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studies that current in one direction is called a direct current (DC). A simple DC electric motor consists of three fundamental components: • a stator—a frame with a coil or permanent magnet to provide a magnetic field • an armature or rotor—a rotating loop of conducting wire on a shaft • a commutator—a split metal ring 608 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 609 As electrons in the current pass through the loop of wire in the armature in a clockwise direction (as seen from above in Figure 12.32), they experience a motor effect deflecting force. When you apply the left-hand rule for magnetic force, electrons on the left side of the loop experience a deflecting force upward, and electrons on the right side of the loop experience a deflecting force downward. The combined effect of both forces results in a rotation of the loop in a clockwise direction. Fm stator N armature e S e brush Fm commutator Figure 12.32 In a simple DC electric motor, the brushes provide a sliding contact between the wires from the battery and the armature. The magnetic field exerts an upward force on the left side of the wire loop and a downward force on the right side, causing the armature to rotate clockwise. Concept Check Describe the changes that must be made to the apparatus, shown in Figure 12.32, to cause the armature to rotate counterclockwise. If the rotation of the loop is to continue, the direction of the motion of the electrons in the loop must change every half-rotation. To accomplish this, the armature is connected to a commutator. A commutator is a split metal ring that is fastened to both ends of the loop of wire in the armature. Each half of the metal ring acts as a contact to the terminals of a power supply. Every half-rotation, the leads of each side of the armature contact a different terminal, changing the direction of the electron movement. Once connected to a steady supply of moving electrons, the armature continues to rotate in one direction. This is the principle of a simple electric motor. The Generator Effect (Electromagnetic Induction) In 1996, NASA did an experiment that involved a satellite attached by a conducting tether wire to a NASA space shuttle orbiting in space around Earth (Figure 12.33). Researchers found that the combination generated a current of about 1 A through
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the wire. The experiment was of particular significance for space scientists because it showed that Figure 12.33 A satellite tethered to a NASA space shuttle Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 609 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 610 this procedure could provide a method of generating the electric energy necessary to power all the electrical components on a space vehicle. This example is a useful and important application of a scientific phenomenon, but this phenomenon can also produce harmful effects in some situations. For example, engineers constructing the 1280-km north–south gas pipeline from Prudhoe Bay to Valdez in Alaska (Figure 12.34(a)) had to take precautions to eliminate the currents of electricity, called telluric currents, in the pipeline. These currents are caused by fluctuations in Earth’s magnetic field. Special magnesium anodes were installed underground along the pipeline to ground it and eliminate the possibility of electrical sparks. (a) (b) Figure 12.34 (a) A pipeline in Alaska; (b) An airplane in flight Similarly, certain grounding conditions must be incorporated in the construction of an airplane to eliminate the current generated by the wings of an airplane in flight through Earth’s magnetic field. These currents could affect the operation of all electrical components on the aircraft (Figure 12.34(b)). How are these examples related? What physical phenomenon is generating the current? The examples described above all involve conductors moving through magnetic fields. The scientific explanation of how they generate electricity began with investigations over 200 years ago. Faraday’s and Henry’s Discoveries Most scientific discoveries are the result of many years of research and investigations. The process is often convoluted and results are often accidental. However, as you have learned, some scientific discoveries are a result of the symmetry of nature. This symmetry led Coulomb and Faraday to conclude that electrical and magnetic forces could be determined using inverse-square relationships similar to Newton’s universal law of gravitation. Similarly, this symmetry in nature, and Oersted’s discovery that electricity could produce magnetism, led scientists to predict that magnetism could produce electricity. Experiments conducted in 1831 by Michael Faraday in England and Joseph Henry (1797–1878) in the United States demonstrated this effect. e LAB For a probeware activity that demonstrates the principle of electromagnetic induction, go to www.pearsoned.ca/ school/
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physicssource. 610 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 611 In a simplified version of their experiment, shown in Figure 12.35, a magnet is moved toward a coil of conducting wire connected to a sensitive galvanometer. When the magnet approaches the coil, the galvanometer’s needle deflects in one direction, indicating that a current is being produced in the coil of wire. This current is called an induced current, which is produced by a generated voltage. When the magnet is pulled away from the coil, the galvanometer deflects in the opposite direction, indicating that the induced current in the coil is in the opposite direction. When the magnet is stationary, no current is induced. If the magnet were held stationary while the coil of wire was moved back and forth, similar induced currents would be produced. Evidently, it does not matter whether the magnet or the coil of wire moves, as long as there is relative motion between a coil of conducting wire and an external magnetic field. In their conclusions, Faraday and Henry stated that when a piece of conducting wire cuts through magnetic field lines, an induced current is produced. The production of electricity by magnetism is called the generator effect or electromagnetic induction. Figure 12.36(a) shows a piece of conducting wire being moved perpendicularly upward through an external magnetic field. As a result, electrons in the wire also move perpendicularly upward. Use the left-hand rule for magnetic force: If the wire is moving upward (thumb) through the external magnetic field (fingers), then each electron experiences a motor-effect force (palm). Electrons will gather at one end of the wire with stored electric energy from the work done on the system in moving the wire. Thus, one end of the wire has an accumulation of electrons with stored electric energy while the other end has a deficiency of electrons (Figure 12.36(a)). If this wire is part of an external circuit, as in Figure 12.36(b), the induced voltage causes a current to flow through the external wire. A difference in electric potential drives electrons through an external circuit, from a region of high electric potential to a region of lower electric potential. N S galvanometer Figure 12.35 When a magnet is moved toward a loop of wire connected to a galvanometer, the galvanometer needle deflects. This indicates that an induced current is being produced in the coil of wire. generator effect or electromagnetic
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induction: production of electricity by magnetism Project LINK How would you apply the principle of electromagnetic induction and the operation of commutators to construct a DC generator wire movement Fm B induced current (a) wire movement induced current Fm B A electron flow (b) Figure 12.36 A current can be induced in a wire by moving the wire through a magnetic field. Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 611 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 612 12-6 Inquiry Lab 12-6 Inquiry Lab Magnetic Fields and Moving Conductors — Demonstration Question What factors influence the effect produced when there is relative motion between an external magnetic field and a conducting wire? Materials and Equipment 2 bar magnets 3 different sizes of coils of conducting wire (the diameter of the coils should allow a bar magnet to be inserted) galvanometer that can be projected onto a screen using an overhead projector galvanometer N S N S N S Figure 12.37 Procedure 1 Set up the apparatus as shown in Figure 12.37. 2 Slowly push one bar magnet at a uniform speed into the largest coil. Then pull the bar magnet out in the opposite direction at the same speed. Observe the deflection of the galvanometer’s needle in both cases. 3 Repeat step 2 with the other end of the magnet. Observe the deflection of the galvanometer’s needle in both cases. Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork 4 Repeat step 2, using two bar magnets. Observe the magnitude and direction of the galvanometer’s deflection. 5 Repeat step 2, using two bar magnets at a faster speed through the largest coil. Observe the magnitude and direction of the galvanometer’s deflection. 6 Repeat step 2, using two bar magnets at the same constant speed through the medium and the smaller coils of conducting wire. Observe the magnitude and direction of the galvanometer’s deflection. Analysis 1. How does the direction of the movement of the magnet affect the direction of the deflection of the galvanometer? 2. How does the polarity of the magnet affect the direction of the deflection? 3. Describe how each of the following factors influences the magnitude of the deflection of the galvanometer: (a) speed of the magnets through the conducting wire (b) strength of the external magnetic field (c) number of loops in
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the coil of conducting wire 4. What is the effect of relative motion between a conducting wire and a magnetic field? 5. Does it make any difference if the magnet or the conducting wire is moved? 6. What are the factors that determine the magnitude and direction of the induced current when there is relative motion between a conducting wire and an external magnetic field? 7. Based on your observations from this activity, design an experiment that demonstrates the effect of a uniform magnetic field on a moving conductor. 612 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 613 9. A battery supplies a current of 5.20 mA to a circuit. Determine the quantity of charge that flows through the circuit in 2.00 s. 10. Two conducting wires parallel to each other carry currents in opposite directions. Using the appropriate hand rule, determine whether the wires will attract or repel each other. 11. A wire 50 cm long and carrying a current of 0.56 A is perpendicular to an external magnetic field of 0.30 T. Determine the magnitude of the magnetic force on the wire. Extension 12. Could a simple electric generator be converted to a simple electric motor? Suggest any alterations that must be made in the design. e TEST To check your understanding of current-carrying conductors and magnetic fields, follow the eTest links at www.pearsoned.ca/school/physicssource. 12.3 Check and Reflect 12.3 Check and Reflect Knowledge 1. What are the factors that affect the magnetic force on a moving charge through an external magnetic field? 2. What are the factors that affect the magnetic force on a charge moving through a conducting wire in an external magnetic field? 3. What is the relationship between amperes and coulombs? 4. In the operation of a simple electric motor and simple electric generator, identify (a) a similarity (b) a difference 5. What symmetry in nature did Faraday and Henry apply in their discovery of electromagnetic induction? 6. What is the function of a split-ring commutator in the operation of a simple DC motor? 7. How do the electrons in a loop of wire in a generator gain energy? Applications 8. A wire lying perpendicular to an external magnetic field carries a current in the direction shown in the diagram below. In what direction will the wire move due to the resulting magnetic force? S N S N I Chapter 12 Properties of electric and magnetic fields
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apply in nature and technology. 613 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 614 12.4 Magnetic Fields, Moving Charges, and New and Old Technologies info BIT Michael Faraday built the first electric motor in 1821. This motor had a stiff wire hanging from a stand. The lower end of the wire was immersed in a cup of mercury with a bar magnet upright in the middle. When current from a battery flowed through the wire, it rotated around the magnet. From the old technologies of the simple beginnings of electric motors, electric meters (such as galvanometers and ammeters), loudspeakers, and electromagnets to the new technologies of magnetohydrodynamic (MHD) propulsion systems and magnetic resonance imaging (MRI), the science of the production of magnetism by electricity plays a significant role in our everyday lives. Although examples of some of these technologies have been described in previous sections, following are other examples of old and new technological applications of this principle. Table 12.2 describes old and new technologies that use moving charges or current-carrying conductors to produce magnetic fields that can interact with external magnetic fields to produce powerful magnetic forces. Table 12.2 Loudspeakers and MHD Propulsion Old Technology New Technology Magnetohydrodynamic (MHD) Propulsion paper cone water intake oppositely charged plates Loudspeakers electron flow x x x B F S N S voice coil Figure 12.38 A simplified diagram of a loudspeaker The operating principle of most loudspeakers is that currentcarrying wires produce magnetic fields that can exert magnetic forces. In the design of the loudspeaker shown in Figure 12.38, a coil of wire, called a voice coil, surrounds the north pole of a very powerful external magnet at the back of the speaker. When your sound system sends an electric signal to the coil, a current is produced in the coil, which produces a magnetic field. As a result, the coil experiences a magnetic force due to the interaction of its magnetic field with the external magnetic field. Depending on the direction of the current in the coil, the magnetic force of attraction or repulsion causes the coil to slide to the left or right. The direction of the current is determined by the electric signal produced by the sound system. As the voice coil slides back and forth, it causes the paper cone to vibrate in or out, creating sound waves as it pushes on the air in front of the cone. The electric signal from the sound system is thus
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converted to a mechanical sound wave in air. 614 Unit VI Forces and Fields superconducting electromagnet jet of water Figure 12.39 MHD uses magnetic fields as a propulsion system for seagoing vessels. The MHD propulsion system is an experimental system for seagoing vessels to replace conventional propeller systems. MHD uses magnetic fields to produce a jet of water for propulsion. Figure 12.39 is a simplified diagram of this type of system. A powerful superconducting magnet surrounds a thruster tube containing seawater. This magnet produces a magnetic field perpendicular to the tube’s length. Inside the tube, electrodes produce a current of ions, perpendicular to the magnetic field, across the tube from the dissolved salts in seawater. As a result of the perpendicular movement of the ions through an external magnetic field, a magnetic force is exerted on the ions, causing them to deflect along the length of the tube. This movement of the water through the tube provides the necessary thrust to propel the vessel. An advantage of MHD propulsion systems is that they have no mechanical moving parts and thus require minimal maintenance. 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 615 Generator Effect Applications The discovery that moving a conducting wire through an external magnetic field generates an induced current in the conductor (generator effect) also led to many important technological applications. From the old technologies of the simple generators, induction coils, and transformers to the new technologies of infant breathing monitors and others, applications of the scientific principle of the production of electricity from magnetism are found everywhere in our lives. Table 12.3 describes two of these applications. Table 12.3 Induction Coils, Transformers, and SIDS Monitors Old Technology New Technology Induction Coils switch SIDS Monitors galvanometer primary coil iron battery secondary coil Figure 12.40 A simplified diagram of Faraday’s induction coil Figure 12.41 Monitors are designed to detect changes in a baby’s breathing. A change in the current in the primary coil produces a changing magnetic field in the iron core. This changing magnetic field produces an induced current in the secondary coil, causing the needle on the galvanometer to deflect. Such coils can induce current in a wire that has no direct connection to the power supply. Figure 12.40 shows a simplified version of Michael Faraday’s original induction coil. In sudden infant death syndrome (SIDS), an infant stops breathing with no apparent cause. One type of SIDS
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monitor uses induced currents to measure an infant’s breathing (Figure 12.41). A coil of wire attached to one side of the infant’s chest carries an alternating current, which produces a magnetic field. This alternating field cuts another coil taped to the other side of the chest and induces an alternating current in this other coil. As the chest moves up and down, the strength of the induced current varies. These variations are monitored. A Motor Is Really a Generator, Which Is Really a Motor You have analyzed and studied the motor effect and the generator effect as separate phenomena in this chapter. However, the symmetry of nature suggests that related phenomena are really variations of the same effect. Since electricity can produce magnetism and magnetism can produce electricity, then perhaps the technologies that are derived from these phenomena are also similar. Is a motor really that different from a generator? Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 615 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 616 12-7 Inquiry Lab 12-7 Inquiry Lab The Curious Relationship Between Motors and Generators Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork Question What is the relationship between the motor effect and the generator effect? Analysis 1. Copy Table 12.5 into your notebook. Use the data from Table 12.4 to complete the calculations in Table 12.5. Materials and Equipment Table 12.5 Calculations for 12-7 Inquiry Lab Magnitude of Acceleration of the Magnet Through the Copper Pipe d a 2 2 t (m/s2) Magnitude of Net Force Causing the Downward Acceleration of the Magnet (Fa ma) (N) Magnitude of the Weight of the Magnet (Fg mg) (N) 2. What is the magnitude of the upward force on the falling magnet? 3. Identify and explain where the generator effect is occurring in this experiment. 4. Identify and explain where the motor effect is occurring in this experiment. 5. Do the generator effect and the motor effect complement each other as the magnet falls through the copper pipe? Explain your answer. 1 100-cm length of copper pipe (internal diameter approximately 1.4 cm) 1 cylindrical rare earth magnet (less than 1.4 cm in diameter) 1 metre-stick 1 stopwatch 1 scale Procedure 1 Copy Table 12.4 into your notebook. Table 12.4 Data for 12-7 Inquiry Lab Mass of
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Magnet Length of Copper (m)(kg) Pipe (d)(m) Average Time Taken for the Magnet to Fall Through the Pipe (t)(s) 2 Measure the mass of the magnet on the scale, and record it in Table 12.4. 3 Measure the length of the copper pipe, and record it in Table 12.4. 4 Holding the pipe in a vertical position, drop the magnet from the exact top of the pipe. Measure the time for the magnet to reappear out the bottom. Record this time in the table. 5 Repeat the procedure in step 4 several times to obtain an average value for the time taken for the magnet to drop the length of the pipe. 616 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 617 An Accidental Discovery Simple DC electric motors and electric generators have three similar components: • an external magnetic field • a loop of conducting wire • a commutator At the 1873 Vienna Exhibition, the Belgian inventor Zénobe-Théophile Gramme (1826–1901) demonstrated a compact and efficient generator that he had designed. A steam engine provided the power to run the generator. A workman mistakenly connected the output of the generator to a second generator in the display. The shaft of the second generator began spinning even though it was not connected to the steam engine. Gramme immediately realized that the second generator was operating as a motor powered by the first generator. Gramme and his colleagues then moved the generators several hundred metres apart and connected them with long wires. The American writer Henry Adams (1838–1918) described the importance of Gramme’s demonstration: “Suddenly it became clear that ELECTRICITY could now do heavy work, transporting power through wires from place to place.” M I N D S O N Perpetual Motion? Suppose that a motor and generator are connected to the same shaft and wired such that the output of the generator powers the motor. If you spin the shaft, the generator supplies energy to the motor, which turns the shaft. The generator then produces more energy to run the motor. Explain why this process cannot continue indefinitely. Lenz’s Law If you did the 12-7 Inquiry Lab, you discovered what happens when you drop a magnet down a metal tube. When a conductor cuts the magnetic field lines of a falling magnet, it generates an induced current in the conducting pipe (the generator effect). However, the induced current moves
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in a circular motion around the circular pipe, so it creates its own vertical magnetic field, inside the metal tube (the motor effect). The direction of the magnetic field can be directed either upward or downward. The direction of the magnetic field that is produced by the circular induced current in the pipe can have one of the following orientations: • It will attract the magnet and cause it to fall faster, thus generating a greater induced current. • It will repel and oppose the motion of the magnet, causing it to fall much slower. Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 617 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 618 The law of conservation of energy requires that you can never get more out of a system than you put into it. So, the direction of the new magnetic field will always oppose the motion of the magnetic field of the original magnet. This is the principle of Lenz’s law, which states: The direction of a magnetically induced current is such as to oppose the cause of the current. For example, if a magnet falls with its north pole directed downward, then the magnetic field produced by the induced current in the conducting pipe will have its north pole pointing upward to repel and oppose this motion. Concept Check Copy Figure 12.42 into your notebook. Apply Lenz’s law by sketching the direction of the induced current in the metal tube and the resulting orientation of the magnetic field in the tube. How would the induced current and magnetic field directions change if the falling magnet’s south pole were directed downward? S N Figure 12.42 Dropping a magnet down a metal tube induces a current in the tube. Figure 12.43 shows a similar situation. As the north pole of a magnet approaches a coil of wire, the induced current generated in the coil produces a north pole to repel and oppose the approaching magnet. S N N S G Figure 12.43 Lenz’s law helps us explain that the direction of current induced in the coil has a magnetic field that exerts a force on the bar magnet that opposes the magnet’s motion. M I N D S O N Lenz’s Law Balance a dime on its edge on a smooth table. Carefully bring a magnet as close as 1 mm to the face of the dime. Quickly pull the magnet away. What happens to the dime? How is this behaviour an application of Lenz�
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�s law? The principle behind Lenz’s law also hinders the operation of electric motors and generators. For an electric motor to operate, an electric current must first be supplied through a conducting loop of wire in a magnetic field, causing the motor effect, so the loop will rotate. 618 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 619 However, as the loop rotates, the conducting wire cuts the magnetic field lines, causing the generator effect. The generator effect induces a current in the loop of wire. The direction of the induced current must be in an opposing direction to the direction of the original current that was supplied. Similarly, to operate a generator, movement of a conducting wire in a magnetic field must be supplied, which will induce a current. However, as soon as the induced current moves through a conductor in a magnetic field, a force on the conducting wire will be produced that opposes the original force and hinders the movement of the conducting wire. e WEB Research the relationship between Lenz’s law and the operation of most vending machines. Write a short report analyzing the operation of vending machines, and describe whether they operate on the principles of Lenz’s law and the generator effect or Lenz’s law and the motor effect. In your description, include the term “eddy currents.” Begin your search at www.pearsoned.ca/ school/physicssource. THEN, NOW, AND FUTURE Nanotechnology Since the start of the Industrial Revolution and with advances in technology, machines have become increasingly smaller. Scientists at University of California’s Berkeley National Laboratory have developed the smallest synthetic electric motor ever made. Essentially, it is an electric rotor spinning on an axle 2000 times smaller than the width of a human hair. Imagine tiny electric motors so small that one motor could ride on the back of a virus, or thousands could fit in the period at the end of this sentence. This is the world of nanotechnology, which is an umbrella word that covers many areas of research and deals with objects in nanometres, or a billionth of a metre. How do nanomachines work? As you have learned, conventional electric motors use electromagnets or strong external magnets to spin rotors made of loops of wire. The spinning of a rotor provides mechanical energy to do work. In the electric motor of nanotechnology, transistors act as switches to move negative and
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positive charges around a circle of electrodes. The charges jump around the stator electrodes, that are measured Figure 12.44 A rosette nanotube causing an electrically charged rotor to spin around and rotate a nanotube shaft. This spinning rotor provides mechanical energy, similar to a conventional electric motor. The nanomotor can perform only small functions, such as moving a second hand on a watch, but it has many advantages: It spins without gears or bearings; it is unaffected by gravity or inertia; and it can run for a long time with no breakdowns. Another application of nanotechnology is the rosette nanotube, like the one shown in Figure 12.44. This type of tube was developed at the National Institute of Nanotechnology of the National Research Council, located at the University of Alberta. The nanotube is made of molecules that assemble themselves into this distinctive shape. Possible future applications of these tubes include: nanowires in molecular electronics, drug delivery systems within the body, and environmentally friendly oil sands upgrading additives. The potential of nanotechnology in electronics is now beginning to be realized and applied to many different fields. Imagine, for example, nanorobots that can be injected into the body to attack viruses and cancer cells. In the future, it may even be possible to construct molecules of oil and gas, reducing our reliance on fossil fuels. The science of nanotechnology is the science of the future. Questions 1. What are some advantages of nanotechnology? 2. Describe how a nanomotor works. 3. What could nanotubes be used for? Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 619 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 620 12.4 Check and Reflect 12.4 Check and Reflect Knowledge 1. Identify two technological applications that employ (a) the motor effect (b) the generator effect 2. What are the three basic components of an electric motor and generator? 3. What is Lenz’s law? Applications 4. For an electric motor: (a) Describe what you must supply to start the operation of the device. (b) Describe what you get out of the operation of the device (motor effect). (c) Using Lenz’s law, explain how the operation of the motor also produces the generator effect to hinder its own operation. 5. For an electric generator: (a) Describe what you must supply
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to start the operation of the device. (b) Describe what you get out of the operation of the device (generator effect). (c) Using Lenz’s law, explain how the operation of the generator also produces the motor effect to hinder its own operation. 6. Explain why you will feel a force of repulsion if you attempt to move a magnet into a coil of wire. 7. The north pole of a magnet is pulled away from a copper ring, as shown in the diagram below. What is the direction of the induced current in the ring? N S Extensions 8. An electric motor requires a current in a loop of wire. However, as the loop rotates, it generates a current which, according to Lenz’s law, must be in an opposing direction. Why must the induced current be in an opposing direction? 9. A hair dryer operates on a very small current. If the electric motor in the hair dryer is suddenly prevented from rotating, the dryer overheats. Why? e TEST To check your understanding of magnetic fields, moving charges, and new and old technologies, follow the eTest links at www.pearsoned.ca/ school/physicssource. 620 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 621 CHAPTER 12 SUMMARY Key Terms and Concepts law of magnetism magnetic field electromagnet solenoid ferromagnetic domain motor effect force current Key Equations F qvB m F IlB m Conceptual Overview ampere commutator generator effect electromagnetic induction Lenz’s law The concept map below summarizes many of the concepts and equations in this chapter. Copy and complete the map to have a full summary of the chapter. definition types Earth domain theory bar magnet U-shaped magnet magnetization by induction law of magnetism generator effect Lenz’s law 2 like poles 2 unlike poles magnetic forces and fields cause moving charges motor effect moving charges in magnetic fields direction of deflection magnitude of deflection Fm qvB charge wire moving charges in 2 current-carrying wires definition of ampere Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 621 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 622 CHAPTER 12 REVIEW Knowledge 1. (12.1) State the major contribution of each of the following scientists to the study of magnet
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ism: (a) William Gilbert (b) Hans Christian Oersted (c) André Ampère (d) Michael Faraday 2. (12.1) State the definition of (a) a magnetic field (b) the direction of a magnetic field 3. (12.1) Compare a magnetic vector arrow at a point near a magnet and a magnetic field line around a magnet. 4. (12.1) How was it determined that there had to be two different types of magnetic poles? 5. (12.1) Sketch the magnetic field lines around each of the following objects and describe the differences in the magnetic fields of (a) a bar magnet and Earth (b) a current-carrying straight piece of conducting wire and a current-carrying coil of conducting wire 6. (12.2) Identify the sources of the two magnetic fields required to produce the motor effect force on a moving charge. 7. (12.2) Describe the deflection of a moving charge, through an external magnetic field, if the direction of the initial motion of the charge is (a) parallel to the external magnetic field lines (b) perpendicular to the external magnetic field lines (c) at an angle to the external magnetic field lines 8. (12.2) Where is a magnetic bottle formed? 9. (12.3) Describe a difference between a galvanometer and an ammeter. 10. (12.3) State two definitions for a current of one ampere. 11. (12.4) Identify two technologies that use the principles of (a) the motor effect (b) the generator effect 12. (12.4) What is the principle of Lenz’s law? 622 Unit VI Forces and Fields Applications 13. Does every charged object necessarily have a positive and a negative charge? Does every magnetized object necessarily have a north and a south pole? Justify your answers. 14. If the direction of the magnetic field outside a magnet is from the north to the south pole, what is the direction of the magnetic field within the magnet? 15. Using domain theory, describe how an iron nail can become magnetic by (a) the process of magnetization by induction (b) the process of magnetization by contact 16. How will the magnetic force on a moving charged particle change if (a) only the charge is doubled? (b) the magnetic field is doubled and the speed is halved? (c) the mass of the
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charge is doubled? 17. Use domain theory to explain the difference between a permanent and a temporary magnet. 18. You are told that a straight piece of copper wire has a steady current in it. Given only a compass, describe how you can find the direction of the current in the wire. 19. A drinking straw with a green grape at one end is suspended by a string from a hanging support. When either end of a magnet is brought close to the grape, repulsion occurs. Describe a possible reason for this effect. 20. An electron in a TV tube is moving at 7.00 106 m/s perpendicular to a magnetic field of magnitude 0.0880 T in the tube. What is the magnetic deflecting force on the electron? 21. A proton travelling at 35 to an external magnetic field of magnitude 0.0260 T experiences a force of magnitude 5.50 1017 N. (a) Calculate the speed of the proton. (b) Calculate the kinetic energy of the proton in joules (J) and electron volts (eV). 22. What speed must an alpha particle maintain if it is to remain suspended, relative to Earth’s surface, as it travels on a tangent to Earth’s surface and perpendicularly through Earth’s magnetic field of 50.0 T? 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 623 23. An alpha particle travelling with a speed of 4.30 104 m/s enters a uniform magnetic field of 0.0300 T. Determine the magnetic force on the particle if it enters the field at an angle (a) perpendicular to the magnetic field (b) 30.0 to the magnetic field (c) parallel to the magnetic field 24. A magnetic field is used to bend a beam of electrons. What uniform magnetic field is required to bend a beam of electrons moving at 1.2 106 m/s in a circular arc of 0.25 m? 25. Two parallel current-carrying wires are observed to attract each other. What is the source of the force of attraction? How could you demonstrate that the force of attraction is not electrostatic attraction? 26. The magnetic force between two magnets was measured as the distance between the magnets was varied. The following information was obtained: Separation, r ( 102 m) Magnitude of Force, F (N) 5.00 10.00 15.00 20.00 25
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.00 4.02 1.01 0.45 0.25 0.16 (a) Draw a graph of the magnetic force as a function of separation distance. (b) From the shape of the graph, what is the relationship between force and separation? 27. An airplane is flying east over Earth’s magnetic north pole. As a result of its motion, one wing was detected as having more electrons than the other. Explain why this phenomenon occurs. Identify which wing will have more electrons. 28. A power line carries a current of 500 A. Find the magnetic force on a 100-m length of wire lying perpendicular to Earth’s magnetic field of 50.0 T. Extensions 29. A magnet is dropped through two similar vertical tubes of copper and glass. Which tube will allow the magnet to fall faster? Explain your answer. 30. A disk magnet on a table has two steel balls in contact with it on either side. The steel balls are slowly moved toward each other while still in contact with the disk magnet. As they move, they repel each other. Describe why the steel balls are attracted to the disk magnet, but repel each other. Consolidate Your Understanding Create your own summary of properties of magnetic and electric fields by answering the questions below. If you want to use a graphic organizer, refer to Student Reference 3: Using Graphic Organizers. Use the Key Terms and Concepts listed on page 621 and the Learning Outcomes on page 580. 1. Create a flowchart to identify technological devices that use electric fields, magnetic fields, or a combination of the two fields to control moving charges. 2. Write a paragraph comparing the effects of electric or magnetic fields on moving charges. Share your report with a classmate. Think About It Review your answers to the Think About It questions on page 581. How would you answer each question now? e TEST To check your understanding of the properties of electric and magnetic fields, follow the eTest links at www.pearsoned.ca/school/physicssource. Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 623 12-PearsonPhys30-Chap12 7/28/08 9:46 AM Page 624 UNIT VI PROJECT Building a Model of a Direct Current Generator Scenario The search for better electrical energy production began with the pioneering work of Faraday and Henry in the 1800s and continues today. Over half of the energy consumed in our world is electrical energy and the average consumption per
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person is increasing every year, so more efficient methods of electrical energy production are being sought all the time. All DC electrical generators consist of three major parts: coils of wire wrapped around a core to make the armature, a commutator, and an external magnetic field. As you learned in this unit, the operating principle behind generators is the movement of a conducting wire through external magnetic field lines so that a voltage is generated in the wire. This in turn induces a current in an external line. All DC generators operate in this fashion. The only difference among generators is the source of the mechanical energy required to turn the turbines that rotate the coil of wires in the magnetic field. In some places, this is the energy of falling water or tides, while in others it is the energy of moving steam from the combustion of fossil fuels or nuclear reactions. Recently, interest has grown in using wind energy to turn the turbines that operate a generator. A single wind generator can produce about 10 MW of electrical power, which is sufficient for a single small farm. The purpose of this project is to research and investigate the operation of a wind-powered electrical generator and to build a model of a wind-powered DC generator capable of generating enough electricity to operate a mini-bulb. Planning Form a team of four or five members, and decide on and plan the required tasks to complete the project. These tasks may include researching the design of a simple generator, obtaining the necessary materials, constructing the model of the generator, preparing a written report, and presenting the project to the entire class. Assessing Results Assess the success of your project based on a rubric* designed in class that considers: research strategies thoroughness of the experimental design effectiveness of the experimental technique effectiveness of the team’s public presentation Materials • insulated copper wire • iron core • split-ring commutator • external magnets • connecting wires • mini-bulb with support base • stiff paper to construct a turbine • balsa wood for the axle • household fan to produce wind Procedure 1 Using the Internet, library, or other resources, research the operation of a simple DC electrical generator and create a design of the model that you will construct. Pay special attention to the commutator required for DC generation. 2 Construct a working model of the generator that can provide the electrical energy to light a mini-bulb. In your model, investigate the factors that determine the magnitude of the generated voltage. 3 Prepare a report explaining the design and the specific functions of all the components. Thinking Further 1
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. What modifications did you make in the construction of your model of a generator that affected the magnitude of the generated voltage? 2. What other type of commutator could you have used in the design of your generator? What type of current would be induced by this commutator? 3. 4. Identify at least three risks and three benefits of a wind-powered electrical generator. Is wind-powered electrical generation a viable and desirable method of electrical energy generation for the future? Explain your answer. *Note: Your instructor will assess the project using a similar assessment rubric. 624 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 625 UNIT VI SUMMARY Unit Concepts and Skills: Quick Reference Concepts Chapter 10 Modern theory of electrostatics Coulomb’s law Summary Resources and Skill Building Physics laws can explain the behaviour of electric charges. 10.1 Electrical Interactions Substances can be classified as conductors, insulators, semiconductors, and superconductors. Objects may be charged through the processes of friction, conduction, and induction. 10-1 QuickLab 10-2 Inquiry Lab 10.2 Coulomb’s Law Coulomb’s law states that the electrical force acting on charged objects depends on the charges and the distance between the charges. 10-3 Inquiry Lab Examples 10.1, 10.2 Vector analysis Electrostatic forces can be solved in one- and two-dimensional situations using vector analysis. Examples 10.3–10.6 Chapter 11 Electric field theory describes electrical phenomena. Fields Electric field Electric field lines Electric potential energy Electric potential 11.1 Forces and Fields Fields are used to explain action at a distance. An electric field is a three-dimensional region of influence surrounding every charge. Electrostatic force affects another charge placed in the field. The electric field is a vector quantity that has magnitude and direction. 11.2 Electric Field Lines and Electric Potential Electric field lines can depict the electric fields around different types of charged objects. Electric potential energy is the amount of work done on a charged object to move it from infinity to a position in an electric field. Electric potential energy can be calculated. Electric potential is the amount of electric potential energy stored per unit charge and can be calculated. Electric potential difference When a charge moves from a location where it has one electric potential to a location where it has another electric potential, the charge experiences an electric potential difference. Motion of a charge in an electric field 11.
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3 Electrical Interactions and the Law of Conservation of Energy When a charge is placed in an electric field, it experiences a force that causes it to accelerate in the direction of the field. The acceleration of the charge is different in a non-uniform field surrounding a point charge than the acceleration of the charge in the uniform field between charged plates. Work done by the system on the charge increases the charge’s potential energy, which can be converted to other forms of energy. Chapter 12 Properties of electric and magnetic fields apply in nature and technology. 11-1 QuickLab, 11-2 Inquiry Lab Examples 11.1–11.4 Minds On activities, eSIM Examples 11.5, 11.6 Example 11.7 Example 11.8 Examples 11.10–11.12 Magnetic fields Cause of magnetism 12.1 Magnetic Forces and Fields Magnetic fields are three-dimensional regions of magnetic influence surrounding every magnet in which other magnets or magnetic substances are affected by magnetic forces. Magnetic fields are vector fields and can be depicted by magnetic field lines. The cause of magnetism is motion of charges and can be explained using the domain theory. If the motion of charges is straight, the magnetic field is circular. If the motion of charges is circular, the magnetic field is straight within the loop. The direction of the magnetic field lines can be described using hand rules. 12-1 QuickLab, 12-2 QuickLab Figure 12.10 Magnetizing objects Objects can be magnetized through contact or induction. Concept Check Motor effect on a moving charge 12.2 Moving Charges and Magnetic Fields A charge moving perpendicularly through an external magnetic field experiences a magnetic force due to two magnetic fields, which can be calculated. This motor effect force can explain the operation of electric motors and other technologies. The magnitude of the motor effect force can be calculated and its direction can be determined using hand rules. 12-3 Inquiry Lab Examples 12.1, 12.2 Motor effect on two current-carrying wires 12.3 Current-carrying Conductors and Magnetic Fields A current-carrying conductor that is perpendicular to an external magnetic field experiences a magnetic force that can be calculated. Generator effect A conductor moving perpendicular to an external magnetic field can produce electricity. Applications of the generator effect and the motor effect 12.4 Magnetic Fields, Moving Charges, and New and Old Technologies The generator effect and the motor effect are used in many technologies. Example 12.4 12-4 QuickLab 12-5 Design a Lab 12-6 Inquiry Lab 12-
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7 Inquiry Lab Lenz’s law Lenz’s law explains how a motor is really a generator and a generator is really a motor. Minds On Unit VI Forces and Fields 625 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 626 UNIT VI REVIEW Vocabulary 1. Use your own words to define the following terms, concepts, principles, or laws. Give examples where appropriate. ampere charge migration charge shift charging by induction commutator conduction conductor coulomb Coulomb’s law current domain electric field line electric potential (voltage) electric potential difference electric potential energy electromagnet electromagnetic induction electron volt electrostatics ferromagnetic field generator effect grounding induction insulator law of conservation of charge law of magnetism Lenz’s law magnetic field motor effect force net charge plasma semiconductor solenoid source charge superconductor test charge 626 Unit VI Forces and Fields Knowledge CHAPTER 10 2. Explain why silver is a better conductor of electricity than rubber. 3. State a technological advantage of developing materials that are superconductors. 4. A negatively charged rubber rod is brought near a small metal ball hanging from an insulated thread. The metal ball is momentarily grounded, and then the ground and the rubber rod are removed. Identify the procedure used to charge the metal ball, and determine the final charge on the metal ball. 5. Describe a similarity and a difference between (a) charging by friction and charging by conduction (b) charge shift and charge migration 6. In each of the following examples, identify the charge on each object and state the method of charging the object. (a) An ebonite rod is rubbed with fur and then is held near a neutral metal sphere. (b) A glass rod is rubbed with silk and then is touched to a neutral metal sphere. 7. During the rubbing process of charging objects, one object gains a net negative charge. What can you conclude about the charge on the other object? Explain why. 8. How do the following factors affect the electrostatic force of attraction between two charged objects? (a) amount of charge on each object (b) distance between the centres of the two objects (c) sign of the charge on each object CHAPTER 11 9. State the superposition principle as it applies to vector fields. 10. Why must a test charge be small when it is used to determine the direction of an electric field around a larger primary charge? 12
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-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 627 11. Draw the electric field lines around 22. Justify the following statement: “A charge (a) a negative point charge (b) a positive charge and a negative point charge in the same region (c) a negatively charged cone-shaped object 12. Why is there no net electric field inside a charged hollow sphere? 13. Explain the following statement: “It is impossible to shield against gravitational fields, but it is possible to shield against electric fields.” 14. Draw a diagram showing two small, equally charged, positive spheres a small distance apart. On your diagram, identify a point A where the electric field is zero and a point B where the electric potential is zero. 15. Describe how electric potential energy changes for positive and negative charges as they move, relative to the reference point at infinity. 16. Two oppositely charged parallel plates are connected to a 120-V DC supply. What happens to the magnitude of the electric field between the plates if the distance between the plates decreases? moving perpendicular to a uniform external magnetic field experiences a force but does not change its speed.” 23. Show how a charge moving through an external magnetic field experiences a deflecting force because of the interaction of two magnetic fields. 24. A positively charged disk is spinning in a clockwise direction as seen from above. What is the direction of the magnetic field? Use a diagram to help you describe its shape. 25. Describe the shape and direction of the magnetic field around a negatively charged dart as it travels directly away from you toward a target on a wall. 26. What factors affect the magnetic force on a charge moving through an external magnetic field? 27. State a difference and a similarity between the motor effect and the generator effect. 28. Describe how Lenz’s law affects the operation of (a) a motor (b) a generator 17. Describe the following for a point a small distance from a positively charged sphere: Applications (a) (b) the magnitude and direction of the electric field at this point the electric potential energy of another small positive charge placed at this point (c) the electric potential at this point 18. Describe the relationship between the work done in moving a charge from one region in an electric field to another region, and the energy gained by the charge. What law governs this relationship? 29. An insulator and a conductor are each contacted by a negatively
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charged rubber rod. Describe the distribution of charge on each object. 30. Describe how Earth’s surface can become positively charged during a thunderstorm. In your description, include the terms “charging by friction” and “charging by conduction.” 31. Assume you have only a negatively charged ebonite rod. Describe a procedure for charging a neutral electroscope CHAPTER 12 (a) positively (b) negatively 19. Draw a diagram of the magnetic field around a bar magnet and Earth. (a) List the similarities between the two fields. (b) List the differences between the two fields. 20. Explain why a single magnetic pole cannot exist by itself. 21. Describe two simple demonstration techniques used to outline the magnetic field around a magnet. 32. Why does a negatively charged ebonite rod initially attract a piece of thread and then eventually push it away? 33. When you touch a Van de Graaff generator, your hair stands on end. Explain your answers to the following questions: (a) Are you being charged by friction, conduction, or induction? (b) Will the same effect occur if you are grounded? Unit VI Forces and Fields 627 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 628 34. A negatively charged ebonite rod is brought 43. A football-shaped hollow conducting object is near a neutral pith ball hanging on an insulating thread. Describe what happens to the charges in the pith ball and the resulting effect (a) before the rod and the pith ball touch (b) after the rod and the pith ball touch 35. A straight length of conducting wire, lying horizontal to the surface of Earth, delivers a current in a direction from south to north. Describe the deflection of a compass needle held directly over the conducting wire. 36. A negatively charged foam plastic ball is hanging from an insulated thread. Another negatively charged ball is brought near, on the same horizontal plane, and the hanging ball swings away so that the supporting thread makes an angle of 30° to the vertical. (a) Draw a free-body diagram depicting the tension force by the string, the force of gravitational attraction, and the electrical force of repulsion on the hanging ball. If the system is in equilibrium, identify the force that balances the gravitational force on the hanging ball and the force that balances the electrostatic force of repulsion on the hanging ball. (b
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) 37. Explain how the properties of selenium are essential in the operation of a photocopier. 38. Explain the difference between charge shift and charge migration during the process of charging by induction. 39. The cell membrane of a neuron may be thought of as charged parallel plates. The electric potential difference between the outside and the inside of the membrane is about 0.70 V. If the thickness of the membrane is 5.0 109 m, calculate (a) the magnitude of the electric field between the outside and the inside of the membrane (b) the amount of work necessary to move a single sodium ion, Na1, with a charge of 1.6 1019 C, across the membrane 40. In a chart, compare the similarities and differences between Newton’s law of universal gravitation and Coulomb’s law of electrostatics. 41. Determine the distance between two electrons if the mutual force of repulsion acting on them is 3.50 1011 N. 42. A neutral small hollow metal sphere is touched to another metal sphere with a charge of 3.00 102 C. If the two charges are then placed 0.200 m apart, calculate the electrostatic force acting on the two spheres after they touch. 628 Unit VI Forces and Fields charged negatively. (a) Draw the object and then draw the charge distribution on the surface of the object. (b) Draw the electric field lines surrounding the object. (c) Where will the intensity of the electric field appear to be the greatest? (d) Explain how this effect can be used to describe the operation of a lightning rod. 44. A car with a vertical antenna is driven in an easterly direction along the equator of Earth. (a) Describe how a current is induced in the antenna. (b) Determine the direction of the induced current. 45. A small charge of 2.0 C experiences an electric force of 3.0 105 N to the left when it is placed in the electric field of another larger source charge. Determine the strength of the electric field at this point. 46. Calculate the kinetic energy gained by a proton that is allowed to move between two charged parallel plates with a potential difference of 2.0 104 V. What maximum speed could the proton acquire? 47. Two spheres with charges of 4.00 C and 3.00 C are placed 0.500 m apart. At what point between the two charges must a third charge of 2.
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00 C be placed so that the net electrostatic force acting on this charge is zero? 48. Two oppositely charged parallel plates have a voltage of 2.5 104 V between them. If 1.24 J of work is required to move a small charge from one plate to the other, calculate the magnitude of the charge. 49. A wire that is 0.30 m long, lying perpendicular to an external magnetic field of magnitude 0.50 T, experiences a magnetic force of 0.11 N. Determine the current in the wire. 50. A charge of 3.0 106 C is placed 0.50 m to the right of another charge of 1.5 105 C. Determine (a) (b) the net electric field at a point midway between the two charges the electrostatic force of attraction acting on the two charges 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 629 51. Touching a Van de Graaff generator can result in a painful shock. If a small length of conducting wire is bent into an L-shape and taped to the ball of the generator, so that a shaft of wire projects outward, you will not get a shock when you touch the ball. (a) How does the ball of the generator initially acquire a negative charge? (b) What process of charging objects is involved in the painful shock you receive initially? (c) How does the shaft of wire that is taped to the ball prevent a shock? (d) What is the function of the conducting strips projecting from the airplane wing tip shown in the photo below? 52. A small foam plastic ball with a mass of 0.015 kg and coated with a conducting material is suspended by a string 0.75 m long. If the ball is initially given a charge of magnitude 1.5 108 C, and another charged ball with a charge of magnitude 2.5 108 C is brought near, the hanging ball swings to a position 1.0 cm from its equilibrium position. Calculate the electrostatic force of repulsion acting on the two charges. Draw a free-body diagram. 53. The largest electric field that can exist between two oppositely charged plates with an air gap between them is 3.00 106 N/C. If this electric field limit is exceeded, then a discharge of charge occurs between the plates, resulting in a spark. If the voltage between the plates is 500 V, what is the minimum distance between the plates before a spark occurs? 54.
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An electron with a mass of 9.11 1031 kg and a charge of magnitude 1.60 1019 C is 5.29 1011 m from a proton with a mass of 1.67 1027 kg and a charge of magnitude 1.60 1019 C. Calculate (a) the gravitational force of attraction between the two masses (b) the electrical force of attraction between the two charges (c) how many times greater the electrical force is than the gravitational force 55. An alpha particle, a proton, and an electron, travelling at the same speed, enter regions with external fields. Compare the motion of these particles as they travel perpendicularly through the same (a) magnetic field (b) electric field (c) gravitational field 56. Given ebonite and glass rods and strips of fur and silk, describe the procedure to charge an electroscope negatively by (a) conduction (b) induction (include the grounding step) 57. A particle with a mass of 2.00 1026 kg and a charge of magnitude 6.40 1019 C is fired horizontally along the surface of Earth in an easterly direction. If the particle passes through a metal detector with a magnetic field of 2.00 104 T in a northerly direction, at what speed must the charged particle be travelling so that the magnetic force counteracts the gravitational force of Earth on the particle at this position? What is the charge of the particle? 58. A helium ion (He2) with a charge of 3.20 1019 C and a mass of 6.65 1027 kg is injected with a speed of 3.00 106 m/s perpendicularly into a region with a uniform magnetic field of 3.30 T. (a) Calculate the magnitude of the deflecting force on the helium ion. (b) If this deflecting force causes the helium ion to travel in a circular arc, what is the radius of the arc? 59. Sphere A, with a charge of 3.50 C, is 4.35 cm to the left of sphere B, with a charge of 2.44 C. Calculate the net electrostatic force on a third sphere C, with a charge of 1.00 C, if this sphere is placed (a) midway on a line joining charges A and B (b) at a point 2.50 cm to the right of charge B (c) at a point 2.50 cm directly down from charge B 60. Calculate the potential difference required to accelerate
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a deuteron with a mass of 3.3 1027 kg and a charge of magnitude 1.6 1019 C from rest to a speed of 8.0 105 m/s. Unit VI Forces and Fields 629 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 630 61. A uniform electric field of 7.81 106 N/C exists between two oppositely charged parallel plates, separated by a distance of 3.20 mm. An electron, initially at rest and with a mass of 9.11 1031 kg, is injected into the electric field near the negative plate and accelerates toward the positive plate. The electron passes through a “hole” in the positive plate and then travels perpendicularly through an external magnetic field with a magnitude of 1.50 T. (a) What is the voltage between the oppositely charged plates? (b) Calculate the maximum speed acquired by the electron between the plates. Ignore relativistic effects. (c) Determine the magnetic force on the electron as it passes through the magnetic field. (d) Describe the motion of the electron through the electric field and then through the magnetic field. 62. A bar magnet is moved toward a coil of wire that is connected to a sensitive galvanometer, as shown in the figure below. Extensions 63. Explain why computer hard disks are encased in metal. 64. Explain why it is safe to remain inside a vehicle during a lightning storm. 65. Are gravitational, electrical, or magnetic forces responsible for the formation of a black hole in space? Explain your answer. 66. Why must technicians who work on very sensitive electronic equipment be grounded? 67. Describe why Earth’s magnetic field creates a “magnetic bottle.” 68. Charged particles in cosmic rays from the Sun are trapped in Earth’s magnetic field in two major radiation belts, called Van Allen radiation belts. The first belt is about 25 500 km and the other is about 12 500 km above the surface of Earth. Explain why spacecraft must avoid these radiation belts. 69. Compare the motion of a negatively charged particle as it travels through a gravitational, electric, or magnetic field in a direction (a) perpendicular to the field N S (b) parallel to and in the same direction as the field G (a) Explain what happens to the galvanometer readings as the north pole of the bar magnet approaches the coil of wire. (b) Describe how Lenz�
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�s law influences the movement of the magnet toward the coil of wire. (c) Explain what happens to the galvanometer readings as the north pole of the magnet is pulled out of the coil of wire. (d) Describe how Lenz’s law influences the movement of the magnet away from the coil of wire. 70. Why is lightning more likely to strike pointed objects than rounded objects on the ground? 71. Describe the process by which the steel beams in a high-rise building can become magnetized. Why does this effect not happen in homes built with wood beams? 72. High-voltage power lines operate with voltages as high as a million volts. Explain why a bird can perch on a power line with no effect, but must be careful not to touch two nearby power lines. 73. Explain the difference between the dip angle and the angle of declination of Earth’s magnetic field. How do these two angles affect the operation of a directional compass in Alberta? 630 Unit VI Forces and Fields 12-PearsonPhys30-Chap12 7/24/08 3:35 PM Page 631 74. An electron is at rest. Can this electron be set in motion by a magnetic field? Explain your answer. 75. Given only a loop of wire connected to a sensitive galvanometer, describe a simple experiment that could be conducted to prove the existence of a changing magnetic field near a high-voltage power line. 76. The image of the magnetotactic bacterium, as seen under a microscope, reveals a row of magnetite crystals within its cellular structure. Make a hypothesis regarding the purpose of this row of magnetite crystals, and design and describe an experiment to test your hypothesis. (a) Plot a graph to determine the relationship between ⏐B⏐and r. (b) What is the relationship between ⏐B⏐and r? (c) What quantities need to be graphed in order to straighten the graph? (d) Complete a new table of values to straighten the graph. (e) Plot a new graph with the variables needed to straighten the graph. (f) Use the graph-slope method and the data from the graph to determine the value of. (g) Use the formula-data substitution method to determine the value of. Skills Practice 77. Draw a Venn diagram to review the similarities and differences between electric and magnetic fields. 78. Design an experiment to determine the direction
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of gravitational, magnetic, and electric fields. 79. Construct a concept map for solving a two- dimensional electrostatic force problem involving three charges at the corners of a triangle. 80. Design an experiment to show that electrostatic forces vary with the inverse square law. Self-assessment 82. Describe to a classmate which field concepts and laws you found most interesting when studying this unit. Give reasons for your choices. 83. Identify one topic pertaining to fields studied in this unit that you would like to investigate in greater detail. 84. What concept in this unit did you find most difficult? What steps could you take to improve your understanding? 81. A current-carrying wire has a magnetic field 85. Assess how well you are able to explain electric potential energy and electric potential. Explain to a classmate how you determine a reference point. e TEST To check your understanding of forces and fields, follow the eTest links at www.pearsoned.ca/school/physicssource. surrounding it. The strength of this magnetic I field can be found using the formula ⏐B⏐, r 2 where ⏐B⏐is the magnitude of the magnetic field surrounding the wire, in teslas, is the constant of permeability for space, in Tm/A, I is the current in the wire, in amperes, and r is the distance to the wire, in metres. An experiment is performed to establish the value of the constant,. The current was constant through the wire at 5.00 A, and the strength of the magnetic field was measured at various distances from the wire. The data recorded are given in the table below. Distance from the wire, r (m) Magnitude of the magnetic field, |B| (T) 0.100 0.200 0.300 0.400 0.500 6.28 3.14 2.09 1.57 1.26 Unit VI Forces and Fields 631 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page 632 U N I T VII Electromagnetic Electromagnetic Radiation Radiation (a) (b) (c) This series of photos of a supernova remnant shows the various types of electromagnetic radiation that are being emitted from the supernova remnant. The large image is a composite. Images (a) and (b) were taken by the Chandra space telescope (high-energy and low-energy X ray), image (c)
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was taken by the Hubble space telescope (visible part of the spectrum), and image (d) by the Spitzer space telescope (infrared). 632 Unit VII 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page 633 (d) Unit at a Glance C H A P T E R 1 3 The wave model can be used to describe the characteristics of electromagnetic radiation. 13.1 What Is Electromagnetic Radiation? 13.2 The Speed of Electromagnetic Radiation 13.3 Reflection 13.4 Refraction 13.5 Diffraction and Interference C H A P T E R 1 4 The wave-particle duality reminds us that sometimes truth really is stranger than fiction! 14.1 The Birth of the Quantum 14.2 The Photoelectric Effect 14.3 The Compton Effect 14.4 Matter Waves and the Power of Symmetric Thinking 14.5 Coming to Terms with Wave-Particle Duality and the Birth of Quantum Mechanics Unit Themes and Emphases • Diversity and Matter • Nature of Science • Scientific Inquiry Focussing Questions The study of electromagnetic radiation and its behaviour requires interpretation of evidence to form theories and models. As you study this unit, consider these questions: • What roles do electricity and magnetism play in electromagnetic radiation? • Does electromagnetic radiation have a wave or a particle nature? • What experimental evidence is required to decide whether electromagnetic radiation has a wave or a particle nature? Unit Project From Particle to Quantum • When you have finished this unit you will be able to use experimental evidence concerning electromagnetic radiation to describe our current understanding of light. You will be able to craft a multi-media presentation of highlights in the development of this understanding. Unit VII Electromagnetic Radiation 633 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page 634 C H A P T E R 13 Key Concepts In this chapter, you will learn about speed and propagation of electromagnetic radiation reflection, refraction, diffraction, and interference total internal reflection and Snell’s Law Learning Outcomes When you have completed this chapter, you will be able to: Knowledge describe how accelerating charges produce EMR explain the electric and magnetic fields associated with the propagation of EMR explain how the speed of EMR can be investigated and calculated describe reflection, refraction, total internal reflection, and simple optical systems describe diffraction, interference, and polarization and how this supports the wave model of light Science, Technology, and
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Society explain that scientific knowledge may lead to the development of new technologies and new technologies may lead to scientific discovery explain that scientific knowledge is subject to change as new evidence is found and as laws and theories are tested and subsequently restricted, revised, or reinforced based on new findings and new technologies 634 Unit VII The wave model can be used to describe the characteristics of electromagnetic radiation. Figure 13.1 A Fresnel lens captures light from a lamp and redirects it into a concentrated beam. This 19th-century lens technology used many prisms to produce a light that could be seen for great distances. When you open your eyes in the morning, the first thing you see is light. In fact, it is the only thing you see. In general, what we perceive with our eyes is a combination of many colours, each with varying brightness, but all coming together through the optical system of our eye. It is light or, more broadly speaking, a form of electromagnetic radiation (EMR), which produces an image in the human eye. But what is EMR? How is it produced? How is it transmitted and at what speed? These questions are historically significant, guiding the direction of research and debate over the ages. Finding answers has helped us understand the fundamental principles of our universe and has enabled advances in technology in areas such as lenses (Figure 13.1), fibre optics, and digital devices. Radios, lasers, global positioning systems, and compact discs (CDs) are examples of devices that depend on an understanding of EMR. The nature of light has long been a topic of intrigue and debate. Early Greek scientific thought was based on the work of Aristotle and Euclid, who concerned themselves with the physical and geometric mechanisms of visual perception. A more modern debate ensued as detailed evidence began to be collected about the wave and/or particle nature of light. Isaac Newton (1642–1727) and Christiaan Huygens (1629–1695) defined this debate with evidence from early experiments. Newton put forward the particle or corpuscular theory, while Huygens supported the wave model. Today, both the particle and wave models of light have some validity and contribute to our present understanding, described by the quantum model of light. 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page 635 In this chapter, you will see that visible light is only one form of EMR. You will also discover that the concepts of electrical charge and magnetic fields, explored in
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Chapters 10–12, come together when you consider EMR. You will investigate the generation, speed, and propagation of EMR, and see how light can be reflected, refracted, diffracted, and polarized. In each instance, the wave and particle nature of light is revealed, helping us to understand these complex phenomena and apply them in new technologies. 13-1 QuickLab 13-1 QuickLab Investigating Light as a Form of EMR Problem How can the properties of light be revealed? Materials light source 2 polarizing filters Procedure 1 Place a light source in front of one polarizing filter and observe whether the light is able to penetrate the filter.??? Figure 13.2 Think About It 2 Turn the filter 90 degrees as shown in Figure 13.2 and observe whether the light is able to penetrate the filter. 3 Place the filters at 90 degrees to one another and observe whether the light is able to penetrate the filters. 4 Slowly rotate one of the filters and observe whether the light is able to penetrate both filters at a variety of angles and positions. Questions 1. Can visible light pass through a single polarizing filter, regardless of how it is oriented? 2. Can visible light pass through two filters when they are aligned at 90 degrees to one another? 3. Describe one characteristic of EMR indicated by your observations of light in this QuickLab. 4. Do the observations made in this investigation explain what EMR is, or do they simply reveal one characteristic of EMR? 1. Do your observations from 13-1 QuickLab give information on whether electromagnetic radiation has wave or particle characteristics? Explain. 2. What experimental evidence would be needed to decide whether electromagnetic radiation has a wave or a particle nature? Discuss your answers in a small group and record them for later reference. As you complete each section of this chapter, review your answers to these questions. Note any changes in your ideas. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 635 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page 636 electromagnetic radiation: radiant energy in the form of a wave produced by the acceleration of electrons or other charged particles. EMR does not require a material medium; can travel through a vacuum. frequency: the number of cycles per unit of time wavelength: the distance between adjacent points that vibrate in phase with one another in a wave info BIT Seeing is a photochemical process that is sensitive to certain wavelengths of electromagnetic radiation
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. When EMR is absorbed by the tissues in the human eye, a compound called retinal changes in physical form from bent to straight. The retinal molecule, in turn, is connected to a membrane-bound protein called opsin forming the complex molecule called rhodopsin. When the retinal molecule changes its form, it separates from the rhodopsin and the opsin triggers a nerve cell to signal the brain that light has been seen. 13.1 What Is Electromagnetic Radiation? Electromagnetic radiation (EMR) is radiant energy, energy that travels outward in all directions from its source. There are different types of EMR, some very familiar, and others rarely mentioned outside scientific discussions. Electromagnetic radiation includes AM/FM radio waves, microwaves, heat, visible light (red to violet), ultraviolet radiation, X rays, and gamma rays. Electromagnetic radiation types are identified based on their frequency, wavelength, and source. The energy sources that produce EMR vary greatly, from nuclear reactions in the Sun, which generate gamma radiation, to chemical reactions in the human body that generate infrared radiation (heat). EMR is produced by the acceleration of charged particles resulting in transverse waves of changing electric and magnetic fields that can travel through space without the need of a material medium. All forms of EMR travel at the same speed, commonly referred to as the speed of light (c, in a vacuum), equal to 3.00 108 m/s. EMR does not always travel directly from the source to the observer. As a result, it can be observed in one of two ways: directly from the source indirectly, as reflected or transmitted radiation Burning magnesium (Figure 13.3) allows both methods of observing radiation. Other objects, such as the text you are reading now, do not produce radiation. You are able to read the text because it is reflecting radiation (light) from another source. If you are wearing glasses, this radiation is transmitted through your lenses. The evidence obtained in 13-1 QuickLab indicates one unique characteristic of electromagnetic radiation. However, visible light, radiant energy that the eye can detect, only makes up a small portion of the spectrum of EMR present in our universe. Concept Check Figure 13.3 When magnesium burns, you can observe radiation directly from the reaction and indirectly from the radiation that is reflected from the smoke. Can infrared radiation reflect off objects and be observed by the human body? Explain and give examples. 636 Unit VII Electromagnetic Radiation 13-
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PearsonPhys30-Chap13 7/24/08 3:43 PM Page 637 electromagnetic spectrum: all types of EMR considered in terms of frequency, wavelength, or energy e WEB To learn more about the different types of EMR and the trends seen across the spectrum, follow the links at www.pearsoned.ca/school/ physicssource. Types of Electromagnetic Radiation The electromagnetic spectrum is the term applied to all the types of EMR considered together in terms of frequency, wavelength, or energy. All parts of the spectrum are found, with varying intensity, in our natural environment. We are most familiar with the visible spectrum since we sense it directly with our eyes. The infrared spectrum is sensed as heat and the ultraviolet spectrum includes radiation that can damage living cells, often causing a physiological response such as sunburn. Other parts of the spectrum may be present as background radiation. Natural background radiation originates from two primary sources: cosmic radiation and terrestrial sources. Cosmic radiation from deep space interacts with atoms in the atmosphere producing X rays and unstable isotopes. Terrestrial sources of gamma radiation include radioactive isotopes of uranium, radon, potassium, and carbon. Also, aboveground nuclear tests of the 1940s–1960s combined with nuclear accidents such as Chernoybl have scattered a substantial amount of radioactive material within our environment. Figure 13.4 shows the overlap of frequencies and wavelengths among the different forms of EMR in the electromagnetic radiation spectrum. 104 108 1012 1016 1020 1024 frequency f (Hz) radio waves microwaves infrared visible light ultraviolet X rays gamma rays 3 x 104 m 3 m 3 x 10–4 m 3 x 10–8 m 3 x 10–12 m wavelength λ (m) a) Electromagnetic spectrum 800 nm 700 nm 600 nm 500 nm 400 nm wavelength λ (m) infrared red yellow blue b) Visible spectrum orange green violet ultraviolet Figure 13.4 The electromagnetic radiation spectrum showing the visible range The relative energy of the different types of EMR varies with frequency across the spectrum. Table 13.1 compares the various sources and characteristics of the radiation found in the EMR spectrum. e WEB To learn more about the medical uses of different types of EMR, follow the links at www.pearsoned.ca/school/ physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 637 13-PearsonPhys30-Chap13 7/24/08 3:43 PM Page
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638 Table 13.1 The Electromagnetic Spectrum: Characteristics Method of Production Characteristics Problems oscillation of electrons in an electric circuit like an antenna oscillation of electrons in special tubes and solid state devices motion of particles, transitions of valence electrons in atoms and molecules higher-energy transitions involving valence electrons in atoms long wavelength allows a large amount of diffraction making it useful for longdistance communication, e.g., PC broadband shorter wavelength reduces diffraction for short-distance communication; frequency matches the natural resonant frequency of water molecules; used in microwave ovens and cell phones causes object absorbing it to become warm; used for remote sensing, night vision scopes, and identification of sources of heat reflects off small objects, making them visible; diffracts around very small objects, making them invisible requires government regulations to control transmission and avoid interference may be linked to some forms of cancer; causes damage to living tissue due to heating of water molecules within tissues significant exposure can burn tissue limits the size of objects that can be seen even higher-energy transitions involving valence electrons in atoms easily absorbed by objects; causes fluorescence of some materials, tanning in humans; kills bacteria may cause sunburn; prolonged exposure can cause mutations and cancer in humans Type of Electromagnetic Radiation Radio and Radar f 104 1010 Hz 104 102 m relative energy: very low Microwaves f 109 1012 Hz 101 104 m relative energy: low Infrared f 1011 4.0 1014 Hz 103 7.5 107 m relative energy: low Visible f 4.0 1014 7.5 1014 Hz 7.5 107 4.0 107 m relative energy: medium Ultraviolet f 7.5 1014 1017 Hz 4.0 107 109 m relative energy: high X ray f 1017 1020 Hz 109 1012 m relative energy: very high penetrates most matter and is absorbed by denser material (like bone or metal); destroys carcinogenic or mutant cells; used for medical imaging in humans and in industry penetrates matter very deeply; destroys carcinogenic or mutant cells on a local scale; used to probe the structure of matter and in industrial imaging penetrates matter very deeply; study of cosmic rays allows investigators to formulate ideas about the universe can cause mutations and cancer in humans can cause radiation sickness and death can cause radiation sickness and death transitions of electrons in an atom or the sudden acceleration of high-energy free electrons Gamma f 1019 1024 Hz 1011 1016 m relative energy: extremely high decomposition of unstable nuclei,
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either spontaneously or by the sudden negative accelerations from highenergy particle accelerators Cosmic f 1024 Hz and greater 1016 m and less relative energy: extremely high bombardment of Earth’s atmosphere by extremely high-energy particles from space 638 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 639 Competing Models of Electromagnetic Radiation Historically, investigators have tried to explain transmission, reflection, refraction, and absorption, and the other characteristics common to all types of EMR by using models. The historical particle model describes EMR as a stream of tiny particles radiating outward from a source. A particle is a discrete unit of matter having mass, momentum (and thus kinetic energy), and the ability to carry an electric charge. The particle model of EMR is the simplest of all the models. It is supported by the facts that EMR propagates in straight lines, can be reflected, and can be absorbed. The pool ball (particle) in Figure 13.5 exhibits particle characteristics. It can travel in a straight line, obey the law of reflection when it bounces off a side rail, and be absorbed by the table when it drops into a pocket. info BIT The Particle Theory of Light, also known as the Corpuscular Theory of Light, was put forward in Newton’s Opticks, published in 1704. particle model: describes EMR as a stream of tiny particles radiating out from a source wave model: describes EMR as a stream of transverse waves radiating out from a source Figure 13.5 A pool ball on a pool table exhibits the particle characteristics of linear movement, reflection, and absorption. Figure 13.6 A water wave is a transverse wave that transfers energy in the form of a disturbance. waterbed sponge A second model, the wave model, describes EMR as a stream of transverse waves radiating outward from a source. As you learned in Chapter 8, a wave is a transfer of energy in the form of a disturbance. The energy transfer usually occurs in, but is not limited to, a material medium like water. A water wave travels in a straight line, reflects from surfaces, and can be absorbed (Figure 13.6). For example, water waves, which continually reflect inside a waterbed, transfer unwanted energy that can be absorbed by a sponge (Figure 13.7). EMR, however, does not require the presence of a medium. The modern wave model of light
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has its origins in the 17th century when the Dutch mathematician and scientist Christiaan Huygens argued that light consisted of waves. He suggested light waves could interfere to produce a wave front, travelling outward in a straight line. At the time, however, Huygens’ wave theory was overshadowed by the immense scientific popularity of Newton’s particle theories. The wave model gained further support from evidence produced by the work of Thomas Young (Figure 13.8) and his now-famous two-slit experiment of 1801. The details and implication of the experiment are set out in section 13.5, but in general, Young was able to show that a beam of light, when split into two beams and then recombined, shows interference effects that can only be explained by assuming that light has wavelike properties. Figure 13.7 Waves in a waterbed exhibit the wave characteristics of linear movement, reflection, and absorption. Figure 13.8 Thomas Young (1773–1829) Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 639 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 640 e WEB To learn more about Young’s experiment and its relationship to the models of EMR, follow the links at www.pearsoned.ca/school/ physicssource. If light behaves as a particle, then one would expect to observe a bright region wherever the light emanating from the slits reaches a distant screen (Figure 13.9(a)). But Young obtained a result similar to that shown in Figure 13.9(b), where a pattern of light of varying intensity is observed after the light passes through the two slits. This pattern is similar to two-source interference patterns in water waves, as seen in section 8.3. Young’s experiment thus presented strong evidence for the wave model of light. schematic representation of particle theory prediction schematic representation of wave theory prediction (a) (b) screen screen Figure 13.9 When light is incident on two small slits, it is diffracted as it passes through each slit and an interference pattern is observed that supports the wave model of light. a) A schematic representation of the particle theory prediction b) A schematic representation of the wave theory prediction Planck and Einstein By the end of the 19th century, both the particle and wave models of EMR were supported by scientific evidence. In 1900, Max Planck (1858
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–1947) proposed a radically new model to explain the spectrum of radiation emitted from a perfectly black object. In a mathematical derivation Planck assumed that all of the vibrating molecules, “oscillators,” in the black body could vibrate with only specific, discrete amounts of energy. In doing so, he had to ignore the continuous distribution of energy of classical physics and introduce the concept of quanta, or discrete packets of energy. Planck was awarded the Nobel Prize for Physics in 1918, “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta.” In 1905 Einstein extended this quantum theory by proposing that light is emitted in quantized, tiny, massless particles, which are now called photons. Planck’s original theory thus evolved into the currently accepted quantum model of light, which is a combination of both the particle and wave models. This model describes light and all other electromagnetic radiation as discrete bundles or “packets” of energy. Each packet, or photon, has both particle and wave characteristics. In the quantum model, EMR has two aspects of behaviour, one being info BIT A quantum is a discrete unit of energy. The term originates from the Latin quantus, which means “how much.” “Quanta” is the plural of “quantum.” The Quantum Theory was so revolutionary that its formulation distinguishes the shift from classical physics to what we now call modern physics. photon: (from the Greek word meaning “light”); a discrete packet of energy associated with an electromagnetic field quantum model: light and all other EMR are discrete bundles of energy, each of which has both particle and wave characteristics 640 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 641 wavelike and the other being particle-like. Quantum theory encompasses these two types of behaviour while, at the same time, challenging both the classical wave and particle models. Planck’s quantum idea challenged the wave model by proposing that EMR does not deliver energy in a continuous form as a wave would, but rather, it delivers energy in small bundles. At the same time, his idea challenged the particle model by limiting the energy of the particles to certain discrete values, a condition not possible for a Newtonian particle. Concept Check Consider the wave model of light and the particle model of light. 1. Which of the
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following best describes the word “model”? (a) simplified description of a complex entity or process; (b) a representation of something on a smaller scale; (c) both a and b. 2. Explain your answer from part 1 and provide several examples of models that are used in a similar way. 3. The EMR spectrum includes many classes of radiation. Describe the wave characteristics used to classify each type of radiation on the spectrum. How are these characteristics related to the energy of the radiation? Maxwell’s Electromagnetic Theory, 1865 In 1865, James Clerk Maxwell (1831–1879) proposed his Electromagnetic Theory, which synthesized earlier ideas and theories with the results of experiments to provide a theoretical framework for future studies. Maxwell proposed the idea that a changing electric field produces a changing magnetic field and that the interaction between these fields propagates as a wave through space. In his theory Maxwell linked concepts of electricity and magnetism so that we now call them “electromagnetic.” Maxwell based his theory on key phenomena observed by earlier investigators of electricity and magnetism. You have already met these ideas in Unit 6, but we will restate them here to appreciate the importance of Maxwell’s contribution. The Concept of the Electric Field As Faraday had proposed and as you saw in Chapter 11, an electric field surrounds any charged particle and an electrostatic force will act on another charged particle when it is placed in that field. The electric field strength can be calculated and electric field lines drawn to illustrate the region of influence. Electric field lines begin and end at a charge (Figure 13.10) and the number of field lines at any closed surface is determined by the total charge at the surface. This concept allows the idea of interaction between particles at a distance, even though there is no contact between them. e WEB To learn more about Max Planck, begin your search at www.pearsoned.ca/ school/physicssource. Project LINK How will you present the observations and theoretical concepts that culminated in the ideas of Planck and Einstein? electric field: a three-dimensional region of electrostatic influence surrounding a charged object magnetic field: a threedimensional region of magnetic influence surrounding a magnet Figure 13.10 Electric field lines are used to illustrate the region of influence between two oppositely charged particles. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 641 13-PearsonPhys30-Chap13 7/24
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/08 3:44 PM Page 642 S N Figure 13.11 The number of magnetic field lines that exit a magnetic material is equal to the number of magnetic field lines that enter a magnetic material — forming a closed loop. capacitor: two conductors, holding equal amounts of opposite charges, placed near one another without touching Maxwell’s Equations: a series of equations that summarized the relationships between electricity and magnetism and predicted the existence of electromagnetic waves and their propagation through space info BIT Maxwell was working with a system of units for electromagnetic theory, referred to as the CGS system. This system was derived from the base units of centimetre, gram, and second. The CGS system has largely been replaced by the SI system, based on the metre, kilogram, and second. The Concept of the Magnetic Field A magnetic field (as discussed in Section 12.1) is a three-dimensional region of magnetic influence surrounding a magnet. Magnetic field lines form a closed loop and represent the direction and magnitude of the magnetic field (Figure 13.11). Relationships Between Electricity and Magnetism Electrical current in a conductor produces a magnetic field perpendicular to the current (Oersted, Unit VI, page 587) and the strength of the magnetic field depends on the magnitude of the current (Ampere, Unit VI, page 607). This relationship is known as Ampere’s Law. Inversely, moving a conductor connected in a circuit through a magnetic field induces an electric current. The magnitude of the electrical current is directly related to the rate of change in the magnetic field (Faraday and Henry, Unit VI, page 611). This relationship is known as Faraday’s Law. Putting the Laws of Electromagnetism Together From this basis and through his work with capacitors, Maxwell developed the laws of electromagnetism. Maxwell’s work with capacitors showed that the electric field produced by a capacitor could have the same effect as a moving charge. In other words, a changing electric field will produce a changing magnetic field in the same manner as a changing electric current can produce a changing magnetic field. This information was extremely important because it showed that a conductor is not necessary for an electromagnetic wave to exist. Maxwell put these ideas together with incredible ingenuity using calculus, which is beyond the scope of this text. His theory is embodied in a series of equations known as Maxwell’s Equations. To visualise the relationship between the electric and magnetic fields, we will proceed step
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wise. Begin with a plane transverse wave that shows an electric field of varying strength (Figure 13.12). Next, consider two of the electric field lines on the transverse wave, an upward one at one location and the corresponding downward one at another. Imagine a path connecting the tips of these two electric field vectors, as shown by the dotted line in Figure 13.13. You can see that the electric field changes direction continuously from one direction to the opposite direction along a "closed path." E E direction of propagation of wave direction of propagation of wave E Figure 13.12 Plane transverse wave showing the electric field lines. Figure 13.13 A path connecting two oppositely directed electric field lines on the transverse wave shows the change in the electric field. 642 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 643 Now, consider the magnetic field that exists perpendicular to the electric field along this closed path. When the electric field is set in motion, the magnetic field will change along with the electric field, and similarly along a closed path. An electromagnetic wave will propagate in a direction perpendicular to both fields (Figure 13.14). The changing electric and magnetic fields will propagate, or radiate, through space in the form of a wave — an electromagnetic wave (Figure 13.15). Maxwell proposed that the electromagnetic wave consists of periodic variations in the electric and magnetic field strengths and that these variations occur at right angles to one another as the wave propagates. B E direction of propagation of wave Figure 13.15 Three-dimensional view of an electromagnetic wave Maxwell’s Predictions Maxwell’s equations not only correctly predicted the existence of electromagnetic waves, but also allowed him to make some predictions about the waves’ properties. 1. Electromagnetic waves are produced whenever an electric charge is accelerating. Therefore, as an electric charge oscillates, electrical energy will be lost, and an equivalent amount of energy will be radiated outward in the form of oscillating electric and magnetic fields. 2. When the electric charge is accelerated in periodic motion, the frequency of oscillation of the charge will correspond exactly to the frequency of the electromagnetic wave that is produced. 3. All electromagnetic waves will travel at a speed of 310 740 000 m/s and obey the universal wave equation (c f) relating speed, frequency, and wavelength. (Note that Maxwell’s theoretical prediction was not far from today’s currently accepted value
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of 3.00 108 m/s for the speed of light in a vacuum.) 4. The oscillating electric and magnetic fields will always be perpendicular to each other and perpendicular to the direction of propagation of the wave. 5. Electromagnetic waves should show all the phenomena associated with transverse waves: interference, diffraction, refraction, and polarization. It is Maxwell’s last prediction that supports the wave model of EMR and relates his predictions to experimental evidence. Interference, diffraction, polarization, and refraction, as they relate to the wave model of EMR, will be explored in sections 13.4 and 13.5. E B direction of propagation of wave Figure 13.14 The magnetic field lines (in red) allow us to visualize the magnetic field that exists perpendicular to both the electric field and the direction of the wave. electromagnetic wave: periodic variation in perpendicular electric and magnetic fields, propagating at right angles to both fields e WEB To learn more about ways of representing the relation between electric and magnetic fields, follow the links at www.pearsoned.ca/school/ physicssource. Compare and contrast the representations you find. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 643 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 644 Producing Electromagnetic Radiation — The Story of Accelerating Charge In 1887, the German physicist Heinrich Hertz (1857–1894) set up an experiment designed both to produce and to detect EMR. In his experimental apparatus, Hertz used a radiator, consisting of a pair of wires attached to both a high-voltage induction coil and a pair of capacity spheres. The wires were separated by a small gap and, given a sufficient quantity of opposite charge on each wire, a current would oscillate back and forth across the gap at a frequency of 109 Hz. With each oscillation a spark was produced when the moving charge ionized the air molecules on its way from one wire to the other. From Maxwell’s equations Hertz knew theoretically that this rapidly moving electric charge should produce EMR. A short distance away from the radiator a collector plate containing a small loop of wire, the antenna, was observed to detect the effect of EMR. While the radiator was in operation and when the radiator and the antenna were tuned to the same frequency, a spark was observed at the antenna indicating a potential difference and an electric current (Figure 13
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.16). 3. Electromagnetic waves create electric current in antenna loop; produces small spark in spark gap. 2. Spark produces electromagnetic waves. 1. Induction coil produces high voltage. e WEB To learn more about details of Hertz’s experiment, follow the links at www.pearsoned.ca/school/ physicssource. Figure 13.16 Hertz’s apparatus consisted of a high-voltage induction coil, a radiator that produces sparks, and an antenna loop. Relating Theory and Practice in Hertz’s Experiment The word “changing” appears a number of times in Maxwell’s original proposal. As Maxwell understood it, a “changing” electric field was crucial to creating an electromagnetic wave. This is where the induction coil was important in Hertz’s experimental design. The induction coil rapidly changed the electric field across the spark gap. When this electric field reached a sufficiently high value, the electrons in the wire “jumped” from one electrode to the other. As the charge was rapidly transferred, the electric field underwent a rapid change that caused a changing magnetic field, which then caused a changing electric field, and so on. An electromagnetic wave was produced and it radiated outward in all directions. 644 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 645 In Hertz’s experiment, when the receiver was tuned to the same frequency as the radiator, the induced current flow in the antenna oscillated at a frequency identical to that of the changing electric field in the radiator. This was conclusive evidence that Hertz’s device had indeed produced the EMR that was being observed at his antenna. Furthermore, he was able to measure the velocity of the waves by using a zinc reflector plate to produce a standing wave and moving a ring antenna within the standing wave. He could determine the magnitude and direction of the wave’s components and therefore the wavelength. Given the frequency of the radiator, the velocity of the wave could be calculated using the universal wave equation (v f). Hertz had produced and measured the wavelength and velocity of non-visible EMR for the first time. Two years later, in 1889, radio pioneers used this method to transmit the first radio waves across the English Channel. By 1901, the first radio waves were transmitted across the Atlantic Ocean from Cornwall, England to St. John’s, Newfoundland and a
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new age of technology had dawned. The production of EMR was one of the greatest scientific achievements of the 19th century, ushering in new possibilities and technologies that are commonplace today. M I N D S O N Going Wireless e WEB To learn more about Marconi and details of the first trans-Atlantic radio transmission, follow the links at www.pearsoned.ca/school/ physicssource. Wireless communication systems that transmit data are commonplace today. 1. Create a list of wireless data transmission technologies. 2. What process must be common to all wireless transmission devices? 3. What variable is used to control the acceleration, and hence the energy, of the accelerating electrons in a wireless device? 4. One radio transmission tower transmits at 5.0 kV. Another transmits at 10.0 kV. Which tower has a greater signal strength and why? 5. Cell phones communicate with cell towers that are dispersed throughout urban cities and transportation corridors. Explain the relationship between the distance that separates a cell phone from the nearest cell tower and the operating power of the cell phone. 6. Many studies have been conducted to investigate and determine if there is a relationship between brain cancer and cell phone usage. Why would one suspect a relationship between cell phone use and brain cancer? Although modern radio signals are generated without a spark, the technology operates in a way similar to Hertz’s original experiment. Radio waves are generated by rapidly changing the electric potential, or voltage, in the radiator tower. The oscillating voltage produces an oscillating electric field in the radiator tower (Figure 13.17). microphone radio signal (radio frequency) RF amp amp sound waves audio signal (audio frequency) AF mixer radiator tower Figure 13.17 Schematic of a simple radio transmitter audio wave (AF) carrier wave (RF) amplitude modulation (AM) Figure 13.18 Amplitude modulation. The audio signal and the carrier signal are mixed by modifying the amplitude of the carrier signal. audio wave (AF) carrier wave (RF) frequency modulation (FM) Figure 13.19 Frequency modulation. The signal is combined with a carrier wave to create a resultant wave with constant amplitude, but varying frequency. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 645 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 646 THEN, NOW, AND FUTURE From Analog Radio to Digital Wireless Analog Radio Technology The first application of Hertz’s discovery was made
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by Guglielmo Marconi (1874–1937). He recognized the potential for transmitting information using electromagnetic waves, by coding the information into dots and dashes, like the Morse code already used in telegraphy. Today, the dots and dashes have been replaced by an analog signal, which uses a continuous spectrum of values. In radio, sound waves in the range of 20–20 000 Hz (audible to humans) are converted into a weak electrical signal in an analog form. This wave is called the audio frequency signal (AF). The weak audio frequency is then amplified and sent to a mixer. In the mixer, a radio frequency (RF) called a carrier signal is added to the audio signal to produce an analog wave that contains two sets of information: the audio signal and the carrier signal. The carrier signal frequency is determined by the government and is unique to that station for that broadcast area. It is the carrier signal that you select when you tune your radio to the station. Finally, the mixed signal is amplified and delivered to the radiator tower, where the movement of electrons through the radiator wire will produce the corresponding form of electromagnetic wave (Figure 13.17). Television signals follow the same process, except that the video and audio portions are transmitted separately. Radio reception is accomplished by tuning the radio to the carrier signal and then removing, or demodulating, the wave. This reveals the audio signal, which is amplified and delivered to the speakers. AM or FM Radio? The mixing of the audio signal and the carrier signal occurs in one of two ways. The first is called amplitude modulation or AM (Figure 13.18). The second method is called frequency modulation or FM. The audio signal and the carrier signal are mixed by modifying the frequency of the carrier signal (Figure 13.19). Figure 13.20 Digital Wireless Cell Phones The cell phone and text messaging technology of today operates on a very different set of principles. When a cell phone is switched on, it registers with the mobile telephone exchange using a unique digital identifier. Once it has been identified, it stays connected to a cellular network of ground-based stations, commonly referred to as cellular tower base stations, which can be found on radio towers, buildings, in church steeples, or any location which is free of physinterference. ical obstacles and Figure 13.20 shows one such cellular tower base station. A cell phone communicates with the cell tower base station using radio waves. In turn the base station relays all information to
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and from the cell phone to another subscriber of the same cell phone network or through an interconnected public switched telephone network that uses fibre optic and copper land lines to transmit the data. When in use, a cell phone converts the analog voice signal of the person talking into a stream of digital data, which is received by the nearest cell tower base station that may be anywhere from 0.8 –13 km away. Each cell tower site has a lowpower microwave broadcast which can be picked up by cell phones. As the user moves, the cell phone constantly monitors the signals being received from various nearby cell tower stations, switching from one tower to another in such a way that the signal strength remains as high as possible. The stream of digital data is unique to each type of network technology and each operator is assigned a different radio frequency so that several networks can operate simultaneously at the same location and they will not interfere with one another. Wireless Technology for the Future Future uses of cellular phone technology are emerging in the market place as the networks evolve and consume broadcast frequencies formerly occupied by some television channels. New entertainment features will include: streaming video; music downloads; podcasts; smooth speech recognition and language translations; ebook features that use projection; barcode readers that direct users to Internet urls; global positioning systems and mapping combined with accelerometers to measure position and movements of the cell phone user. Questions 1. Suggest why cell phone user data, including geographical position and movement, could be considered an invasion of privacy. 2. How can multiple cell phone networks and traditional radio broadcasts all exist in the same geographical location and not interfere with one another? 3. Compare the carrier frequency of radio broadcasts with the unique digital identifier of a cell phone. What common purpose do they both serve? 4. Explain why a cell phone only works in certain areas and how changes in the signal strength will be observed by the cell phone user as he or she moves. 646 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 647 Concept Check 1. Consider two electric field lines on a transverse electromagnetic wave. Explain how a changing electric field will produce a changing magnetic field. 2. Why can electromagnetic radiation propagate through empty space, in the absence of a conductor? 3. An electromagnetic wave is produced if an electric charge accelerates. Explain how these two phenomena are linked. 13.1 Check and Reflect 13.1 Check and Reflect Knowledge 1. List two ways that electromagnetic radiation is detected. 2. List the different
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types of EMR in the spectrum and the ways in which they are classified. 3. Define EMR using the (a) particle model (b) wave model (c) quantum model 4. Describe the evidence that Thomas Young obtained from the two-slit experiment and explain how it supported the wave model of light. 10. If the magnetic field lines held within a closed path are constant and unchanging, will there be an electric field along the closed path? Explain. 11. If an electric charge is not moving, will it produce a magnetic field? Explain. 12. Explain why it is necessary for the technician to leave the room or wear protective clothing when a patient is being X rayed. 13. Describe how an antenna is affected by an electromagnetic wave. 14. Describe one method used to produce EMR with a known frequency. 5. Explain how the observations made by Extensions Hertz support the electromagnetic theories of Maxwell. 6. How did Maxwell’s study of capacitors add to his theories? 7. Why is the process of “changing” critical to the production of EMR? Applications 8. A high-voltage spark gap produced sparks with a frequency of 2.5 103 Hz. What is the frequency of the EMR observed at an antenna located nearby? 9. What properties need to be considered when choosing the material to build an antenna for a radio receiver? 15. Explain, with the aid of a transverse standing wave diagram, how the current in a ring antenna (such as the one used by Hertz) will be affected as the antenna is moved along the axis of the standing wave. How could this data be used to determine the wavelength of the standing wave? 16. Draw a schematic that shows, in general terms, the role of a satellite in delivering digital, high-definition, television signals between the signal provider and the customer’s home receiver. e TEST To check your understanding of the nature and production of electromagnetic radiation, follow the eTest links at www.pearsoned.ca/school/physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 647 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 648 info BIT Maxwell predicted the speed of EMR before Hertz experimentally verified it. Maxwell knew that electric and magnetic fields are not independent of one another and predicted a speed of propagation by dividing the constants, ke and km
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, which describe electric and magnetic forces between moving charged particles. Using today’s currently accepted values of ke and km, we can show that the magnetic field and the electrical field are related by a particular speed. 8.99 109 N 2 m 2 C s2 1.00 107 N 2 C 2 m 8.99 1016 s2 k e k m 8.99 1016 2.99 108 m m2 s2 s 13.2 The Speed of Electromagnetic Radiation In addition to producing and detecting electromagnetic radiation in his 1887 experiment, Hertz was able to show that the radiation was travelling at approximately 3.0 108 m/s. This was an empirical verification of a theoretical speed predicted by Maxwell years earlier, as described on page 643: All electromagnetic waves will travel at a speed of 310 740 000 m/s and obey the universal wave equation (c f) relating speed, frequency, and wavelength. Empirical Determination of the Speed of Electromagnetic Radiation Light travels so quickly that there is no apparent time difference between turning on a light and the light reaching your eyes – even when the light source is many kilometres away. Galileo (1564–1642) found this out in his attempt to measure the speed of light. His experiment was very basic: Galileo and an assistant stood, each with a covered lantern, on adjacent hills separated by about a kilometre. Galileo would uncover his lantern and as soon as his assistant observed Galileo’s lantern, he would uncover the second lantern so that Galileo would see it. Galileo used his pulse rate to measure the time difference. The time interval is actually extremely small – Galileo didn’t know it at the time, but light could travel 2 km in about 0.000 066 67 s, a time you couldn’t even measure with a digital stopwatch. Galileo realized that using his pulse as a timepiece was not appropriate. Earth longer time Sun Earth shorter time Io Jupiter Earth’s orbital diameter approximate Figure 13.21 Earth’s orbital diameter causes the eclipse of Io to occur at different times because of the extra distance the light must travel when Earth is farthest from Jupiter. 648 Unit VII Electromagnetic Radiation Olaus Rømer (1644–1710) and Christiaan Huygens used one of the four easily visible moons of Jupiter to calculate the speed of light. The period of revolution for Io (Jupiter’s brightest moon) had been accurately determined using large amounts
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of astronomical data collected over many years. Roemer knew that the moon Io should be eclipsed and disappear behind Jupiter at regular, periodic intervals (every 42.5 h). However, he discovered that Io appeared to be eclipsed later than scheduled when Earth was farther away from Jupiter and earlier than scheduled when Earth was closer to Jupiter. The time difference between the longest and shortest periods was 22 min (Figure 13.21). 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 649 Huygens reasoned that this discrepant behaviour could be due to the difference in the time taken for the light to travel from Jupiter to Earth. Following this reasoning, he estimated Earth’s orbital diameter and then determined the speed of light based on the following calculations. Orbital diameter 3.0 1011 m v d t Time difference 22 min 1320 s 3.0 1011 m 1320 s 2.3 108 m/s His calculations produced a value of 2.3 108 m/s — astoundingly close to the currently accepted value of 3.0 108 m/s. The value was so large it was rejected by the scientific community of the time and was not accepted until after both Huygens and Roemer were dead. In 1848, Armand Fizeau became the first person to successfully measure the speed of light at the surface of the planet. His apparatus consisted of a rotating toothed wheel, a light source, some lenses, and a mirror. In his experiment, light was allowed to pass through one of the gaps on the toothed wheel and travel toward the mirror (located on an adjacent hilltop 8.63 km away) where it was reflected back toward the spinning wheel (Figure 13.22). The toothed wheel was rotated such that the reflected light was blocked by a tooth in the wheel as it turned, and the observer would not be able to see the source light. Using the rotational frequency of the spinning wheel, Fizeau was able to determine the time it took the light to travel 8.63 km and back (17.26 km round trip) and therefore to determine the speed at which the light was travelling to the distant mirror and back. His experimentally determined value for the speed of light was 3.15 108 m/s — only 5% more than the currently accepted value. toothed wheel mirror glass observer path A path B light source observation point water in water out Figure 13
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.22 Fizeau’s original apparatus for measuring the time it takes light to travel the distance (8.63 km) and back Figure 13.23 Fizeau’s apparatus for measuring the speed of light in water light source Three years later Fizeau conducted an investigation to determine if the speed of light was affected by a moving medium such as water. In this experiment, water was pumped in opposite directions through two parallel tubes (Figure 13.23). Light from a single source was directed through both tubes, following paths A and B, of identical lengths, to an Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 649 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 650 observation point. Fizeau argued that if the speed of the light waves changed as a result of travelling through the water tubes, the waves arriving from path A and B would produce an interference pattern at the observation point. A device called an interferometer was used to observe this interference. The experiment proved that light would travel at different speeds in different media. The interference pattern produced was also very strong evidence that light was a wave. octagonal set of mirrors observer telescope path of light Mount Wilson 35 km light from source curved mirror Mount St. Antonio Figure 13.24 Michelson’s experimental apparatus to measure the speed of light Practice Problems 1. Students using a 12-sided set of rotating mirrors in an experiment similar to Michelson’s determine the speed of light to be 2.88 108 m/s. The mirrors are located 30.0 km from the fixed mirror. What is the frequency of rotation if 1/12th of a rotation occurs before the light returns? 2. Light reflected from an 8-sided set of rotating mirrors travels 15.0 km to a distant fixed mirror and back. At what frequency is the set rotating if it has turned 1/8th of a rotation before the light returns? 3. An 8-sided set of mirrors, similar to Michelson’s, rotating with a frequency of 500 Hz, is located 36.0 km away from a fixed mirror. If the returning light is observed in the system, at what speed is the light travelling? Answers 1. 400 Hz 2. 1.25 103 Hz 3. 2.88 108 m/s 650 Unit VII Electromagnetic Radiation Michelson’s Experiment Building on Fizeau’s 1848 experimental design and the work
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of others, Albert Michelson performed an experiment in 1905, using a rotating set of mirrors instead of a toothed wheel. A very intense light source was directed at an 8-sided, rotating set of mirrors, which reflected the light toward a curved mirror located 35 km away. After travelling 35 km, the light was reflected back toward the rotating mirrors. If the rotating mirrors had made 1/8th of a rotation just as the light returned, the returning light could be observed in a telescope (Figure 13.24). For the returning light to be observed in the telescope, the rotating mirrors had to turn at a very precise frequency. Michelson knew the light had to travel a round trip distance of 70 km, and if the rotating mirrors had made 1/8th of a rotation (or a multiple of an 1/8th rotation) between the time the light left and then returned, the light could be observed in the telescope. All Michelson needed to measure was the frequency of rotation and he could determine the time required for 1/8th of a rotation. When his mirror rotated at 32 000 rpm (533 Hz), he could observe the light in the telescope. Example 13.1 The set of rotating mirrors in Michelson’s experiment was rotating at 533 Hz and the curved mirror was located 35.0 km away. Show how Michelson determined the speed of light from these data. Given distance between rotating and curved mirrors 35.0 km 3.50 104 m f 533 Hz Required speed of light (c) Analysis and Solution c d t The light must travel from the set of mirrors to the curved mirror and back again, so the round-trip distance is twice the given distance. d 2 3.50 104 m 7.00 104 m 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 651 For the light to follow the path illustrated in Figure 13.24, the rotating set of mirrors must complete a minimum of 1/8th of a rotation while the light is travelling between the rotating mirrors and the curved mirror. Therefore, the minimum travel time (t) is equal to 1/8th of the period of rotation1.876 103 s) 8 d c t 4 m 0.0 7 4 s 0.3 1 1 4 5 0 2 1 Hz 533 1.876 103 s 2.345 104 s 2.98 108 m/s Paraphrase According to Michelson’s observations, the speed
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of light in air was found to be 2.98 108 m/s. 13-2 QuickLab 13-2 QuickLab Measuring the Speed of EMR Problem Microwaves are a type of electromagnetic radiation and travel at the speed of light. They have a defined wavelength and frequency when they travel in a vacuum or in air. The universal wave equation v frelates the speed of the wave to its frequency and wavelength, where f is the frequency of the radiation, and is the wavelength. Each microwave oven generates microwaves at a given frequency, usually 2450 MHz. If the wavelength of the microwave can be measured, its speed can be determined using the universal wave equation. Materials microwave oven microwave-safe dish marshmallows Procedure 1 Pack a solid layer of marshmallows in a casserole or microwave-safe dish. 4 Remove the dish from the oven and measure the distance between adjacent soft spots. 5 Determine the average separation distance between several soft spots. This distance is equal to half the wavelength of a microwave. 6 Calculate the wavelength by multiplying the average separation distance by two. 7 Record the frequency of the microwave, which will be indicated on the back of the microwave oven. Questions Use the wavelength and frequency of the microwave to answer the following questions. 1. What is the speed of the microwaves? 2. What is the percent error of your calculated value, when compared with the currently accepted value for the speed of light (3.00 108 m/s)? 3. What are several possible sources of error that 2 Remove the turntable from the microwave oven. could affect your calculation? 3 Place the dish in the oven and cook until the marshmallows begin to melt in four or five locations. 4. Compare the wavelength of microwaves with that of radio waves and visible light waves. Why are they called microwaves? Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 651 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 652 13.2 Check and Reflect 13.2 Check and Reflect Knowledge 1. Why was Galileo’s original experiment to determine the speed of light unsuccessful? 2. Explain why, in a Michelson-type experiment, the rotating mirrors have to turn at a very precise frequency in order for light to reach a stationary observer. 3. Explain why the periodic motion of Jupiter’s moon Io appears to be constantly changing when observed from Earth. Applications 4. In measuring the
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speed of light, the difference in eclipse times for Jupiter’s moon Io, is measured. If an eclipse occurs 24 min later than expected, and Earth’s orbital diameter is 3.0 1011 m, calculate the speed of light. 5. A communications satellite is in orbit around Earth at an altitude of 2.00 104 km. If the satellite is directly above a groundbased station, how long does it take a signal to travel between the satellite and the station? 6. Using a similar approach to Michelson, a student sets up a 64-sided set of rotating mirrors, 8.00 km away from a fixed mirror. What minimum frequency of rotation would be required to successfully measure the speed of light? 7. A 16-sided set of rotating mirrors is used to measure the time it takes light to travel a certain distance. At what frequency does the mirror need to rotate such that it makes 1/16th of a rotation in the time it takes light to travel 3.5 km and back again? 8. An 8-sided set of rotating mirrors rotates at 545 Hz in an experiment similar to that of Michelson. How far away should the fixed mirror be placed in order to correctly measure the speed of light? 652 Unit VII Electromagnetic Radiation 9. The speed of light was measured to be 2.97 108 m/s using a 16-sided set of rotating mirrors and a fixed mirror separated by 5.00 103 m. At what frequency was the mirror rotating? 10. Students who measure the speed of light using an experimental design similar to that of Michelson with an 8-sided set of rotating mirrors, make the following observations when light passes through the apparatus: (a) rotating mirror frequency 1.00 103 Hz (b) distance between fixed and rotating mirror 17.5 km Determine the speed of light based on the students’ recorded observations. 11. A 64-sided set of rotating mirrors is turning at 340 Hz. If a fixed mirror is located 6.55 km away from the rotating mirror and the light is reflected correctly, what value for the speed of light would be obtained? Extensions 12. Global positioning satellites (GPS) are used to pinpoint individual locations on the surface of Earth. A minimum of three GPS satellites are required, each with synchronized atomic clocks. Each satellite simultaneously sends a time signal to a GPS receiver on the surface. The time signal of each satellite is compared with the time on the receiver’s clock. (a) Explain how
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the exact distance between each satellite and the receiver can be determined. (b) Explain how the receiver’s location relative to all three satellites indicates an exact coordinate on Earth’s surface. e TEST To check your understanding of speed and propagation of electromagnetic radiation, follow the eTest links at www.pearsoned.ca/school/physicssource. 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 653 13.3 Reflection On a windless day, the smooth surface of a lake will reflect light and produce stunning images of the surrounding landscape and sky. But introduce the slightest disturbance, a gust of wind or a morning rain shower, and the image becomes distorted and blurred. With enough disturbance, the image on the surface of the water completely disappears. Figure 13.25 a) Reflection in a smooth lake b) Diffuse reflection due to windy conditions How do the images in Figure 13.25 form? Light travels in straight lines through a medium that is uniform. This characteristic is termed rectilinear propagation. The light falling on a flat, smooth, reflecting surface, such as a mirror, undergoes specular or regular reflection. In the case of the smooth lake, the crisp, clear images that form are the result of many parallel incident rays that reflect as parallel reflected rays (Figure 13.26). In the case of the rough lake, the blurred images that form are the result of many parallel incident rays that are scattered after reflecting in many different directions. This behaviour is called diffuse or irregular reflection (Figure 13.27). rectilinear propagation: movement of light in straight lines through a uniform medium Figure 13.26 Ray diagram of specular (regular) reflection Figure 13.27 Ray diagram of diffuse (irregular) reflection The Law of Reflection Particular terms are used to describe reflection. These terms are illustrated in the ray diagram shown in Figure 13.28 on the next page. Regular reflection will occur when an incident light ray contacts a polished, reflecting surface. The incident ray makes contact at the point of incidence and then it leaves the reflecting surface and becomes the reflected ray. If, at the point of incidence, an imaginary line, called the ray diagram: a diagram to show a result of a light ray interacting with a surface Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 653 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 6
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54 e SIM Find out more about the law of reflection from the use of simulations. Follow the eSim links at www.pearsoned.ca/school/ physicssource. plane mirror: a smooth, flat, reflecting surface law of reflection: the angle of reflection is equal to the angle of incidence and is in the same plane mirror plane θ i θ r incident ray reflected ray Figure 13.29 The Law of Reflection image I mirror O point object Figure 13.30 How two different observers see the image of an object located in front of a plane mirror virtual image: an image from which light rays appear to come; cannot be formed on a nonreflective surface or screen real image: an image from which light rays come; can be formed on a diffusely reflecting surface or screen normal line (N), is drawn perpendicular to the surface, then the angle formed between the incident ray and the normal line is called the angle of incidence. The angle formed between the reflected ray and the normal line is called the angle of reflection. point of incidence reflecting surface angle of incidence θ i angle of reflection θ r incident ray reflected ray N normal Figure 13.28 Ray diagram and terminology for regular reflection Light rays that are incident upon a plane mirror at 90 will be reflected directly back to the source. In this case, the incident and reflected rays are parallel to the normal line so that the angle of incidence and the angle of reflection are both zero. In a similar way, when a light ray contacts a mirror at an angle of 25 relative to the normal line, it will reflect at an angle of 25 relative to the normal line. In all cases, the angle of incidence will be equal to the angle of reflection. This is called the law of reflection (Figure 13.29). The angle of reflection is equal to the angle of incidence and is in the same plane. Image Formation in a Plane Mirror Look at the image of an object formed in a plane, flat mirror in Figure 13.30. What do the dashed lines represent? Are they real rays coming from a real object? The dashed lines are an extension of the reflected rays, causing your brain to “believe” that they are rays, originating from an object that is located behind the mirror. But this cannot be so. The light rays appear to be coming from an image, but this image is not real. The image formed in a plane mirror in this fashion is therefore called a virtual image. For example, if you were to hold a piece
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of paper behind the mirror where the image appears to be, you would not see the image on the paper. By contrast, a real image is one that can be formed on a diffusely reflecting surface, such as a piece of paper. A projector is a familiar device that produces real images on a screen. Ray diagrams can be used to show how images form with a reflecting surface, such as a mirror. 654 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 655 13-3 QuickLab 13-3 QuickLab Drawing a Ray Diagram Problem To construct a ray diagram of an image formed in a plane, flat mirror. Materials observer N ray 1 object mirror Figure 13.31 Procedure 1 Choose one point on the object, such as the tip of the candle flame, in Figure 13.31. 2 Draw an incident ray from the selected point on the object (adding a small normal line where the ray is incident on the surface of the mirror) (Figure 13.31). 3 Based on the law of reflection, the reflected ray leaves the mirror surface such that the angle of incidence is equal to the angle of reflection. Draw the reflected ray. 4 Use a dashed line to extend the reflected ray beyond the back surface of the mirror. 5 On your diagram, repeat steps 2 to 4 for a ray originating from the same location on the object, but at a slightly different angle. 6 Note that the viewer must be located where both reflected rays would enter the eye. 7 Repeat steps 2 to 6 and locate the point where the two dashed lines converge (meet). This is the image location for the first selected point on the object. Questions 1. How could you test whether the image formed in the mirror was real or virtual? 2. Is the image magnified or the same size as the object? 3. Will the image appear right side up (erect) or upside down (inverted)? 4. Could you use a ray diagram to determine if the word “HELP” would appear backwards in a mirror? Explain. M I N D S O N Image in a Mirror G F E D C O B A Figure 13.32 Woman in a Mirror Figure 13.32 shows a woman looking in a mirror. 1. How far behind the mirror does her image form, if she is standing 50.0 cm in front of the mirror? Explain how the ray diagram is used to determine this distance. Is the image real or virtual
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? 2. 3. According to the ray diagram, two sections of the mirror could be removed and her entire face would still be visible. Which two sections are not needed? Explain your reasoning. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 655 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 656 magnification: the relationship of the size of the image to the size of the object image attitude: the orientation characteristic of an image, whether erect or inverted Image Characteristics When an object is placed in front of a mirror, the image can be described by four general characteristics (Table 13.2). The magnification relates the size of the image to the size of the object. The image can be enlarged, diminished, or the same size as the object. The image attitude describes the image as being either erect (upright) or inverted (upside down) relative to the object. This is termed vertical inversion. Horizontal, or left/right, inversion can also occur. The position describes where the image forms relative to the surface of the mirror. The type of image distinguishes between real and virtual images. A real image is one that can be projected on a screen or surface, and a virtual image is one that can only be seen, or photographed. Table 13.2 Image Characteristics Image Characteristic Description magnification same size, enlarged, diminished attitude position type erect or inverted displacement from mirror surface real or virtual Image characteristics are important for the proper application of signage and warning labels. For example, the word “ambulance” is written in reverse on the front of the vehicle so that other drivers will be able to read the writing in their rear-view mirrors when the ambulance is approaching. M I N D S O N Image Characteristics Figure 13.33 shows (a) an image of an ambulance and (b) its image in a rear-view mirror. 1. What image characteristic is addressed by printing the word “ambulance” in reverse? 2. While standing in front of a plane, flat mirror, raise your right hand. Which hand in your image was raised? In a plane, flat mirror, do images appear backwards horizontally or upside down vertically or both? Explain. 3. 4. Can an image in a plane mirror ever appear “upside down”? Figure 13.33 (a) The front of an ambulance (b) The image it makes in a rear-view
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mirror 656 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 657 Image Formation in a Curved Mirror Curved mirrors come in a variety of shapes, the most common being spherical. Spherical mirrors, like the ones used for store security, have a unique geometrical shape (Figure 13.34). This shape is derived from a sphere. Imagine a hollow sphere, with a polished mirror surface on the inside and out. hollow sphere polished inside and out convex reflecting surface concave reflecting surface Figure 13.34 A spherical store security mirror Figure 13.35 A curved mirror can be formed from a hollow sphere. By removing a section of the sphere, you produce a double-sided spherical mirror with a concave reflecting surface on one side and a convex reflecting surface on the other (Figure 13.35). The concave reflecting surface is curved inwards. This surface is also referred to as a converging mirror since it causes parallel light rays to converge, or come together, after being reflected. The convex reflecting surface is curved outwards. This surface is also referred to as a diverging mirror since it causes parallel light rays to spread out or diverge after being reflected. As with plane-mirror image formation, ray diagrams are useful to determine how images form from curved surfaces. Ray diagrams for curved surfaces, as in Figure 13.36 for example, are more complex and involve an expanded set of terminology. converging mirror: a concave reflecting surface that causes parallel light rays to converge after being reflected diverging mirror: a convex reflecting surface that causes parallel light rays to spread out after being reflected 1. centre of curvature (C) – the point in space that would represent the centre of the sphere from which the curved mirror was cut 2. radius of curvature (r) – the distance from the centre of curvature to the mirror surface 3. vertex (V) – the geometric centre of the PA curved mirror surface ray 2 ray 1 C ray 3 r F V f 4. principal axis (PA) – an imaginary line drawn through the vertex, perpendicular to the surface of the curved mirror at this point 5. principal focal point (F) – the point where light rays that are parallel to and close to the principal axis converge, or appear to diverge from, after being reflected Figure 13.36 A ray diagram for a converging mirror 6. focal length (f) – the distance from the vertex to the
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focal point, measured along the principal axis. The focal length is related to the radius of curvature by f r2. This means that as the radius of curvature is reduced, so too is the focal length of the reflecting surface. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 657 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 658 Drawing Ray Diagrams for Curved Mirrors The four general image characteristics of magnification, attitude, position, and type, which were applied to plane mirrors, can also be applied to curved mirrors. For curved mirrors, the normal line at the point of incidence is the same as an extension of the radius of the spherical surface drawn at this point. The law of reflection still describes how all the rays will be reflected. To understand and predict how images are produced from curved mirrors, we will need to use three light rays, the law of reflection, and the mirror’s focal point, centre of curvature, and vertex. Two rays from a point on an object locate the corresponding point on the image and the third ray will verify its location (Figure 13.37). Consider each of the three rays: Incident Ray 1 travels from a point on the object parallel to the principal axis. Any ray that is parallel to the principal axis will reflect through the focal point on a converging mirror, or appear to have originated from the focal point on a diverging mirror. Incident Ray 2 travels from a point on the object toward the focal point. Any ray that passes through the focal point on a converging mirror, or is directed at the focal point on a diverging mirror, will be reflected back parallel to the principal axis. PA PA F C Incident Ray 3 travels from a point on the object toward the centre of curvature. Any ray that passes through the centre of curvature on a converging mirror, or is directed at the centre of curvature on a diverging mirror, will be reflected directly back along the incident path. At what incident angle must this light ray hit the mirror surface in order to be reflected straight back along the original path? These three rays alone will allow you to predict and verify the location and characteristics of the image. Notice that some conventions are used when drawing these ray diagrams: 1. objects are often drawn as erect arrows 2. real rays are drawn as solid lines 3. virtual rays (that only appear to exist behind the mirror) are drawn as dashed lines Figure 13
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.38 (a) shows that the converging mirror produces a real image that is inverted and diminished. Figure 13.38 (b) shows that the diverging mirror produces a virtual image that is erect and diminished. Why is one image real and the other virtual? For curved mirrors, a real image is formed where the reflected light rays converge or meet. At this location, a focussed image would appear on a sheet of paper or a screen if it were located in the exact location where the light rays meet. If the screen were to be moved slightly, the image would appear Figure 13.37 Reflection of three rays from spherical mirrors ray 1 ray 3 ray 2 C F V a) A concave, converging mirror ray 1 ray 2 ray 3 V b) A convex, diverging mirror PHYSICS INSIGHT All three incident rays obey the Law of Reflection. The resulting rules for drawing ray diagrams are just the application of geometry. 658 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 659 blurred because the reflected rays would not be converging perfectly at the screen’s new location. For the diverging mirror, the reflected rays appear to be originating from behind the mirror, but if a screen were located there, the incident light rays would not reach it (the rays would be blocked by the mirror). Figure 13.38 Steps for drawing a ray diagram in converging and diverging spherical mirrors e WEB To learn more about ray diagrams for curved mirrors and lenses, follow the links at www.pearsoned.ca/school/ physicssource. ray 1 ray 3 ray 2 object C F V PA image 3 2 1 a) A concave, converging mirror 1 ray 1 ray 2 3 2 ray 3 object image b) A convex, diverging mirror Characteristics of an Image in a Curved Mirror If you look at the image of your face in a polished spoon and bring the spoon closer and closer to your nose, you will demonstrate that with curved mirrors, the object’s distance from the mirror’s vertex has an effect on the characteristics of the image produced. For example, what happens to an image as the object is brought closer to a converging mirror surface? In Figure 13.39(a), the object is located outside the centre of curvature, and a real, inverted, and diminished image appears. If the object is brought closer to the vertex of the mirror
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, such that it is inside the focal length, as in Figure 13.39(b), then the image becomes virtual, erect, and enlarged. Figure 13.39 Object location affects image characteristics. 1 2 3 mirror object C F V PA image a) A converging mirror with object located outside C (real, inverted, diminished) 2 mirror 3 C 1 F object V image PA b) A converging mirror with object located inside F (virtual, erect, enlarged) Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 659 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 660 Table 13.3 Image Characteristics for Converging Mirrors Object Position Image Characteristic distant real, inverted, and diminished, close to F outside centre of curvature real, inverted, and diminished, between F and C at the centre of curvature, C real, inverted, and same size, at C between focal point and centre of curvature real, inverted, and enlarged, beyond C at focal point, F undefined (no image forms), at infinity between focal point and vertex virtual, erect, and enlarged, behind mirror Table 13.3 summarizes the image characteristics for a concave, converging mirror. You can verify the characteristics by placing the concave side of a spoon in front of your nose and slowly moving it outward. Note that your image will disappear briefly just as the distance between your nose and the spoon reaches the focal point of the spoon. In a similar way, the image characteristics produced by a convex, diverging mirror can be investigated by using the other surface of the spoon. 13-4 QuickLab 13-4 QuickLab Converging and Diverging Mirrors Problem How do image characteristics in converging and diverging mirrors change as you change the object distance? Materials polished spoon drawing materials Procedure 1 Place a polished metal spoon on your nose and slowly pull it away from your face while watching the image. 4 Complete ray diagrams for the image of your nose inside F, at F, at C, and at a distance, with respect to the spoon as a diverging mirror. (Remember that a diverging mirror has a virtual focal length, so F and C are located on the opposite side of the spoon to your face.) Questions 1. What happens to the image as the object is brought closer to the surface of a converging mirror? 2. What happens to the image as the object is brought closer to the surface of a diverging mirror
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? 3. What characteristics do all images formed in a 2 Complete ray diagrams for the image of your nose diverging mirror share? inside F, at F, at C, and at a distance, with respect to the spoon as a converging mirror. (Use a small upright arrow to represent your nose in the ray diagram.) 3 Reverse the spoon so that you are looking at the other side and move it away from your nose again. 4. Your image disappears when your face is at the focal point of the mirror. Sketch a ray diagram for a converging mirror with the object located at the focal point to explain why an image should not appear when the object is located at the focal point. 5. Why is it not possible to put an object at F for a convex mirror? 660 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 661 M I N D S O N Seeing Is Believing—Or Is It? A parabolic mirror is shaped so that incident light rays that travel parallel to the principal axis will be reflected to a central point, or focus. Optical illusions can be produced by arranging two parabolic mirrors facing one another. Use a ray diagram to explain how a three-dimensional, real image, is produced in Figure 13.40. Hint: The object is located at the focal point of the top mirror. Therefore, all rays that originate from the object will be reflected straight down from the top mirror. Assume the bottom mirror has the same focal length as the top mirror. real image appears here mirror object inside bottom mirror Figure 13.40 Two parabolic mirrors facing one another, with a hole cut around the vertex of one mirror Equations for Curved Mirrors The general rules for drawing ray diagrams of curved mirrors, in combination with the law of reflection, allow us to generate several equations for use with curved mirrors. These equations can be used to determine the characteristics of the image. In any ray diagram, two rays can be used to determine the image characteristics. In Figure 13.41, there are two similar triangles (AOV, above, and DIV, below). Both triangles have an identical angle () and right angles at the principal axis. Thus, DI AO DV AV O ho C A D hi I F θ θ V B f do di where hi ho This translates into di do ho is the height of the object, hi is the height of the image, do is the distance between the
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mirror vertex and the object, and di is the distance between the mirror vertex and the image. Figure 13.41 Triangles AOV and DIV are similar triangles; triangles FVB and FAO are also similar triangles. When we use the ray OFB, the triangles FVB and FAO (shaded light blue) are also similar. Therefore, AO VB AF VF Since • AO ho • VB hi • AF do • VF f f Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 661 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 662 mirror equation: the equation that relates the focal length of a curved mirror to the image and object distances We can write this relationship as: Since hi ho di do the inverse is also true: Combining the equations gives: Dividing both sides by do and rearranging: We obtain this equation: ho hi ho hi do di d o did o 1 di f do f do di f do f 1 do d o d of 1 f 1 do This is the mirror equation, which relates the focal length of a curved mirror to the image and object distances. Sign Conventions for Use with the Mirror Equation As we have seen previously, there are real and virtual images, which can form either in front of or behind the curved mirror. When using the mirror equation, it is therefore necessary to follow a sign convention, which can distinguish the type of image formed. • Real objects and images have positive distances (measured from vertex). • Virtual objects and images have negative distances (measured from vertex). • Erect images and objects have a positive height. • Inverted images and objects have a negative height. • Converging mirrors have a real principal focal point and the focal length is positive. • Diverging mirrors have a virtual principal focal point and the focal length is negative. With this sign convention, a real image formed by a converging mirror will have a negative height (inverted attitude) while the object height is positive (erect attitude). Both the object and image distances will be positive. Magnification (m) is the ratio of the image height to the object height. A negative sign must be added to the equation for magnification to agree with the sign convention above. m hi ho di do An erect image has a positive magnification and an inverted image has a negative magnification. Concept Check 1. If the object distance is extremely large, approaching infinity, 1
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becomes zero. Based on the mirror equation, where will the d o image form in relation to the focal length of the mirror? 2. For a plane, flat mirror, 1 approaches zero. Based on the mirror f equation, where will the image form in relation to the object? 662 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 663 13-5 Problem-Solving Lab 13-5 Problem-Solving Lab Using a Converging Mirror to Heat Water (Determining the Focal Length of a Converging Mirror) Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork Recognize a Need A converging mirror can be used to focus the radiant energy of the Sun onto a specific point to capture energy, to heat water, or to generate electrical energy using a photovoltaic cell. The Problem Design an efficient apparatus that uses radiation to heat water. Criterion for Success Observing the maximum increase in water temperature in a given time period. Brainstorm Ideas To design an efficient water heater that uses sunlight for energy, you will need to consider the following questions: • Where would a water tank be placed in relation to a converging mirror in order to maximize the amount of radiation that it could absorb? • What colour should the water tank be? • What effect will the size of the mirror have on the water temperature? • What positive and negative effects would result from insulating the water tank? Build a Prototype Construct a miniature version of your water heater using a 50-mL volumetric flask, a thermometer, and a converging mirror mounted on a base of clay. Test and Evaluate Determine the focal length of the converging mirror using a heat lamp, a metre-stick, and a sheet of paper (Figure 13.42). Recall that rays that travel parallel to the principal axis will be reflected back through the focal point. 1. Place the heat lamp a great distance away so that most of the incident rays travel parallel to the principal axis. Adjust the position of the paper until the light is focussed into a small region. Be careful not to block the light from reaching the mirror. You may need to adjust the position of the mirror to avoid this — try tilting it slightly downward and turning the room lights off to see the image clearly. 2. Record the distance between the paper and the vertex of the mirror. This is the approximate focal length of the mirror
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and it is also the image distance. 3. Measure the distance between the lamp and the mirror. This is the object distance. Using the mirror equation, with the image and object distances, calculate and verify the approximate focal length of the mirror. heat lamp paper do di converging mirror clay Figure 13.42 Locating the focal point 4. Using the heat lamp in place of sunlight, measure the increase in water temperature for a given time period when the water tank is located beyond the focal point, at the focal point, and inside the focal point of the mirror. 5. Determine which position results in the greatest increase in water temperature and explain why. Communicate Using presentation software, create a brochure that outlines the basic operating principles of your water heater and explains the role of radiant energy in its operation. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 663 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 664 Example 13.2 An object is located 20.0 cm from a converging mirror that has a focal length of 15.0 cm. (a) Where will the image form relative to the mirror vertex? (b) If the object is 5.00 cm high, determine the attitude, height, and magnification of the image. 20.0 cm Given do f 15.0 cm (converging mirror has a positive focal length) ho 5.00 cm (assume that the object is erect) Required image distance (di) image height (hi) magnification (m) Analysis and Solution (a) To locate the image, use the relationship between focal length and object distance, the mirror equation: 1 f 1 di 1 do 1 f 1 di 1 do Now, solve for di: 1 1 1 20. 15. 0 cm 0 cm d i di 60.0 cm or di d do of f (20.0 cm)(15.0 cm) (20.0 cm) (15.0 cm) 60.0 cm (b) To determine the attitude, height, and magnification, use the equation for magnification: m hi ho di do Now, solve for m: m 60.0 cm 20.0 cm 3.0 Practice Problems 1. A diverging mirror of focal length 10.0 cm produces an image of an object located 20.0 cm from the mirror. Determine the image distance and the magnification. Is the image real or virtual? 2. Determine the image distance,
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magnification, and attributes for the following: (a) a converging mirror with a focal length of 12.0 cm with an object 6.0 cm from the mirror (b) a diverging mirror of focal length 5.00 cm with an object 10.0 cm from the mirror (c) a diverging mirror of focal length 10.0 cm with an object 2.0 cm from the mirror 3. A 5.0-cm-high object is placed 2.0 cm in front of a converging mirror and the image is magnified 4. Where does the image form and what is the focal length of the mirror? 4. A 4.0-cm-high object is placed 15.0 cm from a concave mirror of focal length 5.0 cm. Determine the image characteristics using a ray diagram and the mirror equation. 5. Light from a distant planet is incident on a converging mirror. The image of the planet forms on a screen 45.0 cm from the vertex of the mirror. Find the focal length of the mirror and the image characteristics. Answers 1. di 6.67 cm; m 0.333, virtual 2. (a) di 12 cm; m 2.0, virtual, erect, enlarged (b) di 3.33 cm; m 0.333, virtual, erect, diminished (c) di 1.7 cm; m 0.83, virtual, erect, diminished 8 cm; f 1.6 cm 7.5 cm; m 0.50 ; 3. di 4. di 2.0 cm; real, inverted, diminished hi 5. f 45.0 cm; real, inverted, diminished 664 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 665 Then solve for hi: hi ih d o d o (60.0 cm)(5.00 cm) (20.0 cm) 15.0 cm Paraphrase (a) The real image forms 60.0 cm in front of the mirror vertex. (b) It is inverted and magnified 3 times to a height of 15.0 cm. 13.3 Check and Reflect 13.3 Check and Reflect Knowledge 1. Explain why the normal lines in Figure 13.27 are drawn in a variety of directions. 2. Create a concept map using ray diagram terminology including: principal axis, focal point, centre of curvature, and vertex. 3. Explain the difference between “
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specular” and “diffuse” reflection. 4. Compare “virtual” images with “real” images. 5. Compare “converging” and “diverging” mirrors. 6. Describe the path of the three rays that can be used to determine the characteristics of an image formed in a curved mirror. 7. Why does a diverging, convex mirror have a “virtual” focal point and not a “real” focal point? Applications 8. An object 4.0 cm high is located 10.0 cm in front of a concave mirror. If the image produced is erect, virtual, and 5.0 cm high, what is the focal length of the mirror? 9. Draw a ray diagram for a converging mirror and determine the image characteristics when the object is located at: (a) 2.0 f (b) 0.50 f (c) 3.0 f 10. Draw a ray diagram for a diverging mirror and determine the image characteristics when the object is located at: (a) 0.50 f (b) 1.0 f (c) 1.5 f 11. Some flashlights and headlights use concave mirrors to help generate a light beam. If the light source is positioned at the focal point of the mirror, would all the reflected rays travel outward, parallel to the principal axis? Explain your answer. Include a ray diagram. Extension 12. A reflecting telescope uses a curved mirror to produce the image at the eyepiece. Research the telescope’s design and function. Sketch a ray diagram to explain how the converging mirror is used in the telescope design. e TEST To check your understanding of reflection in plane and curved mirrors, follow the eTest links at www.pearsoned.ca/school/physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 665 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 666 13.4 Refraction A paddle appears bent when you put it in water (Figure 13.43); the Sun appears to change shape when it sets; a rainbow appears when it rains; and a mirage occurs over the hot desert sand, or even over a hot Alberta highway during the day. These are all visual effects created by refraction, a change in the direction of a light wave due to a change in its speed as it passes
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from one medium to another. When light rays pass at an angle from air to water, they immediately change direction. However, not all of the light will be refracted. Light rays are partially reflected and partially refracted when they pass from one medium to the next. Figure 13.44 illustrates this phenomenon.. The process of refraction is described by an incident ray and a refracted ray. The angle between the normal line (drawn perpendicular to the medium interface) and the incident ray is called the angle of incidence. The angle between the normal line and the refracted ray is called the angle of refraction. When the light ray passes from air to water, it is bent or refracted toward the normal line. And conversely, if a light ray travels from water into air, it is bent away from the normal line. The bending of the light ray depends on the refractive index of the medium in which the light travels. The refractive index of a medium is related to the effect on the speed of light when it travels within that medium. A vacuum does not impede or slow down light travelling through it and therefore, the speed of light (c) is defined as its speed in a vacuum. Air slows the light down a small amount and thus has a slightly greater refractive index than a vacuum. Water slows the light down to a greater extent and has a greater refractive index than air. This retarding effect was observed in Fizeau’s experiment on the speed of light in water (page 649). The amount of refraction is related to the magnitude of the change in the speed of light as it passes from one medium to another; the greater the change in speed, the greater the amount of refraction. angle of reflection reflected ray normal angle of incidence Mathematically, the refractive index of a medium (n) can be described as a ratio that compares the speed of light in a vacuum (c) to the measured speed of light in the medium (v). n c v Figure 13.43 A paddle as it appears partially submerged in water refraction: a change in the direction of a light wave due to a change in its speed as it passes from one medium to another refractive index: a ratio comparing the speed of light in a vacuum to the measured speed of light in a given material incident ray less refractive more refractive angle of refraction point of incidence refracted ray Figure 13.44 Ray diagram of partial reflection and partial refraction 666 Unit VII Electromagnetic Radiation 13-Pear
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sonPhys30-Chap13 7/24/08 3:44 PM Page 667 Table 13.4 lists the refrac- tive indexes for some common substances. Since the speed of light in air is very close to the speed of light in a vacuum, the refractive indexes for a vacuum and for air are considered to be the same. The following general rules are based on refractive indexes: • When light passes from a medium with a high refractive index to one with a low refractive index, it is refracted away from the normal line. • When light passes from a medium with a low refractive index to one with a high refractive index, it is refracted toward the normal line. Table 13.4 Absolute Refractive Indexes (for Sodium Yellow Light, 589 nm) Medium vacuum air ice water ethanol glycerin quartz glass crown glass light flint glass Index of Refraction 1.0000 1.0003 1.31 1.33 1.37 1.47 1.47 1.52 1.58 Lucite (plexiglass) 1.52 ruby zircon diamond 1.54 1.92 2.42 Snell’s Law of Refraction Although the phenomenon of refraction had been observed for centuries, it was not until 1621 that the Dutch mathematician, Willebrord Snell (1580–1626), identified the exact relationship between the angle of incidence and the angle of refraction. Snell’s Law states that: sin i sin r a constant i / sin This relationship indicates that for any angle of incidence greater than zero, the ratio sin r will be constant for any light ray that passes through the boundary between the two media. In the case of an air-water interface, the constant is 1.33, which corresponds to the refractive index of water. In fact, for any ray that passes from air ( air) into a second medium ( r, refractive index n), Snell’s Law may also be written in a simple form as: sin air sin r n As long as the first medium is air, the Snell’s constant and the index of refraction for the second medium are one and the same thing. If the first medium is not air, then the general form of Snell’s Law applies. In the general form, the angle of incidence ( i) is replaced with 1 and the angle of refraction ( 2. The index of refraction for the first
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medium is denoted as n1 and the index of refraction r) is replaced with PHYSICS INSIGHT On passing through a medium, each wavelength of EMR has a unique refractive index. For this reason, Absolute Refractive Indexes must be quoted for a specific wavelength, as in Table 13.4. Snell’s Law: For any angle of incidence greater than zero, the ratio sin r is a constant for any light ray passing through the boundary between two media. i/sin info BIT A mirage (derived from the Latin term “mirare,” meaning “to wonder at”) is a naturally occurring phenomenon caused by refraction. Often, mirages appear as large bodies of water on hot desert sand, or small puddles on highways. What appears as water is actually an image of the sky being refracted back up from the hot air just above the road surface. Figure 13.45 Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 667 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 668 e SIM Find out more about refraction through the use of simulations. Follow the eSIM links at www.pearsoned.ca/ school/physicssource. for the second medium is denoted as n2. In all cases, the subscript “1” refers to the incident medium and the subscript “2” is reserved for the refracting medium. The general form of Snell’s Law is written as: sin 1 sin 2 n2 n1 n1 sin 1 n2 sin 2 The general form of Snell’s Law applies in all cases, regardless of the indexes of refraction of the media and/or the direction in which the light travels. Compare the general form with the simple form above. If the first medium is air, or a vacuum, so that n1 is 1.00, substituting this value into the general form results in the simple form of Snell’s Law. Example 13.3 Practice Problems 1. Light passes from a diamond into air. The angle of refraction as the light emerges from the diamond is 25. What was the angle of incidence? 2. Light travelling from air into a transparent material is incident at an angle of 20 and refracted at an angle of 17. Determine the index of refraction for the transparent material. 3. A ray of light
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passes from air into ruby at an incident angle of 15. Calculate the angle of refraction. 4. A ray of light, travelling in air, is incident on an unknown sample at an angle of 20. If the angle of refraction is 15, determine the index of refraction for the unknown sample. Answers 1. 10 2. 1.2 3. 9.7 4. 1.3 Yellow light travels from water into crown glass. The light rays are incident on the crown glass at an angle of 35. Calculate the angle of refraction as the light enters the crown glass. Given 1 35 Required the angle of refraction ( 2) Analysis and Solution From Table 13.4, 1.33 • Index of refraction of water, the first medium, is n1 • Index of refraction of crown glass, the second medium, is n2 1.52 Therefore, the general form of Snell’s Law applies. n1 sin 1 sin 2 2 n2 sin 2 in n1 s 1 n 2 sin1(1.33 si ( ) 2 5. 1 30 n35) Paraphrase The angle of refraction for the light travelling from water into crown glass is 30, when the angle of incidence is 35. 668 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 669 Snell’s Law, Refraction, and Wavelength incident wave fronts λ2 2 d ct D refracted wave fronts Figure 13.46 Wave fronts are refracted as they pass from air into water. vacuum speed c less refractive medium n 1.00 more refractive medium n > 1.00 Can we explain the phenomenon of refraction, and by association, Snell’s Law, by considering light as a wave? Recall that the wave model describes light as a stream of transverse waves radiating outward from a source. Let us assume that the light waves are indeed transverse waves of electromagnetic radiation. As each wave front moves from one medium to the next, there is no change in frequency. In other words, the wave fronts do not “pile up” at the boundary between the two media. The number of waves arriving at the boundary is equal to the number of waves leaving the boundary for any given time interval, so the incident and refracted waves have the same frequency. However, the wavelength changes at the boundary because the speed changes. According to the universal
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wave equation (v f), if the speed changes and the frequency remains constant, then the wavelength must change as well. To visualize this, consider light waves travelling from air into water as illustrated in Figure 13.46: medium speed v The wavelength in air is given by 1 v1/f; The wavelength in water is given by 2 v2/f. The wave fronts travel faster in air than they do in water (Fizeau’s 1 is experiment, page 649). It follows that v1 is greater than v2, and greater than 2. An enlarged view of two wave fronts, as shown in Figure 13.47, reveals another relationship. When we use similar triangles, the angle of incidence 2 are shown within the coloured right triangles and they share the same hypotenuse, x. 1 and the angle of refraction Therefore, the trigonometric ratio for the sine of an angle when applied to each triangle gives: 1 x opposite hypotenuse sin 1 sin 2 opposite hypotenuse 2 x Since x 1 sin 1 therefore and also x 2 sin 2 1 sin 1 2 sin 2 This equation can also be written as: sin 1 sin 2 1 2 incident ray incident wavefront less optically dense medium more optically dense medium refracted wavefront θ 2 refracted ray Figure 13.47 Enlarged view of two wave fronts Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 669 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 670 When the universal wave equation is used, the wavelength can also be written in terms of the wave speed as follows. sin 1 sin 2 sin 1 sin 2 v 1 f v 2 f v1 v2 Therefore, Snell’s Law can be expanded to read: sin 1 sin 2 v1 v2 1 2 n2 n1 By measuring the incident and refracted angles, velocity, and wavelength, this relationship can be tested. Example 13.4 Practice Problems 1. Determine the speed of light in the following materials: (a) water (b) ethanol (c) ruby (d) crown glass 2. Light with a wavelength of 737 nm enters quartz glass at an angle of 25.0. Determine the angle of refraction and the wavelength of the light in the quartz glass. 3. Light enters an unknown crystal from air with a wavelength of 500 nm. If the wavelength of the light in the crystal is found to be 450 nm
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, what is the refractive index of the crystal? Answers 1. (a) 2.26 108 m/s (b) 2.19 108 m/s (c) 1.95 108 m/s (d) 1.97 108 m/s 2. 16.7, 501 nm 3. 1.11 670 Unit VII Electromagnetic Radiation A ray of yellow light with a wavelength of 570 nm travels from air into diamond at an angle of 30. Determine the following: (a) the speed of light in the diamond (b) the wavelength of the light as it travels in the diamond Given 1 570 nm 5.70 107 m Required the speed of light in diamond (v2) the wavelength of light travelling in diamond ( 2) Analysis and Solution v1, the speed of light in air 3.00 108 m/s From Table 13.4, • Index of refraction of air, the first medium, is n1 • Index of refraction of diamond, the second medium, 1.00 is n2 2.42 n i The general form of Snell’s Law applies Therefore, n1v1 n2 v2 2 n1 1 n2 1.00 (3.00 108 ms) 2.42 1.00 (5.70 107 m) 2.42 1.24 108 ms 2.36 107 m 236 nm Paraphrase (a) The speed of light in the diamond is 1.24 108 m/s. (b) The wavelength of light in the diamond is 236 nm. 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 671 13-6 Inquiry Lab 13-6 Inquiry Lab Determining the Refractive Index of a Variety of Materials Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork In this experiment, known values of the refractive indexes of water and ethanol are verified. rays from ray box Question What are the refractive indexes of water and alcohol (ethanol)? Variables Manipulated variable: angle of incidence Responding variable: angle of refraction Controlled variables: refracting substance, wavelength of light 360 0 10 10 20 20 30 30 less refractive (air) more refractive (liquid) 320 40 310 50 300 60 30 210 20 200 10 190 0 180 10 170 3 0 15 60 120 50 130 40 140 semicircular container Materials and Equipment polar coordinate paper graphing paper water ethanol single-sl
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it ray box or laser semicircular plastic dish Procedure 1 Design a data table or spreadsheet with the headings: angle of incidence (i), angle of refraction (r), sin angle of incidence (sin i), sin angle of refraction (sin r), and ratio sin i / sin r. 2 Fill the semicircular dish with water and place it on the polar coordinate paper such that the 0–180° line is perpendicular to the centre of the flat side of the dish. This will make the 0–180 line the normal line (Figure 13.48). 3 Direct a single ray of light along the normal line (0). Record the angle of refraction. (This should be zero if the apparatus is set up correctly.) 4 Record the angles of refraction for 10 to 70, increasing the angle by 10º for each step. 5 Complete the data table or spreadsheet calculations and plot a graph of sin angle of incidence vs. sin angle of refraction. Calculate the slope of this graph. 6 Repeat steps 2 to 5 using ethanol instead of water in the semicircular dish. Figure 13.48 Semicircular dish on polar coordinate paper with several incident ray angles shown Analysis 1. (a) What is the value of the slope for the graph when water was used as the refracting substance? (b) What is the value of the slope when ethanol was used as the refracting substance? 2. According to your slope calculations, which material, water or ethanol, has a higher index of refraction? 3. Calculate the percent error for each index of refraction using the absolute index of refraction given for each substance in Table 13.4. 4. Comment on the sources of error that could have occurred in this experiment. 5. Why is it important that the incident light ray contacts the semicircular dish at the centre? 6. Why is it important that a “semicircular” dish is used, instead of a rectangular dish? 7. Which material is more effective in changing the direction of light when light enters it? 8. Predict what would happen if the semicircular dish were replaced with a semicircular block of glass or Lucite®. If you have time, test your prediction. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 671 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 672 27° 42° 59° 20° 40° 30
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° 50° 50° 0° 0° light source air n 1.00 water n 1.33 Total Internal Reflection If you look down a long straight tube, you should be able to see what is at the other end. If the tube has a bend in it, a set of mirrors could be used to reflect the light through to your eye. This is how the periscope on a submarine works. But what if the tube has multiple bends in it? What could be done to ensure the light is able to travel through the entire length of the tube? Could you line the inside walls with mirrors and angle the light so that it reflects from side to side all along the tube, entering at one end and emerging at the opposite end? Figure 13.49 Increasing the incident angle for light leaving a water-air interface leads to total internal reflection. In theory, this is exactly what happens in the optical fibres that facilitate the data and telephone communication of computer networks. The process is based on total internal reflection. Recall that when light travels from one medium to another, some of the light is partially reflected and some is partially refracted. When light travels from a medium with a high refractive index, like glass, to a medium with a low refractive index, like air, more of the light is reflected than it would be at an interface where its speed decreases. As illustrated by Figure 13.49, when light travels from the water with a high refractive index to the air with a low refractive index, it bends away from the normal line. As the angle of incidence is increased, the angle of refraction also increases, bending farther and farther from the normal line, and eventually reaching the maximum angle of 90. Beyond this angle, refraction ceases and all incident light will be reflected back into the high-index medium (in this case, the water). Therefore, not all the light that approaches a water-air boundary will be refracted. The light approaching the boundary at large angles will be reflected back into the water. This is why underwater pool lights, for example, are not visible at all angles from above. At particular angles, they cannot be seen. When the angle of refraction is 90, the incident angle will have a value unique to the two media that form the interface. This unique angle of incidence is called the critical angle. For light travelling from a medium with a high refractive index to a medium with a low refractive index, the critical angle is determined by assuming that the angle
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of refraction is 90. Concept Check 1. Is light refracted toward or away from the normal line when passing from a medium with a low refractive index like air, into a medium with a high refractive index like water? 2. Based on your answer above, can total internal reflection occur when light travels from a low-index medium like air, into a high-index medium like water? Explain why or why not. total internal reflection: reflection of all incident light back into a medium of higher refractive index due to the inability to refract light beyond the maximum angle of 90 critical angle: for any two media, the size of the incident angle for which the angle of refraction is 90 info BIT The sparkle and glitter associated with diamonds is caused by total internal reflection. The high index of refraction of the diamond and the skill of the jeweller in cutting and finishing are used to trap and focus light, which we observe as sparkles. Polar Bear Diamonds™, mined, cut, and polished in the Northwest Territories, are recognized for their quality worldwide. Figure 13.50 672 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 673 Example 13.5 What is the critical angle for the quartz glass-air interface? Given The angle of refraction at the critical angle is defined to be r 90.0. Required the critical angle for this interface ( critical) Analysis and Solution From Table 13.4, 1.00 • Index of refraction of air, the first medium, is n1 • Index of refraction of quartz glass, the second medium, is n2 1.47 Use the form of Snell’s law for the critical angle: n2 sin critical sin critical critical n1 sin 90.0 90.0 n n1 si n 2 90.0 sin1 n n1 si n 2 90.0 sin1 in s 1.00 7.4 42.9 1 Paraphrase Light at an incident angle of 42.9 or greater will be internally reflected at the quartz glass-air interface. Practice Problem 1. Determine the critical angle for the following interfaces: (a) water and air (b) diamond and air (c) diamond and water Answer 1. (a) 48.8 (b) 24.4 (c) 33.3 e MATH To investigate Snell’s Law, refraction, and the concept of the critical angle,
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using a graphing calculator or a spreadsheet program, visit www.pearson.ca/ school/physicssource. cladding Total internal reflection has many applications. Most notable are optical fibres (Figure 13.51). An optical fibre consists of a central core of glass with a refractive index of approximately 1.5, surrounded by a cladding material of a slightly lower refractive index. Such fibres are used to transmit data on computer and communication networks. The data are transmitted via the modulation of laser light travelling through a glass fibre, with virtually none of the energy loss associated with electrical transmission. Some of the advantages of fibre-optic networks include: • economically less expensive than copper wire of equivalent length • thinner, more flexible, and made of non- flammable materials core >θ c >θ c Figure 13.51 An optical fibre showing cladding, core, axis, and critical angle • able to handle a higher data-carrying capacity based on the fibre bundle diameter • less signal degradation and interference between multiple signals on the same fibre as compared with copper networks • the glass fibres are highly transparent so that repeaters (amplifiers) can be many kilometres apart, as opposed to coaxial cable repeaters that must be less than 1 km apart e WEB To learn more about fibre optics including applications in audiovisual equipment and in nanotechnology, begin your search at www.pearsoned.ca/school/ physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 673 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 674 info BIT Buckyballs, also called C60 or buckminsterfullarenes, are soccer-ball-shaped molecules made of 60 linked carbon atoms. These carbon molecules may be formed into nanotubes to transmit data using light. info BIT Binoculars consist of two identical telescopes mounted side by side and aligned to point in the same direction. When the images from each side of the binoculars are viewed simultaneously, the user is able to sense depth and distance that is not possible with a single telescope view. eyepiece Porro prisms objective Figure 13.52 The light path in binoculars The Porro prisms in the binoculars use total internal reflection to direct light through the compact, short body of the binoculars with minimal absorption, producing a high-quality image. F F F Figure 13.53 A
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double Porro prism A double Porro-prism system is used in binoculars to reorient an inverted image while at the same time producing a longer, folded pathway for the light to travel between the objective lens and the eyepiece, producing a greater magnification. Fibre-optic systems also have drawbacks. The complex design of the fibres makes the fibre-optic system relatively expensive to set up and the fibres are also subject to wear and breakage. 13-7 Decision-Making Analysis 13-7 Decision-Making Analysis Fibre Optics: Endless Light The Issue The new knowledge society of today has emerged as a result of our ability to communicate and transmit data on a large scale. And to a very large extent, the infrastructure that supports our communication is based on the principles of total internal reflection and its application in fibre-optic systems. New and innovative advances are being made with fibre optics every day – from nanotechnology to entertainment services. Background Information The flexibility of fibre optics has allowed for applications in a variety of industries. For example, telephone, television, and data networks, originally all separate industries, are merging into one large application, which is supported by a fibre-optic backbone. In medicine, applications include the use of a fibrescope to both illuminate internal organs and capture images of them. For example, to see inside the small intestine, an endoscope, a small, flexible bundle of fibres is inserted down the patient’s throat and through the stomach. Once the fibres reach the small intestine, images can be transmitted through individual fibres while others are simultaneously used to illuminate the tissue. This application prevents the need for more-invasive, risky procedures that can leave the patient with a higher risk of infection and perhaps result in a prolonged hospital stay. Other applications include mechanical imaging. For example, fibre-optic bundles can be used to inspect the interior of long pipes, vessels, and hard-to-reach locations, such as bore holes for oil and water wells, pipelines, and hazardous goods containers. With proper instrumentation devices attached, fibre-optic bundles can be used in dangerous conditions to identify gases, pressures, temperatures, and concentrations. Nanotechnology is an emerging application for fibre optics. In this industry, new nanomaterials, such as C60 ( buckyballs), are being investigated to expand the power and application of data transmission using light. Analyze and Evaluate 1. Research applications of fibre optics using the Internet, research journals,
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and periodicals. 2. Group all the applications by industry and identify what you believe to be the most important applications in each industry. 3. For each of the most important applications, identify the social, political, economic, and environmental impact of the technology. 4. New applications of fibre optics are emerging in nanotechnology. Identify these applications and describe what technological advances could be expected as a result of merging nanotechnology with fibre-optic technology. 5. Prepare a multimedia presentation that demonstrates advances that have been made in a number of industries as a result of fibre-optic applications. Predict what future applications and social issues will evolve as new technologies begin to merge with fibre optics. 674 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 675 Prisms — Dispersion of White Light Where have you seen a rainbow? Everyone has seen one in the sky, but what about in a diamond, or perhaps on the wall near a hanging crystal in a window. There are many items that produce a rainbow of colours, but all the rainbows have something in common. They all originate from white light. The investigation of such effects was undertaken by Newton in 1666. He placed a transparent prism in a beam of sunlight passing through his window shutter at Cambridge University. On the opposite wall, an elongated band of colours appeared and he called this a spectrum (Figure 13.54). In this spectrum, Newton noted the colours red, orange, yellow, green, blue, indigo, and violet, in the order in which they appeared. info BIT It may be that Newton noted seven colours in the spectrum, in part to go along with the views of society at that time: numerology regarded 7 as a “special” number. Six colours are more commonly used, as in Figure 13.55 and Table 13.5. spectrum: the bands of colours making up white light; in order, red, orange, yellow, green, blue, violet dispersion: separation of white light into its components Figure 13.54 The complete spectrum produced by white light passing through a prism Newton set out to determine which of two things was true: either the colours of the spectrum are added to the white light by the prism, or the prism separates all the colours from the white light. To test this, Newton set up two prisms, the first one exposed to white light, producing a spectrum of colours (Figure 13.55), and the second one
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, only exposed to the red (monochromatic) light coming from the first prism. The second prism did not produce any more colours; only the red light emerged. As a second test, Newton placed a converging lens into the path of the spectrum of light and observed the resulting white light as an image on a sheet of white paper. slit aperture white light prism Based on his observations, Newton concluded that white light is made up of all the colours in the spectrum and the prism was simply separating the colours from one another. The separation of white light into its components is called dispersion. screen red orange yellow green blue violet visible spectrum Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 675 Figure 13.55 Dispersion of white light by a prism 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 676 The recomposition of the spectrum, therefore, should produce white light, as Newton was able to demonstrate in his second experiment. Recomposition can also be demonstrated by painting all the colours of the spectrum in certain proportions on a disc and spinning the disc at a very high speed (Figure 13.56). At sufficient speed, the disc appears white. Is dispersion consistent with the wave model of light? If we consider light as a wave, each of the colours in the spectrum has a unique wavelength, as listed in Table 13.5. As light enters a medium with a high refractive index, the wavelength is reduced but the frequency is unchanged. The speed of a wave is the product of its frequency and wavelength as described by the universal wave equation (v f ). Therefore, the light must slow down as it enters the highindex medium. Because the refractive index is related to the speed of a wave, it is therefore also related to the wavelength of the wave when the frequency is constant. n o o wavelength in vacuum wavelength in medium This relationship between refractive index and wavelength, for a constant frequency, means that the refractive index is different for each wavelength of light that passes through the same medium. Incident light with a smaller, shorter wavelength will slow down and refract to a greater extent than light with a larger, longer wavelength. As each wavelength of light refracts at a slightly different angle, the wavelengths will separate, producing a continuous spectrum. The wave model of light, therefore, is consistent with the phenomenon of dispersion. Figure 13.56 Recomposition of the spectrum using a high-speed disc
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Table 13.5 Wavelength of each colour in the spectrum e SIM To learn more about the effects of prisms through the use of simulations, follow the eSim links at www.pearsoned.ca/school/ physicssource. Colour violet blue green yellow orange red Wavelength (nm) 400-450 450-500 500-570 570-590 590-610 610-750 676 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 677 13-8 QuickLab 13-8 QuickLab Dispersion Using a Prism Problem Is white light made up of all the colours we can see? Materials 2 identical prisms convex lens intense light source (flashlight or white light ray box) white paper Procedure 1 Using one prism, observe the colour spectrum on a piece of white paper. A dark room and a very intense white light source will work best. The convex lens may be placed between the light source and prism to intensify the beam of white light if the spectrum is not immediately clear and visible. 2 Position the second prism in the spectrum produced by the first one. Rotate the second prism and position the white paper so that you can observe the light that emerges from the second prism. 3 Explore a variety of positions and orientations with both prisms. Questions 1. What is the best way to make a colour spectrum using a single prism? Draw a picture showing the flashlight, prism, convex lens, paper, and the colour spectrum. 2. List the colours of the spectrum from most refracted to least refracted. 3. Compare the wavelengths of each colour to the amount of refraction observed. Describe the relationship between wavelength and amount of refraction. 4. Are the colours of the spectrum added to, or separated from, white light when white light passes through a prism? Support your answer with observations made using two prisms. 5. The spectrum produced by a prism is similar to that observed as a rainbow. Research and explore the similarities between the spectrum produced by a prism and that which is observed as a rainbow. Thin Lenses Projection systems connected with computers and movie players, and optical systems including microscopes and telescopes, use lenses that refract light in order to generate images. A typical thin lens is a circular piece of transparent material with a spherically shaped surface that refracts, or changes the direction of light that passes through it. The uniformly curved surface will refract light rays to varying ext
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ents depending on where they contact the lens. Rays that are incident near the edge of the lens will be refracted at larger angles than those that are incident near the centre, where the two faces of the lens are almost parallel. For a converging lens, rays that travel parallel to the principal axis will be refracted inward, intersecting at the principal focus, F. For a diverging lens, rays that travel parallel to the principal axis will be refracted outward, appearing as though they have originated at a virtual principal focus, F. The distance from each lens to F is the focal length, f (Figures 13.57 and 13.58). Each type of lens has a secondary focus, F at a distance of f, on the opposite side of the lens. converging lens: a lens that refracts rays travelling parallel to the principal axis inward to intersect at the principal focus diverging lens: a lens that refracts rays travelling parallel to the principal axis outward to appear as though they have originated at a virtual principal focus Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 677 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 678 PHYSICS INSIGHT A ray that travels through the optical centre is indeed refracted. This causes the refracted ray to emerge parallel to, but laterally shifted from, the incident ray. For a thin lens, this lateral shift is so small that it is assumed that the ray is not refracted. O F principal focus f focal length Figure 13.57 Converging lens F virtual principal focus PA optical centre F O lateral shift Figure 13.58 Diverging lens f focal length Figure 13.59 Lateral shift Drawing Ray Diagrams for Thin Lenses A ray diagram is a useful tool for predicting and understanding how images form as a result of light rays emerging from a curved lens. Ray diagrams for lenses are similar to the ray diagrams used with curved mirrors; only two rays are needed to predict the image location, and a third is used as verification (Figure 13.60). 1 3 optical centre image object F 2 F 2 3 1 object 1 2 3 F image F Figure 13.60 Ray diagrams for (a) converging, and (b) diverging lenses 678 Unit VII Electromagnetic Radiation Ray 1 travels parallel to the principal axis and is refracted such that it emerges and passes through (or appears to have originated from) the principal focus, F. Ray 2
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travels through (or appears to be directed toward) the secondary focus, F, and is refracted such that it emerges and travels parallel to the principal axis. Ray 3 travels straight through the optical centre of the lens and is not bent. The ray diagram is not only used to identify the location of an image relative to the lens, but it can also illustrate the other three image attributes of type, attitude, and magnification, that have previously been described for curved mirrors. Relative to the object, the image produced by a thin lens can be real or virtual, inverted or erect, and enlarged or diminished. 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 679 Example 13.6 2F and 2F represent two points each at a distance of 2f (twice the focal length) from a lens on either side of the lens. Draw a diagram to determine the image attributes for an object located at the following positions relative to a converging lens. a) beyond 2F b) at F c) between the lens and F Given Object position relative to the converging lens Required Ray diagram showing image location and attributes Analysis and Solution Construct a ray diagram and visually inspect the image location relative to the converging lens and the image attributes.. object 2F F image F 2F Figure 13.61 (a) object 2F F F 2F Figure 13.61 (b) image Practice Problems 1. Using a ray diagram, determine the image attributes for the following: (a) an object located between 2F and F relative to a converging lens (b) an object located between 2F and F relative to a diverging lens (c) an object located between F and a diverging lens Answers 1. (a) Image is beyond 2F, real, inverted, and enlarged. (b) Image is between the lens and F, virtual, erect, and diminished. (c) Image is between the lens and F, virtual, erect, and diminished. 2F F object F 2F Figure 13.61 (c) Paraphrase (a) The image is located between F and 2F, and is real, inverted, and diminished. (b) No image is formed. (c) The image is located between F and 2F, and is virtual, erect, and enlarged. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 679 13-PearsonPhys30-Chap13 7/24/08 3
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:44 PM Page 680 e WEB Equations for Thin Lenses To learn more about thin lenses, follow the links at www.pearsoned.ca/ school/physicssource. An equation relating object distance, image distance, and the focal length of a curved lens can be derived using an analysis nearly identical to that for curved mirrors. In Figure 13.62, the green triangle XOF and the blue triangle ZYF are similar. Therefore, ho object X F O F Y hi image Z do f di Figure 13.62 Thin lens ray diagram OX OF ho f hi ho hi ho di do di do YZ YF hi f di f di f di do f di f di f f f Dividing both sides by di, and simplifying, gives the thin lens equation. thin lens equation: the equation that relates object distance, image distance, and focal length of a curved lens 1 di d i d ido 1 do d i d i f 1 di 1 f Notice that this equation is identical to the mirror equation. A sign convention will be used here also, to distinguish between real and virtual distances as well as to identify erect and inverted images. Sign Conventions for the Thin Lens Equation • All distance measurements are relative to the optical centre of the lens. • Positive distances are used for real objects and images. • Negative distances are used for virtual images. • Positive image and object heights are upward relative to the principal axis. • Negative image and object heights are downward relative to the principal axis. • Converging lenses have a positive focal length. • Diverging lenses have a negative focal length. Results with a thin lens are similar to those with a curved mirror in that an erect image has a positive magnification and an inverted image has a negative magnification. Therefore, the same magnification equation that is used for curved mirrors is also used for thin lenses. m hi ho di do 680 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 681 Example 13.7 A 2.5-cm-high object is placed 10.0 cm from a diverging lens of focal length 5.0 cm. Determine the image distance, height, and attributes using the thin lens equation. Verify your answer with a ray diagram. Given ho do 2.5 cm 10.0 cm f 5.0 cm (The lens is a diverging lens.) Required image distance (di) height (hi) image characteristics Analysis
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and Solution The thin lens equation can be used to determine the image distance, image height, and attributes. A ray diagram can verify the answer hi 1 1 1 hi 10. 5.0 0 cm d i cm ih d o d o (3.33 cm)(2.5 cm) 10.0 cm 3 0 cm 10. di cm 10.0 3 3.3 cm 0.83 cm Paraphrase The image distance is 3.3 cm and height is 0.83 cm, indicating the image is virtual, erect, and diminished. The answers are verified by the ray diagram in Figure 13.63. object 2F F image F 2F Figure 13.63 Practice Problems 1. A 3.00-cm-high object is located 15.0 cm from a converging lens with a focal length of 10.0 cm. (a) How far is the image from the lens? (b) How high is the image? (c) Describe the image characteristics and verify them using a ray diagram. 2. A 10.0-cm-high candle is placed 100.0 cm from a diverging lens with a focal length of 25.0 cm. Determine the following using a ray diagram and the thin lens equation: (a) the image location from the lens (b) the image height (c) the type of image formed 3. A projector uses a converging lens to create a focussed image on a screen located 5.00 m away. The image is generated from a slide located 7.50 cm from the lens. (a) What is the focal length of the lens? (b) Determine the magnification of the image. Answers 1. (a) di (b) hi (c) image is real, inverted, enlarged 30.0 cm 6.00 cm 20.0 cm 2.00 cm 2. (a) di (b) hi (c) image is virtual 3. (a) f 7.39 cm (b) m 66.7 X Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 681 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 682 13-9 Inquiry Lab 13-9 Inquiry Lab Comparing Keplerian and Galilean Telescopes Required Skills Initiating and Planning Performing and Recording Analyzing and Interpreting Communication and Teamwork In this investigation the functional differences between a Keplerian telescope, with two converging lenses,
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and a Galilean telescope, with one converging lens and one diverging lens, are explored. Question What are the differences between Keplerian and Galilean telescopes? Variables manipulated: type of lens used responding: magnification of the image controlled: focal length of double concave and double convex lenses Materials and Equipment long focal length converging lens short focal length converging lens short focal length diverging lens lens holders • • • • • optical bench for measuring distances • • paper, ruler, pencil, and tape lamp Procedure for Galilean Telescope 1 Place the long focal length converging lens at one end of the optical bench. This will serve as the objective lens for both telescopes. 2 If the long focal length is unknown, place a lamp in front of the lens and measure the distance between the lamp and the lens. Next, slide a sheet of paper or screen along the optical bench until a bright spot appears (image of the lamp). Record the distance between the paper and the lens. Calculate the focal length of the objective lens using the lens equation, and image and object distances. 3 Place the short focal length diverging lens halfway between the converging lens and its principal focus as illustrated in Figure 13.64. objective lens eyepiece fo light from object Figure 13.64 Galilean telescope 682 Unit VII Electromagnetic Radiation 4 Draw a series of 5 vertical arrows, 1 cm apart, on a sheet of paper. Tape the sheet to a distant wall so that it will be visible in the telescope. 5 Look through the eyepiece with one eye and compare the image with the sheet seen with an unaided eye. 6 Estimate how many vertical arrows (viewed with the unaided eye) would fit in between two of the vertical arrows observed in the telescope. Record this as the estimated magnification. Procedure for Keplerian Telescope 7 Remove the diverging lens from the optical bench. 8 If the focal length of the short focal length converging lens is not known, place a lamp in front of the lens and slide a sheet of paper or screen along the optical bench until a bright spot appears. Record the focal length for the short focal length lens. 9 Position the short focal length converging lens such that the distance between the two lenses is slightly less than the sum of the focal lengths, as shown in Figure 13.65. fo objective lens fe eyepiece light from object Figure 13.65 Keplerian telescope 10 Estimate how many vertical arrows (viewed with the unaided eye) would fit
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in between two of the vertical arrows observed in the telescope. Record this as the estimated magnification. Analysis 1. Prepare a table that compares the attitude, magnification, and brightness of the image in each telescope. 2. Complete a ray diagram to show how the image is formed in each telescope. 3. Suggest a different application for each telescope. Compare the telescopes’ efficiency in looking at relatively close terrestrial objects, like mountains, or far away extraterrestrial objects, such as the Moon. 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 683 13.4 Check and Reflect 13.4 Check and Reflect Knowledge 1. Is light bent toward or away from the normal line when it passes from a lowindex medium to a high-index medium? 2. How is the index of refraction measured for a particular medium? 3. Construct a concept map using the following terms: incident ray, refracted ray, angle of incidence, angle of refraction, normal line. 4. What is the difference between the simple and general forms of Snell’s Law? 5. Suggest an experimental design to determine the critical angle for a Lucite®-air boundary. 6. List five advantages of a fibre-optic network when compared to traditional electric networks used for data transmission. 7. How does a prism separate all the colours of white light? 8. Explain the effect of wavelength on the index of refraction for a particular medium, like glass. How does this lead to dispersion? 15. Using a ray diagram, determine the image attributes for an object located at 2F relative to a converging lens. 16. A camera with a converging lens (f 4.50 cm) is used to take a picture of a 25.0-m-high tree that is 50.0 m from the camera. How tall is the image? Is it erect or inverted? 17. Red light of 700 nm and blue light of 475 nm are both incident on a Lucite® block. Which colour will be refracted to a greater extent? 18. The focal length of the converging lens in a computer projector is 8.00 cm. An LCD panel positioned inside the projector serves as the object for the lens. (a) If the LCD panel is located 8.10 cm from the lens inside the projector, how far away should the screen be placed so that a clear image is produced? (b) If the LCD panel is 1.75 cm high, how large is
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the image on the screen? (c) If the screen is moved 3.0 m closer to the projector, how far must the object now be from the lens in order to generate a focussed image? Applications 9. What is the speed of light in water (n 1.33)? Extensions 19. Explain why jewellery crystals, such as 10. A light ray is incident on a block of quartz diamonds, sparkle. glass (n 1.47) at an angle of 35.0. Determine the angle of refraction. 11. When a ray of light passes from water (n 1.33) into a Lucite® block (n 1.52) it is refracted at an angle of 28.0. Determine the angle of incidence. 12. What is the critical angle for a Lucite®-air interface? 13. A light ray travelling in water approaches the water-air boundary at an angle of 50. What happens to the light ray at the boundary? 14. Light with a wavelength of 540 nm enters a ruby crystal (n 1.54) at an angle of 25.0. Determine the angle of refraction and the wavelength of the light in the ruby. 20. Explain why you can start a fire with a magnifying glass. 21. If a diver is underwater and looks up, a circular “hole” appears on the surface of the water directly above the diver. Other than this hole, the surface appears as a mirror. Explain how this happens. e TEST To check your understanding of refraction, fibre optics, dispersion, and lenses, follow the eTest links at www.pearsoned.ca/school/physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 683 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 684 Figure 13.66 Christiaan Huygens Huygens’ Principle: a model of wave theory, which predicts the motion of a wave front as being many small point sources propagating outward in a concentric circle at the same speed as the wave itself 13.5 Diffraction and Interference In section 8.3 you learned that there are many forms of interference patterns produced by mechanical waves that travel in a medium like water. And more importantly, you learned that the interference pattern contains information about the properties of the waves that created the pattern. In this section, we will investigate
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interference patterns produced by electromagnetic radiation and analyze the patterns to further our understanding of the wave model of light. Huygens’ Principle Robert Hooke proposed the Wave Model of Light in his Micrographica of 1665. The first major improvement to this model was made by Christiaan Huygens (Figure 13.66), twenty years later. A Dutch physicist, Huygens contributed to the study of astronomy by advancing techniques in lens grinding and by discovering Saturn’s rings and its largest satellite, Titan. He is also credited with producing the pendulum clock, originally proposed by Galileo. With regard to wave theory, Huygens described an elegant, conceptual model that predicted the motion of a wave front. This model is known as Huygens’ Principle. A wave front consists of many small point sources of tiny secondary waves, called wavelets, which propagate outward in a concentric circle at the same speed as the wave itself. The line tangent to all the wavelets constitutes the wave front. To visualize Huygens’ Principle, consider a point source of light that emits electromagnetic waves outward, in a concentric circle. At an instant in time, the wave front will form the line t1. According to Huygens’ Principle, all the points along line t1 become secondary sources, producing wavelets that radiate outward to form the wave front at a future time t2. Line t2 is the tangent to all the wavelets that make up the wave front a short time later (Figure 13.67). The same analysis can be performed on a straight wave front, as shown in Figure 13.68. t1 t2 point source of EMR point source of EMR t2 t1 circular wave front straight wave front Figure 13.67 Circular wave front emitted by a light source Figure 13.68 Straight wave front emitted by a distant light source 684 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 685 Concept Check Use Huygens’ Principle to predict the wave front shapes that occur after the straight waves pass through the openings shown in Figure 13.69 (a) and (b). (a) barrier (b) barrier t t e SIM Explore Huygens’ Principle through simulations. Follow the eSim links at www.pearsoned.ca/school/ physicssource. Figure 13.69 At each
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opening, one or more secondary point sources exist. Draw the line tangent across the wavelets that would exist a short time after the secondary point sources pass through the opening. Explain what has happened to the shape and direction of the wave front after it passed through the barriers in Figure 13.69 (a) and (b). According to Huygens’ Principle, a periodic straight wave will continue to propagate in a straight line until it encounters a barrier. If the wave front encounters a small opening or aperture in the barrier, the wave front will change shape and direction. This process is called diffraction. Young’s Experiment Huygens’ wave theory can explain many of the properties of electromagnetic radiation, including reflection and refraction, but initially, the scientific reputation of Newton and his belief in the particle model of light dominated the scientific community. However, in 1801, an experiment by Thomas Young provided significant evidence in support of the wave model of light. Problems Involved in the Design of the Experiment In the years leading up to Young’s groundbreaking work, scientists studying light were attempting to observe an interference pattern that was similar in nature to that of two point sources. They believed that if light were indeed a wave, then two side-by-side light sources should produce an interference pattern similar to that observed with water in a ripple tank (section 8.3). In these early light experiments, the light from the two sources was incident on a nearby screen that was observed for evidence of an interference pattern. What was not well understood at the time was that waves of high frequency and short wavelength result in a very short distance between nodal lines, the regions of destructive interference that appear dark. As we know today, light waves have an extremely high frequency and short wavelength, so the distance between nodal lines would be so small it could diffraction: the change in shape and direction of a wave front as a result of encountering a small opening or aperture in a barrier, or a corner interference: an effect resulting from the passage of two like waves through each other e SIM Learn more about Young’s classic experiment through simulations. Follow the eSim links at www.pearsoned.ca/ school/physicssource. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 685 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 686 Figure 13.70 top down view interference pattern sunlight pinhole card not be
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observed using traditional means. In addition, to produce an interference pattern, the waves must be in phase. When there are two incandescent lights, each source emits light in random bursts not necessarily in phase with the other source. These bursts of light have a variety of wavelengths that make up all the colours in the spectrum. Therefore, the interference pattern would not be constant, but would vary rapidly over time, making any observation of a pattern more difficult. screen (a) Interference fringes (b) Photo of two-slit pattern from Young’s experiment info BIT The double-slit experiment works for water waves too! When a single wave front is incident on a barrier with two holes it will produce an interference pattern similar to the one observed by Young. This effect is explored in Unit IV. Figure 13.71 Interference pattern produced by water waves passing through two holes in a barrier antinode: a point of interaction between waves, at which only constructive interference occurs; in an interference pattern, antinodes occur at path difference intervals of whole wavelengths The Experiment Thomas Young solved both of these problems and successfully observed an interference pattern produced by light. Young conducted his experiment using a pinhole in a window shutter and a card he described as “a slip of card, about one-thirtieth of an inch in breadth (thickness).” The card was positioned edgewise into a horizontal sunbeam directed into the room by using a mirror. The sunbeam had a diameter slightly greater than the thickness of the card. When the card was positioned edgewise in the centre of the sunbeam it split the sunbeam into two coherent beams separated by a very small distance (Figure 13.70(a)). The effect was equivalent to light passing through two slits that were very close together. The small distance of separation between the two beams of light expanded the interference pattern on the screen so that the distance between nodal lines was large enough that it could be observed. Light coming from both sides of the card was in phase and the wave fronts could create a fixed interference pattern of light and dark bands, called interference fringes, on the screen (Figure 13.70 and also Figure 13.9, page 640). Bright fringes or antinodal lines were regions of constructive interference and dark fringes or nodal lines were regions of destructive interference. Young’s double-slit experiment provided the strong evidence needed for acceptance of the wave model of light. The Interference Pattern
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To understand how the interference pattern is created, consider three different points. The first point is the central antinode (Figure 13.72). This point occurs at the centre of the pattern, along the perpendicular bisector. The perpendicular bisector is an imaginary straight line that runs from the midpoint of a line joining the two slits to the area of constructive interference on the screen. This area of constructive interference may be called the central antinode, central bright fringe, or central maximum. At the central antinode, both waves travel the same distance from the slits and arrive at the screen in phase. Constructive interference will be observed as a bright band. path difference: 3.00 λ 3.00 λ 0 λ screen path length 3.00 λ d } constructive interference at central antinode Figure 13.72 A path difference of zero at the central antinode results in constructive interference and a bright band on the screen. path length 3.00 λ l 686 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 687 The second point is the first node, also called the first dark fringe or the first minimum. At this location, one wave travels a distance of 1 farther than the other so that the waves arrive out of phase, and 2 destructive interference is observed as a dark band (Figure 13.73). The third point is another antinode or bright fringe. At this location, one wave travels a distance of 1 farther than the other, causing the waves to arrive at the screen in phase once again (Figure 13.74). This pattern of bright and dark fringes repeats as we move out in both directions from the central, perpendicular bisector. path difference: 3.75 λ 3.25 λ λ 1 2 path length 3.75 λ path difference: 4.50 λ 3.50 λ 1 λ path length 4.50 λ d d } destructive interference at 1st node path length 3.25 λ l path length 3.50 λ l } constructive interference at n 1 antinode Figure 13.73 A path difference of 1, at the first node results in destructive 2 interference and a dark band on the screen. Figure 13.74 A path difference of, at the first antinode results in constructive interference and a bright band on the screen. Mathematical Analysis of Young’s Experiment Young’s
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experiment and interference patterns in general contain mathematical information about the waves that create them. For instance, the interference pattern that Young observed can be used to determine the wavelength of the light that created it. dark nodes (destructive interference) bright central antinode bright antinodes (constructive interference) 4 3 2 1 bisector 1 P 2 3 4 screen 10 λ 9 λ S1 S2 d Figure 13.75 Two point sources, separated by a short distance, produce antinodal lines of constructive interference and nodal lines of destructive interference. Point sources, S1 and S2, from the same original source, are separated by a short distance, d, and produce identical waves that are in phase. node: a point of interaction between waves, at which only destructive interference occurs; in an interference pattern, nodes occur at path difference intervals of one-half wavelength PHYSICS INSIGHT Compare Figures 13.74 and 13.75. The absolute path lengths (expressed in number of wavelengths) shown in Figure 13.74 and 13.75 are different, yet the waves are still causing constructive interference at the point where they meet. How can this be? If the distance separating the sources and the screen is constant, then the path length (expressed in number of wavelengths) to the antinodes can and will be different for different wavelengths of light. As long as the “difference” in path lengths is a whole number multiple of the wavelength of the light, constructive interference will occur where the waves meet and this will be observed as an antinode. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 687 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 688 path length: the distance between a point source and a chosen point in space difference in path length: the difference between two path lengths, each measured from a different origin and extending to a common point in space An equal number of antinodal lines, or lines of constructive interference, radiate outward on either side of the perpendicular bisector of the line joining S1 and S2. Beginning at the perpendicular bisector, antinodal lines on each side are numbered 1, 2, 3, etc., resulting in a duplicated series (Figure 13.75). For our analysis, we will chose a point, P, on the n 1 antinodal line. The path length from S1 to P and from S2
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to P can be measured in multiples of wavelengths. From Figure 13.75, the path S1P is 10 and the path S2P is 9. The difference in path length from point P to the two sources is 10 9 1 Therefore, for any point along the first antinodal line (on either side of the bisector), the following relationship is true: S1P S2P 1 Recall from section 8.3 that waves that are 1 out of phase will constructively interfere. This explains why the antinodal lines are bright. A similar analysis for a point, P2, on the n 2 antinodal line generates a similar relationship 2 S1P2 – S2P2 An analysis of points on the n 3 and n 4, etc., antinodal lines follows a similar pattern, generating the following relationship between path difference and wavelength along the antinodal lines. S1Pn S2Pn n, where n 1, 2, 3, 4,... (1) If the wavelength is sufficiently large, the difference in path length can be calculated as in Equation (1). Light waves have extremely small wavelengths; therefore, another method of calculating path length becomes appropriate. Difference in Path Length for Waves of Short Wavelength Pn (close) X S1 S2 S1 d X θn S2 d Pn far away Consider the triangles illustrated in Figure 13.76, showing the path length from two sources to point Pn. For any point along the antinodal line Pn, the difference in path length will be equal to the segment S1X. When Pn is selected to be very far away, the paths are nearly parallel. In such a case, S1-X-S2 forms a right triangle allowing the difference in path length to be expressed in terms of angle n, Figure 13.76 Two point sources, separated by a short distance, d, with path lengths to point Pn sin n S1X d sin n difference in path length d 688 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 689 Constructive interference For constructive interference or antinodal lines, substitute the generalization for the difference in path length (Equation 1 above): sin n n d d sin n n for n 1, 2, 3, 4,... (2) Equation (2)
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can be used to determine the wavelength of the light used to produce the interference pattern, where n is the number of the nth antinodal line and d is the distance between the two sources. For a point Pn located far away from the two point sources, the angle n is equivalent to the angle, located between the perpendicular bisector and the straight line drawn between the midpoint of the two sources and the point Pn. The angle is known as the angle of diffraction (Figure 13.77). angle of diffraction: the angle formed between the perpendicular bisector and the straight line to a nodal or antinodal point on the interference pattern screen Pn bisector Destructive interference The analysis for destructive interference, which occurs along the nodal lines, is done in an almost identical way. Recall that destructive interference occurs when the path difference between waves is a half number of wavelengths. Therefore, the generalization for the difference in path length becomes (n 1 ) 2 d d sin n (n 1 ) 2 for n 1, 2, 3, 4,... (3) sin n where n is the number of the nth nodal line relative to the perpendicular bisector, d is the distance between the two sources, and the angle n is equivalent to the angle between the perpendicular bisector and the straight line drawn between the midpoint of the two sources and the point Pn. θα X S1 θ n S2 C (midpoint) d Figure 13.77 When Pn is far away, line S1Pn is approximately parallel to CPn, making n and equal. is the angle of diffraction. Example 13.8 Monochromatic light is incident on two slits separated by 0.30 mm, and the first bright fringe (n 1 antinode) is located at an angle of 0.080 from the central antinode. What is the wavelength of the light? Given d 0.30 mm 3.0 104 m n n 1 0.080 Required Wavelength () Practice Problems 1. Light of an unknown wavelength is incident on two slits separated by 0.20 mm. The second bright fringe is located at an angle of 0.26 from the central antinode. What is the light’s wavelength? 2. Blue light of 460 nm is incident on two slits that are 0.55 mm apart. What is the angle of diffraction for the third antinodal line? Analysis and
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Solution The angle given is relative to the central antinode, which occurs along the perpendicular bisector; therefore, the angle given is relative to the perpendicular bisector. The first bright fringe is a region of constructive Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 689 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 690 3. The second nodal line of an interference pattern occurs at 0.095 relative to the central antinode. The two slits are separated by 0.40 mm. What is the wavelength and colour of light producing this pattern? Answers 1. 4.5 107 2. 0.14 3. 4.4 107 m, violet interference, where the path difference must be one full wavelength different. Therefore, Equation (2) above for constructive interference applies: d sin n n (3.0 104 m) sin 0.080 1 4.2 107 m Paraphrase Monochromatic light with a wavelength of 4.2 107 m will produce a bright fringe at an angle of 0.080 from the central antinode. info BIT When the angle is small (10º) sin is nearly equal to tan as illustrated in the calculated values for both shown below: sin Angle tan sin / tan 1 2 5 0.017452 0.017455 0.9998 0.03490 0.03492 0.9994 0.08716 0.08749 0.9962 10 0.1736 0.1763 0.9847 distance between central antinode and nth antinodal line Pn xn screen Finding Wavelength and the Angle of Diffraction Under Experimental Conditions In experimental settings, it is often difficult to measure the angle of diffraction because this angle is very small relative to a point that is very far away from the two slits. It is easier to obtain a value for sin by determining the ratio x/l, as shown in Figure 13.78. Here, x is the distance between the central antinode (where the perpendicular bisector intersects the screen) and the antinodal fringe. The length of the perpendicular bisector, l, is the distance from the midpoint between the slits to the screen, where the interference pattern is observed. In Figure 13.78, x/l tan, but when l is much greater than x, the ratio of x/l is very small (generally
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less than 0.2) making tan nearly equal to sin (see Infobit). Therefore, it is acceptable to assume sin x/l in this case. By replacing sin with the ratio x/l, we arrive at the following equation for antinodal (bright) fringes: x d l n xd nl Applying the same analysis to nodal (dark) fringes gives: x d l 1 n 2 xd 1 l n 2 l (distance to the screen) n λ S1 θ C d θ S2 Figure 13.78 Determining sin using x and l 690 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 691 Example 13.9 A student measuring the wavelength of light emitted by a krypton gas sample directs the light through two slits separated by 0.264 mm. An interference pattern is created on a screen 3.0 m from the slits and the distance between the second bright (n 2 antinode) fringe and the central antinode is measured to be 1.18 cm. What is one of the wavelengths of light emitted by the krypton gas sample? Given d 0.264 mm 2.64 104 m l 3.0 m x 1.18 cm 1.18 102 m n 2 Required wavelength () Analysis and Solution The bright fringe is a region of constructive interference, where the path difference must be a whole number of wavelengths. Solve for the wavelength by using the equation: xd nl (1.18 102 m)(2.64 104 m) (2)(3.0 m) 5.2 107 m Paraphrase The krypton gas sample emits light with a wavelength of 5.2 107 m. Practice Problems 1. Monochromatic light is incident on two slits separated by 0.15 mm. An interference pattern is observed on a screen 5.0 m away. The distance between the 3rd dark fringe and the central antinode is 4.50 102 m. What is the wavelength of the light? 2. Monochromatic light is incident on two slits separated by 3.00 105 m. The distance between antinodes is 3.10 102 m. If the screen is 1.50 m from the slits, what is the light’s colour and wavelength? 3. A student used light of wavelength 5.00 107 m and found that the
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distance between the third node and the central antinode was 1.00 101 m. If the screen was located 1.20 m away from the slits, how far apart are the slits? Answers 1. 5.4 107 m 2. 6.20 107 m, red 3. 1.50 105 m Poisson’s Bright Spot Using an analysis similar to that in Example 13.9, Young calculated the wavelength of various colours of light and announced his results in 1807. He was still overshadowed by Newton’s reputation and support for the particle model of light. Thus, Young was not taken seriously by the scientific community until 1818, when Augustin Fresnel, a French physicist (Figure 13.79), proposed another mathematical wave theory. This new theory formed a critical turning point in the debate between the wave and particle models of the time. A mathematician by the name of Simon Poisson argued that the equations in Fresnel’s theory could be used to predict a unique diffraction pattern that should be produced when light is incident on a small round disc. Poisson showed that Fresnel’s equations should produce a central bright fringe at the centre of a shadow cast by a solid round disc when it is illuminated by a point source of monochromatic light (Figure13.80(a)). This predicted result was similar to the central bright fringe observed in Young’s double-slit experiment. Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 691 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 692 Fresnel’s equations showed that the light that is diffracted around the edge of the disc should constructively interfere at the centre of the shadow. Poisson was unable to observe this bright spot experimentally and was of the opinion that he had refuted the wave model of light. solid disc shadow equal path length bright spot of constructive interference Figure 13.79 Augustin Fresnel (1788–1827). Fresnel’s analysis of diffraction provided the theoretical groundwork for the transverse wave model of light. He also designed the lenses that were used in lighthouses of the 19th century and in overhead projectors, solar collectors, and beacon lights of the 21st century. diffraction grating: a sheet of glass or plastic etched with a large number of parallel lines. When light is incident on the grating, each line or slit acts as one individual light
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source. Figure 13.81 Diffraction grating info BIT The concentric circles cut into a CD when it is made make it behave as a diffraction grating. So, the rainbow pattern you see when you look at a CD is the product of interference caused by diffraction. Figure 13.80 Poisson’s Bright Spot (a) Constructive interference at the bright spot (b) Photograph of Poisson’s Bright Spot Poisson’s prediction of a bright spot was retested by Dominique Arago in 1818, and this time, the bright spot was verified (Figure 13.80 (b)). The bright spot is known as “Poissson’s Bright Spot” because, even though Poisson was a supporter of the particle model of light, he had predicted the spot’s existence if the wave model of light was correct. Mainly as a result of this verification of Poisson’s Bright Spot, by 1850 the wave model of light was generally accepted by the scientific community. The model was then successfully applied to many of the properties of light. Diffraction Gratings Young’s experiment used only two small point sources of light that were in phase. A diffraction grating (Figure 13.81) has a very large number of equally spaced, parallel lines that act as individual light sources. When light is incident on a multi-slit diffraction grating, an interference pattern, similar to that of a double slit, is produced on a distant screen. But there are several key differences. First, the large number of lines in a diffraction grating can deliver more light energy to the distant screen, increasing the brightness of the interference pattern. Second, the antinodal, bright fringes are more defined, being sharper and narrower. And third, when line separation is very small, the separation between the lines (d) is inversely proportional to the distance between the fringes (x) in the interference pattern according to d xd nl nl x Therefore, the extremely small separation distance between lines on a grating will cause an increase in the separation between the fringes in the pattern. For these reasons, a diffraction grating is a very precise apparatus for investigating the wavelength of light. When a diffraction grating is used with monochromatic light, the interference pattern will have the same colour as the wavelength of light used to produce it. When full-spectrum, white light is used, each
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antinode will appear as a rainbow because each wavelength is diffracted Figure 13.82 692 Unit VII Electromagnetic Radiation 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 693 at a slightly different angle. The relationship between the wavelength and the angle is: n d sin n as was derived earlier for Young’s double-slit experiment. Diffraction gratings are produced by etching a large number of parallel lines on a sheet of glass or plastic. Each grating is defined by the number of lines per centimetre etched on it. Tens of thousands of lines per centimetre are common. The distance between lines is the inverse of the number of lines per centimetre, so as the number of lines increases, the distance between any two lines decreases accordingly. Diffraction in Nature — Solar and Lunar Coronas When the Sun is rising (Figure 13.83) or an overexposed photo of the Moon is taken (Figure 13.84), a pattern of rings appears. These are examples of diffraction in the atmosphere. When light from the Sun or Moon enters the atmosphere it encounters uniformly sized droplets of water and ice crystals. Diffraction occurs as the light bends around the edges of the particles at varying degrees depending upon the wavelength of the light. A series of coloured rings surrounding the astronomical object results. e LAB For a probeware activity, go to www.pearsoned.ca/ school/physicssource. e SIM Investigate diffraction gratings through the use of simulations. Follow the eSim links at www.pearsoned.ca/ school/physicssource. e WEB To learn more about diffraction effects in nature, follow the links at www.pearsoned.ca/ school/physicssource. Example 13.10 Figure 13.83 A rising Sun Figure 13.84 A ring around the Moon A diffraction grating has 1000 lines/cm. When light passes through the grating, an interference pattern is produced on a screen 4.00 m away. The first-order bright fringe is 19.2 cm away from the central antinode. What is the wavelength and colour of the light? Given d 1 cm / 1000 lines 1 103 cm/line 1 105 m/line x 19.2 cm 1.92 101 m n 1 l 4.00 m Required wavelength of light () Analysis and Solution The separation between lines is the inverse of the lines per cent
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imetre given for the diffraction grating. Line separation is very small, and the first-order bright fringe is n 1 because it is an antinode, or line of constructive interference. Therefore, this equation applies: Practice Problems 1. An unknown wavelength of light is incident on a diffraction grating with 2500 lines/cm. The distance between the central antinode and the 3rd dark node is 20.0 cm when the screen is located 50.0 cm from the grating. Determine the wavelength of the light. 2. How many lines/cm are there in a diffraction grating if the 3rd dark fringe is located 5.00 cm from the central antinode when the screen is located 60 cm from the grating? Assume 500 nm. 3. A diffraction grating has 1000 lines/cm. When red light ( 750 nm) is incident on the grating, what will be the separation between bright antinodes on a screen 3.0 m away? Chapter 13 The wave model can be used to describe the characteristics of electromagnetic radiation. 693 13-PearsonPhys30-Chap13 7/24/08 3:44 PM Page 694 Answers 1. 6.40 107 m 2. 6.7 102 lines/cm 3. 23 cm xd nl (1.92 101 m)(1 105 m) (1)(4.00 m) 4.80 107 m 480 nm Paraphrase Based on the interference pattern, the incident light is 480 nm, which corresponds to the colour blue. 13-10 Inquiry Lab 13-10 Inquiry Lab Determining the Wavelength of Red, Green, and Blue Light This investigation will verify the known wavelengths of different colours of light. According to the visible spectrum, red, green, and blue light have wavelengths in the following ranges: • red: 650–750 nm • green: 500–550 nm • blue: 450–500 nm Question What are the wavelengths of red, green, and blue light? Variables manipulated: the distance between the diffraction grating and the screen will be indirectly manipulated responding: the position of the first-order antinodes for red, blue, and green light on the “apparent” screen controlled: single-filament incandescent light source and diffraction grating Materials and Equipment thin-film diffraction grating two metre-sticks single-filament lamp masking tape and pen Required Skills Initiating and Planning Performing and Recording Analy
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