Download Pearson Physics Electric Charges and Forces Chapter Contents

10/22/2016
Chapter 19 Lecture
Chapter Contents
Pearson Physics
• Electric Charge
• Electric Force
• Combining Electric Forces
Electric Charges
and Forces
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Electric Charge
Electric Charge
• Matter is made of electric charges, and electric
charges exert forces on one another.
• The effects of electric charge have been known
since at least 600 B.C.
• The Greeks noticed that when rubbed with animal
fur, amber—a solid, translucent material formed
from the fossilized resin of extinct trees—attracts
small, lightweight objects.
• The figure below shows the charging process as well as
the effect a charged amber rod has on scraps of paper.
• Electric charge comes in two distinct types. This may be
demonstrated with two charged amber rods and a
charged glass rod.
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Electric Charge
Electric Charge
• In the figure below, a charged amber rod is
suspended from a string. When another charged
amber rod is brought near the suspended rod, it
rotates away, indicating a repulsive force.
• As the figure below indicates, when a charged
glass rod is brought close to the suspended
amber rod, the amber rod rotates toward the
glass, indicating an attractive force.
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Electric Charge
Electric Charge
• It follows that the charges on the amber and
glass must be different. These different types of
charge are opposites, as in the familiar
expression "opposites attract."
• We know today that the two types of electric
charge found on amber and glass are the only
types of electric charge that exist.
• In 1747, Benjamin Franklin (1706–1790)
proposed that the charge on glass be called
positive (+) and the charge on amber be called
negative (−).
• Atoms are electrically neutral. Each atom
contains a small, dense nucleus with a positive
charge that is surrounded by a "cloud" of
electrons with an equal negative charge.
• Two types of particles are found in the nucleus:
one is positively charged and the other is
electrically neutral.
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Electric Charge
Electric Charge
• Simplified representations of an atom are shown in the
following figure.
• Since electrons have a negative charge, the
charge on an electron is –e. This is one of the
defining properties of the electron. The other
defining property of the electron is its mass, me:
me = 9.11 x 10−31 kg
• In contrast, the charge on a proton—one of the
main constituents of the nucleus—is exactly +e.
Therefore, the total charge on atoms, which
have an equal number of electrons and protons,
is precisely zero.
•
All electrons have exactly the same charge. The charge
on an electron is defined to have a magnitude e equal to
1.6 x 10−19 C, where C stands for coulomb, the SI unit of
charge.
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Electric Charge
Electric Charge
• The mass of the proton, which is about 2000
times larger than the mass of an electron, is
mp = 1.673 x 10−27 kg
• The neutron is the other main constituent of the
nucleus. As its name implies, the neutron has
zero charge. Its mass is slightly larger than that
of a proton:
mn = 1.675 x 10−27 kg
• Since electrons always have the charge –e and
protons always have the charge +e, it follows
that all objects must have a total charge that is
an integer multiple of e.
• The fact that electric charge comes in integer
multiples of e is referred to as charge
quantization.
• Charge quantization is key to understanding the
behavior of atoms and molecules, for the
addition or removal of even a single electron is a
significant event for an atom or molecule.
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Electric Charge
Electric Charge
• A coulomb is a large amount of charge. Since
the charge on an electron has a magnitude of
only 1.6 x 10−19 C, it follows that the number of
electrons in a coulomb is
1 C/1.6 x 10−19 C = 6.25 x 1018 electrons
•
As we have seen, electric charge can be transferred between objects
simply by rubbing fur across an piece of amber. This transfer of charge
is illustrated in the figure below.
•
Before charging, the fur and amber are both neutral. During the
rubbing process some electrons are transferred from the fur to the
amber, giving the amber a negative charge.
• A lightning bolt can deliver 20–30 coulombs of
charge. A more common unit of charge is the
microcoulomb, µC, where 1 µC = 10−6 C.
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Electric Charge
Electric Charge
• At the same time the fur acquires a positive
charge.
• At no time during the process is charge ever
created or destroyed. This is an example of one
of the fundamental conservation laws of physics:
Electric charge is conserved. This means
that the total electric charge in the
universe is constant.
• It should be noted that when charge is
transferred from one object to another, it is
generally due to movement of electrons.
• In a typical solid the nuclei of the atoms are fixed
in position. The outer electrons of these atoms,
however, are weakly bound and easily
separated.
• As a piece of fur rubs across amber, for example,
some of the electrons that were originally a part
of the atoms in the fur are separated from those
atoms and deposited onto atoms in the amber.
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Electric Charge
Electric Charge
• An atom that gains or loses electrons is called
an ion. More specifically, atoms that lose
electrons become positive ions, and atoms that
gain electrons become negative ions. This
transfer process is referred to as charging by
separation.
• When two materials are rubbed together, the
magnitude and sign of the charge each material
acquires depend on how strongly that material
holds onto its electrons.
• For example, if silk is rubbed against glass, the
silk acquires a negative charge. If silk is rubbed
against amber, however, the silk becomes
positively charged.
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Electric Charge
Electric Charge
• Transferring charge by rubbing objects together
is a type of charging by separation known as
triboelectric charging.
• This type of charging can be understood by
referring to the following table. The larger the
number of plus signs associated with a material
in the table, the more readily it gives up electrons
and becomes positively charged. Similarly, the
larger the number of minus signs associated with
a material, the more readily it acquires electrons
and becomes negatively charged.
• In general, when two
materials in the table
are rubbed together,
the one higher in the
list becomes positively
charged and the one
lower in the list
becomes negatively
charged.
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Electric Charge
Electric Charge
• Charge separation occurs not only when one
object is rubbed against another, but also when
objects collide. For example, collisions of
crystals of ice in a rain cloud cause charge
separation that can results in bolts of lightning
that bring the charges together.
• The rotating blades of a helicopter become
charged due to the collisions between the
blades and dust particles in the air.
• The charged blades give off sparks that are visible at
night (see figure below).
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• Similarly, particles in the rings of Saturn are constantly
undergoing collisions and becoming charged. The
Voyager spacecraft recorded electric discharges, similar
to lightning bolts on Earth.
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Electric Charge
Electric Charge
• In addition, the faint radial lines, or spokes, that
extend across the rings of Saturn (see figure
below) are the result of electric forces between
charged particles.
• We know that charges of opposite sign attract. It is also
possible, however, for a charged rod to attract small
objects that have zero total charge. The mechanism
responsible for this attraction is called polarization.
• To see how polarization works, consider the figure
below.
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Electric Charge
Electric Charge
• When a positively charged rod is brought close
to a neutral object, the atoms at the surface of
the object distort, producing excess negative
charge on the surface. The induced charge is
referred to as a polarization charge.
• Because the polarization charge is opposite that
on the rod, there is an attractive force between
the rod and the object.
• Of course, the same conclusion is reached if we
consider a negative rod held near a neutral
object.
• It is for this reason that both charged amber and
charged glass attract neutral objects—even
though their charges are opposite.
• As the figure below indicates, a negatively
charged balloon can attract a stream of water,
even though the water molecules are electrically
neutral.
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Electric Charge
Electric Charge
• When one end of an amber rod is rubbed with
fur, the rubbed portion becomes charged, and
the other end remains neutral. The charge does
not move from one end to the other.
• Materials like amber, in which charges are not
free to move, are called insulators. Most
insulators are nonmetallic substances, and most
are also good thermal insulators.
• In contrast, a conductor is a material that allows
charges to move freely from one location to
another. Most metals are good conductors.
• The figure below provides examples of
insulators and conductors.
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Electric Charge
Electric Charge
• When an uncharged metal sphere is touched by
a charged rod, some charge is transferred at the
point of contact [figure (a)]. Because like
charges repel and because charges move freely
through a conductor, the transferred charge
quickly spreads out and covers the entire
surface of the sphere [figure (b)].
• The insulating base prevents charge from
flowing from the sphere into the ground.
• On a microscopic level, the difference between
conductors and insulators is that the atoms in
conductors allow one or more of their outermost
electrons to become detached. These detached
electrons, often referred to as conduction
electrons, can move freely throughout the
conductor.
• Insulators, in contrast, have very few, if any, free
electrons. In an insulator the electrons are bound
to their atoms and cannot move from place to
place within the material.
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Electric Charge
Electric Force
• Since the flow of electric charge can be dangerous to
people, insulating gloves like those shown in the figure
below are important to the safety of electrical workers.
• Not only do opposite charges attract and like
charges repel, but the strength of the attraction
or repulsion depends on the magnitude of the
charges.
• The force between electric charges, which we
refer to as the electric or electrostatic force, also
depends on the separation between the charges.
• Consider two electric charges q1 and q2. What
Coulomb discovered is that if you double charge
q1, the force doubles. If you double charge q2,
the force again doubles.
• Materials that have properties intermediate between
those of a good conductor and those of a good insulator
are referred to as semiconductors.
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Electric Force
Electric Force
• Thus, the electrostatic force depends on the
product of the magnitudes of the charges. That is,
F depends on |q1||q2|
• Thus, doubling either charge doubles the force.
• Coulomb also discovered that the electric force
becomes weaker as the charges are moved
farther apart. In fact, he found that if you double
the separation between the charges, the force
drops off by a factor of 4. Thus, the electrostatic
force depends on the inverse square of the
distance between the charges.
• Coulomb combined these observations into a law.
Coulomb's law relates the strength of the electrostatic
force between point charges to the magnitude of the
charges and the distance between them:
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• In this equation, the constant k has the following value:
k = 8.99 x 109 N·m2/C2.
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Electric Force
Electric Force
• Summarizing: The magnitude of the electric force is given
by Coulomb's law. The electric force acts along the line
connecting the two charges. In addition, we know that like
charges repel and opposite charges attract.
• These properties are illustrated in the figure below, where
force vectors are shown for pairs of charges of various
signs.
• Newton's third law applies to each of the cases
shown in the preceding figure. For example, the
force exerted on charge 1 by charge 2 is always
equal in magnitude and opposite in direction to
the force exerted on charge 2 by charge 1.
• The following figure illustrates the "opposites
attract, likes repel" rule in a dramatic way.
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Electric Force
Electric Force
• The Van de Graaff generator charges the
student's hair, giving each strand of hair the
same charge. The like charges repel, creating
the ultimate bad hair day.
• It's interesting to compare Coulomb's law for the
electric force and Newton's law for the force of
gravity. The equations are as follows:
Coulomb's law
F = k|q1||q2|/r2
Newton's law of gravity F = km1m2/r2
• In each case the force decreases as the square
of the distance between the objects. In addition,
each force depends on the product of two
magnitudes of a physical quantity. For electric
force the physical quantity is the charge; for
gravity it is the mass.
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Electric Force
Electric Force
• Because the electric force can be attractive or
repulsive, the total electric force between neutral
objects, such as the Earth and Moon, is
essentially zero. Basically, the attractive and
repulsive forces cancel one another out.
• This is not the case with gravity, however.
Gravity is always attractive, exerting a larger
total force on larger astronomical bodies. Thus,
the total gravitational force between the Earth
and the Moon is not zero.
• Gravity is also behind the formation of black holes—objects
whose gravity is so strong that not even light can escape
from them. If a star or other object comes too close to a
black hole, it will be pulled into the hole, as illustrated in the
figure below.
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Electric Force
Electric Force
• The electric force rules at the atomic level,
where gravity plays essentially no role. To see
why, let's compare the electric and gravitational
forces between a proton and an electron in a
hydrogen atom.
• Using Newton's law of gravity, the gravitational
force between the proton and electron can be
shown to be 3.36 x 10−47 N.
• Using Coulomb's law, the electric force is found
to equal 8.22 x 10−8 N.
• Taking the ratio of these two forces, we find that
the electric force is 2.26 x 1039 times greater!
• Another indication of the strength of the electric
force is given in the following Quick Example.
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Combining Electric Forces
Combining Electric Forces
• The electric force, like all forces, is a vector
quantity.
• When a charge experiences forces due to two or
more other charges, the total force on it is the
vector sum of the individual forces.
• Thus, the total force acting on a given charge is
the sum of the individual forces between just two
charges at a time, with the force between each
pair of charges given by Coulomb's law.
• As an example of combining electric forces,
consider the system shown in the figure below.
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• In this case the total force on charge 1, F1, is the
vector sum of the forces due to charge 2 and
charge 3:
F1 = F12 + F13
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Combining Electric Forces
Combining Electric Forces
• Thus, the electric forces combine by vector
addition to give the total force. This addition
process is referred to as the superposition of
forces.
• In the following Guided Example, we apply
superposition to three charges in a line.
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Combining Electric Forces
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Combining Electric Forces
• When the individual forces do not act along a
straight line, we start by drawing arrows
representing the individual force vectors. The
sum of these vectors gives the total force.
• The sum can be found using components or
graphically by placing the individual force
vectors head to tail, head to tail, and so on.
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Combining Electric Forces
Combining Electric Forces
• Although Coulomb's law is stated for point charges, it
can be applied to spherical charge distributions as well.
• For example, suppose a sphere has a charge Q spread
evenly over its surface. If a point charge q is outside the
sphere, a distance r from its center, then the force
between the point charge and the sphere is simply
F = k|q||Q|/r2
• Thus, for charges outside a sphere, a spherical charge
distribution behaves as if all the charge were
concentrated at its center. Charges inside the sphere
experience zero force.
• When a charge Q is spread evenly over the
surface of a sphere, it is convenient to specify the
amount of charge per unit area on the sphere. The
charge per area is referred to as the surface
charge density, σ.
• The surface charge density is defined by the
following equation:
σ = Q/A
• It follows that if a sphere has an area A and a
surface charge density σ, its total charge is
Q = σA
• The dimensions of σ are charge per area, or C/m2.
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Combining Electric Forces
• The following example examines the force on a
point charge exerted by a spherical charge
distribution.
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Chapter 20 Lecture
Chapter Contents
Pearson Physics
• The Electric Field
• Electric Potential Energy and Electric Potential
• Capacitance and Energy Storage
Electric Fields
and Electric Energy
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The Electric Field
The Electric Field
• An electrically charged object sets up a force field
around it; this force field is known as an electric field.
• To help visualize an electric field, look at a group of
grass seeds suspended in a fluid (see figure below).
• In figure (a) there is no net electric charge, and
hence no electric field. The seeds point in
random directions.
• In figure (b), the seeds line up in the direction of
the electric field. Each seed experiences an
electric force, and the force causes it to align
with the field.
• The standard way to draw electric fields is
shown in the figure on the next slide. Here a
positive charge +Q is shown at the center of
figure (a) and a negative charge –Q is shown at
the center of figure (b).
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The Electric Field
The Electric Field
• The direction of an electric field is away from a
positive charge and toward a negative charge.
• A small positive test charge (+q0) at location A in
the preceding figure experiences a force that is
in the same direction as E.
• A small negative test charge (−q0) at location B
experiences a weaker force (since it's farther
away from the central charge) that is in the
opposite direction from E.
• Because the force on a positive charge is in the
same direction as the electric field, we always
use positive test charges to determine the
direction of E.
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The Electric Field
The Electric Field
• You've just seen the connection between the direction of
the electric field and the direction of the electric force.
How do we determine the magnitude of the electric field?
• By definition, the magnitude of the electric field is the
electric force per charge:
• In this definition it is assumed that the test charge is
small enough that it does not disturb the position of any
other charges in the system.
• You will sometimes be given the electric field E at a
given location and be asked to determine the force a
charge q experiences at that location. This can be done
as follows:
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The Electric Field
The Electric Field
• The following
example
illustrates how
the magnitude
and direction
of the force on
a charge may
be determined.
•
Perhaps the simplest example of an electric field is the field
produced by a point charge. Figure (a) below shows a point charge
at the origin.
•
If a small test charge q0 is placed at a distance r from the origin, the
force it experiences is directed away from the origin and has a
magnitude given by Coulomb's law:
F = kq1q0/r2
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The Electric Field
The Electric Field
• Applying the definition of the electric field, E = F/q0, we
find that the magnitude of the electric field is
E = F/q0 = kq/r2
• As you can see, the electric field due to a point charge
decreases with the inverse square of the distance. In
general, the electric field a distance r from a point charge
q has the following magnitude:
• The electric field points away from a positive
point charge. And as the figure below shows, the
electric field points toward a negative point
charge.
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The Electric Field
The Electric Field
• The following example illustrates how the electric field
due to a point charge is determined.
• The electric field due to a point charge
decreases rapidly as the distance from the
charge increases. The field never actually goes
to zero, however. On the other hand, the electric
field increases as the distance gets closer to
zero. Thus, the closer you get to an electric
charge, the stronger its electric field.
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The Electric Field
The Electric Field
•
• The total electric field at the point P is the vector
sum of the fields due to the charges q1 and q2.
Notice that E1 and E2 point away from the
charges q1 and q2, respectively and have the
same magnitude, E: E = kq/r2.
• To find the total electric field, Etotal, we use
components.
• In the y direction, E1,y = –E sinθ and E2,y = +E
sinθ. It follows that the y component of the total
electric field is zero:
Etotal,y = E1,y + E2,y = –E sinθ + E sinθ = 0
•
When a system consists of several charges, the total electric field is
found by superposition—that is, by calculating the vector sum of the
electric fields due to the individual charges.
As an example, let's calculate the total electric field at point P due to
two charges as shown in the figure below.
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The Electric Field
The Electric Field
• Similarly, we determine the x component of Etotal:
Etotal,x = E1,x + E2,x = E cosθ + E cosθ = 2E cosθ
• A second
example of
how the total
electric field is
determined is
given below.
• Thus, the total electric field at P is in the positive
x direction. The magnitude of the total electric
field is equal to 2E cosθ.
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The Electric Field
The Electric Field
• Many aquatic creatures are capable of producing
electric fields. For example, some freshwater fish
in Africa can use their specialized tail muscles to
generate an electric field. They are also able to
detect variations in this field as they move
through their environment. This assists them in
locating obstacles, enemies, and food.
• Much stronger fields are produced by electric
eels and electric skates. The electric eel
Electrophorus electricus generates an electric
field strong enough to kill small animals and to
stun larger animals.
• The following set of rules provides a consistent
method for drawing electric field lines:
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The Electric Field
The Electric Field
• The following are examples of how these rules are
applied.
• In the figure below, the electric field lines all start at the
positive charge, point radially outward, and go to infinity.
In addition, the lines are closer together near the charge.
• The next figure shows the field produced by a charge of
−2q. In this case, the direction of the field lines is
reversed—they start at infinity and end on the negative
charge. In addition, the number of lines is doubled, since
the magnitude of the charge has been doubled.
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The Electric Field
The Electric Field
• Electric fields tend to form specific patterns depending
on the charges involved. A few such patterns, for various
combinations of charges, are shown in the figure below.
• In figure (a), some field lines start on one charge
and terminate on another. Notice also that the
field lines are close together, indicating that the
electric field is intense between the charges.
• In contrast, the field is weak between the
charges in figure (b), where the field lines are
widely spaced.
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The Electric Field
The Electric Field
• The charge combination of +q and –q in figure
(c) is known as an electric dipole. The total
charge of a dipole is zero, but because the
positive and negative charges are separated,
the electric field does not vanish. Instead, the
field lines form loops that are characteristic of a
dipole.
• Dipoles are common in nature. Perhaps the
most familiar example is the water molecule,
which is positively charged at one end and
negatively charged at the other.
•
A simple but particularly important field picture results when charge
is spread uniformly over a very large (essentially infinite) plate, as
illustrated in the figure below.
•
The electric field is uniform in this case, in both direction and
magnitude. The field points in a single direction—perpendicular to
the plate. Most remarkably, the magnitude of the electric field
doesn't depend on the distance from the plate.
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The Electric Field
The Electric Field
•
If two plates with opposite charge are placed parallel to each other
and separated by a finite distance, the result is a parallel-plate
capacitor. An example is shown in the figure below.
• Conductors contain an enormous number of electrons
that are free to move about. This simple fact has some
rather interesting consequences. For one, any excess
charge placed on a conductor moves to its outer surface,
as is indicated in the figure below.
•
The field in this case is uniform between the plates and zero outside
the plates. This case is ideal, which is exactly true for infinite plates
and a good approximation for large plates.
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The Electric Field
The Electric Field
• In this way the individual charges are spread as
far apart from one another as possible.
• On a conducting sphere, excess charge placed
on the sphere distributes itself uniformly on the
surface. None of the excess charge is within the
volume of the conductor.
• The distribution of charge on the surface of a
conductor guarantees that the electric field
within the conductor is zero. This effect is
referred to as shielding. Shielding occurs
whether the conductor is solid or hollow.
• Shielding is put to use in numerous electrical
devices, which often have a metal foil or wire
mesh enclosure surrounding the sensitive
electrical circuits.
• Related to shielding is the fact that electric field
lines always contact a conductor at right angles
to its surface. In addition, the field lines crowd
together where a conductor has point or a sharp
projection, as illustrated in the following figure.
The result is an intense electric field at a sharp
metal point.
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The Electric Field
• The crowding of field lines at a point is the basic principle
behind the operation of lightning rods. During an
electrical storm the electric field at the tip of a lightning
rod becomes so intense that electric charge is given off
into the atmosphere. In this way a lightning rod
discharges the area near the house, thus preventing
lightning from striking the house, which would transfer a
large amount of charge in one sudden blast.
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The Electric Field
• Because electric forces act a distance, it is
possible to charge an object without touching it
with a charged object.
• The charging of an object without direct contact
is referred to as charging by induction.
• The following figure illustrates the steps involved
in charging an object by induction.
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The Electric Field
•
First, a negatively charged rod is brought close to the sphere as in
figure (a). The charged rod induces positive and negative charge on
opposite sides of the conducting sphere. At this point the sphere is
still electrically neutral.
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The Electric Field
• The sphere is then grounded using a conducting
wire [figure (b)]. Negative charges repelled by
the rod enter the ground. (In general, grounding
refers to the process of connecting a charged
object to the Earth with a conductor and is
indicated by the symbol .)
• With the charged rod still in place, the grounding
wire is removed. This traps the net charge on
the sphere [figure (c)].
• The charged rod is then removed. The sphere
retains a charge with a sign that is opposite that
on the charged rod [figure (d)].
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
•
• Since the charges attract one another, you have
to exert a force and do work to pull your hands
apart. This work is stored in the electric field as
electric potential energy.
• If you release the charges, they speed up as
they race toward each other, converting their
electric potential energy into kinetic energy.
•
Both electric and gravitational forces can store mechanical work in
the form of potential energy. Mechanical work that is stored as
electrical energy is referred to as electric potential energy.
Suppose you have a positive charge in one hand and a negative
charge in the other, as is shown in the figure below.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
•
•
A positive charge q placed in the uniform electric field E shown in
the figure below experiences a downward electric force of
magnitude F = qE.
•
If the charge is moved upward through a distance d, the electric
force and the displacement are in opposite directions. Therefore, the
work done by the electric force is negative and equal in magnitude
to the force times distance: W = −qEd.
If they charges are both positive, they will repel one another. Moving
two charges that repel each other closer together requires
mechanical work. This work will be stored as electric potential
energy, as is shown in the figure below. If the charges are released,
they fly apart from one another, converting electric potential energy
to kinetic energy.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
•
• Suppose the electrical potential energy of a
charge q changes by the amount ∆PE. By
definition, we say that the electric potential, V, of
the charge changes by the amount PE/q.
• The electric potential is basically electric
potential energy per charge. The electric
potential is generally referred to as voltage
because it is measured in a unit called the volt.
•
From the equation that relates work and potential energy,
∆PE = −W, we see that the system's change in electric potential
energy is given by ∆PE = −W or ∆PE = qEd.
Notice that the electric potential energy increases in this case. This
is like increasing the gravitational potential energy by raising a ball
against the force of gravity, as is indicated in the figure below.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• You are probably familiar with voltage in the
form of 120-V electricity in your home or 1.5-V
batteries for your camera.
• The volt is named in honor of Alessandro Volta
(1745–1827), who invented a predecessor to the
modern battery. The volt has the units of energy
(J) per charge (C):
1 V = 1 J/C
• Equivalently, 1 joule of energy is equal to 1
coulomb times 1 volt:
1 J = (1 C)(1 V)
• It follows that a 1.5-V battery does 1.5 J of work
for every coulomb of charge that flows through it;
that is, (1 C)(1.5 V) = 1.5 J. In general, the
change in electric potential energy, ∆PE, as a
charge q moves through an electric potential
(voltage) difference ∆V is ∆PE = q∆V.
• The following example illustrates how the
change in electric potential energy is found.
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Electric Potential Energy and
Electric Potential
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Electric Potential Energy and
Electric Potential
• In general, a high-voltage system has a lot of electric
potential energy. The figure below shows the situation
for charges of opposite sign. When the charges are
widely separated, the voltage is high. If these charges
are released, a lot of electrical energy is converted into
kinetic energy.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• For the case of charges with the same sign, like
those in the figure below, the situation is
reversed. Charges close together correspond to
high voltage because they fly apart at high
speed when released.
• There is a straightforward and useful connection
between the electric field and electric potential.
• To obtain this relationship, we will apply the definition
∆V = ∆PE/q to the case of a charge that moves through
a distance d in the direction of the electric field, as is
shown in the figure below.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• The work done by the electric field in this case
is simply the magnitude of the electric force
F = Eq, times the distance, d:
W = qEd
• Therefore, the change in electric potential is
∆V = ∆PE/q = −W/q = −(qEd)/q = −Ed
• Solving for the electric field, we find the following:
•
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To summarize, the electric field depends on the rate of change of
the electric potential with position. In terms of a gravitational
analogy, you can think of the electric potential, V, as the height of
the hill and the electric field, E, as the slope of the hill. This analogy
is illustrated in the figure below.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
•
• The following
example illustrates
how (a) the electric
field in a capacitor
can be determined
and (b) how the
change in electric
potential energy of a
charge moved
between the plates of
the capacitor can be
calculated.
•
As the figure below illustrates, the electric potential decreases in the
direction of the electric field. In addition, the electric potential doesn't
change at all in the direction perpendicular to the electric field.
For the case shown, the electric field is constant. As a result, the
electric potential decreases uniformly with distance.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• Energy conservation applies to a charged object
in an electric field. As a result, the sum of the
object's kinetic and electric potential energies
must be the same at any two points, say A and B:
½mvA2 + PEA = ½mvB2 + PEB
• This equation applies to any conservative force.
Notice, however, that the PE term in the equation
depends on the type of conservation force
involved.
• For a uniform gravitational field, the potential
energy is PE = mgy.
• For an ideal spring, the potential energy is
PE = ½kx2.
• For an electrical system, the potential energy is
PE = qV.
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• Charges that are free to move in an electric field
will accelerate.
– Positive charges accelerate in the direction of
decreasing electric potential.
– Negative charges accelerate in the direction
of increasing electric potential.
• In both cases, the charge moves to a region of
lower electric potential energy—like a ball rolling
downhill to a position where it has lower
gravitational potential energy.
• We have learned that the electric field a distance
r from a point charge is given by
E = kq/r2
• Similarly, it can be shown that the electric
potential at a distance r from a point charge is
the following:
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• Notice that the electric potential is zero at an
infinite distance, r = ∞. Also, the potential is
positive for a positive charge and negative for a
negative charge.
• The electric potential is a number (scalar) and
therefore it has no associated direction.
• If a charge q0 is in a location where the potential
is V, the corresponding electric potential energy
is
PE = q0V
• For the special case where the electric potential
is due to a point charge q, the electric potential
energy is as follows:
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Electric Potential Energy and
Electric Potential
Electric Potential Energy and
Electric Potential
• The following example illustrates how the
potential of a point charge is determined.
•
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As mentioned earlier, the sign of the electric potential V depends on
the sign of the charge in question. This relationship is illustrated in
the figure below.
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Electric Potential Energy and
Electric Potential
Capacitance and Energy Storage
• The figure shows the electric potential near (a) a
positive charge and (b) a negative charge. In the
case of the positive charge, the potential forms a
potential hill. The negative charge produces a
potential well.
• Like many physical quantities, the electric
potential obeys a simple superposition principle.
Therefore, the total electric potential due to two
or more charges is equal to the algebraic sum of
the potentials due to the individual charges.
• The algebraic sign of each potential must be
taken into account.
• A common way for electrical systems to store
energy is in a device known as a capacitor.
• A capacitor gets its name from the fact that it
has a capacity to store both electric charge and
electrical energy.
• Capacitors are an important element in modern
electronic devices. No cell phone or computer
could work without capacitors.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• In general, a capacitor is nothing more than two
conductors, referred to as plates, separated by a
finite distance.
• When the plates of a capacitor are connected to
the terminals of a battery, they become charged.
One plate acquires a positive charge, +Q, and
the other plate acquires an equal and opposite
negative charge, −Q.
• To be specific, suppose a certain battery
produces a potential difference (or voltage) V
between its terminals. When this battery is
connected to a capacitor, a charge of magnitude
Q appears on each plate.
• The ratio of the charge stored to the applied
voltage—that is, the ratio Q/V—is called the
capacitance, C.
• The greater the charge Q for a given voltage V,
the greater the capacitance of the capacitor.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• Summarizing,
• From the relation C = Q/V we see that the units
of capacitance are coulombs per volt. In the SI
system this combination of units is referred to as
the farad (F), in honor of the English physicist
Michael Faraday (1791–1867). In particular,
1 F = 1 C/V
• Just as the coulomb is a rather large unit of
charge, so too is the farad a rather large unit of
capacitance. Typical values for capacitance are
in the picofarad (1 pF = 10−12 F) to microfarad
(1 µF = 10−6 F) range.
• In this equation, Q is the magnitude of the
charge on either plate and V is the magnitude of
the voltage difference between plates. By
definition, the capacitance is always a positive
quantity.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• A bucket of water provides a useful analogy
when thinking about capacitors, as shown in the
figure below.
• For this analogy we make the following
identifications:
– The cross-sectional area of the bucket is the
capacitance, C.
– The amount of water in the bucket is the
charge, Q.
– The depth of the water is the potential
difference, V, between plates.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• In terms of this analogy, charging a capacitor is like
pouring water into a bucket. If the capacitance is large,
it's like having a wide bucket. In this case, the bucket
holds a lot of water when it has a given level of water. A
narrow bucket with the same water level holds much less
water. This can be seen in the figure below.
• Similarly, a capacitor
with a large
capacitance holds a
lot of charge (water)
for a given applied
voltage (water level).
• The following
example illustrates
how the relationships
C = Q/V and E =
−∆V/d may be
applied.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• The two main factors that determine the
capacitance of a capacitor are plate area and
plate separation.
• If the area of the plates is increased, the
capacitance goes up. It's just like the analogy
with a bucket of water—a capacitor with a large
plate area is like a bucket with a large crosssectional area.
• If the plate separation is decreased, the
capacitance increases. The reason is that a
smaller separation between plates reduces the
potential difference between them. This means
that less voltage is required to store a given
amount of charge—which is another way of
saying that the capacitance is larger.
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Capacitance and Energy Storage
Capacitance and Energy Storage
•
• As mentioned before, capacitors store more than
just charge—they also store energy. It can be
shown that the total energy, PE, stored in a
capacitor with charge Q and potential difference
V is
PE = ½QV
• Therefore, increasing a capacitor's charge or
voltage increases its stored energy.
•
•
The dependence of capacitance on plate separation is useful in a
number of interesting applications.
Each key on a computer keyboard is connected to the upper plate of
a parallel plate capacitor, as illustrated in the figure below.
When you press on a key, the separation between the plates of the
capacitor decreases. This increases the capacitance of the key. The
circuitry of the computer detects this change in capacitance and
determines which key has been pressed.
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Capacitance and Energy Storage
Capacitance and Energy Storage
• The energy stored in a capacitor can be put to a
number of practical uses.
• A camera's flash unit typically contains a
capacitor with a capacitance of about 400 µF.
When fully charged to a voltage of 300 V, this
capacitor contains roughly 15 J of energy.
Because of the rapid release of energy—
discharge takes less than a millisecond—the
power output of a flash unit is impressively
large—about 10–20 kW.
• A defibrillator uses a capacitor to deliver a shock to a
person's heart, restoring it to normal function. Capacitors
can have the opposite effect as well. It is for this reason
that they can be quite dangerous, even in electrical
devices that are turned off and unplugged from the wall.
• For example, a typical TV set or computer monitor
contains a number of capacitors. Some of these
capacitors store a significant amount of charge for a long
period of time. If you reach into the back of an unplugged
television set, there is a danger that you might come in
contact with the terminals of a capacitor, resulting in a
shock.
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Chapter 21 Lecture
Chapter Contents
Pearson Physics
• Electric Current, Resistance, and Semiconductors
• Electric Circuits
• Power and Energy in Electric Circuits
Electric Current
and Electric Circuits
Prepared by
Chris Chiaverina
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• All electric circuits have one thing in
common—they depend on the flow of electric
charge.
• When electric charge flows from one place to
another, we say it forms an electric current. The
more charge that flows, and the faster it flows,
the greater the electric current.
• In general, electric charge is carried through a
circuit by electrons.
• Suppose an amount of charge ∆Q flows past a
given point in a wire in the time ∆t. The electric
current, I, in the wire is simply defined as the
amount of charge divided by the amount of time.
• The following equation is used to determine the
current flowing in a wire.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• The unit of current is the ampere (A), or amp for
short. It is named for the French physicist
André-Marie Ampère (1775–1836).
• A current of 1 amp is defined as the flow of
1 coulomb of charge in 1 second:
1 A = 1 C/s
• A 1-amp current is fairly strong. Many electronic
devices, like cell phones and digital music players,
operate on currents that are a fraction of an amp.
• The following Conceptual Example illustrates
how the current depends on both the amount of
charge flowing and the amount of time.
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Electric Current, Resistance, and
Semiconductors
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Electric Current, Resistance, and
Semiconductors
• The following example shows that the number of
electrons flowing in a typical circuit is extremely
large. The situation is similar to the large
number of water molecules flowing through a
garden hose.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• When charge flows through a closed path and
returns to its starting point, we say that the closed
path is an electric circuit.
• In a type of circuit known as a direct-current
circuit, or DC circuit, the current always flows in
the same direction. Circuits that run on batteries
are typically DC circuits.
• Circuits with currents that periodically reverse their
direction are referred to as alternating-current
circuits, or AC circuits. The electricity provided by
a wall plug in your house is AC.
• Although electrons move fairly freely in metal
wires, something has to push on them to get
them going and keep them going. It's like water
in a garden hose; the water flows only when a
force pushes on it. Similarly, electrons flow in a
circuit only when an electrical force pushes on
them.
• Figure (a) below shows
that there is no water
flow if both ends of the
garden hose are held
at the same level.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Figure (b) shows that water flows from the end where the
gravitational potential energy is high to the end where it is
low. The difference in gravitational potential energy
between the two ends of the hose results in a force on the
water—which in turn produces a flow. A battery performs
a similar function in an electric circuit.
• A battery uses chemical reactions to produce a
difference in electric potential between its two
ends, which are referred to as the terminals. The
symbol for a battery is
.
• A battery's positive terminal has a high electrical
potential and is denoted with a plus (+) sign; the
negative terminal has a low electric potential and
is denoted with a minus sign (−).
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• When a battery is connected to a circuit, electrons
move in a closed path from one terminal of the
battery through the circuit and back to the other
terminal of the battery. The electrons leave from
the negative terminal of the battery and return to
the positive terminal.
• The situation is similar to the flow of blood in your
body. Your heart acts like a battery, causing blood
to flow through a closed circuit of arteries and
veins in your body.
• The figure below shows a simple electrical system
consisting of a battery, a switch, and a lightbulb
connected together in a flashlight.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• The circuit diagram in figure (b) below shows that
the switch is open—creating an open circuit.
When a circuit is open, no charge can flow. When
the switch is closed, electrons flow through the
circuit and the light glows.
• The figure below shows a mechanical equivalent
of the flashlight circuit. The person lifting the
water corresponds to the battery, the paddle
wheel corresponds to the lightbulb, and the
water is like the electric charge.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• The difference in electric potential between the
terminals of the battery is the electromotive force,
or emf. Symbolically, the electromotive force is
represented by the symbol ε (the Greek letter
epsilon). The unit of emf is the same as that of
electrical potential, namely, the volt.
• The electromotive force is not really a force.
Instead, the emf determines the amount of work a
battery does to move a certain amount of charge
around a circuit.
• To be specific, the magnitude of the work done
by a battery with the emf ε as charge ∆Q moves
from one terminal to the other is
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• The following example illustrates how the charge
that passes through a circuit and the work done by
the battery moving that charge can be determined.
• When drawing an electric circuit, it's helpful to include an
arrow to indicate the flow of current. By convention, the
direction of the current in an electric circuit is the direction
in which a positive test charge would move.
• In typical circuits, the charges that flow are actually
negatively charged electrons. As a result, the flow of
electrons and the current arrow point in opposite directions,
as indicated in the figure below.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• As surprising as it may seem, electrons move rather
slowly through a wire. Their path is roundabout because
they are involved in numerous collisions with the atoms
in the wire, as indicated in the figure below.
• At this speed, it would take an electron about
3 hours to go from a car's battery to the headlights.
However, we know that the lights come on almost
immediately. Why the discrepancy?
• While the electrons move with a rather slow
average speed, the influence they have on one
another, due to the electrostatic force, moves
through the wire at nearly the speed of light.
• A electron's average speed, or drift speed, as it is called,
is about 10−4 m/s—that's only about a hundredth of a
centimeter per second!
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Electrons flow through metal wires with relative
ease. In the ideal case, the electrons move with
complete freedom. Real wires, however, always
affect the electrons to some extent.
• Collisions between electrons and atoms in a
wire cause a resistance to the electron's motion.
This effect is similar to friction resisting the
motion of a box sliding across a floor.
• To move electrons against the resistance of a
wire, it is necessary to apply a potential
difference between the wire's ends.
• Ohm's law relates the applied potential
difference to the current produced and the wire's
resistance. To be specific, the three quantities
are related as follows:
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Ohm's law is named for the German physicist
Georg Simon Ohm (1789–1854).
• Rearranging Ohm's law to solve for the resistance,
we find
R = V/I
• From this expression, it is clear that resistance has
units of volts per amp. A resistance of 1 volt per
amp defines a new unit—the ohm. The Greek letter
omega (Ω) is used to designate the ohm. Thus,
1 Ω = 1 V/A
• A device for measuring resistance is called an
ohmmeter.
• A resistor is a small device used in electric
circuits to provide a particular resistance to
current. The resistance of a resistor is given in
ohms, as shown in the following Quick Example.
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• In an electric circuit, a resistor is signified by a
zigzag line, 222.. , as a reminder of the zigzag
path of the electrons in the resistor.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• The following chart summarizes the elements of
electric circuits, their symbols, and their physical
characteristics.
• A wire's resistance is affected by several factors.
• The resistance of a wire depends on the material from
which it is made. For example, if a wire is made of
copper, its resistance is less than if it is made from iron.
The resistance of a given material is described by its
resistivity, ρ.
• A wire's resistance also depends on it length, L, and its
cross-sectional area, A. To understand these factors,
let's consider water flowing through a hose. If the hose is
very long, its resistance to the water is correspondingly
large. On the other hand, a wide hose, with a greater
cross-sectional area, offers less resistance to the water.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Combining these observations regarding the
factors that affect a wire's resistance, we can
write the following relationship:
• As a wire is heated, its resistivity tends to increase. This
effect occurs because atoms that are jiggling more
rapidly are more likely to collide with electrons and slow
their progress through the wire.
• The following table summarizes the four factors that
affect the resistance of a wire.
• The units of resistivity are ohm-meters (Ω·m),
and its magnitude varies greatly with the type of
material. Insulators have large resistivities;
conductors have low resistivities.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Though Ohm's law is an excellent approximation
for metal wires and the resistors used in electric
circuits, it does not apply to all materials. Materials
known as semiconductors are an important
exception to Ohm's law.
• Elements such as germanium and silicon are
insulators in their pure form. However, when
impurities are added—which is referred to as
doping—these substances can conduct electricity.
Doping produces two types of semiconductors.
• If a small amount of arsenic is added to
silicon—say, one arsenic atom per million silicon
atoms—the silicon becomes a conductor. The
arsenic-doped silicon conducts electricity because
electrons break free from the arsenic atoms and
move freely through the material.
• Silicon doped in this way is referred to as an
n-type semiconductor because current is carried by
negative (n) electrons.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Silicon also becomes a semiconductor when it is
doped with gallium instead of arsenic. In this case,
however, the gallium atoms take electrons from the
silicon atoms, forming positively charged "holes"
that can carry current. Because positive (p) holes
carry the current, this type of material is referred to
as a p-type semiconductor.
• Unlike a typical resistor, a semiconductor has a
lower resistance when its temperature increases.
This is because an increase in temperature makes
it easier for electrons to move, and this produces
more current. The result is a decrease in
resistance.
• Semiconductors can be used to make a variety of
electronic devices. The simplest semiconducting device,
the diode, consists of a p-type semiconductor joined to an
n-type semiconductor. A diode is shown in the figure
below.
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• The basic property of a diode is that it allows current to
flow in one direction, but not the other.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• For example, when the positive terminal of a
battery is attached to the p-type semiconductor in
an ideal diode, as in the figure below, the current
flows with zero resistance. In this case, we say
that the diode is forward biased.
• On the other hand, if the positive terminal of a
battery is connected to the n-type semiconductor
of an ideal diode, as in the figure below, no
current flows at all. In this case, we say that the
diode is reverse biased.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Because of the one-way nature of diodes, they
find uses in electric circuits.
• One application is the conversion of AC current
(which alternates in direction) to DC current
(which flows in one direction only).
• Another application makes use of the fact that
light is emitted when electrons and holes come
together in a diode. This is the basic process
behind the operation of an LED, light-emitting
diode.
• Another useful semiconductor device is produced
by making a "sandwich" of three layers of
semiconductors. The most common type of
transistor has an n-type semiconductor on either
side of the sandwich and a thin p-type
semiconductor in the middle, as is shown in the
figure below. This is known as an npn transistor.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• Transistors can also be made with the opposite
sequence of semiconductors, resulting in a pnp
transistor.
• The basic function of a transistor is to act as an
electronic switch that controls the flow of current
in a circuit.
• Consider the schematic view of an npn transistor
shown in the figure on the next slide. The three
electrodes of the transistor are the collector, the
base, and the emitter. Of these three electrodes,
it is the base that switches on or off the flow of
current through the other two electrodes.
• You might find it helpful to think of the control of current
by the base electrode as similar to turning a valve in a
large-diameter water pipe. Though it doesn't take much
force to turn the valve, once the valve is opened, a large
volume of water flows through the pipe. Similarly, a small
base current "opens the valve" that allows a large
amount of current to flow from the collector to the
emitter.
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Electric Current, Resistance, and
Semiconductors
Electric Current, Resistance, and
Semiconductors
• In a typical transistor, a current I in the base can control the
flow of current of up to 300I through the other two
electrodes. Therefore, any signal with a changing current
that comes into the base electrode is reflected accurately in
a corresponding change in current flowing from the
collector to the emitter—but amplified 300 times.
• The water valve analogy for a transistor is shown in the
figure below.
• One of the great advantages of transistors is that
a small base current can turn a transistor on, by
allowing current to flow through it, or off, by
preventing the flow of current.
• A device that can switch rapidly is just what's
needed in modern digital computers, whose
language is based on the binary digit (bit), which
takes on the value 1 or 0. Computers represent
these two states by a transistor that is either on
or off.
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Electric Current, Resistance, and
Semiconductors
Electric Circuits
• Many transistors are required in a computer.
Most electronic devices today rely on silicon
wafers, called microchips, that contain
thousands of transistors, diodes, and resistors
connected in elaborate circuits.
• These integrated circuits (ICs) are built up layer
by layer on a silicon wafer by depositing specific
patterns of silicon, gallium, and arsenic, and so
on, to produce the desired arrangement of
n-type and p-type semiconductors.
• Electric circuits often contain a number of
resistors connected in various ways.
• One way resistors can be connected is end to
end. Resistors connected in this way are said to
form a series circuit. The figure below shows
three resistors R1, R2, and R3, connected in
series.
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Electric Circuits
Electric Circuits
• The three resistors acting together have the same
effect—that is, they draw the same current—as a single
resistor, which is referred to as the equivalent resistor, Req.
• This equivalence is illustrated in the figure below.
• When resistors are connected in series, the
equivalent resistance is simply the sum of the
individual resistances.
• In our case, with three resistors, we have
Req = R1 + R2 + R3
• In general, the equivalent resistance of resistors
in series is the sum of all the resistances that are
connected together:
• The equivalent resistor has the same current, I, flowing
through it as each resistor in the original circuit.
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Electric Circuits
Electric Circuits
• The equivalent resistance is greater than the
greatest resistance of any individual resistor.
• In general, the more resistors connected in series,
the greater the equivalent resistance.
• For example, the equivalent resistance of a circuit
with two identical resistors, R, connected in series
is Req = R + R = 2R. Thus, connecting two
identical resistors in series produces an equivalent
resistance that is twice the individual resistances.
• The following example illustrates the functioning
of a series circuit.
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Electric Circuits
Electric Circuits
• Resistors that are connected across the same
potential difference are said to form a parallel
circuit.
• An example of three resistors connected in parallel
is shown the figure below.
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Electric Circuits
Electric Circuits
• In a case like this, the electrons have three parallel paths
through which they can flow—like parallel lanes on the
highway.
• The three resistors acting together draw the same current
as a single equivalent resistor, Req, as indicated in the
figure below.
• When resistors are connected in parallel, the
reciprocal of the equivalent resistance is equal
to the sum of the reciprocals of the individual
resistances. Thus, for our circuit of three
resistors, we have
1/Req = 1/R1 + 1/R2 + 1/R3
• In general, the inverse equivalent resistance is
equal to the sum of all of the individual inverse
resistances:
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Electric Circuits
Electric Circuits
• As an example of parallel resistors, consider a
circuit with two identical resistors, R, connected in
parallel. The equivalent resistance in this case is
1/Req = 1/R + 1/R
1/Req = 2/R
• Solving for the equivalent resistance gives
Req = ½R. Thus, connecting two identical resistors
in parallel produces an equivalent resistance that
is half of the individual resistances.
• A similar calculation shows that three resistors,
R, connected in parallel produces an equivalent
that is one-third of the original resistances,
or Req = ⅓R.
• These results show a clear trend, namely, the
more resistors connected in parallel, the smaller
the equivalent resistance.
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Electric Circuits
Electric Circuits
• In general, the equivalent resistance of a parallel
circuit is less than or equal to the smallest
individual resistance. What happens if one of the
individual resistances is zero?
• In this case, the equivalent resistance is also
zero, because Req is less than or equal to the
smallest individual resistance, and a resistance
can't be negative.
• This situation, referred to as a short circuit, is
illustrated in the figure below. In a short circuit,
all the current flows through the path of zero
resistance.
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Electric Circuits
Electric Circuits
• The following example illustrates the functioning
of a parallel circuit.
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Electric Circuits
Electric Circuits
• The rules that apply for series and parallel resistors can be
applied to a variety of interesting circuits that aren't purely
series or parallel.
• The circuit in the figure below contains a total of four
resistors, each with resistance R, connected in a way that
combines series and parallel features. Because the circuit
is not strictly series or parallel, we can't directly calculate
the equivalent resistance.
• What we can do, however, is break the circuit into smaller
subcircuits, each of which is purely series or purely parallel.
For example, we first note that the two vertically oriented
resistors on the right are in parallel with one another; hence
they can be replaced with their equivalent resistance R/2.
• The next step is to replace these two resistors with R/2.
This yields the circuit shown below.
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Electric Circuits
Electric Circuits
• Notice that this equivalent circuit consists of three
resistors in series, R, ½R, and R. The equivalent
resistance of these resistors is equal to their sum,
Req = R1 + R2 + R3 = 2.5R.
• Therefore, the equivalent resistance of the original
circuit is 2.5R, as indicated in the figure below.
• By considering the resistors in pairs or groups that
are connected in parallel or in series, you can
reduce the entire circuit to one equivalent circuit.
This method is applied in the following example.
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Electric Circuits
© 2014 Pearson Education, Inc.
Electric Circuits
• The current flowing through a circuit, or the potential
difference between two points in a circuit, can be measured
directly with a meter.
• The device used to measure current is an ammeter. An
ammeter is designed to measure the flow of current through
a particular portion of a circuit.
• For example, you might want to know the current flowing
between points A and B in the circuit shown in the figure
below.
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Electric Circuits
Electric Circuits
• To measure this current, the ammeter must be
added to the circuit in such a way that all the
current flowing from A to B also flows through the
meter. This is done by connecting the meter in
series with the other circuit elements between A
and B, as is shown in the figure below.
• If the ammeter has a finite resistance—which is
the case for any real meter—then its presence in
a circuit alters the current it is intended to
measure. Thus, an ideal ammeter would have
zero resistance. Real ammeters, however, give
accurate readings as long as their resistance is
much less than the other resistances in the
circuit.
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Electric Circuits
Electric Circuits
• A voltmeter is a device used to measure the
potential difference between any two points in a
circuit. To measure the voltage between two
points, for example, points C and D in the figure
below, the voltmeter is placed in parallel at the
appropriate points.
• Because a small current must flow through the
voltmeter in order for it to work, the meter reduces
the current flowing through the circuit. As a result,
the measured voltage is altered from its ideal
value. Thus, an ideal voltmeter would have infinite
resistance.
• Real voltmeters give accurate readings as long as
their resistance is much greater than other
resistances in the circuit.
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Electric Circuits
Power and Energy in Electric Circuits
• Sometimes the functions of an ammeter, a voltmeter,
and an ohmmeter (a meter to measure resistance) are
combined in a single device called a multimeter. An
example of a multimeter is shown in the figure below.
• The power delivered by an electric circuit increases
with both the current and the voltage. Increase
either, and the power increases.
• When a ball falls in a gravitational field, there is a
change in gravitational potential energy. Similarly,
when an amount of charge, ∆Q, moves across a
potential difference, V, there is a change in
electrical potential energy, ∆PE, given by
∆PE = (∆Q)V
• Adjusting the settings on a multimeter's dial allows a
variety of circuit properties to be measured.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• Recalling that power is the rate at which energy
changes, P = ∆E/∆t, we can express the electric
power as follows:
P = ∆E/∆t = (∆Q)V/∆t
• Knowing that the electric current is given by
I = (∆Q)/∆t allows us to write an expression for the
electric power in terms of the current and voltage.
• Thus, the electric power used by a device is equal
to the current times the voltage. For example, a
current of 1 amp flowing across a potential
difference of 1 V produces a power of 1 W.
• The following example provides another example
of how the electric power is calculated.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• The equation P = IV applies to any electrical
system. In the special case of a resistor, the electric
power is dissipated in the form of heat and light, as
shown in the figure, where the electric power
dissipated in an electric space heater.
• Applying Ohm's law, V = IR, which deals with resistors,
we can express the power dissipated in a resistor as
follows:
P = IV = I(IR) = I2R
• Similarly, solving Ohm's law for the current, I = V/R, and
substituting that result gives an alternative expression for
the power dissipated in a resistor:
P = IV = (V/R)V = V2/R
• All three equations for power are valid. The first, P = IV,
applies to all electrical systems. The other two
(P = I2R and P = V2/R) are specific to resistors, which is
why the resistance, R, appears in those equations.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• The following example shows how currents and
resistances are related.
• The power dissipated by a resistor is the result of
collisions between electrons moving through the circuit
and the atoms making up the resistor.
• The potential difference produced by the battery causes
conduction electrons to accelerate until they bounce off
an atom, causing the atoms to jiggle more rapidly.
• The increased kinetic energy of the atoms is reflected as
an increased temperature of the resistor. After each
collision, the potential difference accelerates the
electrons again, and the process repeats. The result is
the continuous transfer of energy from the conducting
electrons to the atoms.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• The filament of an incandescent lightbulb is
basically a resistor inside a sealed, evacuated
tube. The filament gets so hot that it glows, just
like the heating coil on a stove or the coils in a
space heater.
• The power dissipated in the filament determines
the brightness of the lightbulb. The higher the
power, the brighter the bulb. This basic concept is
applied in the example on the next slide.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• The local electric company bills consumers for the
electricity they use each month. To do this, they
use a convenient unit for measuring electric
energy called the kilowatt-hour.
• Recall that a kilowatt is 1000 W, or equivalently,
1000 J/s. Similarly, an hour is 3600 s. Combining
these results, we see that a kilowatt-hour is equal
to 3.6 million joules of energy:
1 kWh = (1000 J/s)(3600 s) = 3.6 x 106 J
• The figure below shows the type of meter used
to measure the electrical energy consumption of
a household, as well as the typical bill.
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Power and Energy in Electric Circuits
Power and Energy in Electric Circuits
• The following example illustrates how the cost of
electrical energy is calculated.
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Chapter 23 Lecture
Chapter Contents
Pearson Physics
• Electricity from Magnetism
• Electric Generators and Motors
• AC Circuits and Transformers
Electromagnetic
Induction
Prepared by
Chris Chiaverina
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© 2014 Pearson Education, Inc.
Electricity from Magnetism
Electricity from Magnetism
• When Hans Oersted observed that an electric
current produces a magnetic field, it was pure
serendipity. In contrast, Michael Faraday
(1791–1867), an English chemist and physicist,
was aware of Oersted's results, and purposely
set out to see if a magnetic field could produce
an electric field. His ingenious experiments
showed that such a connection does exist.
• Faraday found that a changing magnetic field
produces an electric current, but a magnetic
field that doesn't change has no such effect.
Faraday set out to study this type of behavior.
• The following figure shows a simplified version
of Faraday's experiment.
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• As the figure indicates, two electric circuits are
involved. The first, called the primary circuit,
consists of a battery, a switch, a resistor, and a
wire coil wrapped around an iron bar.
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Electricity from Magnetism
Electricity from Magnetism
• When the switch is closed on the primary circuit,
a current flows through the coil, producing a
strong magnetic field in the iron bar.
• The secondary circuit also has a wire coil
wrapped around the same iron bar, and this coil
is connected to an ammeter that detects any
current in the circuit. There is no battery in the
circuit, and no direct physical contact between
the two circuits. What does link the circuits,
instead, is the magnetic field in the iron bar.
• When the switch is closed on the primary circuit,
the magnetic field in the iron bar rises from zero
to some finite amount, and the ammeter in the
secondary coil deflects to one side briefly and
then returns to zero. As long as the current in the
primary circuit is maintained at a constant value,
the ammeter in the secondary circuit gives a zero
reading.
• If the switch on the primary circuit is then opened,
so the magnetic field drops again to zero, the
ammeter in the secondary circuit deflects briefly
in the opposite direction and then returns to zero.
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Electricity from Magnetism
Electricity from Magnetism
• These observations can be summarized as
follows:
– The current in the secondary circuit is zero as
long as the magnetic field in the iron bar is
constant. It does not matter whether the
constant value of the magnetic field is zero or
nonzero.
– When the magnetic field in the secondary coil
increases, a current is observed to flow in one
direction in the secondary coil. When the
magnetic field in the secondary coil decreases,
a current is observed to flow in the opposite
direction.
• It is important to note that the current in the
secondary coil appears without any physical
contact between the primary and secondary coils.
• For this reason, the current in the secondary coil
is referred to as an induced current. The process
of inducing an electric current in a circuit by using
a changing magnetic field is known as
electromagnetic induction.
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Electricity from Magnetism
Electricity from Magnetism
• Because an induced current behaves the same as
a current produced by an electromotive force (emf)
supplied by a battery, we say that the changing
magnetic field creates an induced emf in the
secondary circuit.
• As far as the circuit is concerned, the changing
magnetic field has the same effect as a battery.
• Faraday observed that the magnitude of the
induced emf is proportional to the rate of change of
the magnetic field—the more rapidly the magnetic
field changes, the greater the induced emf.
• Any means of changing the magnetic field is as effective as
changing the current in the primary.
• The figure below shows a common classroom
demonstration of induced emf.
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• In this case, there is no primary circuit; instead, the
magnetic field is changed by simply moving a bar magnet
toward or away from a coil connected to an ammeter.
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Electricity from Magnetism
Electricity from Magnetism
• When the magnet is moved toward the coil, the
meter deflects in one direction; when it is pulled
away from the coil, the meter deflects in the
opposite direction. There is no deflection when
the magnet is held still.
• Understanding electromagnetic induction
requires a new concept—magnetic flux. The word
flux basically means "flow." For example, the flux,
or flow, of air through a window is directly related
to the direction of the wind and the crosssectional area of the window.
• Similarly, magnetic flux is a measure of the
number of magnetic field lines that pass through
a given area.
• A magnetic field perpendicular to a surface gives
a high flux, and the larger the surface area, the
greater the flux. A magnetic field parallel to a
surface gives zero flux.
• Figure (a) on the next slide shows a magnetic
field B crossing a surface area, A, at right angles.
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Electricity from Magnetism
• The magnetic flux, Ф, in this case is simply the magnitude
of the magnetic field times the area:
Ф = BA
• If the magnetic field is parallel to the surface—like wind
blowing parallel to an open window—then no field lines
cross through the surface. As figure (b) on the next slide
shows, the magnetic flux in this case is zero: Ф = 0.
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Electricity from Magnetism
• In general, only the component of B that is perpendicular
to a surface contributes to the magnetic flux. The magnetic
field in figure (c) crosses the surface at an angle θ relative
to the normal.
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Electricity from Magnetism
Electricity from Magnetism
• Therefore, the magnetic field's perpendicular
component is B cos θ. The magnetic flux, then, is
simply B cos θ times the area, A:
• Magnetic flux depends on the magnitude of the
magnetic field B, its orientation with respect to
the surface, θ, and the area of the surface, A.
A change in any of these variable results in a
change in flux.
• The following example illustrates the effect of
changing the orientation of a wire loop in a
region of constant magnetic flux.
• The SI unit of magnetic flux is the weber (Wb),
named after physicist Wilhelm Weber (1804–1891).
It is defined as follows: 1 Wb = 1 T·m2.
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Electricity from Magnetism
© 2014 Pearson Education, Inc.
Electricity from Magnetism
© 2014 Pearson Education, Inc.
Electricity from Magnetism
Electricity from Magnetism
• The next example illustrates motions of a wire
loop that produce a change in flux.
• Faraday found that the secondary coil experiences an
induced emf only when the magnetic flux changes with
time. In general, the rate at which the magnetic flux
changes is defined as follows:
rate of change of magnetic flux =
change in magnetic flux/change in time = ∆Ф/∆t
• If there are N loops in a coil, the induced emf is given by
Faraday's law of induction:
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Electricity from Magnetism
Electricity from Magnetism
• The negative sign in Faraday's law indicates that
the induced emf opposes the change in
magnetic flux.
• If only the magnitude of the emf is of concern,
then the following equation may be used:
|ε| = |∆Ф|/|∆t|
• Notice that Faraday's law gives the emf that is
induced in a coil or loop of wire. The current that
is induced as a result of the emf depends on the
resistance of the circuit—just as in the case of a
battery connected to a resistor. This is shown in
the following example.
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Electricity from Magnetism
© 2014 Pearson Education, Inc.
Electricity from Magnetism
• A familiar example of Faraday's law in action is the
dynamic microphone. This type of microphone uses a
stationary magnet and a wire coil attached to a movable
diaphragm, as illustrated in the figure below.
• Sound waves move a coil of wire in a microphone,
changing the magnetic flux through the coil. The result is
an induced emf that is amplified and sent to speakers.
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Electricity from Magnetism
Electricity from Magnetism
• Nature often reacts in a way that opposes change. For
example, if you compress a gas, the pressure of the gas
increases—and opposes the compression.
• A similar principle applies to induced electric currents. It
is known as Lenz's law, and was first stated by Estonian
physicist Heinrich Lenz (1804–1865).
• Lenz's law states that an induced current always flows in
a direction that opposes the change that caused it.
• Lenz's law is the reason for the negative sign in
Faraday's law. It indicates that the induced current
opposes the change in magnetic flux.
• To see how Lenz's law works, consider a bar
magnet that is moved toward a conducting loop,
as in figure (a) below.
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Electricity from Magnetism
• If the north pole of the magnet approaches the
loop, a current is induced that tends to oppose
the motion of the magnet. To be specific, the
current in the loop creates a north pole of a
magnet. This produces a repulsive force acting
on the magnet, opposing the motion.
• On the other hand, suppose the magnet is
pulled away from the loop, as in figure (b) on the
next slide. The induced current is in the opposite
direction, creating a south pole and a
corresponding attractive force—which again
opposes the motion.
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Electricity from Magnetism
• The following example serves to illustrate Lenz's law.
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Electricity from Magnetism
Electricity from Magnetism
• Lenz's law also applies to a decreasing magnetic field.
Figure (a) below shows a conducting loop in a constant
magnetic field. If the magnetic field decreases in
magnitude, as is shown in figure (b), the induced current
produces a magnetic field that acts to maintain the
original magnetic strength in the loop.
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© 2014 Pearson Education, Inc.
Electricity from Magnetism
Electricity from Magnetism
• An object moving through a magnetic field may experience
an induced current. As an example, consider the situation
shown in figure (a) below.
• The magnetic field is constant in this system, but
the magnetic flux through the loop still changes
with time. A decrease in the area enclosed by the
loop, caused by the downward motion of the rod,
causes the magnetic flux to decrease.
• The motion of the rod produces an emf, called a
motional emf. The magnitude of the motional emf
is
ε = vBL
• Notice that the emf depends directly on the speed
of the rod, its length, and the strength of the
magnitude field through which it moves.
• A metal rod of length L is shown moving in the vertical
direction with a speed v through a region with a constant
magnetic field B. The rod is in frictionless contact with two
vertical wires, which allows a current to flow in a loop
through the rod, the wires, and the lightbulb.
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Electricity from Magnetism
Electricity from Magnetism
• According to Lenz's law, the direction of the motional
emf—and thus the direction of the induced current—must
oppose changes caused by the motion of the rod. To see
how this works, consider figure (b) below.
• The following example illustrates how the direction of the
induced current in a ring first falling through, and then
falling out of, a region with a magnetic field is determined.
•
The direction of the current induced by the rod's downward
motion is counterclockwise, because this direction
produces an upward force on the rod, opposing its
downward motion.
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Electricity from Magnetism
© 2014 Pearson Education, Inc.
Electricity from Magnetism
• The figure below shows a sheet of metal falling from a
region with a magnetic field to a region with no field.
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• In the portion of the sheet that is just leaving the field, a
localized circulating current known as an eddy current is
induced in the metal. This current retards the motion of
the metal sheet, having an effect much like a frictional
force.
• The friction-like effect of eddy currents is the basis for
magnetic braking. Magnetic braking is used in everything
from exercise bicycles to roller coasters.
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Electric Generators and Motors
Electric Generators and Motors
• An electric generator is a device designed to
convert mechanical energy to electrical energy.
• The mechanical energy used to drive a
generator can come from many different
sources. Examples include falling water in a
hydroelectric dam, expanding steam in a coalfired power plant, and a gasoline-powered motor
in a portable generator.
• All generators use the same basic operating
principle—mechanical energy moves a
conductor through a magnetic field to produce a
motional emf.
• Rotating a wire loop or coil in a magnetic field to
change the magnetic flux allows the
electromagnetic induction process to continue
indefinitely.
• Thus, rotating a coil of wire through a magnetic
field is a way to transfer energy from mechanical
motion to an electric emf and current.
• To see how this works, imagine a wire coil of
area A located in the magnetic field between the
poles of a magnet, as illustrated in the figure on
the next slide.
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Electric Generators and Motors
• As mechanical work rotates the coil with an angular speed
ω, the emf produced in it is given by Faraday's law. In the
case of a rotating coil, it can be shown that Faraday's law
gives the following result:
ε = NBA sinωt
© 2014 Pearson Education, Inc.
© 2014 Pearson Education, Inc.
Electric Generators and Motors
• This result is plotted in the figure below. Notice
that the induced emf in the coil alternates in sign,
which means that the current in the coil alternates
in direction. For this reason, this type of generator
is referred to as an alternating current generator
or, simply, an AC generator.
• The maximum emf occurs when sin ωt = 1. Thus,
εmax = NBAω
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Electric Generators and Motors
Electric Generators and Motors
• This result is applied in the next example.
© 2014 Pearson Education, Inc.
© 2014 Pearson Education, Inc.
Electric Generators and Motors
Electric Generators and Motors
• A current-carrying loop in a magnetic field experiences a
torque that tends to make it rotate. If such a loop is
mounted on an axle, as shown in the figure below, the
magnetic torque can be used to operate machinery.
• Instead of doing work to turn a coil and produce
an electric current, as in a generator, an electric
motor uses an electric current to produce
rotation of a loop or coil, which then does work.
• An electric motor transforms energy from electric
emf and current into mechanical motion. It
follows that an electric motor is basically an
electric generator run in reverse.
• This device converts electric energy to mechanical work.
A device that converts electric energy into mechanical
energy is called an electric motor.
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AC Circuits and Transformers
• Electricity comes in two types—direct current
and alternating current. Each has benefits and
drawbacks. Alternating current is particularly
useful in the home, in part because it works so
well with devices called transformers that
change the voltage.
• A simplified AC circuit diagram for a lamp is
shown in the figure on the next slide. The bulb is
represented by a resistor with equivalent
resistance R and the wall socket is shown as an
AC generator, represented by a circle enclosing
one cycle of a sine wave.
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AC Circuits and Transformers
• The voltage delivered by an AC generator is plotted in
figure (a) below.
• Notice that the graph has the shape of a sine curve. In
fact, the mathematical equation for the voltage is
V = Vmax sin ωt
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AC Circuits and Transformers
AC Circuits and Transformers
• In household circuits, the angular frequency is
ω = 2πf, with f = 60 Hz. The maximum voltage,
Vmax, is the largest value of the voltage during a
cycle.
• Because the voltage in an AC circuit depends on
the sine function, we say that it has a sinusoidal
dependence.
• The current in a resistor in an AC circuit is
I = Imax sinωt
• The value of the maximum current is given by
Ohm's law:
Imax = Vmax/R
• Thus, the current in an AC circuit also has a sinusoidal
dependence. This result is plotted on the graph in the
figure below.
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• The voltage and current for a resistor reach their maximum
values at the same times. This means that the voltage and
current are in phase with one another. Other circuit
elements, like capacitors and inductors, have different
phase relationships between the current and voltage.
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AC Circuits and Transformers
AC Circuits and Transformers
• Notice that both the voltage and the current in
the previous two figures have average values of
zero. Thus, the average values of AC quantities
give very little information. A more useful type of
average, or mean, is the root mean square, or
rms for short.
• To see the significance of a root mean square,
start by taking the square of the AC current:
I 2 = I 2max sin2ωt
• This result is plotted in the figure below.
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AC Circuits and Transformers
AC Circuits and Transformers
• As the graph indicates, the current squared is
always positive and varies symmetrically
between 0 and I 2max.
• Now, we take the square root of the average so
that the final result is a current rather than a
current squared. This yields the rms value of the
current:
• These results are applied in the following example.
• The same reasoning applies to the rms value of
the voltage in an AC circuit. Therefore,
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AC Circuits and Transformers
AC Circuits and Transformers
• The average power in an AC circuit depends on
the root mean square values of the voltage and
current.
• Recall that the power dissipated in a resistor is
P = I 2R
• The current in an AC circuit is changing
constantly with time, and therefore the power is
also changing with time. So what is the average
power in the circuit?
• To find the average power dissipated in a
resistor, we recall the average of the current
squared is
. Using this result, we find
• We've also seen that the maximum current is
related to the rms current by the relation
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© 2014 Pearson Education, Inc.
AC Circuits and Transformers
AC Circuits and Transformers
• Therefore, the average power in an AC circuit can be
written as follows:
Pav = I 2rmsR
• Similar conclusions apply to the other power formulas as
well. For example, recall that the power dissipated in a
DC circuit can be written as P = V 2/R or as P = IV. To
find the average power in an AC circuit, we take those
equations and simply change the current to the rms
current and the voltage to the rms voltage:
Pav = V2rms /R
Pav = IrmsVrms
• Thus, the average power in an AC circuit is
given by the same equations used in DC
circuits—we just replace the DC current and
voltage with the corresponding rms values.
• The example on the next slide illustrates how the
rms voltage, rms current, and average power
are calculated in an AC circuit.
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AC Circuits and Transformers
© 2014 Pearson Education, Inc.
AC Circuits and Transformers
© 2014 Pearson Education, Inc.
AC Circuits and Transformers
AC Circuits and Transformers
• It is easy to forget that household electrical
circuits pose potential dangers to homes and
their occupants. Fortunately, there are many
devices available to ensure electrical safety.
• Fuses—In the case of a fuse, the current in a
circuit must flow through a thin metal strip
enclosed within the fuse. If the current exceeds
a predetermined amount (typically 15 A), the
metal strip becomes so hot that it melts and
breaks the circuit. Thus when a fuse "burns out,"
it is an indication that too many devices are
operating on that circuit.
• Circuit Breakers—Circuit breakers like the one in
figure (a) below provide protection in a way
similar to a fuse by means of a switch that
incorporates a bimetallic strip.
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AC Circuits and Transformers
– When the bimetallic strip is cool, it closes the
switch, allowing current to flow. When the strip
is heated by a large current, however, it
bends enough to open the switch and the stop
the current. Unlike a fuse, which cannot be
used after it burns out, a circuit breaker can
be reset when the bimetallic strip cools and
returns to its original shape.
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AC Circuits and Transformers
– A polarized plug provides protection by ensuring
that the case of an electrical appliance, which is
connected to the wide prong, is at low potential.
– Furthermore, when an electrical device with a
polarized plug is turned off, the high potential
extends only from the wall outlet to the switch,
leaving the rest of the device at zero potential.
AC Circuits and Transformers
• Polarized Plugs—The first line of defense against
accidental shock is the polarized plug (see figure (b)
below), on which one prong is wider than the other prong.
– The corresponding wall socket will accept the plug in
only one orientation, with the wide prong in the wide
receptacle. The narrow receptacle of the outlet is wired
to the high-potential side of the circuit; the wide
receptacle is connected to the low-potential side, which
is essentially at ground potential.
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AC Circuits and Transformers
• Grounded Plugs—The next line of defense against
accidental shock is the three-prong grounded plug shown
in figure (c) below.
– In this plug, the rounded third plug is connected directly
to ground when plugged into a three-prong receptacle.
In addition, the third prong is wired to the case of an
electrical appliance.
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AC Circuits and Transformers
– If something goes wrong within the appliance, and a
high-potential wire comes into contact with the case,
the resulting current flows through the third prong,
rather than through the body of a person who
happens to touch the case.
• GFCI Devices—A even greater level of protection is
provided by a device known as a ground fault circuit
interrupter (GFCI), shown in figure (d) below.
AC Circuits and Transformers
– The basic operating principle of an interrupter is
illustrated in the figure below.
– Notice that the wires carrying an AC current to the
protected appliance pass through a small iron ring.
When the appliance operates normally, the two wires
carry equal amounts of current in opposite directions—in
one wire the current goes to the appliance, and in the
other the current returns from the appliance.
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AC Circuits and Transformers
– Each of the wires produces a magnetic field,
but because their currents are in opposite
directions, the magnetic fields are in opposite
directions as well. As a result, the magnetic
fields of the two wires cancel.
– If a malfunction occurs in the appliance— say
a wire frays and contacts the case—current
that would ordinarily return through the power
cord may pass through the user's body
instead and into the ground.
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AC Circuits and Transformers
– In such a situation, the wire carrying current to
the appliance immediately produces a net
magnetic field within the iron ring that varies
with the frequency of the AC generator. The
changing magnetic field in the ring induces a
current in the sensing coil wrapped around
ring, and the induced current triggers a circuit
breaker in the interrupter. This cuts the flow of
current to the appliance within a millisecond,
protecting the user.
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AC Circuits and Transformers
AC Circuits and Transformers
• It is often useful to change the voltage from one
value to another in an electrical system. For
example, high-voltage power lines may operate
at voltages as high as 750,000 V, but before the
electric power can be used in homes it must be
stepped down (lowered) to 120 V. In other
situations voltages need to be stepped up.
• The electrical device that changes the voltage in
an AC circuit is called a transformer.
• A simple transformer is shown in the figure below.
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• Here an AC generator produces an alternating current in
the primary (p) circuit at the voltage Vp. The primary
circuit includes a coil with Np loops wrapped around an
iron core. The iron core intensifies and concentrates the
magnetic flux and ensures, at least to a good
approximation, that the secondary (s) coil experiences
the same magnetic flux as the primary coil.
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AC Circuits and Transformers
AC Circuits and Transformers
• The secondary coil has Ns loops around an iron core and
is part of a secondary circuit that may operate a CD
player, a lightbulb, or some other device.
• To relate the voltage of the primary and secondary
circuits, we apply Faraday's law of induction to each of
the coils. The result, after some straightforward algebra,
is the transformer equation:
• This equation relates the voltages and the number
of loops in the two circuits. Solving the transformer
equation for the voltage in the secondary circuit
yields
Vs = Vp(Ns/Np)
• The transformer equation shows that if the number
of loops in the secondary coil is less than the
number of loops in the primary coil, the voltage is
stepped down to a lower value. Similarly, if the
number of loops in the secondary coil is higher,
the voltage is stepped up to a higher value.
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AC Circuits and Transformers
AC Circuits and Transformers
• There is a tradeoff between voltage and current in
a transformer. This follows from the law of
conservation of energy.
• Because energy must always be conserved, the
average power in the primary circuit must be the
same as the average power in the secondary
circuit.
• Since power can be written as P = IV, it follows
that
IpVp = IsPs
• Isolating the currents Is and Ip on one side of the
equation and the voltages Vs and Vp on the other
side yields the following equation:
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AC Circuits and Transformers
AC Circuits and Transformers
• This version of the transformer equation shows
an important relationship, namely, if a
transformer increases the voltage by a given
factor, it decreases the current by the same
factor. Similarly, if it decreases the voltage, it
increases the current.
• For example, suppose the number of loops in a
secondary coil of a transformer is twice the
number of loops in the primary coil. This
transformer doubles the voltage in the
secondary circuit, Vs = 2Vp, and at the same
time halves the current, Is = Ip/2.
• The following example illustrates an application
of the transformer equation.
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AC Circuits and Transformers
AC Circuits and Transformers
• A transformer depends on a changing magnetic flux to
create an induced emf in the secondary coil. If the
current is constant—as in a DC circuit—there is no
induced emf, and the transformer ceases to function.
• This is an important advantage that AC circuits have
over DC circuits and one reason why most electrical
power systems operate with alternating current.
• Transformers play an important role in the transmission
of electrical energy from the power plants that produce it
to the communities and businesses where it is used.
• When electrical energy is transmitted over large
distances, the resistivity of the wires that carry the current
becomes significant. If a wire carries a current I and has a
resistance R, the power dissipated as heat is P = I 2R.
• One way to reduce this energy loss is to reduce the
current. A transformer that steps up the voltage of a
power plant by a factor of 20 will at the same time reduce
the current by a factor of 20, which reduces the power
dissipation by a factor of 202 = 400. When the electricity
reaches the location where it is to be used, step-down
transformers lower the voltage to a level such as 120 V
or 240 V.
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