Chapter 24
Electric Currents and DC Circuits
“What we call physics comprises that group of natural
sciences which base their concepts on measurements;
and whose concepts and propositions lend themselves
to mathematical formulation. Its realm is accordingly
defined as that part of the sum total of our knowledge
which is capable of being expressed in mathematical
terms.” Albert Einstein
24.1 Electric Current
Before 1800 the study of electricity was limited to electrostatics, the study of
charges at rest. It was impossible to obtain large amounts of electric charge for any
continuous period of time. In 1800, Alessandro Volta (17451827), an Italian
physicist, invented the “Voltaic pile,” later to be called an electric battery. The
battery converted chemical energy into electrical energy thereby supplying a
potential difference and relatively large quantities of charge that could flow from
the battery for relatively long periods of time. With the invention of the battery,
applications of electricity grew by leaps and bounds.
Let us first consider the motion of electric charges when a battery is used to
supply a potential difference between two parallel plates. The space between the
plates is either a vacuum or is filled with air. The negative plate has a small hole in
it to allow the introduction of an electron into the uniform electric field, as shown in
figure 24.1. The electron immediately experiences a force in the electric field, given
by
F = qE = −eE
Figure 24.1 Motion of an electron in air or a vacuum.
where e is the charge on the electron. The motion of the electron can be determined
by the techniques discussed in section 21.10. The electron will experience the
acceleration:
a =
F
= −
eE
m m
Chapter 24 Electric Currents and DC Circuits
The position and velocity of the electron are given by the kinematic equations
r = v
0
t +
1
at
2
2
and
v = v
0
+ at
If a battery is now connected across the ends of a length of wire, the flow of
electrons is much more complicated because of the molecular structure of the wire
itself. The atoms of the metal wire are arranged in a lattice structure, as shown in
figure 24.2(a). Metals can be thought of as an array of positive ions surrounded by a
Figure 24.2 Motion of an electron in a wire.
more or less uniform sea of negatively charged electrons that help hold the ions in
place. The outer electron of the atom in the metal wire is loosely bound, and in the
lattice structure moves about freely. The free electron of each atom moves about in
a random manner within the lattice structure due to the constant interaction
between the moving electron and each fixed positive ionized atom it sees as it moves
about. These electrons resemble the random motion of gas molecules and are
sometimes referred to as the electron gas. They have no net directed motion along
the wire. When a potential difference is applied across the ends of the wire, as in
figure 24.2(b), an electric field is set up within the wire. The electron in this field
starts to move, but the motion is not the simple motion of the electron in figure 24.1
because of the constant interactions with the positive ions of the lattice. The
electron slowly drifts with an average velocity v
d
, the drift velocity, in the expected
242
Chapter 24 Electric Currents and DC Circuits
direction as seen in figure 24.2(c). Because of the constant interaction with the
atoms of the lattice structure, it takes almost 30 s in some metals for the electron to
drift about 1 cm. This is extremely slow as compared to the motion that would be
experienced by the electron in figure 24.1.
Because of this complexity of the motion of the electron within the solid wire,
we will make no further analysis of the motion of the electron by Newton’s laws and
the kinematic equations; we will adopt a much simpler procedure. When a potential
difference is applied across the ends of a wire and electrons start to drift within the
wire, we will say that an electric current exists in the wire. The electric current is
defined as the amount of electric charge that flows through a cross section of the wire
per unit time. We write this mathematically as
q
I
t
=
(24.1)
The SI unit of current is the ampere, named after the French physicist, André
Marie Ampère (17751836). An ampere of current is a flow of one coulomb of charge
per second. That is,
1 ampere = 1
coulomb
second
We abbreviate this as
1 A = 1 C/s
Because one coulomb of charge represents 6.242 × 10
18
electronic charges, the
ampere is the flow of 6.242 × 10
18
electrons per second. Sometimes we use a smaller
unit of current, the milliampere, abbreviated mA.
1 mA = 10
−3
A
Because electric current was defined before any knowledge of the existence of
electrons, the original current was assumed to be a flow of positive charges. (It is
interesting to note that it was Benjamin Franklin [17061790] an American
statesman and scientist who first called the charges positive and negative. He
assumed it was the positive charges that moved.) Although it is now known that
this is incorrect for the conduction through a solid, it is usual to continue with this
historical convention and define the current to be a flow of positive charges from a
position of high potential to one of low potential. This current is called
conventional current to distinguish it from the actual electron flow. The flow of
electrons is called the electron current. In all the cases considered, a flow of
positive charges in one direction (conventional current) is completely equivalent to a
flow of negative charges in the opposite direction (electron current). In gases and
electrolytic liquids, both positive and negative charges flow in opposite directions,
but both contribute to the positive current.
243
Chapter 24 Electric Currents and DC Circuits
The flow of charges in a wire is represented in what is called a circuit
diagram, and the simplest one is shown in figure 24.3. This circuit consists of the
Figure 24.3 A circuit diagram.
battery and a very long wire, which is connected to both terminals of the battery.
The battery, the supplier of a potential difference, is shown as the two horizontal
lines. The longer line represents the positive terminal of the battery, the point of
highest potential, and is labeled with a positive sign (+). The shorter line represents
the negative terminal of the battery and is labeled with a negative sign in the
diagram (−). This negative terminal is arbitrarily chosen to be the point of zero
potential. The line from the positive terminal to the negative terminal is the
external circuit and here represents the length of wire connected to the two
terminals of the battery. The potential difference maintained by the battery is
labeled V in the figure.
The flow of charge in the circuit is analogous to dropping a ball from a height
h and is shown in figure 24.4. At the height h, the ball has a positive potential
Figure 24.4 Analogy of mechanical and electrical motion.
energy with respect to the ground, which is at zero potential energy. The ball falls
from a position of high potential energy to the ground, as shown in figure 24.4(a).
The electrical circuit of figure 24.3 is pulled apart in figure 24.4(b) to show the
analogy more clearly. In the electrical circuit, a positive charge leaves the positive
terminal of the battery where it has the potential V, and ‘’falls’’ through the
connecting wire to the ‘’ground’’ or zero of potential. This analogy between the
mechanical case and the electrical case is even clearer if you recall that we defined
the potential as the potential energy per unit charge. So when a positive charge
flows or ‘’falls’’ from high potential to low potential it is falling from a position of
high potential energy to a position of low potential energy. Whenever a potential
difference exists between any two points in a circuit, positive charge flows from the
244
Chapter 24 Electric Currents and DC Circuits
point of high potential to the point of low potential. The resulting current is called a
direct current (DC) since the charges flow in only one direction.
24.2 Ohm’s Law
If a wire is connected to both terminals of a battery, charges will flow. To determine
the current produced by these flowing charges, a device called an ammeter is placed
in the circuit and is shown as A in figure 24.5(a). The ammeter is a device that
displays the amount of current in a circuit. Another device, called a voltmeter, is
Figure 24.5 Experimental determination of Ohm’s law.
connected across the battery and is shown as V in figure 24.5(a). The voltmeter
reads the potential difference between any two points in an electrical circuit. In
figure 24.5(a) the voltmeter reads the potential difference across the terminals of
the battery. A picture of a typical laboratory meter is shown in figure 24.5(c).
For a particular battery maintaining a potential difference V
1
, a current I
1
is
recorded by the ammeter. When a larger battery of potential difference V
2
replaces
battery V
1
, a larger current I
2
is observed in the circuit. By successively using
different batteries, we observe different currents. If we plot the current recorded by
the ammeter in the circuit against the applied battery voltage, we obtain a linear
relationship, as shown in figure 24.6. This implies that the current in the circuit is
Figure 24.6 Ohm’s law.
directly proportional to the applied voltage, that is,
245
Chapter 24 Electric Currents and DC Circuits
I ∝ V (24.2)
To make an equality out of this proportionality, we introduce a constant of
proportionality (1/R) and the proportionality 24.2 becomes the equation
V
I
R
=
(24.3)
where R is called the resistance of the wire. Equation 24.3 is called Ohm’s law after
Georg Simon Ohm (17871854), a German physicist who discovered the relation in
1827. Ohm’s law states that the current in a circuit is directly proportional to the
applied potential difference V and inversely proportional to the resistance R of the
circuit. The SI unit of resistance is the ohm, designated by the Greek letter Ω,
where
1 ohm = 1
volt
= 1 V/A
ampere
(For completeness, we should note that there are some materials that do not obey
Ohm’s law. That is, a graph of the current versus the voltage is not a straight line.
For such materials, the resistance is not a constant, but varies with voltage.
However, the resistance for any voltage can still be defined as the ratio of the
voltage to the current. We will not be concerned with such materials in this book.)
Every wire has some resistance and the resistance of a circuit is shown in the
circuit diagram in figure 24.5(b) as a saw tooth symbol that is labeled as R. The
wire, or any other resistive material, is called a resistor. Ohm’s law enables us to
determine the current in a circuit when the resistance of the circuit and the applied
voltage are known.
Example 24.1
Finding the current by Ohm’s law. A 6.00V battery is applied to a circuit having a
resistance of 10.0 Ω. Find the current in the circuit (see figure 24.7).
Figure 24.7 Simple application of Ohm’s law.
Solution
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Chapter 24 Electric Currents and DC Circuits
The current in the circuit, found by Ohm’s law, equation 24.3, is
I =
V
=
6.00 V
= 0.600
V
R 10.0 Ω V/A
= 0.600 A
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Example 24.2
Finding the resistance by Ohm’s law. A potential difference of 12.0 V is applied to a
circuit and a current of 0.300 A is observed. What is the resistance of the circuit?
The resistance of the circuit, found from the rearrangement of Ohm’s law, is
Solution
R =
V
=
12.0 V
= 40.0 Ω
I 0.300 A
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24.3 Resistivity
The concept of the resistance of a wire can be more easily understood by looking at
the lattice structure of the wire as in figure 24.2(c). When a potential difference is
applied to the ends of the wire the electrons slowly drift along the wire. The greater
the length of the wire, the longer it takes for the electrons to drift along the wire.
Since the current is the flow of charge per unit time, a longer period of time implies
a smaller current in the wire. This reduced current is viewed from Ohm’s law as an
increase in the resistance of the wire. Therefore, the resistance of a wire is directly
proportional to the length l of the wire, that is,
R ∝ l (24.4)
The larger the crosssectional area of the wire the greater are the number of
electrons that can pass through it per unit time. Therefore, the greater the area, the
larger the current in the wire. Viewed from Ohm’s law, equation 24.3, this
increased current implies a smaller value of resistance. Hence, the resistance of a
wire is inversely proportional to the crosssectional area of the wire, that is,
247
Chapter 24 Electric Currents and DC Circuits
R ∝
1
(24.5)
A
We can combine proportionalities 24.4 and 24.5 into the one proportionality
R ∝
l
(24.6)
A
To make an equation out of this, we need a constant of proportionality. The
constant of proportionality should depend on the material that the wire is made of,
because the electrons in the wire constantly interact with the atoms of the lattice.
Different atoms exert different forces on the free electrons, and hence affect the
drift velocity of the electrons. The proportionality constant is called the resistivity ρ,
and proportionality 24.6 becomes the equation
l
R
A
ρ
=
(24.7)
Equation 24.7 says that the resistance of a wire is directly proportional to its
resistivity and its length, and inversely proportional to its crosssectional area. A
table of resistivities for various materials is shown in table 24.1. The SI unit of
resistivity is an ohm meter, abbreviated Ω m. Everything else being equal,
Table 24.1
Resistivities (ρ) of Some Different Materials at 20
0
C
and Mean Temperature Coefficient of Resistivity (α)
Material
ρ, Ω m
α
0
C
−1
Aluminum
Brass
Carbon (graphite)
Copper
Gold
Iron
Lead
Mercury
Platinum
Silver
Tungsten
Amber
Bakelite
Glass
Hard Rubber
Mica
Wood
2.82 × 10
−8
7.00 × 10
−8
3500 × 10
−8
1.72 × 10
−8
2.44 × 10
−8
9.71 × 10
−8
20.6 × 10
−8
98.4 × 10
−8
10.6 × 10
−8
1.59 × 10
−8
5.51 × 10
−8
5.00 × 10
14
2 × 10
5
 2 × 10
14
10
13
 10
14
10
13
 10
16
10
11
 10
15
10
8
 10
11
3.9 × 10
−3
2.00 × 10
−3
−0.5 × 10
−3
3.93 × 10
−3
3.40 × 10
−3
5.20 × 10
−3
4.30 × 10
−3
8.90 × 10
−3
3.90 × 10
−3
3.80 × 10
−3
4.50 × 10
−3
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Chapter 24 Electric Currents and DC Circuits
materials with relatively small values of resistivity ρ, such as metals, have small
resistances and hence make good conductors of electricity. Materials with large
values of ρ, the nonmetals, have large resistances and therefore make poor
conductors. Poor conductors are, however, good insulators. A good conductor has a
resistivity of the order of 10
−8
Ω m, whereas a good insulator has a resistivity value
of the order of 10
13
10
15
Ω m. Good conductors of electricity are also good conductors
of heat. This is because the highly mobile electrons are also carriers of thermal
energy in conductors. For the same reason, good electrical insulators are also good
thermal insulators.
Example 24.3
The resistance of a spool of wire. Find the resistance of a spool of copper wire, 500 m
long with a diameter d of 0.644 mm.
The crosssectional area of the wire is given by
Solution
A =
πd
2
4
(
)
2
3
0.644 10 m
4
π
−
×
=
= 3.26 × 10
−7
m
2
The resistance of the wire spool, found from equation 24.7, is
R = ρ
l
A
( )
×
Ω 10 ×=
−
−
27
8
m 103.26
m 500
m 1.72
= 26.4 Ω
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Example 24.4
The resistance of an amber rod.
Find the resistance of an amber rod 30.0 cm long by
2.00 cm high and 2.50 cm thick, as shown in figure 24.8.
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Chapter 24 Electric Currents and DC Circuits
Figure 24.8
The resistance of an amber rod.
The resistance from one end of the rod to the other, found from equation 24.7, is
Solution
( )
(
)
( )( )
14
0.300 m
5.00 m
0.0200 m 0.0250 m
l
R
A
ρ= = × 10 Ω
= 3.00
×
10
17
Ω
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24.4 The Variation of Resistance with Temperature
It is found experimentally that the resistivity of a material is not constant but
rather varies with temperature. A graph of the variation of the resistivity with
temperature for a metal is shown in figure 24.9(a). Notice that the graph is a curve,
which means that the variation is not linear. However, a straight line can be drawn
that approximates the curve for a range of temperatures, as shown in figure 24.9(b).
The slope
m
of this straight line is given as
m
=
∆ρ
=
ρ − ρ
0
∆
t
t
−
t
0
where ρ is the resistivity of the material at the temperature
t
and ρ
0
is the
resistivity of the material at the temperature
t
0
. Rearranging the equation we get
ρ − ρ
0
=
m
(
t
−
t
0
)
Solving for the resistivity ρ, gives
ρ = ρ
0
[1 +
m
(
t
−
t
0
)] (24.8)
ρ
0
Let us now define a new constant α, called the
mean temperature coefficient of
resistivity,
as
α =
m
ρ
0
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Chapter 24 Electric Currents and DC Circuits
Figure 24.9
The variation of resistivity with temperature.
Thus, by measuring the slope of the curve in the area of interest, and dividing by
the resistivity ρ
0
at the reference temperature
t
0
, we obtain the mean temperature
coefficient of resistivity α. The value of α for various materials is shown in table
24.1. Notice that since the slope
m
is a ratio of resistivity to temperature, the
temperature coefficient of resistivity α , which is equal to that slope divided by the
resistivity, has units of
0
C
−
1
. With this new temperature coefficient of resistivity, we
can write equation 24.8 as
ρ = ρ
0
[1 + α (
t
−
t
0
)] (24.9)
Equation 24.9 gives the resistivity of a material at the temperature t when the
resistivity of the material ρ
0
is known at the reference temperature t
0
.
Example 24.5
Temperature dependence of resistivity.
The resistivity of copper at 20.0
0
C is 1.72
×
10
−8
Ω m. Find its resistivity at 200
0
C.
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Chapter 24 Electric Currents and DC Circuits
The temperature coefficient of resistivity for copper, found from table 24.1, is 3.93
×
10
−3
0
C
−
1
. The resistivity of copper at 200
0
C, found from equation 24.9, is
Solution
ρ = ρ
0
[1 + α (
t
−
t
0
)]
= (1.72
×
10
−8
Ω m)[1 + (3.93
×
10
−3
0
C
−
1
)(200 − 20.0)
0
C ]
= 2.94
×
10
−8
Ω m
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Because the resistivity of a wire changes with temperature, the resistance of that
wire also changes with temperature. If we multiply both sides of equation 24.9 by
the length of the wire and divide by the crosssectional area of the wire, we obtain
ρ
l
= ρ
0
l
[1 + α(
t
−
t
0
)] (24.10)
A
A
However, ρ
l
/
A
is equal to the resistance
R
from equation 24.7. Therefore, we can
write equation 24.10 as
(
)
0
1R R t tα
0
= + −
(24.11)
Equation 24.11 gives the resistance of a resistor
R
, at the temperature
t
, if the
resistance
R
0
, at the reference temperature
t
0
, is known.
Example 24.6
Temperature dependence of resistance.
If the resistance of a copper resistor is 50.0 Ω
at 20.0
0
C, find its resistance at 200
0
C.
The temperature coefficient of resistivity for copper, found from table 24.1, is 3.93
×
10
−3
0
C
−1
. The resistance of the copper resistor at 200
0
C, found from equation 24.11,
is
Solution
R
=
R
0
[1 + α(
t
−
t
0
)]
= (50.0 Ω)[1 + (3.93
×
10
−3
0
C
−1
)(200 − 20.0)
0
C]
= 85.4 Ω
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Chapter 24 Electric Currents and DC Circuits
We should note that not all materials have the same kind of variation of
resistivity with temperature. A group of materials known as
semiconductors
have a
temperature variation as shown in figure 24.9(c).
When the resistivity of metals is plotted against the absolute temperature, a
very strange phenomenon occurs at very low temperatures, as shown in figure
24.9(d). At a certain temperature, known as the critical temperature
T
c
, the
resistivity, and hence the resistance of the material, drops to zero. The material is
then said to be superconducting, and the material is called a superconductor. This
phenomenon was first discovered by Kamerlingh Onnes in the Netherlands in 1911.
He found that for mercury, the resistivity effectively dropped to zero when the
temperature was reduced to 0.05 K. Researchers have since found newer materials
that become superconducting at much higher temperatures. By the 1960s the
critical temperature for superconduction was found to be as high as 20 K. Steady
research into newer materials has raised the critical temperature even further. By
the end of 1986 the critical temperature of 40 to 50 K had been reached with an
oxide of barium, lanthanum, and copper. By March 1987 superconductivity had
been made to occur at temperatures above 90 K.
The reason for the great importance of superconductivity lies in the fact that
once the resistance of a material drops to zero, the energy dissipated in the
resistance also drops to zero and the current continues to exist forever. (We will see
that the energy dissipated in a resistor is given by
I
2
R
. If
R
is zero, no energy is
lost.) If newer superconducting materials can be found that have still higher critical
temperatures, a time will come when superconductivity will be a practical new
technology that will revolutionize the entire electrical industry. The cost of
delivering electricity will be greatly reduced by making essentially lossless
transmission lines. Some superconducting thin films will be used in new devices for
electronic components in computers. Electric motors will run with a minimum of
energy. The change will be as great as it was with the discovery of the transistor
and the subsequent use of the microchip in electrical components.
24.5 Conservation of Energy and the Electric Circuit 
Power Expended in a Circuit
Consider the simple resistive circuit in figure 24.10. The power supplied by the
battery is the work it does per unit time, that is
P
=
W
(24.12)
t
But the work done within the battery is done by chemical means, and its final
result is to move a positive charge
q
from the negative terminal inside the battery to
the positive terminal within the battery. That is, a charge
q
had to be ‘’lifted’’ from
2413
Chapter 24 Electric Currents and DC Circuits
Figure 24.10
Conservation of energy in an electric circuit.
the point of zero potential to the point of higher potential
V
within the battery.
Recall from the definition of the potential that the potential is the potential energy
per unit charge,
V
=
PE
=
W
(21.19)
q
q
where
W
is the work that must be done to give the charge its potential energy. From
equation 21.19, the work done within the battery is simply
W qV
=
(24.13)
The power supplied by the battery is found from equations 24.12 and 24.13 as
P
=
W
=
qV
(24.14)
t
t
But
I
=
q
(24.1)
t
the current coming out of the battery. Combining equation 24.1 with equation 24.14
gives the power supplied by the battery as
P
IV
=
(24.15)
The conservation of energy applied to the circuit can be expressed as
Energy supplied to the circuit = Energy consumed in the circuit (24.16)
Since the power is energy per unit time, if we divide both sides of equation 24.16 by
the time
t
, we get
Power supplied to the circuit = Power consumed in the circuit (24.17)
2414
Chapter 24 Electric Currents and DC Circuits
Therefore, the power consumed in the circuit is
Power consumed = Power supplied =
IV
If the circuit contains only a resistor, then the voltage across the resistor is given by
Ohm’s law as
V
=
IR
. Therefore, the power consumed or dissipated in the resistor is
P
=
IV
=
I
(
IR
)
2
P
I R=
(24.18)
Equation 24.18 gives the rate at which energy is dissipated in the resistor with
time. This energy that is lost as the charge “falls” through the resistor shows up as
heat in the resistor, and is usually referred to as
Joule heat.
We can also express
equation 24.18 as
P
=
V
2
(24.19)
R
by substituting
I
=
V
/
R
into equation 24.18.
Recall from chapter 7, the unit of power is the watt, where
1 watt = 1
joule
= 1 J/s
second
Checking the units for the power supplied to a circuit we get
P
=
IV
= ampere volt =
(coulomb)(joule)
=
joule
= watt
second coulomb second
which we can abbreviate as
1 W = 1 A V = (1 C/s)(1 J/C) = 1 J/s = 1 W
Thus, in electrical circuits, we represent the unit for power as
watt = ampere volt = A V
because it is equivalent to the previous definition.
When the charge q leaves the battery, it is at the potential V. As it
‘’
falls
’’
through the resistor, figure 24.10(b), it loses energy as it falls to a position of lower
potential. Therefore, we say that there is a potential drop across the resistor.
The potential drop across the resistor is given by Ohm’s law as
V
IR
=
(24.20)
2415
Chapter 24 Electric Currents and DC Circuits
It is sometimes convenient to mark the position of the resistor that is at the highest
potential with a plus (+) sign, and the point of the resistor that is at the lowest
potential with a negative sign (−) to remind ourselves that the charge will fall from
a plus (+) to a minus (−) potential. This is shown in figure 24.10(b).
The mechanical equivalent of the circuit of figure 24.10(a) is shown in figure
24.10(c). A ball of mass
m
is lifted to a height
h
by a person. The person who does
the work lifting the mass is equivalent to the battery. The ball has potential energy
at the top. We assume that the medium that the ball will fall through is resistive.
Therefore, the ball will not fall freely, continually accelerating, but rather will be
slowed down by the friction of the medium until the ball moves at a constant
velocity, its terminal velocity. This is similar to the charge moving at the constant
drift velocity. As the ball falls and loses potential energy, the lost potential energy
shows up as heat generated by the frictional forces between the ball and the
medium. This is similar to the loss of energy of the charge through Joule heating as
it ‘’falls’’ through the resistor.
Example 24.7
A 60W light bulb.
A 60.0W light bulb is screwed into a 120V lamp outlet. (a) What
current will flow through the bulb and what is the resistance of the bulb? (b) What
is the power dissipated in the Joule heating of the wire in the bulb?
a.
Because the power rating of the bulb is known, we can find the current from
equation 24.15 as
Solution
P
=
IV
I
=
P
=
60.0 W
= 0.500
A V
V
120 V V
= 0.500 A
The resistance of the bulb, found from Ohm’s law, is
R
=
V
=
120 V
= 240 Ω
I
0.500 A
b.
The power dissipated in the Joule heating of the wire in the bulb, found from
equation 24.18, is
P
=
I
2
R
= (0.500 A)
2
(240 Ω)
= 60.0 W
Note that the power supplied is equal to the power dissipated as expected.
To go to this Interactive Example click on this sentence.
2416
Chapter 24 Electric Currents and DC Circuits
Example 24.8
A 120W light bulb.
What is the current and resistance of a 120W light bulb
connected to a 120V source?
We find the current from
Solution
I
=
P
=
120 W
= 1.00 A
V
120 V
We determine the resistance of the bulb by Ohm’s law as
R
=
V
=
120 V
= 120 Ω
I
1.00 A
To go to this Interactive Example click on this sentence.
Up to now, we studied a circuit that contained only one resistor. Suppose
there are several resistors in the circuit. Is there a difference in the circuit if these
resistors are connected in different ways? The answer is yes and we will now study
different combinations of these resistors

in particular, resistors in series, resistors
in parallel, and combinations of resistors in series and parallel.
24.6 Resistors in Series
A typical circuit having resistors in series is shown in figure 24.11(a).
The
characteristic of a series circuit is that the same current that flows from the battery
flows through each resistor. That is, the current I is the same everywhere in the series
circuit.
Figure 24.11(b) displays the circuit unfolded, showing how the positive
charge will fall from high potential to low potential. Figure 24.11(c) shows a
mechanical analogue of the circuit of resistors in series. A ball is picked up from the
ground and placed at the top of a threestep stairway. At the top step the ball has
its maximum potential energy. If the ball is given a slight push it rolls off the top
step dropping down to the first step, losing an amount of potential energy PE
1
. This
lost potential energy is first converted to kinetic energy as the ball falls, and
the
kinetic energy is then converted to thermal energy in the collision of the ball with
the step. The ball then rolls off the first step dropping down to the second step,
losing an amount of potential energy PE
2
. It then rolls off the second step, falling
2417
Chapter 24 Electric Currents and DC Circuits
Figure 24.11
Resistors in Series
down to the third and last step at the ground level. This time it loses an amount of
potential energy PE
3
. The law of conservation of energy says that the total energy
given to the ball to place it at the top step must be equal to the total energy that it
loses as it falls from step to step to the ground. That is,
PE
top
= PE
1
+ PE
2
+ PE
3
(24.21)
The electrical circuit, figure 24.11(b), is analogous to this mechanical staircase. The
charge
q
is raised to the potential
V
by the chemical action of the battery. As the
charge ‘’falls’’ through the first resistor, it drops in potential by
V
1
. As it falls
through the second resistor it drops in potential by
V
2
. The charge experiences
another potential drop,
V
3
, as it falls through
R
3
.
By the law of conservation of
energy the potential supplied to the charge by the battery must be equal to the
potential that the charge loses as it falls through the resistors.
Therefore,
V
=
V
1
+
V
2
+
V
3
(24.22)
The voltage drop across each resistor in figure 24.11, given by Ohm’s law, equation
24.20, is
V
1
=
IR
1
V
2
=
IR
2
V
3
=
IR
3
Replacing these equations into equation 24.22 gives
V
=
IR
1
+
IR
2
+
IR
3
(24.23)
2418
Chapter 24 Electric Currents and DC Circuits
Dividing both sides of equation 24.23 by
I
gives
V
=
R
1
+
R
2
+
R
3
(24.24)
I
where
V
is the total potential applied to the circuit and
I
is the total current in the
circuit, so
V
/
I
should, by Ohm’s law, equal
R
, the total resistance of the circuit, that
is
V
=
R
(24.25)
I
From equations 24.24 and 24.25, it is obvious that
1 2
R R R R
3
=
+ +
(24.26)
That is,
the sum of the three resistances in the series circuit is equivalent to one
resistance R called the equivalent resistance.
Figure 24.12(a) is equivalent to the
simpler circuit shown in figure 24.12(b), with
R
, the equivalent resistance, given by
equation 24.26. That is, the three resistors
R
1
,
R
2
, and
R
3
could be replaced by the
one equivalent resistor
R
without detecting any electrical change in the circuit.
Figure 24.12
Equivalent circuit for resistors in series.
Although equation 24.26 was derived for three resistors it is obvious that it holds
for the sum of any number of resistors in series.
The fact that the equivalent resistance of resistors in series is just the sum of
the individual resistances should not be too surprising. Because, if each of the
resistors were a wire of the same material, same crosssectional area, but with
different lengths
l
1
,
l
2
, and
l
3
, respectively, then the length of the wire when the
three resistors are connected in series is just
l
=
l
1
+
l
2
+
l
3
2419
Chapter 24 Electric Currents and DC Circuits
Multiplying each term by ρ/
A
gives
31 2
ll l
l
A A A
A
ρ
ρ ρ ρ= + +
But using equation 24.7,
R
= ρ
l
/
A
gives
R
=
R
1
+
R
2
+
R
3
which is the same result as before, equation 24.26. This derivation may be easier to
see but it does not give the same insight as is obtained by using the law of
conservation of energy.
Example 24.9
Resistors in series.
Resistors
R
1
= 20.0 Ω,
R
2
= 30.0 Ω, and
R
3
= 40.0 Ω are connected
in series to a 6.00V battery. Find the equivalent resistance of the circuit and the
current in this series circuit.
The equivalent resistance, given by equation 24.26, is
Solution
R
=
R
1
+
R
2
+
R
3
= 20.0 Ω + 30.0 Ω + 40.0 Ω
= 90.0 Ω
The current is found by Ohm’s law, with
R
the equivalent resistance, that is,
I
=
V
=
6.00 V
= 0.0667 A
R
90.0 Ω
To go to this Interactive Example click on this sentence.
Example 24.10
Power dissipated in resistors in series.
Find the power supplied and the power
dissipated in each resistor in example 24.9.
The power supplied by the battery is
Solution
2420
Chapter 24 Electric Currents and DC Circuits
P
=
VI
= (6.00 V)(0.0667 A) = 0.400 W
The power dissipated in each resistor is
P
1
=
I
2
R
1
= (0.0667 A)
2
(20.0 Ω) = 0.0890 W
P
2
=
I
2
R
2
= (0.0667 A)
2
(30.0 Ω) = 0.133 W
P
3
=
I
2
R
3
= (0.0667 A)
2
(40.0 Ω) = 0.178 W
The total power dissipated in all resistors is
P
=
P
1
+
P
2
+
P
3
= 0.0890 W + 0.133 W + 0.178 W
= 0.400 W
Note that the power supplied to the circuit is equal to the power dissipated in the
resistors.
To go to this Interactive Example click on this sentence.
Example 24.11
Potential drop across resistors in series.
Find the potential drop across each resistor
of examples 24.9 and 24.10.
The potential drop across each resistor, found by Ohm’s law, is
Solution
V
1
=
IR
1
= (0.0667 A)(20.0 Ω) = 1.33 V
V
2
=
IR
2
= (0.0667 A)(30.0 Ω) = 2.00 V
V
3
=
IR
3
= (0.0667 A)(40.0 Ω) = 2.67 V
Notice that the sum of the potential drops,
V
1
+
V
2
+
V
3
, is equal to 6.00 V, which is
equal to the applied voltage of 6.00 V.
To go to this Interactive Example click on this sentence.
2421
Chapter 24 Electric Currents and DC Circuits
24.7 Resistors in Parallel
A typical circuit with resistors connected in parallel is shown in figure 24.13(a).
Resistors in a
parallel circuit
are connected such that the top of each resistor is
connected to the same point
A
, whereas the bottom of each resistor is connected to
Figure 24.13
Resistors in parallel.
the same point
B
. Therefore, the potential difference between
A
and
B
is the same
as the potential difference across each resistor. We assume that the resistance of
the connecting wires is negligible compared to the resistors in the circuit, and can
be ignored. Therefore, the potential across
AB
is the same as the potential
V
supplied by the battery. Consequently,
the characteristic of resistors connected in
parallel is that the potential difference is the same across every resistor
. That is,
V
=
V
1
=
V
2
=
V
3
(24.27)
When the total current
I
from the battery reaches the junction
A
, it divides into
three parts;
I
1
goes through resistor
R
1
,
I
2
goes through resistor
R
2
, and
I
3
goes
through
R
3
. Because none of the charge disappears,
I
=
I
1
+
I
2
+
I
3
(24.28)
Equation 24.28 is a statement of
the law of conservation of electric charge.
Electric charge can neither be created nor destroyed and hence the electric charges
entering a junction must be equal to the electric charges leaving a junction. Thus, the
electric current entering a junction is equal to the electric current leaving a junction.
At the junction
B
these three currents again combine to form the same total current
I
that entered the junction
A
. The current in each resistor can be found by applying
Ohm’s law to that resistor. That is,
I
1
=
V
1
(24.29)
R
1
2422
Chapter 24 Electric Currents and DC Circuits
I
2
=
V
2
(24.30)
R
2
I
3
=
V
3
(24.31)
R
3
Replacing these values of the currents back into the current equation, 24.28, gives
I
=
V
1
+
V
2
+
V
3
(24.32)
R
1
R
2
R
3
But, the potentials are equal by equation 24.27 (i.e.,
V
=
V
1
=
V
2
=
V
3
). Hence,
equation 24.32 becomes
I
=
V
+
V
+
V
R
1
R
2
R
3
Dividing both sides of the equation by
V
, gives
I
=
1
+
1
+
1
(24.33)
V R
1
R
2
R
3
But the lefthand side of equation 24.33 contains the total current
I
in the circuit,
divided by the total voltage
V
applied to the circuit, and by Ohm’s law is simply
I
=
1
(24.34)
V
R
where
R
is the total resistance of the entire circuit. Equating 24.34 to equation
24.33 gives
1 2
1 1 1 1
R R R R
= + +
3
(24.35)
Equation 24.35 says that the reciprocal of the total resistance of the circuit is
equivalent to the sum of the reciprocals of each parallel resistance. We call R the
equivalent resistance of the resistors in parallel.
The resistor
R
could replace the
three resistors
R
1
,
R
2
, and
R
3
, and the circuit would still behave the same way
electrically.
Although we derived equation 24.35 from a circuit with only three resistors
in parallel, the form is completely general. If there were
n
resistors in parallel, the
equivalent resistance would be found from
1 2 3
1 1 1 1 1
...
n
R R R R R
= + + + +
(24.36)
2423
Chapter 24 Electric Currents and DC Circuits
Example 24.12
Equivalent resistance of resistors in parallel.
Three resistors
R
1
= 20.0 Ω,
R
2
= 30.0
Ω, and
R
3
= 40.0 Ω are connected in parallel to a 6.00V battery, as shown in figure
24.13. Find the equivalent resistance of the circuit.
From equation 24.35 the equivalent resistance is
Solution
1
=
1
+
1
+
1
R R
1
R
2
R
3
=
1
+
1
+
1
=
0.108
20.0 Ω 30.0 Ω 40.0 Ω Ω
R
= 9.23 Ω
Notice that in this calculation 1/
R
= 0.108, and to get the actual value of
R
we need
to take the reciprocal of 0.108, which yields the correct value of the resistance as
9.23 Ω. Failure to take the final reciprocal is a very common student error.
To go to this Interactive Example click on this sentence.
Compare this result with example 24.9 in which the same three resistors
were connected in series. There, the total equivalent resistance was 90.0 Ω when
the resistors were connected in series, whereas here the same resistors connected in
parallel give an equivalent resistance of only 9.23 Ω. It is clear from these two
examples that the way the resistors are connected in a circuit makes a great deal of
difference in their effect on the circuit.
Note that when the resistors are connected in
parallel, the equivalent resistance is always less than the smallest of the original
resistances.
In this case the total resistance, 9.23 Ω, is less than the smallest
resistance of 20.0 Ω. The equivalent circuit of the parallel circuit in figure 24.13(a)
is shown in figure 24.13(b), where
R
is the equivalent resistance, found in equation
24.35.
Example 24.13
The total current in the parallel circuit.
Find the total current coming from the
battery in the example 24.12.
The total current in the circuit, found from Ohm’s law with
R
as the equivalent
resistance of the circuit, is
Solution
2424
Chapter 24 Electric Currents and DC Circuits
I
=
V
=
6.00 V
= 0.650 A
R
9.23 Ω
Notice that the current from the battery is almost ten times greater when the
resistors are connected in parallel then when connected in series. This is easily
explained, since the resistance in parallel is only about onetenth of the resistance
when the resistors are in series.
To go to this Interactive Example click on this sentence.
Example 24.14
The current in each parallel resistor.
Find the current through each resistor in
example 24.12.
Because the voltage across each resistor is 6.00 V, we can find the current from
Ohm’s law applied to each resistor as given in equations 24.29, 24.30, and 24.31.
Namely,
Solution
I
1
=
V
=
6.00 V
= 0.300 A
R
1
20.0 Ω
I
2
=
V
=
6.00 V
= 0.200 A
R
2
30.0 Ω
I
3
=
V
=
6.00 V
= 0.150 A
R
3
40.0 Ω
Note that the current through each resistor is different, but the sum of
I
1
+
I
2
+
I
3
=
0.650 A is equal, to the total current of 0.650 A flowing from the battery in the
circuit. Also note that the resistor with the smallest value of resistance has the
largest value of current passing through it. This is sometimes stated as:
The current
always takes the path of least resistance.
To go to this Interactive Example click on this sentence.
Example 24.15
Power supplied to a parallel circuit.
What is the power supplied to the circuit in
example 24.12?
2425
Chapter 24 Electric Currents and DC Circuits
The power supplied by the battery is
Solution
P
=
IV
= (0.650 A)(6 V) = 3.90 W
To go to this Interactive Example click on this sentence.
Example 24.16
Power dissipated in a parallel circuit.
What is the power dissipated in each resistor
in example 24.14?
The power dissipated in each resistor is
Solution
P
1
=
I
1
2
R
1
= (0.300 A)
2
(20.0 Ω) = 1.80 W
P
2
=
I
2
2
R
2
= (0.200 A)
2
(30.0 Ω) = 1.20 W
P
3
=
I
3
2
R
3
= (0.150 A)
2
(40.0 Ω) = 0.900 W
Again note that the sum of the powers dissipated in each resistor
P
1
+
P
2
+
P
3
= 1.80 W + 1.20 W + 0.900 W
= 3.90 W
is the same as the total power supplied to the circuit by the battery, within round
off error.
To go to this Interactive Example click on this sentence.
24.8 Combinations of Resistors in Series and Parallel
A typical circuit showing a simple combination of resistors connected in series and
parallel is shown in figure 24.14. Let us determine the current through each
resistor and the voltage drop across it. To do so, we use the techniques of sections
24.6 and 24.7. Resistors
R
2
and
R
3
are connected in parallel. Their equivalent
resistance
R
23
, found from equation 24.36, is
2426
Chapter 24 Electric Currents and DC Circuits
Figure 24.14
Combinations of resistors in a circuit.
1
=
1
+
1
R
23
R
2
R
3
=
1
+
1
=
0.0583
30.0 Ω 40.0 Ω Ω
R
23
= 17.2 Ω
The circuit of figure 24.14(a) can be replaced by its equivalent circuit, figure
24.14(b), where
R
23
is in series with
R
1
. The equivalent resistance of
R
1
and
R
23
in
series, found from equation 24.26, is
R
123
=
R
1
+
R
23
= 20.0 Ω + 17.2 Ω
= 37.2 Ω
The circuit of figure 24.14(b) can now be replaced by the equivalent circuit, figure
24.14(c), containing only one resistor, the equivalent resistor
R
123
= 37.2 Ω.
The current from the battery is now easily determined by Ohm’s law as
I
=
V
=
6.00 V
= 0.161 A
R
123
37.2 Ω
Since the resistor
R
1
is in series with the battery, this same current flows through it
(i.e.,
I
1
=
I
= 0.161 A). However, this is not the current through
R
2
and
R
3
because
they are in parallel and the current divides as it enters the two paths. In order to
determine
I
2
and
I
3
we must first determine the voltage drop across
R
2
and
R
3
.
The voltage drop across
R
1
, found from Ohm’s law, is
V
1
=
I
1
R
1
= (0.161 A)(20.0 Ω) = 3.22 V
The total applied potential is equal to the sum of the potential drops, that is,
2427
Chapter 24 Electric Currents and DC Circuits
V
=
V
1
+
V
2
The potential drop
V
2
across the parallel resistors is
V
2
=
V
−
V
1
= 6.00 V − 3.22 V
= 2.78 V
The potential drop
V
3
is also equal to 2.78 V since
R
2
and
R
3
are connected in
parallel. With the potential drop across the parallel resistors known, the current in
each resistor, found from Ohm’s law, is
I
2
=
V
2
=
2.78 V
= 0.0927 A
R
2
30.0 Ω
I
3
=
V
3
=
2.78 V
= 0.0695 A
R
3
40.0 Ω
Note that the sum of the currents
I
2
+
I
3
= 0.0927 A + 0.0695 A = 0.162 A
is equal, within roundoff errors of our calculations, to the total current in the
circuit, 0.161 A.
The power delivered to the circuit is
P
=
IV
= (0.161 A)(6.00 V) = 0.966 W
The power dissipated in each resistor is
P
1
=
I
1
2
R
1
= (0.161 A)
2
(20.0 Ω) = 0.518 W
P
2
=
I
2
2
R
2
= (0.0927 A)
2
(30.0 Ω) = 0.258 W
P
3
=
I
3
2
R
3
= (0.0695 A)
2
(40.0 Ω) = 0.193 W
The sum of the power dissipated in the resistors (0.969 W) is equal, within roundoff
error, to the supplied power of 0.966 W.
24.9 The Electromotive Force and the Internal
Resistance of a Battery
In the early days of electricity, in order to keep an analogy with Newtonian
mechanics, it was assumed that if a charge moved through a wire, there must be
some force pushing on the charge to cause it to move through the wire. This force
acting on the charge was called an
electromotive force.
The battery, the supplier
2428
Chapter 24 Electric Currents and DC Circuits
of this force, was called a
seat of electromotive force,
abbreviated emf, and
pronounced as the individual letters emf. The emf is usually written as the script
letter
E
. Today of course, we know that this emf is really a misnomer. The battery is
supplying a potential difference. However, the name emf still remains. The emf is
measured in volts, since the emf is a potential difference. You will hear comments
such as, a battery has an emf of 6 V, or the generator supplies an emf of 120 V, and
the like.
If there is no current being drawn from the battery, the emf of the battery
and the potential difference between its two terminals are the same. When a
current exists, however, the potential difference between the terminals is always
less than the emf of the battery. The reason for this is that every battery has an
internal resistance, which is a characteristic of the battery and cannot be
eliminated. The internal resistance of the battery acts as though it were a resistor
in series with the battery itself. It is usually represented in a circuit diagram as the
lower case
r
in figure 24.15. When a current exists in the circuit there is a potential
drop, equal to the product
Ir
, across the internal resistance. The potential difference
between the terminals of the battery, becomes
V
=
E
−
Ir
(24.37)
which is, of course, less than the emf of the battery.
1
Figure 24.15
The internal resistance of a battery.
Example 24.17
The potential and emf of a battery.
A battery has an emf of 1.50 V and when
connected to a circuit supplies a current of 0.0100 A. If the internal resistance of the
battery is 2.00 Ω, what is the potential difference across the battery?
1
We should note that if the battery is being charged by an external source, then the terminal
voltage would be given by V =
E
+ Ir.
2429
Chapter 24 Electric Currents and DC Circuits
The potential difference across the battery, given by equation 24.37, is
Solution
V
=
E
−
Ir
= 1.50 V − (0.0100 A)(2.00 Ω)
= 1.48 V
Notice that when the current in the circuit is relatively small, the potential drop
across the internal resistance is also quite small, and the terminal voltage is very
close to the emf of the battery.
If the current in the circuit is much larger, as for example
I
= 0.500 A then
the terminal voltage is
V
=
E
−
Ir
= 1.50 V − (0.500 A)(2.00 Ω)
= 0.500 V
which is very much less than the emf of the battery.
To go to this Interactive Example click on this sentence.
Example 24.18
The emf of a battery connected to a circuit.
A battery has an emf of 1.50 V and an
internal resistance of 3.00 Ω. It is connected as shown in figure 24.15 to a resistance
of 500 Ω. Find the current in the circuit, and the terminal voltage of the battery.
The internal resistance
r
is in series with the load resistor
R
,
so the total resistance
in the circuit is just the sum of the two of them. Using Ohm’s law to determine the
current we have
Solution
I
=
E
=
1.50 V
= 2.98
×
10
−3
A
R
+
r
500 Ω + 3.00 Ω
The terminal voltage across the battery is
V
=
E
−
Ir
= 1.50 V − (2.98
×
10
−3
A)(3.00 Ω)
= 1.49 V
2430
Chapter 24 Electric Currents and DC Circuits
To go to this Interactive Example click on this sentence.
Because the value of
r
is usually very small, relative to the resistance
R
of
the circuit, we neglect it in many problems. In such cases, we assume that the
terminal voltage and the emf are the same. Whether we can make this assumption
or not, depends on the values of
r
and
R
and the accuracy that we are willing to
accept in the solution to the problem. In example 24.18, neglecting
r
gives a current
of 3.00
×
10
−3
A, an error of about 0.7%. If that amount of error is acceptable in the
problem, the internal resistance can be ignored. If that error is not acceptable then
the internal resistance must be taken into account.
Batteries in Series
Since the emf of most batteries is only 1.50 V, we need to place many of them in
series to get a higher emf. If three 1.50V batteries are connected in series, as in
figure 24.16, with the positive terminal of one battery connected to the negative
terminal of the next battery and so on, the emf of the combination is 3(1.50 V) =
Figure 24.16
Batteries in series.
4.50 V.
In general, when batteries are connected in series, the total emf is the sum of
the emf’s of each battery.
That is,
E
=
E
1
+
E
2
+
E
3
+ … +
E
n
(24.38)
By connecting any number of batteries in series, any larger emf may be obtained.
Because each battery is in series, the current through each battery is the same, and
the total internal resistance of the combination is just the sum of the internal
resistance of each battery.
Batteries in Parallel
Three identical batteries, of negligible internal resistance, are connected in parallel
in figure 24.17. The negative terminals of all the batteries are all connected
together, and the positive terminals of all the batteries are all connected to one
another.
Since the batteries are in parallel, the potential difference across the
combination is the same as the emf of each battery,
that is,
2431
Chapter 24 Electric Currents and DC Circuits
Figure 24.17
Identical batteries in parallel.
E
=
E
1
=
E
2
=
E
3
(24.39)
At junction
A
, the three battery currents combine to give the circuit current
I
=
I
1
+
I
2
+
I
3
(24.40)
Because we assume the batteries to be identical (
I
1
=
I
2
=
I
3
), the total circuit
current available is three times the current available from any one battery. By a
suitable combination of batteries in series and parallel, batteries of any voltage and
current can be made.
Example 24.19
Batteries in parallel.
Three identical 6.00V batteries are connected in parallel to a
resistance of 500 Ω, as in figure 24.17. Find the current in the circuit and the
current coming from each battery. Assume the internal resistance of the batteries to
be zero.
The total current
I
in the circuit, found from Ohm’s law, is
Solution
I
=
E
=
6.00 V
= 0.0120 A
R
500 Ω
This total current is supplied by three batteries so that,
I
= 3
I
0
, where
I
0
is the
actual current from any one battery. Therefore, the battery current is
I
0
=
I
=
0.0120 A
= 4.00
×
10
−3
A
3 3
2432
Chapter 24 Electric Currents and DC Circuits
To go to this Interactive Example click on this sentence.
24.10 The Wheatstone Bridge
A standard technique to determine the value of a particular resistor is to connect it
into a circuit, such as in figure 24.18(a). It is usually assumed that the resistance of
the ammeter is negligible, and the resistance of the voltmeter is so large that
negligible current flows through it. Therefore, the current
I
in the resistor is
measured, and the voltage
V
across it is also measured. The value of the resistance
R
is easily determined from Ohm’s law as
R
=
V
(24.41)
I
Figure 24.18
Techniques for measuring voltage and current in a simple circuit.
As long as our assumptions hold, equation 24.41 is valid. Now let’s look a little
closer at our assumptions. If the resistance of the ammeter is anywhere near the
order of magnitude of the resistance
R
, the assumptions break down. Although the
ammeter does measure the current
I
through
R
, there is now a voltage drop
V
A
across the ammeter, and the voltmeter reads the voltage drop
V
=
V
R
+
V
A
The calculated resistance is
R
V V
R
I
+
=
A
(24.42)
which gives a value for
R
greater than the exact value found by equation 24.41
when the effects of the ammeter can be neglected.
To remedy this, we might say, let’s change the measuring technique to the
one shown in figure 24.18(b). With this second technique, the voltmeter reads only
the voltage
V
R
across the resistor. At first glance, it might seem that the problem is
2433
Chapter 24 Electric Currents and DC Circuits
solved. But if we look carefully we note that the current
I
in the circuit splits at
junction
A
, where the resistor and voltmeter are in parallel. A current
I
R
flows
through the resistor while a current
I
v
flows through the voltmeter. The ammeter
reads the current
I
=
I
R
+
I
v
The computed resistance
R
therefore becomes
R
R V
V
R
I I
=
+
(24.43)
But this is less than that computed from the exact value of equation 24.41, when
the effects of the voltmeter can be neglected in the circuit.
This analysis of the measurements of a simple circuit again points out the
importance of the assumptions made in deriving any equation. As long as the initial
assumptions hold, the derived equations hold. When there is a breakdown in the
assumptions, the derived equations are no longer valid.
Note that when the
assumptions of a negligible resistance for the ammeter, and a negligible current in
the voltmeter (very high resistance voltmeter), are correct, equations 24.42 and
24.43 reduce to 24.41.
When the previous assumptions do not hold, it is still possible to make
accurate measurements of resistance by means of a circuit that is called the
Wheatstone bridge,
figure 24.19(a). Its virtue is that it is a null detection method
Figure 24.19
The Wheatstone bridge.
2434
Chapter 24 Electric Currents and DC Circuits
that does not depend on the characteristics of the galvanometer, but uses it merely
to be a sensitive detector of the condition of current versus no current. The values of
resistors
R
1
,
R
3
, and
R
4
, in the circuit are assumed to be known, whereas
R
2
is the
unknown resistor. A battery of emf
E
is applied to the circuit and a current
I
emanates from the battery. At the junction
A
, the current
I
splits into two currents,
I
1
and
I
3
.
I
=
I
1
+
I
3
When current
I
1
reaches the junction
C
it splits into two currents,
I
2
through
R
2
and
I
g
the current through the galvanometer. That is,
I
1
=
I
2
+
I
g
(24.44)
The galvanometer needle deflects indicating that there is a current in it. At junction
D
the galvanometer current
I
g
combines with current
I
3
to form current
I
4
, which
passes through resistor
R
4
. That is,
I
3
+
I
g
=
I
4
(24.45)
At junction
B
, currents
I
2
and
I
4
combine to form the original current
I
.
A current
I
g
exists in the galvanometer, because there is a difference of
potential between points
C
and
D
in the circuit. This difference in potential is
caused by the different currents in the different resistors. The values of these
resistors
R
1
,
R
3
, and
R
4
are varied until the galvanometer reads zero (i.e.,
I
g
= 0).
When this occurs the bridge is said to be balanced. When the galvanometer reads
zero, it means that there is no longer a difference in potential between points
C
and
D
. (Remember a current exists in a conductor when there is a difference in
potential. If there is no current, there is no difference in potential.) The circuit in
figure 24.19(a) can now be replaced by the one in figure 24.19(b). The points
C
and
D
have coalesced electrically into one point, because they are at the same potential.
It is now obvious from figure 24.19(b) that
R
1
is now in parallel with
R
3
and
therefore the voltages across them are equal, that is,
V
1
=
V
3
Therefore, from Ohm’s law,
I
1
R
1
=
I
3
R
3
(24.46)
It is also obvious that
R
2
is now in parallel with
R
4
and hence the voltages across
them are also equal, that is,
V
2
=
V
4
I
2
R
2
=
I
4
R
4
(24.47)
Dividing equation 24.47 by equation 24.46, we obtain
2435
Chapter 24 Electric Currents and DC Circuits
2 2 4 4
1 1 3 3
I R I R
I R I R
=
(24.48)
However, since the bridge is balanced,
I
g
= 0, and equation 24.44 reduces to
I
1
=
I
2
and equation 24.45 to
I
3
=
I
4
Therefore, the currents in equation 24.48 cancel, leaving the simple ratio of the
resistors
2 4
1 3
R R
R R
=
Solving for the unknown resistor
R
2
, we obtain
4
2
3
R
R
R
=
1
R
(24.49)
The Wheatstone bridge is a very accurate means of determining the resistance of an
unknown resistor. A picture of a typical commercial Wheatstone bridge is shown in
figure 24.19(c).
Example 24.20
The Wheatstone bridge.
A Wheatstone bridge is balanced when
R
1
= 5.00 Ω,
R
3
=
10.0 Ω, and
R
4
= 4.00 Ω. Find the unknown resistance
R
2
.
The value of the unknown resistor, found from equation 24.49, is
Solution
(
)
( )
( )
4
2 1
3
4.00
5.00 2.00
10.0
R
R R
R
Ω
= = Ω =
Ω
Ω
To go to this Interactive Example click on this sentence.
24.11 Kirchhoff’s Rules
Quite often we find circuits in which the resistors are not simply in series or
parallel, or their combinations. An example of such a circuit is shown in figure
2436
Chapter 24 Electric Currents and DC Circuits
24.20. The locations of the different batteries in the circuit prevents the resistors
from being in parallel. In order to solve this more complicated circuit problem, two
rules established by the German physicist Gustav Robert Kirchhoff (18241887) are
used.
Kirchhoff’s first rule is: The sum of the currents entering a junction is equal
to the sum of the currents leaving the junction.
That is,
entering junction leaving junction
I I=
∑ ∑
(24.50)
Figure 24.20
A circuit problem solved by Kirchhoff’s rules.
This first rule is a statement of the law of conservation of electric charge.
2
In order to apply the first rule, we must make an assumption about the
currents in the circuit. We assume that current
I
1
emanates from battery
E
1
,
I
2
from
E
2
, and
I
3
from
E
3
, all in the directions shown in figure 24.20. If the actual
directions of the currents are as assumed, the numerical values of the currents
found by the equations will be positive. If the direction of the currents is in the
opposite direction from those assumed, the numerical values of the currents found
by the equations will be negative, indicating that the direction of the currents is
really reversed. Using Kirchhoff’s first rule at junction
A
, gives
I
2
=
I
1
+
I
3
(24.51)
Kirchhoff’s second rule
states that the change in potential around a closed loop is
equal to zero.
This can also be stated in the form,
around any closed loop, the sum of
the potential rises plus the sum of the potential drops is equal to zero.
That is,
2
We should note that Kirchhoff’s first rule is sometimes stated in the equivalent form that the
algebraic sum of the currents toward a junction is zero. That is, currents entering the junction are
positive and currents leaving the junction are negative. In this notation, equation 24.50 would take
the equivalent form ΣI = 0.
2437
Chapter 24 Electric Currents and DC Circuits
closed loop
0V
∆
=
(24.52)
Kirchhoff’s second rule is really another statement of the law of conservation of
energy. It says that
if a positive charge is carried completely around the loop of the
circuit, in either a clockwise or counterclockwise direction, on returning to its
original point, it will have the same value of V with which it started. Hence, there is
no change in potential when the charge returns to the same place where it started
from.
The mechanical analogue is that of a person standing on, let’s say, the fifth
floor of a tenstory building. The person has a certain potential energy at that point.
The person now goes for a walk up and down the stairway. If the person goes up the
stairs he increases his potential energy. If he goes down the stairs he decreases his
potential energy. But when he returns to the fifth floor, from whence he started, he
has the same potential energy that he started with.
Because charge flows from a point of high potential to one at a lower
potential, it is convenient to place a plus (+) sign on the side of a resistor that the
current enters and a negative sign (
−
) on the side of the resistor that the current
leaves, as shown in figure 24.20. If the charge that is carried around the circuit
traverses the loop of the circuit in the direction of the current, then as it passes the
resistor it goes from a position of higher potential to one of lower potential, and
thereby loses potential as it crosses the resistor. That is, there is a potential drop
across the resistor, which we will designate as
−IR
. If the carried charge passes
through a resistor in the direction that is opposite to the direction of the current,
then it is going from a position of lower potential to one of higher potential and
hence it experiences a potential rise across the resistor, which we will designate as
+
IR
.
In traversing a battery, if the carried charge comes out of the positive
terminal of the battery, it is at a positive potential, and this we designate as +
E
. If
it comes out of the negative terminal of the battery, when traversing the circuit, it
has dropped in potential and this we designate as
−
E
. A good mechanical analogy
here would be that the battery is like an elevator in a tenstory building. When the
person enters at the ground floor, he is carried up to the fifth floor by the elevator.
The elevator has caused his potential energy to increase. The stairs are like the
resistors. When the person walks up and down the stairs from the fifth floor, he
gains or loses potential energy. If he enters the elevator at any floor along his
excursion and rides to another floor, he can gain or lose additional potential energy.
With the help of these conventions, we can now apply Kirchhoff’s second rule
to figure 24.20. Let us carry a charge around the lefthand loop in a
counterclockwise direction, starting at the positive terminal of
E
1
.
The sum of the
potential rises and drops as the loop is traversed is
E
1
−
I
1
R
1
−
I
2
R
2
+
E
2
= 0 (24.53)
Note that
E
1
and
E
2
are positive since the charge emanates from the positive
terminals in traversing the loop counterclockwise, and that the products,
I
1
R
1
and
2438
Chapter 24 Electric Currents and DC Circuits
I
2
R
2
, are both negative since the charge dropped in potential as it traversed these
resistors in the counterclockwise direction. The decision to traverse the loop
counterclockwise was completely arbitrary. If we wanted to traverse the loop in a
clockwise direction, the sum of the potential drops and rises would yield
−
E
1
−
E
2
+
I
2
R
2
+
I
1
R
1
= 0 (24.54)
In this case the emf’s are negative because in traversing the loop in a clockwise
direction, the carried charge emerges from the negative terminals of the batteries.
The products of the
IR
s across the resistors are now positive, and are called
potential rises,
because in traversing the loop in a clockwise direction, the carried
charge entered the resistors at their lowest potential (
−
) and emerged at their
highest potential (+). No new information is obtained in this way, however, because
if we multiply each term in equation 24.54 by a negative one, we get back equation
24.53. Therefore, it does not matter whether the loop is traversed in a clockwise or
counterclockwise direction.
Recall one of the fundamental principles of algebra that in the solution of
equations there must be as many equations as there are unknowns in order to solve
for all the unknowns.
At this point, there are three unknowns, the currents
I
1
,
I
2
,
and
I
3
and only two equations 24.51 and 24.53. We need one additional equation to
solve the problem. We can obtain the third equation by applying Kirchhoff’s second
rule to the righthand loop in figure 24.20. This loop will be traversed in a
counterclockwise direction starting at the negative terminal of the battery
E
3
. The
sum of the potential rises and drops around the righthand loop is
−
E
3
−
E
2
+
I
2
R
2
+
I
3
R
3
= 0 (24.55)
Again note that by our convention, in traversing the loop in a counterclockwise
direction, the carried charge emerged from the negative terminals of the batteries,
and hence the emf’s are negative. The products of the
IR
s are positive and are
potential rises because the resistors were entered at the position of their lowest
potential and exited at the position of their highest potential.
There are now three equations in the three unknown currents,
I
1
,
I
2
, and
I
3
,
and we must solve them simultaneously. It is more convenient to place the
numerical values of the emf’s and the resistors into the equations and then solve
them simultaneously. The equations are
Algebraically
Numerically
I
2
=
I
1
+
I
3
I
2
=
I
1
+
I
3
(24.51)
E
1
−
I
1
R
1
−
I
2
R
2
+
E
2
= 0 6.00
−
10.0
I
1
−
20.0
I
2
+ 12.0 = 0 (24.56)
−
E
3
−
E
2
+
I
2
R
2
+
I
3
R
3
= 0
−
6.00
−
12.0 + 20.0
I
2
+ 30.0
I
3
= 0 (24.57)
2439
Chapter 24 Electric Currents and DC Circuits
We will find the solution to the three equations by the method of
substitution. We eliminate the current
I
2
from the equations by substituting
equation 24.51 into both equations 24.56 and 24.57, that is,
6.00
−
10.0
I
1
−
20.0(
I
1
+
I
3
) + 12.0 = 0
−
6.00
−
12.0 + 20.0(
I
1
+
I
3
) + 30.0
I
3
= 0
Combining terms, the loop equations become
18.0
−
30.0
I
1
−
20.0
I
3
= 0 (24.58)
and
−
18.0 + 20.0
I
1
+ 50.0
I
3
= 0 (24.59)
The problem is now reduced to solving two equations for the two unknowns
I
1
and
I
3
. The easiest way to solve these equations is to multiply one of the equations
by a factor that will make one current term in equation 24.59 equal to a current
term in equation 24.58. Then the two equations are either added or subtracted to
eliminate the common term, leaving one equation, in one unknown current, which is
then easily solved. For example, if we multiply both sides of equation 24.59 by 1.50,
one term becomes +30.0
I
1
, and when this equation is added to equation 24.58, the
I
1
term is eliminated. That is,
(1.50)(
−
18.0 + 20.0
I
1
+ 50.0
I
3
) = (1.50)(0)
gives
−
27.0 + 30.0
I
1
+ 75.0
I
3
= 0 (24.60)
Adding this to equation 24.58 gives
18.0
−
30.0
I
1
−
20.0
I
3
= 0 (24.58)
ADD
−
27.0 + 30.0
I
1
+ 75.0
I
3
= 0 (24.60)
−
9.00 + 0 + 55.0
I
3
= 0
Solving for the current
I
3
, we obtain
55.0
I
3
= 9.00
I
3
= 0.164 A
Substituting this value for the current
I
3
back into equation 24.58 allows us to solve
for the current
I
1
. That is,
18.0
−
30.0
I
1
−
20.0(0.164) = 0
I
1
= 0.491 A
At this point the currents
I
1
and
I
3
could be substituted back into equation 24.51 to
find the current
I
2
. However, we will not do this. Instead we will evaluate the
current
I
2
from one of the loop equations and use equation 24.51 as a check on our
2440
Chapter 24 Electric Currents and DC Circuits
arithmetic. The reason for doing this check is that it is very easy to make a mistake
in the algebra and/or the arithmetic in the problem.
Substituting the value of
I
1
just found, back into equation 24.56, yields
6.00
−
10.0(0.491)
−
20.0
I
2
+ 12.0 = 0
Solving for
I
2
,
I
2
= 0.655 A
As a check, place the values of the currents just found back into equation 24.51:
I
2
=
I
1
+
I
3
0.655 A = 0.491 A + 0.164 A = 0.655 A
Thus,
0.655 A = 0.655 A CHECK
The current equation thus acts as a good check on our calculations. The problem is
now solved, since we know the three currents.
It is interesting and informative to plot the change in potential that a
positive charge would encounter as it traversed each battery and resistor as it
moves around the loop. Considering the first loop
E
1
−
I
1
R
1
−
I
2
R
2
+
E
2
= 0
6.00
−
10.0
I
1
−
20.0
I
2
+ 12.0 = 0
6.00
−
10.0(0.491)
−
20.0(0.655) + 12.0 = 0
6.00 V
−
4.91 V
−
13.09 V + 12.0 V = 0 (24.53)
These terms are plotted in figure 24.21. Emerging from the positive terminal of the
Figure 24.21
Plot of the potential drops and rises as the left loop of figure 24.20 is
traversed.
2441
Chapter 24 Electric Currents and DC Circuits
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