Electricity and Magnetism

UNIT E Electricity and Magnetism 364 Unit E

Unit Contents 11 12 13 The principles of conservation of energy and charge apply to electrical circuits. 11.1 Electrical Circuits 11.2 Series Circuits 11.3 Parallel and Mixed Circuits 11.4 Power Consumption Properties of magnetic fields apply in nature and technology. 12.1 Magnetic Forces and Fields 12.2 The Motor Principle 12.3 Using Electromagnetism Electromagnetic induction is used to generate most of the electrical energy used today. 13.1 Using Magnetism to Induce an Electric Current 13.2 The Generator and Electrical Energy Generation 13.3 The Transmission of Electrical Energy Unit Task The first electric motor was invented over 180 years ago. The design of the motor has vastly improved over the years. Today, most motors are inexpensive, reliable, and environmentally friendly. Using concepts learned in this unit, you will research, design, and build a small electric DC motor that can be powered using a 9-V battery. You will build your motor using components found at a hardware store. The function of your motor will be to lift as much weight as possible without stalling. DISCOVERING PHYSICS A city looks beautiful at night. The buildings and signs are lit up and create a wonderful mosaic of colours. Each light, whether it is in a building, on a sign, or a street light, is part of the city s electrical circuit. Electrical engineers help to design and plan the huge electrical circuit that provides electricity to the buildings and homes in a city. The electrical circuit must be able to expand as more houses and buildings are added. It must also be designed to provide electricity to different areas of the city so that a power outage in one area does not affect other areas in the city. Since the electrical needs of different parts of the city vary, the amount of electricity provided must be regulated through the different regions. What principles determine the way in which the electrical circuit of a city is designed? What devices are used to control the electricity as it flows through the city?

CHAPTER 11 The principles of conservation of energy and charge apply to electrical circuits. Learning Expectations By the end of this chapter, you will: Developing Skills of Investigation and Communication use appropriate terminology related to electricity, including: direct current, alternating current, conventional current, electron flow, electrical potential difference, electrical resistance, and power analyze diagrams of series, parallel, and mixed circuits with reference to Ohm s law and Kirchhoff s laws design and build real or computer-simulated mixed direct current (DC) circuits, and explain the circuits with reference to direct current, potential difference, and resistance solve problems involving energy, power, potential difference, and current Electricity is mysterious and fascinating. You cannot see it move, you cannot hold it in your hand, it is not even visible except when there is a spark although it is everywhere. Even though we depend on electricity, most people only think about it when the power goes out or when they get an electrical shock. We do not stop to think about all the electrical devices that make our lives easier. Imagine what your life would be like without TVs, cell phones, computers, MP3 players, cars, refrigerators, electric stoves and gas furnaces, medical imaging devices such as magnetic resonance imaging (MRI) and X-ray machines, and light bulbs (Figure 11.1). The number of electrical devices that surround us is staggering to contemplate. And yet, the first practical method of generating electricity was discovered only 200 years ago when Italian physicist Alessandro Volta (1745 1827) made the first battery. However, whether it is a cell phone or a car, all electrical devices contain many of the same components, and all obey the same physical laws. Understanding Basic Concepts distinguish between conventional current and electron flow distinguish between alternating current (AC) and direct current (DC) explain Ohm s law and Kirchhoff s laws in relation to electricity Figure 11.1 The Eiffel Tower looks very impressive when it is lit up at night. It has over 20 000 light bulbs, over 40 km of wires, and consumes over 120 000 W of power. 366 Unit E P

11.1 Electrical Circuits Section Summary An electrical circuit contains an energy source, conductors, and a load. Ohm s law describes the relationship between electric current, potential difference, and resistance in a circuit. In the late 1700s, Italian scientist Luigi Galvani (1737 1798) performed a dissection on a dead frog. He discovered that when he touched the leg muscles of the frog with two different metals, the muscles contracted. He concluded that these metals somehow released the animal electricity that was stored in the frog. Although Galvani s conclusion was wrong, his discovery turned out to be very important. Alessandro Volta recognized the implications of Galvani s discovery. He showed that, instead of the metals releasing electrical energy stored in the frog, the two metals produced electrical energy, which caused the legs to contract. He experimented with different metals to see how much continuous electrical energy he could create. He conducted his experiments by placing two different metals in contact and then touching them to his tongue, which could detect a small steady flow of electrical energy. After much experimentation, Volta placed a pile of copper and zinc discs on top of one another separated by cardboard discs soaked in a sulphuric acid solution. He attached wires to each end of the pile and, for the first time ever, was able to create a large, continuous flow of electricity. He had created a battery (Figure 11.2). A battery is a device that converts stored chemical potential energy into electrical energy and is capable of providing a steady flow of current electricity. Soon after the invention of the battery, scientists began experimenting with current electricity and circuits. Current electricity is the flow of charged particles along a conductor. A charged particle is a particle that has an electrical charge, such as a proton (positive) or an electron (negative). Circuit Fundamentals To operate, an electrical device requires electrical energy, which is provided by a steady flow of charged particles along a closed loop. This closed loop is called an electrical circuit. The circuit must form a closed loop so that the charged particles charges moving through the conductor can return to the battery. A circuit contains: a source of electrical energy, such as a battery a conductor, such as a wire a load that changes electrical energy into light, sound, heat, or motion As the charged particles travel through a circuit, they carry the electrical energy from the source (battery) to the load, which converts the electrical energy into another form of energy. Note that circuits can be very complex and can contain different types of loads and devices. To simplify drawing circuits, we use circuit symbols. discs zinc copper Figure 11.2 Volta created the first battery in 1800. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 367

Circuit symbols are used to represent the various components in drawings of electrical circuits (Figure 11.3). Table 11.1 shows the symbols of some of the most common components. energy source electrical load switch conducting wires Figure 11.3 A simple circuit containing a battery, a switch, and a load (the light bulb). Table 11.1 Circuit Symbols used in Circuit Diagrams Symbol Component Description wire conductor; provides a path for current flow battery variable DC power source provides electrical energy to the circuit; longer side is the positive terminal provides a variable amount of energy to a circuit ground switch electrical connection to ground that prevents a shock to a person opens or closes the circuit light bulb resistor ammeter type of load that converts electrical energy to light energy general load that converts electrical energy to another form of energy (heat) device that measures the current in a circuit voltmeter device that measures the potential difference in a circuit To understand what happens as a circuit operates, we need to take a closer look at the role of charges and the energy they transmit. Current Highway 401 in the Toronto area is the busiest highway in North America. If you were to stand beside the highway and count the cars as they passed, you would find that about 17 500 cars pass by every hour that is about 4.8 cars a second or over 420 000 cars a day! City planners find it useful to know the number of vehicles that use the roads to develop future roadways. Similarly when looking at electrical circuits, it is useful to know the quantity of charge that passes by a point in the circuit per second. The amount of charge transferred per unit time is referred to as current. 368 Unit E Electricity and Magnetism P

Conventional Current and Electron Flow When scientists first began studying circuits, they assumed that positive charges flowed through the wires in a circuit (Figure 11.4). They called the flow of positive charges in a circuit conventional current. Since unlike charges attract, conventional current is considered to flow from the positive terminal of a battery, around the circuit to the negative terminal. After scientists learned more about the structure of atoms, they concluded that current consists of free electrons, which are negatively charged. During the operation of a circuit, the electrons move from the negative terminal of the battery to the positive terminal. The movement of electrons in a circuit is called electron flow. For the purposes of circuit analysis, it makes no difference whether we talk about conventional current or electron flow. Throughout the rest of this unit, we will stay with the convention and use the term conventional current. Current is represented by the symbol I, and is measured in amperes (A). Since current is the amount of charge that is transferred per unit time, the equation for current is: + + + PHYSICS INSIGHT Conventional current assumes that positive charges (protons) move through the circuit from the positive terminal of the power supply to the negative terminal. This model is incorrect, but has become entrenched over many years. Electron flow assumes that electrons move through the circuit from the negative terminal to the positive terminal. + + + point + + + wire + + + I _ q t where I is the current in amperes (A), q is the amount of charge in coulombs (C), and t is the time in seconds (s). Direct Current (DC) A battery provides a steady flow of current in one direction, known as direct current, or DC. Strictly speaking, DC can fluctuate, but cannot change directions. DC is used in all electrical equipment that requires an adapter when you plug it into a wall outlet or is powered by a battery. Alternating Current (AC) A wall socket provides alternating current. Alternating current, or AC, changes direction periodically. That is, the charges in the wire move back and forth over the same spot and do not actually move from one terminal to another. This type of current is used in the wiring of your house. You will study the reasons why AC is used in chapter 13. charges Figure 11.4 In conventional current, positive charges move from the positive terminal to the negative terminal. PHYSICS INSIGHT A coulomb is equivalent to the charge on 6.25 10 18 electrons or protons. An ampere is a flow of 1 C of charge past a point in a conductor in 1 s. Example 11.1 A battery delivers a charge of 9.00 C in 1.00 min of operation. What amount of current is generated in ma? The SI prefix milli- (m) is equal to 10 3. Given q 9.00 C t 1.00 min 60.0 s Required current (I) Practice Problems 1. A D-cell battery delivers a charge of 200.0 C in 65.0 s. Determine the current produced by this battery. 2. A car battery provides a current of 600.0 A for 2.48 s. Determine the charge provided by the battery. 3. A battery has a total charge capacity of 10 800 C. For how long can this battery deliver a current of 450 ma? P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 369

Answers 1. 3.08 A 2. 1.49 10 3 C 3. 2.4 10 4 s Analysis and Solution I _ q t 9.00 C 60.0 s 0.150 C/s 0.150 A Paraphrase The current delivered by this battery is 0.150 A, or 150 ma. Figure 11.5 An ammeter in a circuit Measuring Current A device called an ammeter is used to measure current. Figure 11.5 shows the proper placement of an ammeter in a circuit. The ammeter is placed into the circuit so that the current flows through it. Since the current cannot bypass the meter, the meter measures the entire current. Figure 11.6 Resistors come in many shapes and sizes to meet different requirements. Resistance Electrical charges move through a circuit with little or no room between them. If there is something that restricts the flow of current in one spot in the circuit, the effect is felt throughout the entire circuit. Resistance is the degree to which the flow of current is opposed in a circuit. A resistor is a device that resists or restricts the flow of current (Figure 11.6). As charges move through a resistor, the resistor removes energy from the charges and converts it to another type of energy. The energy usually takes the form of heat. For example, a light bulb is a resistor that converts the energy of the charges into heat and light. Almost all components offer some resistance in a circuit, even if that is not their primary role. We will assume that the resistance of a component is constant. Suggested Activity The Battery and Potential Difference E1 Quick Lab Overview on page 374 For a circuit to do anything useful, current must flow through it. But current does not move by itself there must be a battery that forces the electrons through the circuit and provides energy to the components. To understand fully how circuits work, we need to take a closer look at the role of the battery. As the charges pass through a battery, it increases their potential energy. We can think of a battery as being similar to water pump a water pump. The pump increases the potential energy of water by lifting it to a certain height (Figure 11.7). The potential energy of the water depends on the mass of the water and the height to which it is lifted. Therefore it is useful to define a reference point where the potential energy is zero, which is at the bottom of the pump. The maximum potential energy is at the top of the pump. For electrical potential energy we will water reservoir take the negative terminal of the battery as the point of zero Figure 11.7 At the bottom of the water pump, the potential energy. potential energy of the water is defined as zero. 370 Unit E Electricity and Magnetism P

Electrical Potential In an electrical circuit, it is not practical to refer to the potential energy of the charges as they move through the circuit. This is because potential energy depends on the quantity of charge, which changes as the circuit operates. For example, an electric motor may use more charge per second under a heavy load than when it has no load. It is more practical to use a measurement that is independent of the amount of charge flowing in the circuit. For this reason, we use the term electrical potential. We define electrical potential as the electrical potential energy per unit charge. Electrical potential is represented by the symbol V, and its units are volts (V). It can be written mathematically as: V E_ q PHYSICS INSIGHT The symbol for a battery is shown below. The longer line is the positive terminal and the shorter line is the negative terminal. where V is the electrical potential in volts (V), E is the electrical potential energy in joules (J), and q is the charge in coulombs (C). Note that the electrical potential is the same whether there are many charges (large current) or few charges (small current) flowing through the circuit. Potential Difference As charges pass through a load in the circuit, they transfer energy to the load. The charges have a greater electrical potential before they pass through the load than after they pass through the load. This change in potential is referred to as the potential difference ( V). Potential difference is measured in volts (V). Potential difference is also referred to as voltage. The potential difference is always measured between two points in the circuit (Figure 11.8). We can determine the change in potential by subtracting the initial potential from the final potential: V V final V initial V _ ( E final q V _ E q ) ( E initial _ q ) where V is the potential difference in volts (V), E is the change in potential energy of the charges as they pass through a load in joules (J), and q is the charge in coulombs (C). A load within a circuit uses energy and decreases the potential. This creates a negative potential difference, which is also called a voltage drop, across the component. A voltage drop implies a loss of energy so a negative sign is not usually used. The combination of all the voltage drops in a circuit will decrease the potential by exactly the same amount as the battery increases the potential. Measuring Potential Difference To measure potential difference, we use a voltmeter. The voltmeter must be placed across two points in the circuit and it will measure the voltage drop across that portion of the circuit. Figure 11.9 shows the proper placement of a voltmeter in a circuit. light bulb light V initial V final V i V f Figure 11.8 The potential of the charges before they pass through a light bulb is greater than after they pass through the light bulb. Some of the energy is converted to light. PHYSICS INSIGHT Do not confuse voltage with volts. Voltage is potential difference and has the symbol ΔV. The volt is the unit for potential difference and has the symbol V. Figure 11.9 A voltmeter is placed across a component. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 371

Example 11.2 Practice Problems 1. A potential difference of 120.0 V is measured across a light bulb. The light bulb is left on for 30 min allowing a charge of 900 C to flow through it. How much energy is converted to light and heat? 2. A charge of 50.0 C has a change in potential energy of 1.00 10 3 J as it flows through a resistor. What is the potential difference across the resistor? Answers 1. 1.08 10 5 J 2. 20.0 V A potential difference of 10.0 V is measured across a resistor in a circuit. If a charge of 20.0 C passes through the resistor, how much electrical energy is dissipated as heat? Given V 10.0 V q 20.0 C Required change in potential energy ( E) Analysis and Solution V _ E q E q V (20.0 C)(10.0 V) 200 CV 200 J Paraphrase The amount of electrical energy converted to heat is 200 J. Suggested Activities E2 Skill Builder Overview on page 374 E3 Inquiry Activity Overview on page 374 Concept Check 1. Explain what current is. 2. Explain the difference between how an ammeter is connected in a circuit with how a voltmeter is connected. 3. What is the difference between direct current and alternating current? Explore More What are the effects of Ohm s law in a simple circuit? Ohm s Law Twenty-seven years after the battery was invented, German scientist Georg Ohm (1787 1854) determined the relationship between potential difference, current, and resistance. During his experiments, Ohm applied different voltages to a resistor and measured the corresponding current and voltage drop across it. Ohm recorded and made a graph of his data (Figure 11.10). Potential Difference vs. Current Figure 11.10 Ohm showed a linear relationship between the potential difference and current. PHYSICS INSIGHT It is common to write Ohm s equation as V IR. However, this equation is not strictly correct because V represents electrical potential. Potential Difference (V) Current (A) The slope of this line is the resistance. Because the relationship between potential difference and current is linear, the resistance does not change. Ohm wrote this relationship mathematically as: R V_ I 372 Unit E Electricity and Magnetism P

where R is the resistance in ohms ( ), V is the potential difference in volts (V), and I is the current in amperes (A). This mathematical equation is known as Ohm s law. Ohm s law is often rewritten as: V IR Note that not all components obey Ohm s law. For example, light-emitting diodes (LEDs), diodes, transistors, and fluorescent lights do not obey Ohm s law. In this unit, we will assume that all circuit components obey Ohm s law. Take It Further Circuits are designed with different kinds of components. Research five components. Be prepared to show the symbol of each component and provide a brief description of its purpose. Example 11.3 A student is asked to determine the value of the resistor in a circuit (Figure 11.11). Table 11.2 shows the data obtained when the current was varied. Table 11.2 Results of Experiment Current (A) 0.00 0.0 1.00 70.0 2.00 155.0 4.00 310.0 4.50 327.5 Potential Difference (V) Given values of potential difference (see Table 11.2) values of current (see Table 11.2) Required the resistance (R) Analysis and Solution The current is the independent variable and the potential difference is the dependent variable. Figure 11.12 shows a graph of the data. The slope of the line of best fit is the resistance. slope _ y V_ x I R Choose two points from the line of best fit. R 262 V 37 V 3.5 A 0.5 A 225 V 3.0 A 75 Potential Difference (V) variable power supply Figure 11.12 Paraphrase The resistor has a resistance of 75. Figure 11.11 ammeter resistor voltmeter Potential Difference vs. Current 400 350 300 250 200 150 100 50 0 0 1 2 3 4 5 Current (A) Practice Problems 1. Determine the resistance of a circuit using the following data. Current (A) 0.00 0.0 1.00 58.0 2.00 108.0 4.00 220.0 4.50 245.0 Potential Difference (V) 2. A student performs an experiment similar to Example 11.3. The student determines the resistor to have a resistance of 130. Copy the following table into your notebook and fill in the missing values. Current (A) 0.00 0.0 1.50 4.00 Potential Difference (V) 292.5 780.0 Answers 1. 55 2. Current: 2.25 A and 6.00 A Potential difference: 195.0 V and 520.0 V P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 373

E1 Quick Lab Creating a Pile Battery Purpose To create a battery using copper and zinc metals Activity Overview In this Quick Lab, you will build a battery using a lemon and two metals. Your teacher will give you a copy of the full activity. Prelab Questions Consider the questions below before beginning this activity. 1. Can an electrical potential be created between two dissimilar metals? Figure 11.13 Activity setup 2. How does the electrical potential difference change when two batteries are connected together? E2 Skill Builder Activity Using an Ammeter and a Voltmeter Activity Overview In this Skill Builder, you will learn how to set up and use an ammeter, which measures current, and a voltmeter, which measures potential difference. The ammeter is connected in line with the resistor. The voltmeter is connected so that the voltmeter s terminals are on either side of the resistor. Your teacher will give you a copy of the full activity. Figure 11.14 Proper circuit setup of an ammeter and a voltmeter DI Key Activity E3 Inquiry Activity Investigating Ohm s Law Question What is the relationship between potential difference, current, and resistance in a simple circuit? 0 10 V power supply ammeter resistor Activity Overview REQUIRED SKILLS Recording and organizing data Drawing conclusions In this activity, you will build a circuit with a resistor and a power source. You will then connect an ammeter and a voltmeter, and measure the current as you increase the voltage. Your teacher will give you a copy of the full activity. Prelab Questions Consider the questions below before beginning this activity. 1. What relationship exists between the voltage, current, and resistance in a simple circuit? Figure 11.15 Circuit for activity voltmeter 2. How is the resistance of a circuit affected when the current through the circuit is changed? 374 Unit E Electricity and Magnetism P

11.1 Check and Reflect Key Concept Review 1. What conditions must be met for an electrical circuit to operate? 2. What advantage did the invention of the battery bring to the study of electricity? 3. Explain the meaning of the symbols I, q, V, R, and t. 4. How does conventional current differ from electron flow? 5. What is DC and how is it different from AC? 6. Explain the term resistance. 7. Explain how a light bulb and a resistor are (a) the same and (b) different. 8. Why is the measurement of electrical potential more versatile than electrical potential energy? 9. Explain how two batteries can have the same potential, but different potential energy. Connect Your Understanding 10. Determine the current generated in a circuit if 18.0 C of charge flow per minute. 11. What amount of charge is stored in a D-cell alkaline battery if it provides a current of 450 ma for 45.6 h? 12. The capacity of a 9-V rechargeable battery is 625 ma h. What is the battery s charge capacity in coulombs? 13. The potential difference across an electric motor is 150.0 V. The potential energy of 600.0 J is converted into kinetic energy. Determine the amount of charge that flowed through the motor. 14. A rechargeable nickel-metal hydride AA battery has a charge capacity of 9000 C. How long can it provide a current of 0.500 A? 15. A light bulb has a potential difference of 120.0 V. If 4.5 10 3 C of charge pass through it, how much energy is converted into light and heat? 16. A resistor generates 1800.0 J of heat when 200.0 C of charge pass through it. What is the potential difference across the resistor? 17. A small electric motor requires a potential difference of 9.00 V and draws a current of 800.0 ma. If it runs for 55.0 s, how much energy does it consume? 18. A student performs a lab where the voltage drop across a resistor is measured as a function of current. Plot a graph from the values given in the following table, and determine the resistance of the circuit. Experimental Data Current, I (ma) 0.00 0.00 200 9.00 400 16.0 600 24.0 800 31.0 1000 40.0 Potential Difference, V (V) 19. A potential difference of 2.80 V exists across a light-emitting diode (LED). If the LED consumes 42.0 J of energy, what amount of charge flowed through it? 20. A set of decorative outdoor lights is left on for 12.0 h. The lights draw a current of 2.00 A, and run on 120.0 V. How much energy will they consume in this time? Reflection 21. What concept in this section did you find most difficult to understand? Why? For more questions, go to P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 375

11.2 Series Circuits Section Summary Kirchhoff s voltage law is the law of conservation of energy for circuits. In a series circuit, the current is constant, the total resistance is the sum of the resistors, and the total voltage of the battery equals the sum of the voltage drops across the resistors. During the dark winter months, many trees in parks and in front of buildings are decorated with strings of decorative lights (Figure 11.16). Sometimes, an entire string of lights will go dark when one light bulb burns out. This happens because the light bulbs have been joined together so that the current flows along one path. Placing all the components in a circuit along one path is called placing them in series. A series circuit has only one path for the charges to follow. Figure 11.17 shows a series circuit. Figure 11.16 Decorative lights are sometimes wired in series. Figure 11.17 Three light bulbs wired in series. Suggested Activity E4 Inquiry Activity Overview on page 379 Current in a Series Circuit In a series circuit, there is only one path for the current to follow. Therefore, the current in all parts of the circuit will be the same. Mathematically this is expressed as: I T I 1 I n 50.0 V I T 2.5 A R 1 5.00 15.0 Figure 11.18 The total resistance of this circuit can be found by adding the resistors. where I T is the total current in the series circuit and n is the last resistor in the circuit. Remember that since a circuit forms a closed loop, the number of charges flowing through the circuit never changes. A break in the path blocks the current throughout the entire circuit. Resistance in a Series Circuit Figure 11.18 shows two resistors in series. Recall that a resistor is a device that restricts the flow of current. Putting two resistors in series with one another further restricts the current flow. 376 Unit E Electricity and Magnetism P

The total resistance in a series circuit is the sum of the resistors. Mathematically this is expressed as: R T R 1... R n where R T is the total resistance in the series circuit and n is the last resistor in the circuit. Using this equation, we can determine the total resistance for the circuit in Figure 11.18: Explore More How does increasing the number of resistors affect the resistance and current of a series circuit? R T R 1 R 2 5.00 15.0 20.0 A Short Circuit in a Series Circuit What would happen to the current if there were no load, or resistance, in a series circuit? Assume that we remove the resistors in Figure 11.18. We can use Ohm s law to calculate the current in the circuit: V IR I V_ R 50.0 V 0 A Ohm s law suggests that the current would be infinite. Practically this cannot happen. Either the power supply/battery will burn out or the wire will heat up and burn out. If this wire were in the wall of a house, a fire could start. Modern houses have circuit breakers in the electrical panel that will turn off the electricity to the circuit when the current gets too large, preventing a fire. A circuit with no load is called a short circuit. Figure 11.19 shows an example of a short circuit in a series circuit. Figure 11.19 This is a short circuit because there is no load. Potential Difference in a Series Circuit What is the potential difference across each resistor in series? In a water pump with two water wheels, the pump lifts the water to a certain height and then the water falls from that height through the wheels (Figure 11.20(a)). Similarly, in a series circuit, the battery increases the potential of the charges by a certain amount, and the components in the circuit must reduce the potential by the same amount (Figure 11.20(b)). (a) water pump water wheel (resistor 1) water wheel (resistor 2) (b) 50.0 V I T 2.5 A R 1 5.00 15.0 water reservoir Figure 11.20 (a) A water pump increases the potential of the water. (b) A battery increases the potential of the charges while the resistors in the circuit decrease the potential of the charges. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 377

The reason for this has to do with the law of conservation of energy. In 1845, German physicist Gustav Kirchhoff (1824 1887) recognized that, in any closed circuit loop, the sum of the potential differences through all the components must be zero. This is referred to as Kirchhoff s voltage law. Kirchhoff s voltage law is written mathematically as: 0 V 1... V n For example, in Figure 11.20(b), the sum of the voltage drops across both resistors should equal the sum of the voltage increases. We will refer to the potential difference caused by the battery as V T. Note that V T is the total voltage increase in the circuit. We can modify Kirchhoff s voltage law slightly and write it mathematically as: V 1... V n where V T is the potential difference provided by the battery. We can use Ohm s law to calculate V 1 and V 2. V 1 I 1 R 1 V 2 I 2 R 2 I T I 1 I 2 2.5 A V 1 (2.5 A)(5.00 ) V 2 (2.5 A)(15.0 ) 12.5 V 37.5 V The voltage drops across R 1 and R 2 are 12.5 V and 37.5 V, respectively. We can check that V 1 and V 2 are correct using Kirchhoff s voltage law. V 1 V 2 12.5 V 37.5 V 50.0 V 200 V Figure 11.21 Question 3 Concept Check 1. Draw a circuit diagram showing a battery and four light bulbs in series. 2. Explain what happens to the current and potential difference in a series circuit. 3. Draw a diagram to represent Figure 11.21 using the water/water pump analogy. Example 11.4 Practice Problems 1. Determine the current, total resistance, and voltage drops in a series circuit in which 10.0 V, R 1 4.0, 10.0, and R 3 6.0. 2. Determine the current, total resistance, and voltage drops in a series circuit in which 12.0 V, R 1 5.0, 15.0, and R 3 100. Determine the current, total resistance, and voltage drops through all the resistors in the circuit shown in Figure 11.22. Given 200 V R 1 20.0 50.0 R 3 30.0 Required current (I) total resistance (R T) voltage drop through all resistors ( V 1, V 2, V 3) Analysis and Solution First, we calculate the total resistance. 200 V Figure 11.22 R 1 20.0 R 3 30.0 50.0 R T R 1 R 3 20.0 50.0 30.0 100 378 Unit E Electricity and Magnetism P

We can calculate the current using Ohm s law. IR T I _ V T R T 200 V 100 2.00 A We can now calculate the voltage drops using Ohm s law. Answers 1. R T 20.0 I T 0.500 A V 1 2.0 V V 2 5.0 V V 3 3.0 V 2. R T 120.0 I T 0.100 A V 1 0.500 V V 2 1.50 V V 3 10.0 V V 1 IR 1 (2.00 A)(20.0 ) 40.0 V V 2 IR 2 (2.00 A)(50.0 ) 100 V V 3 IR 3 (2.00 A)(30.0 ) 60.0 V Paraphrase The total resistance is 100, the current is 2.00 A, and the voltage drops across R 1, R 2, and R 3 are 40.0 V, 100 V, and 60.0 V, respectively. Series Circuit Summary This section introduced three equations that can be used to determine the current, resistance, and potential difference in a series circuit (Table 11.3). Table 11.3 Series Circuit Equations Current I T I 1 I n Current remains the same throughout the entire circuit. Resistance R T R 1... R n The total resistance is the sum of all the resistances in the circuit. Potential Difference V 1... V n The sum of the voltage drops through the circuit is equal to the voltage increase provided by the battery. Take It Further To see how the number of light bulbs in a circuit affects the brightness of the bulbs, use circuit simulation software to create two circuits with a different number of identical light bulbs. Be prepared to present your findings by showing the circuit, the current, and the voltage drop across each light bulb. REQUIRED SKILLS E4 Inquiry Activity Using appropriate equipment and tools Reporting results Measuring Current and Potential Difference in a Series Circuit Question What are the current and voltage drops across the resistors in a series circuit? Activity Overview In this activity, you will measure the current and the potential difference in a series circuit. You will need to correctly connect an ammeter and a voltmeter. Your teacher will give you a copy of the full activity. R 1 R 3 Figure 11.23 Series circuit for activity R 2 Prelab Questions Consider the questions below before beginning this activity. 1. What is the current at different positions in a series circuit? 2. Use Kirchhoff s voltage law to predict what happens to the voltage drops across the components in a closed circuit. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 379

11.2 Check and Reflect Key Concept Review 1. What is a series circuit? 2. Draw a circuit diagram that shows four resistors in series. 3. Explain how Kirchhoff s voltage law is the same as the law of conservation of energy. 4. What happens to the current when there is a short circuit? 5. Explain what causes a short circuit. Connect Your Understanding 6. Explain what happens to a series circuit if one component in the circuit breaks. 7. A series circuit contains three resistors: R 1 is 12.0, R 2 is 18.0, and R 3 is 45.0. A battery provides a potential difference of 100.0 V. (a) What is the total resistance of the circuit? (b) What current flows through the circuit? (c) What is the voltage drop across each resistor? 8. Determine the potential difference across the battery and the three light bulbs shown in the following circuit diagram.? Question 8 I 5.00 A R 1 50.0 9. Determine the value of the third resistor shown in the following circuit diagram. R 1 80.0 R 3 200 100 10. Two resistors are in series. Determine the resistance (in k ) provided by R 1 if 20.0 V, I 5.00 ma, and 1.00 k. The SI prefix kilo- (k) is equal to 10 3. 11. All of the light bulbs shown in the circuits below are identical and have the same resistance. (a) Explain which circuit would have a larger current. (b) In which circuit would the light bulbs glow brightest? Explain your answer. 10 V 10 V Question 11 12. A series circuit is sometimes compared with a closed water pipe. The components of a circuit are similar to components placed along the water pipe, and electrical potential is similar to the pressure in the water pipe caused by a water pump. Explain how this analogy might make sense in terms of potential difference, current, and resistance. Reflection 13. What did you learn about series circuits that you did not know? For more questions, go to 40.0 V 90.0 I 200 ma R 3? Question 9 380 Unit E Electricity and Magnetism P

11.3 Parallel and Mixed Circuits Section Summary Kirchhoff s current law states that the current entering a junction must be equal to the current leaving the junction. In a parallel circuit, the total current equals the sum of the currents through each branch, the total resistance decreases as the number of branches increases, and the voltage is constant across each branch. junctions Studying a series circuit helps to develop an understanding of the fundamentals of electrical circuits. However, in practice, a series circuit has a drawback if one component malfunctions or a wire breaks, the entire circuit stops working. This is the electrical equivalent of a city that has only one road that passes by every house and business. If a traffic jam occurs anywhere along the road, all traffic will stop. The solution is to have many roads that are connected to each other, which allows traffic to branch out in many directions. Electrical circuits are designed in a similar fashion. A parallel circuit is a closed circuit in which the current has more than one path, or branch, to follow (Figure 11.24). The point at which the path splits is called a junction. Current in a Parallel Circuit In a parallel circuit, there are two or more branches for the current to follow. Look at the parallel circuit shown in Figure 11.25. If we assume the flow of conventional current, charges exit the positive terminal of the battery and move through the circuit to junction A. At junction A, the current splits into two paths and recombines at junction B as it flows back to the battery. Gustav Kirchhoff recognized that electrical charge is conserved in any closed electrical circuit. This is because of the law of conservation of charge, which states that the total electrical charge of a closed system remains constant. Kirchhoff was able to describe the law of conservation of charge in terms of current. This is known as Kirchhoff s current law, which states that the current entering junction A water pump must be equal to the current leaving the junction. The same is true at junction B. Kirchhoff s current law is written mathematically as: Figure 11.24 Three light bulbs wired in parallel. The dots in the circuit diagram represent the four junctions. I T I T Figure 11.25 In this parallel circuit, the current has two paths to follow. A B R 1 R 2 water wheel I T I 1... I n where 1 through n are the branches of the circuit. A parallel circuit is similar to a water pump with two water wheels (Figure 11.26). The water splits into two paths and recombines in the reservoir at the bottom. water reservoir Figure 11.26 A water pump with two parallel water wheels. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 381

Resistance in a Parallel Circuit Figure 11.25 on the previous page shows a parallel circuit with two resistors. As the number of resistors in parallel increases, the total circuit resistance decreases. In fact, the total circuit resistance is always less than the lowest resistance found in the branches. The resistance of a parallel circuit is determined by the following equation: 1_ 1_... 1_ R T R 1 R n where R T is the total resistance in the parallel circuit and n is the last resistor in the circuit. Figure 11.27 A short circuit in a parallel circuit Suggested Activity E5 Inquiry Activity Overview on page 388 A Short Circuit in a Parallel Circuit Figure 11.27 shows a short circuit in a parallel circuit. This is an example of a short circuit because the current does not split equally. There is a path for the current to follow that bypasses the resistor. Potential Difference in a Parallel Circuit Recall that Kirchhoff s voltage law states that in any closed circuit loop, the sum of the potential difference (voltage drop) through the resistors must equal the voltage gain across the battery. However, in a parallel circuit, the potential difference through each load is the same and is equal to the voltage increase of the battery. Mathematically this is expressed as: V 1 V n where V T is the potential difference provided by the battery. Example 11.5 Practice Problems 1. Determine the total resistance, total current, and the currents through each branch of the following circuit. 90.0 V Figure 11.29 2. Determine the total resistance, total current, and the currents through each branch of the following circuit. 120 V Figure 11.30 R 1 45.0 R 1 5.00 90.0 10.0 Answers 1. R T 30.0, I T 3.00 A I 1 2.00 A I 2 1.00 A 2. R T 2.73 Ω I T 44.0 A I 1 24.0 A I 2 12.0 A I 3 8.00 A R 3 15.0 Analyze the parallel circuit shown in Figure 11.28 to determine the total resistance, total current, and current through each branch. Given R 1 400 100 240 V Required total resistance (R T ) current through branch 1 (I 1 ) total current (I T ) current through branch 2 (I 2 ) Analysis and Solution We need to determine the total resistance to find the total current. 1_ 1_ 1_ R T R 1 R 2 1_ 1 R T 400 1 100 1_ 0.0125 R 1 T R T 80.0 240 V Figure 11.28 I T R T I T _ V T R T 240 V 80.0 3.00 A R 1 400 100 382 Unit E Electricity and Magnetism P

We can use Ohm s law to find the current through each branch. Remember, the potential difference through each branch is the same and equal to the voltage provided by the battery. V 1 V T V 2 V T V 1 I 1 R 1 I 1 _ V 1 R 1 240 V 400 0.600 A V 2 I 2 R 2 I 2 _ V 2 R 2 240 V 100 2.40 A We can check that I 1 and I 2 are correct using Kirchhoff s current law. I T I 1 I 2 0.600 A 2.40 A 3.00 A Paraphrase The total resistance of the circuit is 80, the total current is 3.00 A, the current through branch 1 is 0.600 A, and the current through branch 2 is 2.40 A. Parallel Circuit Summary This section introduced three equations that can be used to determine the current, resistance, and potential difference in a parallel circuit. Table 11.4 summarizes these equations. Table 11.4 Parallel Circuit Equations Current I T I 1... I n The sum of the currents through each branch must equal the total current of the circuit. Resistance 1_ R T 1_ R 1... 1_ R n The total resistance of the circuit decreases as the number of branches increases. Potential Difference V 1 V n The voltage drops across the loads are the same and are equal to the voltage increase of the battery. Concept Check 1. Draw a circuit diagram showing a battery and four light bulbs in parallel. 2. Explain what happens to the current and potential difference in a parallel circuit. 3. Describe two situations in which a parallel circuit would be preferable to a series circuit. Mixed Circuits Almost all electric devices contain a combination of series and parallel circuits. These circuits are called mixed circuits. For example, the circuits in a car, computer, house, and cell phone are mixed circuits (Figure 11.31). Figure 11.31 A cell phone circuit board contains many components that are wired in series and parallel. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 383

Simplifying a Mixed Circuit 72.0 V R 1 24.0 20.0 R 3 30.0 We can use Ohm s law to analyze a mixed circuit. However, because a mixed circuit contains a combination of series and parallel circuits, it can be challenging to determine the resistance, current, and potential difference. The best approach is to look at the parallel part of the circuit separately from the series part of the circuit. We can also reduce the complexity of the circuit by, wherever possible, replacing several resistors with an equivalent resistor. For example, we can analyze the mixed circuit shown in Figure 11.32. We begin by simplifying the circuit. We can calculate the combined resistance of R 2 and R 3, and substitute an equivalent resistor (R ) in their place. Figure 11.32 In this mixed circuit, R 1 is in series with R 2 and R 3, which are in parallel with each other. 72.0 V R 1 24.0 Figure 11.33 Replacing the two parallel resistors with the equivalent resistor R makes this a series circuit. R 12.0 1_ 1_ 1_ R R 2 R 3 1_ 1 R 20.0 1 30.0 1_ 0.083 R 1 R 12.0 We can redraw the circuit to show that the two parallel resistors have an effective resistance of 12.0. We can use the equivalent resistor R to represent the two parallel resistors in our circuit diagram (Figure 11.33). Since the circuit is now a series circuit, we can calculate the total resistance of the circuit using the sum of the resistors: R T R 1 R 24.0 12.0 36.0 We can use Ohm s law to calculate the total current: I T _ V T R T 72.0 V 36.0 2.00 A We can now use Ohm s law to determine the voltage drop across R 1 and R : V 1 I T R 1 (2.00 A)(24.0 ) 48.0 V V I T R (2.00 A)(12.0 ) 24.0 V R 1 24.0 V 1 48.0 V 20.0 R 3 30.0 72.0 V V 2 24.0 V V 3 24.0 V I T 2.00 A Figure 11.34 The voltage drop across each resistor is known. Since R is really the equivalent resistor for R 2 and R 3, which are in parallel, the voltage drop across each of the parallel branches must be 24.0 V. The original circuit can be redrawn with the voltage drops written beside all the resistors (Figure 11.34). To determine the current through each branch of the mixed circuit, we use the voltage drop through each branch. I 2 _ V 2 I R 3 _ V 3 2 24.0 V 20.0 1.20 A R 3 24.0 V 30.0 0.80 A 384 Unit E Electricity and Magnetism P

Table 11.5 outlines the steps you can use to analyze mixed circuits when the resistances are known. Table 11.5 Simplifying a Mixed Circuit Step Step 1 Procedure Reduce the circuit to a simple series circuit by using equivalent resistors. Explore More How does increasing the number of resistors affect the resistance and current of a mixed circuit? Step 2 Step 3 Step 4 Step 5 Determine the total resistance and total current of the series circuit using the equation for R T for series circuits and Ohm s law. Determine the voltage drop across each resistor in the circuit using Ohm s law. Redraw the original circuit with the voltage drops beside each resistor. Remember that the voltage drop across the parallel resistors will be the same. Determine the current through the parallel resistors using Ohm s law. Example 11.6 Analyze the circuit diagram shown in Figure 11.35 to solve for the current and potential difference through each resistor. Given R 1 400 R 3 1800 R 5 600 500 R 4 1200 200 V 200 V R 5 600 R 1 400 500 R 3 1800 R 4 1200 Required current and voltage drop through each resistor Figure 11.35 Analysis and Solution This mixed circuit must be simplified to a series circuit so that we can solve for total resistance and total current. The first step is to combine R 1 and R 2 into an equivalent resistor called R 1 2. Since these two resistors are in series, the total resistance is the sum of the resistors. We can redraw the circuit diagram to look like Figure 11.36. 200 V R 5 600 R 1 2 900 R 3 1800 R 4 1200 R 1 2 R 1 R 2 400 500 900 Figure 11.36 The next step is to combine the three resistors in parallel into one equivalent resistor called R : 1_ 1 R 900 1 1800 1 1200 1_ 0.00250 R 1 R 400 We can redraw the circuit diagram (Figure 11.37) and determine R T : R T R R 5 400 600 1.00 10 3 We can determine the total current of the circuit using Ohm s law: I T R T I T _ V T R T 200 V 1.00 10 3 0.200 A 200 V Figure 11.37 R 5 600 R 400 P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 385

Practice Problems 1. Determine the current and potential difference through each resistor in Figure 11.38. 10.0 V R 5 75.0 Figure 11.38 2. Determine the current and potential difference through each resistor in Figure 11.39. 120 V Figure 11.39 R 3 4.50 R 4 8.00 R 1 10.0 30.0 R 1 30.0 R 3 100 10.0 R 4 200 Answers 1. I 1 0.0625 A, I 2 0.0625 A, I 3 0.0250 A, I 4 0.0125 A, I 5 0.100 A, V 1 0.625 V, V 2 1.88 V, V 3 2.50 V, V 4 2.50 V, and V 5 7.50 V 2. I 1 1.50 A, I 2 4.50 A, I 3 6.00 A, I 4 6.00 A, V 1 45.0 V, V 2 45.0 V, V 3 27.0 V, and V 4 48.0 V Now we can determine the voltage drop across R and R 5. Remember, the voltage drop across R is the voltage drop across the three parallel branches in the original circuit. V I T R V 5 I T R 5 (0.200 A)(400 ) (0.200 A)(600 ) 80.0 V 120 V We can use the potential difference to determine the current through each branch: V 1 I 1 2 R 1 2 V 3 I 3 R 3 V 4 I 4 R 4 I 1 2 _ V 1 2 R 1 2 80.0 V 900 0.08889 A I 3 _ V 3 R 3 80.0 V 1800 0.04444 A I 4 _ V 4 R 4 80.0 V 1200 0.06667 A Note that, since R 1 is in series with R 2, I 1 2 I 1 I 2. The final step is to determine the voltage drop across the resistors in branch one. The current through branch one is 0.0889 A, and the voltage drop across the entire branch is 80.0 V. Resistors 1 and 2 are in series with each other in this branch, so the sum of their voltage drops must equal 80.0 V. We can determine the voltage drop across the resistors using Ohm s law: V 1 I 1 R 1 V 2 I 2 R 2 (0.08889 A)(400 ) (0.08889 A)(500 ) 35.6 V 44.4 V Paraphrase We can summarize our results in the following table: R 1 R 2 R 3 R 4 R 5 Suggested Activity E6 Inquiry Activity Overview on page 388 30 V R 1 R 2 R 3 R 4 Figure 11.40 R 1 is wired in series with the battery and the light bulbs, which are in parallel with each other. ΔV 35.6 V 44.4 V 80.0 V 80.0 V 120 V I 0.0889 A 0.0889 A 0.0444 A 0.0667 A 0.200 A Application of a Mixed Circuit Design The steps that you take to analyze a mixed circuit depend on the information that you are given and what you need to determine. For example, imagine we have three identical light bulbs. Each one operates with 10 V and has a resistance of 60. The light bulbs are to be wired together so that if one bulb burns out, the other two light bulbs remain working. We also have a 30-V battery and some resistors. If we place the light bulbs in parallel, there will be a potential difference of 30 V across each light bulb, which will cause the bulbs to burn out. One solution is to use a resistor in series with the battery to drop the voltage to 10 V (Figure 11.40). To determine the value of this resistor, we first calculate the resistance of the parallel portion of the circuit only. Since we know the resistance of each bulb, we will use the equation to calculate the resistance of a parallel circuit using the symbol R to represent the parallel portion of the circuit. 386 Unit E Electricity and Magnetism P

1_ 1_ 1_ 1_ R R 2 R 3 R 4 1_ 1_ R 60 1_ 60 1_ 60 _ 1 0.050 1 R R 20 We can redraw the circuit to show that the three light bulbs wired in parallel have an effective resistance of 20. We can use the equivalent resistor R // to represent the three light bulbs in our circuit diagram (Figure 11.41). The circuit has been simplified to a series circuit. Kirchhoff s voltage law states that the sum of the voltage drops must equal the increase in potential of the battery. If the voltage drop across the equivalent resistor ( V ) is 10 V because the light bulbs require 10 V we can calculate the voltage drop across R 1 : 30 V R 1? Figure 11.41 The equivalent resistor, R represents the combined resistance of the three light bulbs. R 20 V 1 V V 1 V 30 V 10 V 20 V Since we are dealing with a series circuit, the current is the same throughout the circuit. In other words, the current through R is the same as the current through R 1. We can calculate the current through R using Ohm s law: V I R I _ V R _ 10 V 20 0.50 A We can now determine the value of R 1 using Ohm s law: V 1 I 1 R 1 R 1 _ V 1 I 1 20 V 0.50 A 40 Therefore, R 1 must have a resistance of 40 if the voltage across the light bulbs is to be 10 V. The effect of R 1 in our mixed circuit is to lower the electrical potential across the light bulbs so the potential difference across the bulbs will be correct. Take It Further There are many types of circuit designs that are fundamental to electronics. For example, a Wheatstone bridge is a common circuit design that has a simple purpose: to find the exact resistance of a resistor. Use a circuit simulation tool to experiment with a Wheatstone bridge. Identify how the circuit is designed and the components are used. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 387

REQUIRED SKILLS E5 Inquiry Activity Measuring Drawing conclusions Measuring Current and Potential Difference in a Parallel Circuit Question What are the current and voltage drops across the resistors in a parallel circuit? Activity Overview In this activity, you will measure the current and the potential difference in a parallel circuit. You will need to correctly connect an ammeter and a voltmeter. You will analyze your results and compare them with the values obtained using the parallel circuit equations. Your teacher will give you a copy of the full activity. Prelab Questions Consider the questions below before beginning this activity. 1. What does Kirchhoff s voltage law predict about the voltage drop through each branch of a parallel circuit? 2. What does Kirchhoff s current law predict about the current through each branch of a parallel circuit? Figure 11.42 Connect the parallel circuit as shown. REQUIRED SKILLS E6 Inquiry Activity Recording and organizing data Reporting results Measuring Current and Potential Difference in a Mixed Circuit Question What are the current and voltage drops across the resistors in a mixed circuit? R 1 Activity Overview In this activity, you will measure the current and the potential difference in a mixed circuit. You will need to correctly connect an ammeter and a voltmeter. You will analyze your results and compare them with your calculated values. Your teacher will give you a copy of the full activity. R 2 R 3 Figure 11.43 A mixed circuit Prelab Questions Consider the questions below before beginning this activity. 1. Which components of this mixed circuit are in series? 2. Which components of this mixed circuit are in parallel? 388 Unit E Electricity and Magnetism P

11.3 Check and Reflect Key Concept Review 1. Copy the following table into your notebook, and fill in the cells with the appropriate equations. Circuit Summary Type of Circuit Series circuit Parallel circuit Potential Difference Resistance Current 2. What is a parallel circuit? 3. Draw a circuit diagram that shows three resistors in parallel. 4. What effect does increasing the number of paths in a parallel circuit have on (a) the total resistance and (b) the total current? 5. If a parallel circuit develops a short circuit in one of the paths, what will happen to the current flow through the other paths? 6. If a parallel circuit contains three paths, each containing resistors of exactly the same value, explain what will happen if a resistor in one of the paths burns out and does not allow current to flow. 7. If two pathways in a parallel circuit have different resistances, will the current in each pathway be the same? Explain your answer. Connect Your Understanding 8. Determine the value of R 2 in the following circuit diagram. 112 V R 1 700? R 3 600 10. Determine the current through the ammeter in the circuit with (a) two light bulbs in parallel and (b) after a third light bulb has been added in parallel. All light bulbs have a resistance of 4.00. (a) (b) Question 10 11. Determine the total current and the current through all the branches of the following parallel circuit. 180 V Question 11 12. Determine the voltage drops and current through all the resistors in the following circuit diagram. 90.0 V 12.0 V 12.0 V Question 12 Reflection R 1 60 k R 1 10.0 R 4 20.0 20 k 30.0 R 3 30 k R 3 30.0 R 4 45 k 13. What new insights or strategies did you develop for solving mixed circuits? Question 8 I 0.720 A 9. Determine the total resistance and the current through the branches of the following circuit. For more questions, go to 75.0 V R 1 1.250 k 1.500 k R 3 2.200 k Question 9 P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 389

11.4 Power Consumption Section Summary Power is the rate at which energy is transferred. The power consumed by an electrical appliance can be determined. PHYSICS INSIGHT Remember that work is equal to change in energy. The equation for power can also be written as: P W_ t PHYSICS INSIGHT It is common practice to omit the from an equation. This is not correct and the power equation should be written as: P VI People often refer to the awesome power of nature. They might talk about a powerful hurricane or the powerful explosion of a volcano. However, in physics, power is defined as the rate at which energy is transferred or the change in energy per unit time. In other words, it is the amount of energy generated or used each second. The unit for power is the watt (W). The equation for power is: P _ E t where P is the power in watts (W), E is the change in potential energy in joules (J), and t is the time in seconds (s). Note that 1 W 1 J/s. An electrical circuit has components, such as resistors or light bulbs, that consume power. The power consumed by a component is the amount of energy it uses every second. For example, a 60-W incandescent light bulb consumes 60 J of energy each second. A battery does not consume energy it produces electrical energy from the chemical compounds contained inside. It is capable of generating power. This means that it can increase the energy of the charges by a certain amount each second. The total power consumed by all the components in the circuit must equal the power generated by the battery or power supply. Power Equations Recall from section 11.1 that the change in energy can be determined by the equation E Vq where E is the change in potential energy in joules (J) of the charges as they pass through a load and q is the charge in coulombs (C). We can substitute this equation into the equation for power: P _ Vq t Since q I t, we can further simplify the equation as: P V(I t) t V(I t) t P VI where P is the power in watts (W), V is the potential difference in volts (V), and I is the current in amperes (A). Note that 1 W 1 V A. We can also combine Ohm s law with the power equation. Table 11.6 shows the derivation of other forms of the power equation. 390 Unit E Electricity and Magnetism P

Table 11.6 Derivation of Power Equations Derivation Equation When to Use Substitute Ohm s law equation for V into the power equation: V IR P VI P (IR)I P I 2 R P I 2 R I and R are known V is not known Explore More How can you arrange four identical resistors in a circuit to consume the most power? The least power? Substitute Ohm s law equation for I into the power equation: V IR I _ V R P V ( _ V R ) _ P V2 R _ P V2 R V and R are known I is not known Example 11.7 Determine the power consumed by the light bulb in Figure 11.44. Given 120 V R L 240 Required power consumed by the light bulb (P L ) Analysis and Solution Since the light bulb is the only component in the circuit, V L 120 V P L _ V L 2 R L (120 V)2 240 60.0 W 120 V Paraphrase The light bulb consumes 60.0 W of power. Figure 11.44 Power Consumption in Circuits R L 240 As you have seen, there are differences between series and parallel circuits. The power consumption depends on whether the circuit is a series or parallel circuit. In a series circuit, the voltage drop across a component depends on the number of components in series with it. As the number of components in a series circuit increases, the resistance also increases, and the power consumed by each component is less. As the number of components in a parallel circuit increase, the total resistance decreases and, as a result, the power consumed by the circuit increases. Practice Problems 1. Determine the power dissipated by the resistor in Figure 11.45. 10.0 V Figure 11.45 2. Determine the resistance and power consumption of the resistor in Figure 11.46. 12.6 V Figure 11.46 I 2.80 A Answers 1. 0.022 W 2. R 4.50 P 35.3 W R 4.50 k R? Suggested Activity E7 Inquiry Activity Overview on page 394 P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 391

Example 11.8 Compare the power consumed by the light bulbs in a series circuit (Figure 11.47) with the power consumed by the light bulbs in a parallel circuit (Figure 11.48). R 1 240 120 V 240 120 V R 1 240 240 R 3 240 R 3 240 Figure 11.47 Figure 11.48 Practice Problems 1. Determine the power consumed by the resistors in Figure 11.49. 80.0 V Figure 11.49 2. Determine the power through each resistor in Figure 11.50. 15.0 kv Figure 11.50 Answers 1. P 1 5.56 W P 2 5.00 W P 3 2.78 W 2. P 1 56.3 kw P 2 37.5 kw P 3 150 kw R 1 200 R 3 100 R 1 4.00 k 180 6.00 k R 3 1.50 k Given 120 V R 1 R 3 240 Required power consumed by the light bulbs in a series circuit and in a parallel circuit (P 1, P 2, P 3 ) Analysis and Solution We will analyze the power consumed by each light bulb in the series circuit. R T R 1 R 3 240 240 240 720 2 P 1 I 1 R 1 2 P 2 I 2 R 2 P 3 I 32 R 3 (0.167 A) 2 (240 ) (0.167 A) 2 (240 ) (0.167 A) 2 (240 ) 6.67 W 6.67 W 6.67 W We can now analyze the power consumed by each light bulb in the parallel circuit: P 1 _ V 1 2 P V 2 _ 2 P 1 R 3 V 2 _ 3 2 (120 V)2 240 (120 V)2 240 I T R T I T _ V T R T 120 V 720 0.167 A R 3 (120 V)2 240 60.0 W 60.0 W 60.0 W Paraphrase Each light bulb consumes 6.67 W of power when placed in series, but 60.0 W of power when placed in parallel. Concept Check 1. What component in a circuit generates power? 2. Which type of circuit would consume more power: a circuit containing two resistors in series or a parallel circuit with the same resistors? 3. Is it possible for a circuit component to consume more energy than is generated by the battery? Explain your answer. 392 Unit E Electricity and Magnetism P

House Wiring and Power Consumption A house is wired so that the outlets are in parallel with each other. Appliances plugged into the wall will be in parallel with each other and will draw the power they need. One appliance will not interfere with the operation of another appliance. In other words, each appliance will always receive the same potential difference: 120 V. Imagine that you have three power outlets in your bedroom. They are wired in parallel so that up to three different appliances can be plugged into the wall. A hair dryer and a computer are plugged into two separate outlets in the room. What is the current through the ammeter in Figure 11.51? The voltage drop across each branch of this circuit is 120 V because it is a parallel circuit. Each branch draws current, and the sum of the currents in these branches equals the total current through the ammeter. To determine the current through the ammeter, calculate the current drawn by the hair dryer (I H ) and computer (I C ). P H VI H I H _ P H V 1200 W 120 V 10.00 A P C VI C I C _ P C V 180 W 120 V 1.50 A The total current drawn by the hair dryer and computer is: I T I H I C 10.00 A 1.50 A 11.50 A Since house wiring can carry up to 15 A safely, running these two appliances at the same time poses no hazard. But what happens if a 600-W vacuum cleaner is plugged into the third outlet and turned on? Calculate the current of the vacuum cleaner (I V ) and then the total current (I T ). I V _ P V V 600 W 120 V 5.00 A I T I H I C I V 10.0 A 1.50 A 5.00 A 16.5 A 120 V The total current will be 16.5 A, which exceeds the maximum current allowed through the wires. To prevent this much current, a circuit breaker is placed into the circuit. A circuit breaker is a switch that opens when a current higher than a certain amount flows through it. In this case the breaker would trip, or open, when a current greater than 15 A flows through the circuit. The circuit breaker protects the wires from carrying too much current and possibly overheating and starting a fire. For most rooms in a house, the maximum power rating of the circuit is 1800 W (120 V 15 A). Some rooms such as the kitchen and laundry room use 240 V and 20 A because the appliances, such as the stove and clothes dryer, consume more power. The maximum power consumption in that case is 4800 W (240 V 20 A). Some houses use fuses instead of circuit breakers. A circuit breaker can be reset and reused. A fuse burns out and can only be used once. P H 1200 W P C 180 W Figure 11.51 Two appliances are plugged into two outlets in this room and turned on: a 1200-W hair dryer and a 180-W computer. The other outlet is not used and no current flows through it. PHYSICS INSIGHT The symbol represents a fuse in a circuit diagram. Take It Further Every house contains different appliances. A typical house has a refrigerator, a stove, and a washer and dryer. Each appliance consumes a different amount of power. Make a list of ten household appliances in your home and research the typical power consumption of each. P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 393

Example 11.9 Practice Problems 1. There are five appliances: an air conditioner (1000 W), a refrigerator (450 W), a washing machine (500 W), a popcorn popper (250 W), and a stereo (20 W). What are the two highest-power appliances that can be plugged into two outlets and be running at the same time? The maximum current capacity of the circuit is 15.0 A and the outlets provide 120 V. 2. Determine the current through each of the ammeters in Figure 11.52. 120 V A 1 Figure 11.52 space heater A 2 A 3 stove P H 900 W P S 700 W Answers 1. air conditioner and the washing machine 2. I 1 13.33 A I 2 7.50 A I 3 5.83 A A student rents a small apartment that has four 120-V outlets wired in parallel. She moves in with six appliances: a kettle (900 W), a vacuum cleaner (600 W), a TV (100 W), a microwave oven (1500 W), a coffee maker (800 W), and a computer (150 W). What combination of four appliances can she operate simultaneously if the maximum current allowed in the circuit is 15.0 A? Given P K 900 W P V 600 W P TV 100 W P M 1500 W P CM 800 W P C 150 W I T 15.0 A V T 120 V Required combination of four appliances that will not exceed the maximum power possible in the circuit Analysis and Solution The total current cannot exceed 15.0 A. Since we know that the wiring of the apartment is in parallel and the potential difference of the circuit is 120 V, we can determine the total power of the circuit. P T V T I T (120 V)(15.0 A) 1800 W The power of four components running simultaneously cannot exceed 1800 W. Possible combinations are: P TV P C P V P CM 100 W 150 W 600 W 800 W 1650 W P TV P C P V P K 100 W 150 W 600 W 900 W 1750 W Paraphrase There are two combinations of four appliances running simultaneously that will not overload the circuit. REQUIRED SKILLS E7 Inquiry Activity Predicting Analyzing patterns Comparing the Power Consumption of Series and Parallel Circuits Question How does the power consumption of a series circuit compare with the power consumption in a parallel circuit? Activity Overview In this activity, you will investigate the power consumption of two resistors when they are wired in series and then in parallel. You will measure the current and voltage drop across each resistor, and determine the power consumption. Your teacher will give you a copy of the full activity. Figure 11.53 Series circuit for the activity Prelab Questions Consider the questions below before beginning this activity. 1. How does the voltage drop compare among branches in a parallel circuit? 2. Does the current in a series circuit change through each component? 394 Unit E Electricity and Magnetism P

11.4 Check and Reflect Key Concept Review 1. Define power. 2. How does the power consumed in a series circuit differ from that consumed in a parallel circuit? 3. Explain, in terms of electrical energy, how a 60-W light bulb differs from a 100-W light bulb. 4. What effect does a lower resistance have on the power consumed by a component in a circuit? 5. (a) What is the purpose of a circuit breaker or fuse? (b) How are they different? Connect Your Understanding 6. A block heater of a car is designed to keep the oil in the engine block warm during cold weather. If a heater has a voltage of 120 V and draws 3.30 A, how much power does it consume? 7. Determine the power consumed (in kw) by the heating coil in the following diagram. I 2.40 A Question 7 8. A power generation facility produces 455 MW of power. If the transmission line has a voltage of 500 000 V, what current does it carry? The SI prefix mega- (M) is equal to 10 6. 9. A typical lightning strike during a thunderstorm can carry 400 ka and have a potential difference of 1.1 GV. How much power does it generate? The SI prefix giga- (G) is equal to 10 9. 10. Determine the power consumed by each resistor in the following circuit. R 1 33.33 heating coil R 6.80 k 11. There are two circuits with two identical resistors. One is a series circuit and one is a parallel circuit. Compare the amount of power used in the series circuit and the parallel circuit. Justify your answer. 12. A kitchen of a house is wired to accept a maximum of 30.0 A. Determine which of the following appliances can be plugged into the fourth outlet in the circuit below and used simultaneously with the other appliances. Power Consumption of Appliances Appliance toaster oven slow cooker blender popcorn maker 120 V Question 12 Power Consumption 1100 W 600 W 300 W 250 W 13. A 60-W incandescent light bulb and a 15-W compact fluorescent light bulb (CFL) produce the same amount of light. Both use 120 V. Determine the current used by the incandescent bulb and the CFL. 14. A 60-W light bulb and a 100-W light bulb are connected in parallel. Which one will be brighter? Reflection dishwasher garburator microwave P D 1500 W P G 450 W P M 1200 W 15. What topic discussed in this section did you find the most interesting? Why? 500 V 100 R 3 200 For more questions, go to Question 10 P Chapter 11 The principles of conservation of energy and charge apply to electrical circuits. 395

CHAPTER 11 CHAPTER REVIEW Key Concept Review 1. Draw a circuit diagram of a circuit that contains an energy source, an ammeter, a light bulb, and a voltmeter. k 2. Distinguish between conventional current and electron flow. k 3. What is the difference between alternating current and direct current? k 4. What device is used to measure current and how is it connected to a circuit? k 5. State two effects a resistor has on a circuit. k 6. Explain how potential difference is different from potential energy. Use an analogy to explain the difference. k 7. What device is used to measure potential difference, and how is it connected to a circuit? k 8. Explain the significance of Ohm s law and Kirchhoff s laws. k 9. Use the proper symbols to draw the following circuits. (a) A battery is connected to two light bulbs in series. k (b) A variable power source is connected to three resistors in parallel. k 10. Distinguish between current in a series circuit and a parallel circuit. k 11. Distinguish between potential difference in a series circuit and a parallel circuit. k 12. Determine the potential difference across the battery in the following circuit. t 14. Determine the power dissipated by the light bulb in the following diagram. a I 5.00 A Question 14 light bulb V 50.0 V Connect Your Understanding 15. Does a circuit component that has no voltage drop across it consume power? Explain your answer. t 16. A rechargeable AA battery has a charge capacity of 2500 ma h. Determine the total charge contained in the battery. t 17. A 4.50-V flashlight draws a current of 400.0 ma when the light is on. If the flashlight is on for 50.0 s, what quantity of electrical potential energy is converted into light and heat? t 18. A computer laptop battery contains a total potential energy of 262 800 J and delivers 14.4 V. Determine how long the laptop can operate if it draws a current of 1.69 A. t 19. The potential difference across a compact fluorescent light bulb is 120.0 V. If 350.0 J of electrical potential energy are converted to light and heat, how much charge flowed through the bulb? t 20. Determine the value of resistor 3 in the following circuit diagram. t R 1 8.00 R 1 30? 15 Question 12 I 0.200 A 200 V Question 20 I 5.00 A R 3? 20.0 13. Write a paragraph explaining how power and energy are related. c 396 Unit E Electricity and Magnetism P