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10/22/2016 Chapter 19 Lecture Chapter Contents Pearson Physics • Electric Charge • Electric Force • Combining Electric Forces Electric Charges and Forces © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 1 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 2 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 3 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 4 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 5 10/22/2016 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). © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 6 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 7 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 8 10/22/2016 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: © 2014 Pearson Education, Inc. • In this equation, the constant k has the following value: k = 8.99 x 109 N·m2/C2. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 9 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 10 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. • 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 © 2014 Pearson Education, Inc. 11 10/22/2016 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. © 2014 Pearson Education, Inc. Combining Electric Forces © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 12 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. Combining Electric Forces • The following example examines the force on a point charge exerted by a spherical charge distribution. © 2014 Pearson Education, Inc. 13 10/22/2016 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 © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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). © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 1 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 2 10/22/2016 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 © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 3 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 4 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 5 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 6 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 7 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 8 10/22/2016 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. © 2014 Pearson Education, Inc. 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)]. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 9 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 10 10/22/2016 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. © 2014 Pearson Education, Inc. Electric Potential Energy and Electric Potential © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 11 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: • © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 12 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 13 10/22/2016 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 14 10/22/2016 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. • © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 15 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 16 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 17 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 18 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 19 10/22/2016 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 © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 1 10/22/2016 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. © 2014 Pearson Education, Inc. Electric Current, Resistance, and Semiconductors © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 2 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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 (−). © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 3 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 4 10/22/2016 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 © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 5 10/22/2016 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! © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 6 10/22/2016 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 7 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 8 10/22/2016 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. © 2014 Pearson Education, Inc. • The basic property of a diode is that it allows current to flow in one direction, but not the other. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 9 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 10 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 11 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 12 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 13 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 14 10/22/2016 Electric Circuits Electric Circuits • The following example illustrates the functioning of a parallel circuit. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 15 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 16 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 17 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 18 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 19 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 20 10/22/2016 Power and Energy in Electric Circuits Power and Energy in Electric Circuits • The following example illustrates how the cost of electrical energy is calculated. © 2014 Pearson Education, Inc. 21 10/22/2016 Chapter 23 Lecture Chapter Contents Pearson Physics • Electricity from Magnetism • Electric Generators and Motors • AC Circuits and Transformers Electromagnetic Induction Prepared by Chris Chiaverina © 2014 Pearson Education, Inc. © 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 1 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 2 10/22/2016 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 3 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 4 10/22/2016 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 5 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 6 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. Electricity from Magnetism • The following example serves to illustrate Lenz's law. © 2014 Pearson Education, Inc. 7 10/22/2016 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. © 2014 Pearson Education, Inc. © 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 8 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 9 10/22/2016 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. © 2014 Pearson Education, Inc. 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 © 2014 Pearson Education, Inc. 10 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 11 10/22/2016 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. © 2014 Pearson Education, Inc. 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 © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 12 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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, © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 13 10/22/2016 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 © 2014 Pearson Education, Inc. © 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 14 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 15 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 16 10/22/2016 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. 17 10/22/2016 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. © 2014 Pearson Education, Inc. • 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. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 18 10/22/2016 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: © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 19 10/22/2016 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. © 2014 Pearson Education, Inc. © 2014 Pearson Education, Inc. 20