conservation of mass
Electrons entering must equal the
electrons leaving
The junction rule states that the total
current directed into a junction must
equal the total current directed out of
the junction.
84. Example 14 Using Kirchhoff’s Loop Rule
Determine the current in the circuit.
( ) ( )
dropspotentialrisespotential
0.8V0.612V24 Ω++Ω= II
A90.0=I
∑ =
i
iV 0
( )321 VVVVbattery ++=
85. Reasoning Strategy
Applying Kirchhoff’s Rules
1. Draw the current in each branch of the circuit. Choose any
direction. If your choice is incorrect, the value obtained for
the current will turn out to be a negative number.
2. Mark each resistor with a + at one end and a – at the other
end in a way that is consistent with your choice for current
direction in step 1. Outside a battery, conventional current is
always directed from a higher potential (the end marked +) to
a lower potential (the end marked -).
3. Apply the junction rule and the loop rule to the circuit,
obtaining in the process as many independent equations as
there are unknown variables.
4. Solve these equations simultaneously for the unknown
variables.
86. 20.10.1. What is the current through the 4-Ω resistor in this circuit?
a) 0.67 A
b) 0.75 A
c) 1.0 A
d) 1.3 A
e) 1.5 A
87. 20.10.2. What is the current through the 1-Ω resistor in this circuit?
a) 2.8 A
b) 3.0 A
c) 3.4 A
d) 4.3 A
e) 4.8 A
88. 20.10.3. Which one of the following equations is not correct relative
to the other four equations determined by applying Kirchoff’s
Rules to the circuit shown?
a) I2 = I1 + I4
b) I2 = I3 + I5
c) 6 V − (8 Ω) I1 − (5 Ω) I2 − (4 Ω) I3 = 0
d) 6 V − (6 Ω) I4 − (5 Ω) I2 − (2 Ω) I5 = 0
e) 6 V − (8 Ω) I1 − (6 Ω) I4 − 6 V − (2 Ω) I5 − (4 Ω) I3 = 0
90. A dc galvanometer. The coil of
wire and pointer rotate when there
is a current in the wire.
91. An ammeter must be inserted into
a circuit so that the current passes
directly through it.
92. If a galvanometer with a full-scale
limit of 0.100 mA is to be used to
measure the current of 60.0 mA, a
shunt resistance must be used so that
the excess current of 59.9 mA can
detour around the galvanometer coil.
93. To measure the voltage between two points
in a circuit, a voltmeter is connected between
the points.
95. Parallel capacitors
∑=
i
iP CC
21 qqq +=
Capacitors in Parallel
Voltage is the same on each side of the circuit
Charges on each capacitor directly add
VCVC 21 += ( )VCC 21 +=
96. 21 VVV +=
Series capacitors
∑=
i iS CC
11
Capacitors in Series
Since there is only one path, charge is the same in all
capacitors regardless of capacitance
Voltage drop of each capacitor directly add
21 C
q
C
q
+=
+=
21
11
CC
q
97. 20.12.1. A parallel plate capacitor is connected to a battery and becomes fully
charged. A voltmeter is used to measure the potential difference across the
plates of the capacitor. Then, an uncharged thin metal plate is inserted into the
gap between the parallel plates without touching either plate. What affect, if
any, does the insertion of the plate have on the potential difference across the
plates?
a) The potential difference will not change.
b) The potential difference will increase to twice its initial value.
c) The potential difference will decrease to one half its initial value.
d) The potential difference will increase to a value that cannot be determined
without having more information.
e) The potential difference will decrease to a value that cannot be determined
without having more information.
98. 20.12.2. Three parallel plate capacitors, each having a capacitance of
1.0 µF are connected in series. The potential difference across
the combination is 100 V. What is the charge on any one of the
capacitors?
a) 30 µC
b) 300 µC
c) 3000 µC
d) 100 µC
e) 1000 µC
103. 20.13.1. In physics lab, Rebecca measured the voltage across an
unknown capacitor in an RC circuit, every ten seconds after a
switch in the circuit that allows the capacitor to discharge is
closed. The capacitor was initially fully charged. Using the
graph, estimate the time constant.
a) 7.5 s
b) 15 s
c) 30 s
d) 45 s
e) 60 s
104. 20.13.2. An RC circuit contains a battery, a switch, a resistor, and a
capacitor – all connected in series. Initially, the switch is open and the
capacitor is uncharged. Which one of the following statements
correctly describes the current in the circuit during the time the
capacitor is charging?
a) The current is increasing with increasing time.
b) The current is constant with increasing time.
c) The current is decreasing with increasing time.
d) The current increases for the first half of the time until the capacitor is
fully discharged, and then decreases during the second half of the time.
e) The current can either increase or decrease with increasing time
depending on the value of the time constant.
