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. How Waves Interact with Matter Waves interact with matter in several ways. The interactions occur when waves pass from one medium to another. The types of interactions are reﬂection, refraction, and diffraction. Each type of interaction is described in detail below. You can see animations of the three types at this URL: http://www.acoustics.salford.ac.uk/schools/teacher/ lesson3/ﬂash/whiteboardcomplete.swf Reﬂection An echo is an example of wave reﬂection. Reﬂection occurs when waves bounce back from a surface they cannot pass through. Reﬂection can happen with any type of waves, not just sound waves. For example, light waves can 24 www.ck12.org Concept 7. Wave Interactions also be reﬂected. In fact, that’s how we see most objects. Light from a light source, such as the sun or a light bulb, shines on the object and some of the light is reﬂected. When the reﬂected light enters our eyes, we can see the object. Reﬂected waves have the same speed and frequency as the original waves before they were reﬂected. However, the direction of the reﬂected waves is different. When waves strike an obstacle head on, the reﬂected waves bounce straight back in the direction they came from. When waves strike an obstacle at any other angle, they bounce back at the same angle but in a different direction. This is illustrated in diagram below. In this diagram, waves strike a wall at an angle, called the angle of incidence. The waves are reﬂected at the same angle, called the angle of reﬂection, but in a different direction. Notice that both angles are measured relative to a line that is perpendicular to the wall. FIGURE 7.1 Refraction Refraction is another way that waves interact with matter. Refraction occurs when waves bend as they enter a new medium at an angle. You can see an example of refraction in the picture below. Light bends when it passes from air to water or from water to air. The bending of the light traveling from the ﬁsh to the man’s eyes causes the ﬁsh to appear to be in a different place from where it actually is. FIGURE 7.2
Waves bend as they enter a new medium because they start traveling at a different speed in the new medium. For 25 example, light travels more slowly in water than in air. This causes it to refract when it passes from air to water or from water to air. Q: Where would the ﬁsh appear to be if the man looked down at it from straight above its actual location? A: The ﬁsh would appear to be where it actually is because refraction occurs only when waves (in this case light waves from the ﬁsh) enter a new medium at an angle other than 90°. www.ck12.org Diffraction Did you ever notice that you can hear sounds around the corners of buildings even though you can’t see around them? The Figure 7.3 shows why this happens. As you can see from the ﬁgure, sound waves spread out and travel around obstacles. This is called diffraction. It also occurs when waves pass through an opening in an obstacle. All waves may be diffracted, but it is more pronounced in some types of waves than others. For example, sound waves bend around corners much more than light does. That’s why you can hear but not see around corners. FIGURE 7.3 For a given type of waves, such as sound waves, how much the waves diffract depends on the size of the obstacle (or opening in the obstacle) and the wavelength of the waves. The Figure 7.4 shows how the amount of diffraction is affected by the size of the opening in a barrier. Note that the wavelength of the wave is the distance between the vertical lines. FIGURE 7.4 26 www.ck12.org Summary Concept 7. Wave Interactions • Three ways that waves may interact with matter are reﬂection, refraction, and diffraction. • Reﬂection occurs when waves bounce back from a surface that they cannot pass through. • Refraction occurs when waves bend as they enter a new medium at an angle and start traveling at a different speed. • Diffraction occurs when waves spread out as they travel around obstacles or through openings in obstacles. Vocabulary • diffraction : Bending of a wave around an obstacle or through an opening in an obstacle. • reﬂection : Bouncing back of waves from a barrier they cannot pass through. • refraction : Bending of waves as they enter a new medium at an angle and change speed. Practice Make
a crossword puzzle of terms relating to wave interactions. Include at least seven different terms. You can use the puzzle maker at the following URL. Then exchange and solve puzzles with a classmate. http://puzzlemaker.disc overyeducation.com/CrissCrossSetupForm.asp Review 1. What is reﬂection? What happens if waves strike a reﬂective surface at an angle other than 90°? 2. Deﬁne refraction. Why does refraction occur? 3. When does diffraction occur? How is wavelength related to diffraction? References 1. Zachary Wilson.. CC BY-NC 3.0 2. Zachary Wilson.. CC BY-NC 3.0 3. Student: Flickr:MaxTorrt; Radio: Flickr:Kansir.. CC BY 2.0 4. Zachary Wilson.. CC BY-NC 3.0 27 CONCEPT 8 www.ck12.org Wave Interference • Deﬁne wave interference. • Compare and contrast constructive and destructive interference. • Explain how standing waves occur. When raindrops fall into still water, they create tiny waves that spread out in all directions away from the drops. What happens when the waves from two different raindrops meet? They interfere with each other. When Waves Meet When two or more waves meet, they interact with each other. The interaction of waves with other waves is called wave interference. Wave interference may occur when two waves that are traveling in opposite directions meet. The two waves pass through each other, and this affects their amplitude. Amplitude is the maximum distance the particles of the medium move from their resting positions when a wave passes through. How amplitude is affected by wave interference depends on the type of interference. Interference can be constructive or destructive. Constructive Interference Constructive interference occurs when the crests, or highest points, of one wave overlap the crests of the other wave. You can see this in the Figure 8.1. As the waves pass through each other, the crests combine to produce a wave with greater amplitude. You can see an animation of constructive interference at this URL: http://phys23p.sl.psu.e du/phys_anim/waves/embederQ1.20100.html Destructive Interference Destructive interference occurs when the crests of one wave overlap the troughs, or lowest points, of another wave. The Figure 8.2 shows what happens. As the waves
pass through each other, the crests and troughs cancel each other out to produce a wave with zero amplitude. You can see an animation of destructive interference at this URL: htt p://phys23p.sl.psu.edu/phys_anim/waves/embederQ1.20200.html 28 www.ck12.org Concept 8. Wave Interference FIGURE 8.1 Standing Waves Waves may reﬂect off an obstacle that they are unable to pass through. When waves are reﬂected straight back from an obstacle, the reﬂected waves interfere with the original waves and create standing waves. These are waves that appear to be standing still. Standing waves occur because of a combination of constructive and destructive interference. You can see animations of standing waves at the URLs below. http://skullsinthestars.com/2008/05/04/classic-science-paper-otto-wieners-experiment-1890/ http://www.physicsc lassroom.com/mmedia/waves/swf.cfm Q : How could you use a rope to produce standing waves? A : You could tie one end of the rope to a ﬁxed object, such as doorknob, and move the other end up and down to generate waves in the rope. When the waves reach the ﬁxed object, they are reﬂected back. The original waves and the reﬂected waves interfere to produce a standing wave. Try it yourself and see if the waves appear to stand still. Summary • Wave interference is the interaction of waves with other waves. • Constructive interference occurs when the crests of one wave overlap the crests of the other wave, causing an increase in wave amplitude. • Destructive interference occurs when the crests of one wave overlap the troughs of the other wave, causing a decrease in wave amplitude. 29 www.ck12.org FIGURE 8.2 • When waves are reﬂected straight back from an obstacle, the reﬂected waves interfere with the original waves and create standing waves. Vocabulary • standing wave : Wave appearing to stand still that forms when a wave and its reﬂected wave interfere. • wave interference : Interaction of waves with other waves. Practice Review wave interference at the following URL. Then do the Check Your Understanding problem at the bottom of the Web page. Be sure to check
your answers. http://www.physicsclassroom.com/class/waves/u10l3c.cfm Review 1. What is wave interference? 2. Create a table comparing and contrasting constructive and destructive interference. 30 www.ck12.org Concept 8. Wave Interference 3. What are standing waves? How do they form? References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Christopher Auyeung.. CC BY-NC 3.0 31 www.ck12.org Wave Speed CONCEPT 9 Objective The student will: • Solve problems involving wavelength, wave speed, and frequency. Vocabulary • wave equation: Relates wavelength and wave speed. Distance is speed times time, x = vt, so wavelength is wave speed times period l = vT. • wave speed: How quickly the peak of each wave is moving forward. The Wave Equation Simple harmonic motion, moving back and forth in place, has amplitude along with period and frequency. Wave motion means that the back-and-forth change is also moving through space. This means that there are two further qualities of a wave. • The wavelength is the distance between two compressions in the direction of motion of the wave, and is represented by the Greek letter lambda, l. For an ocean wave, it would be the distance in meters from the top of one wave to the next. • The wave speed is how quickly the peak of each wave is moving forward, and is represented by v (for velocity, although for our current purpose the direction is not important). These two are related by the period T of the wave. The period is the time it takes for a wave to complete one cycle, which is the time it takes for a peak to move forward one wavelength. The wave equation expresses this. Distance is speed times time, x = vt, so wavelength is wave speed times period l = vT. This can alternately be expressed in terms of frequency. Suppose there are three waves every second. This is frequency f = 3:0 Hz, equivalent to period T = 1 3 s. This means that during one second, three waves come out from the source. Since each peak is one wavelength, l, ahead of the other, this means that during that one second, the lead wave has gone ahead three wavelengths. The distance the wave goes in one second is the wave speed. So, the wave speed is equal to the frequency times the wavelength, v = l
f. Because f = 1 l = vT! v = l T, these are mathematically the same: T = l 1 T = l f The wave equation says distance wavelength is equal to wave speed multiplied by period T. http://demonstrations.wolfram.com/SpeedOfSound/ 32 www.ck12.org Concept 9. Wave Speed FIGURE 9.1 Illustrative Example 1 a. The ripple tank arm in Figure 9.1 has a period of 0.25 s and the length of the arrow in the ﬁgure is 3.76 cm. What is the wavelength of the water waves if the picture is to scale? Answer: There are four wavelengths from the start to the end of the arrow. Wavelength is: l = 3:76cm 4 = 0:94cm b. What is the velocity of the wave? Answer: Since the period is T = 0:25 s, the frequency is, velocity is v = l f = (0:94 cm)(4:0 Hz) = 3:76 cm s. Check Your Understanding f = 1 T = 1 0:25 s = 4:0 Hz or four cycles per second. So the 1. A sound wave travels at the speed of 343:0 m/s through the air at room temperature, 20:0C(68:0F). If the frequency of the sound is 261.6 Hz (a middle-C note), what is the wavelength of the note? Answer = 343:0 m/s 261:6 Hz = 1:311 m 2. X-rays are electromagnetic waves. A particular type of x-ray has a frequency of 3:0 1017 Hz and a wavelength of 1:0 109 m. What is the velocity of this type of x-ray? = 3:0 108 m/s Answer: v = l f = 1:0 109 m)(3:0 1017 1 s This result is true for any electromagnetic radiation traveling in a vacuum, including visible light. 3. Compared to the speed of sound in air at a temperature of 20C, how many times faster is the speed of light through the air at the same temperature? Answer: We’ll assume that the velocity of light vL = 3:0 108 m s is approximately the same through the air as it is 33 www.ck12.org through a vacuum. Number 1 above gave the velocity of sound for a temperature of 20C as vs = 343:
0 m s. Answer: VL = 874; 635:57! 8:7 105. The velocity of light is indeed a good deal greater than the velocity of Vs sound in air. In fact, the velocity of light in vacuum is the greatest velocity that exists. No material object can travel at this velocity! We will discuss these ideas in Chapter 23 (The Special Theory of Relativity). 4. At the moment a lightning ﬂash is seen, a person begins to count off seconds. If he hears the thunder after seven seconds, approximately how far away from the person did the lightning ﬂash originate? Answer : We’ll assume that the sound travels at a velocity of 343 m/s. In seven seconds the sound has traveled x = vt! x = 343 m s (7s) = 2401! 2400 m. 5. Sound travels about 1,500 m/s in water. A destroyer ship locates an enemy submarine using a sonar signal which takes a 4.3 s to travel to, reﬂect, and return to the destroyer. How far is the sub from the destroyer? Answer : The time for the signal to reach the sub is half of the total time of travel,! 4:3 s x = vt! x = 1; 500 m s (2:15 s) = 3225! 3200 m or 3.2 km. 2 = 2:15! 2:2 s. Using References 1. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 34 www.ck12.org Concept 10. Sound Waves CONCEPT 10 • Deﬁne sound. • Describe sound waves and how they are generated. • Identify media through which sound waves can travel. Sound Waves Crack! Crash! Thud! That’s what you’d hear if you were in the forest when this old tree cracked and came crashing down to the ground. But what if there was nobody there to hear the tree fall? Would it still make these sounds? This is an old riddle. To answer the riddle correctly, you need to know the scientiﬁc deﬁnition of sound. Deﬁning Sound In science, sound is deﬁned as the transfer of energy from a vibrating object in waves that travel through matter. Most people commonly use the term sound to mean what they hear when sound waves
enter their ears. The tree above generated sound waves when it fell to the ground, so it made sound according to the scientiﬁc deﬁnition. But the sound wasn’t detected by a person’s ears if there was nobody in the forest. So the answer to the riddle is both yes and no! How Sound Waves Begin All sound waves begin with vibrating matter. Look at the ﬁrst guitar string on the left in the Figure 10.1. Plucking the string makes it vibrate. The diagram below the ﬁgure shows the wave generated by the vibrating string. The moving string repeatedly pushes against the air particles next to it, which causes the air particles to vibrate. The vibrations spread through the air in all directions away from the guitar string as longitudinal waves. In longitudinal waves, particles of the medium vibrate back and forth parallel to the direction that the waves travel. You can see an animation of sound waves traveling through air at this URL: http://www.mediacollege.com/audio/01/sound-wave s.html Q: If there were no air particles to carry the vibrations away from the guitar string, how would sound reach the ear? A: It wouldn’t unless the vibrations were carried by another medium. Sound waves are mechanical waves, so they can travel only though matter and not through empty space. 35 www.ck12.org FIGURE 10.1 A Ticking Clock The fact that sound cannot travel through empty space was ﬁrst demonstrated in the 1600s by a scientist named Robert Boyle. Boyle placed a ticking clock in a sealed glass jar. The clock could be heard ticking through the air and glass of the jar. Then Boyle pumped the air out of the jar. The clock was still ticking, but the ticking sound could no longer be heard. That’s because the sound couldn’t travel away from the clock without air particles to pass the sound energy along. You can see an online demonstration of the same experiment—with a modern twist—at this URL: http://www.youtube.com/watch?v=b0JQt4u6-XI MEDIA Click image to the left for more content. Sound Waves and Matter Most of the sounds we hear reach our ears through the air, but sounds can also travel through liquids and solids. If you swim underwater—or even submerge your ears in bathwater—any
sounds you hear have traveled to your ears through the water. Some solids, including glass and metals, are very good at transmitting sounds. Foam rubber and heavy fabrics, on the other hand, tend to mufﬂe sounds. They absorb rather than pass on the sound energy. Q: How can you tell that sounds travel through solids? A: One way is that you can hear loud outdoor sounds such as sirens through closed windows and doors. You can also hear sounds through the inside walls of a house. For example, if you put your ear against a wall, you may be able to eavesdrop on a conversation in the next room—not that you would, of course. 36 www.ck12.org Summary Concept 10. Sound Waves • In science, sound is deﬁned as the transfer of energy from a vibrating object in waves that travel through matter. • All sound waves begin with vibrating matter. The vibrations generate longitudinal waves that travel through matter in all directions. • Most sounds we hear travel through air, but sounds can also travel through liquids and solids. Vocabulary • sound : Transfer of energy from a vibrating object in longitudinal waves that travel through matter. Practice Watch the video “How Sound Waves Travel” at the following URL. Then explain how sound waves begin and how they travel, using the human voice as an example. http://www.youtube.com/watch?v=_vYYqRVi8vY MEDIA Click image to the left for more content. Review 1. How is sound deﬁned in science? How does this deﬁnition differ from the common meaning of the word? 2. Hitting a drum, as shown in the Figure 10.2, generates sound waves. Create a diagram to show how the sound waves begin and how they reach a person’s ears. FIGURE 10.2 3. How do you think earplugs work? 37 References 1. Guitar string photo by Flickr:jar(); illustration by Christopher Auyeung (CK-12 Foundation).. CC BY 2.0 2. S.L. Ratigan.. CC BY 2.0 www.ck12.org 38 www.ck12.org Concept 11. Frequency and Pitch of Sound CONCEPT 11 Frequency and Pitch of Sound • Deﬁne the pitch of sound. • Relate the pitch of sound to the frequency of sound waves. • Identify
infrasound and ultrasound. A marching band passes you as it parades down the street. You heard it coming from several blocks away. Now that the different instruments have ﬁnally reached you, their distinctive sounds can be heard. The tiny piccolos trill their bird-like high notes, and the big tubas rumble out their booming bass notes. Clearly, some sounds are higher or lower than others. High or Low How high or low a sound seems to a listener is its pitch. Pitch, in turn, depends on the frequency of sound waves. Wave frequency is the number of waves that pass a ﬁxed point in a given amount of time. High-pitched sounds, like the sounds of the piccolo in the Figure 11.1, have high-frequency waves. Low-pitched sounds, like the sounds of the tuba Figure 11.1, have low-frequency waves. For a video demonstration of frequency and pitch, go to this URL: http://www.youtube.com/watch?v=irqfGYD2UKw Can You Hear It? The frequency of sound waves is measured in hertz (Hz), or the number of waves that pass a ﬁxed point in a second. Human beings can normally hear sounds with a frequency between about 20 Hz and 20,000 Hz. Sounds with frequencies below 20 hertz are called infrasound. Infrasound is too low-pitched for humans to hear. Sounds with frequencies above 20,000 hertz are called ultrasound. Ultrasound is too high-pitched for humans to hear. 39 www.ck12.org FIGURE 11.1 Some other animals can hear sounds in the ultrasound range. For example, dogs can hear sounds with frequencies as high as 50,000 Hz. You may have seen special whistles that dogs—but not people—can hear. The whistles produce sounds with frequencies too high for the human ear to detect. Other animals can hear even higher-frequency sounds. Bats, like the one pictured in the Figure 11.2, can hear sounds with frequencies higher than 100,000 Hz! FIGURE 11.2 Q: Bats use ultrasound to navigate in the dark. Can you explain how? A: Bats send out ultrasound waves, which reﬂect back from objects ahead of them. They sense the reﬂected sound waves and use the information to detect objects they can’t see in the
dark. This is how they avoid ﬂying into walls and trees and also how they ﬁnd ﬂying insects to eat. Summary • How high or low a sound seems to a listener is its pitch. Pitch, in turn, depends on the frequency of sound waves. • High-frequency sound waves produce high-pitched sounds, and low-frequency sound waves produce low- pitched sounds. 40 www.ck12.org Concept 11. Frequency and Pitch of Sound • Infrasound has wave frequencies too low for humans to hear. Ultrasound has wave frequencies too high for humans to hear. Vocabulary • infrasound : Sound with a frequency below the range of human hearing (less than 20 hertz). • pitch : How high or low a sound seems to a listener. • ultrasound : Sound with a frequency above the range of human hearing (greater than 20,000 hertz). • wave frequency : Number of waves that pass a ﬁxed point in a given amount of time. Practice At the following URL, complete the interactive module to review and test your knowledge of the frequency and pitch of sound. http://www.engineeringinteract.org/resources/oceanodyssey/flash/concepts/pitch.htm Review 1. What is the pitch of sound? 2. How is the pitch of sound related to the frequency of sound waves? 3. Deﬁne infrasound and ultrasound. References 1. Piccolo: U.S. Navy photo by Chief Mass Communication Specialist David Rush; Tuba: Bob Fishbeck.. Public Domain 2... Public Domain 41 CONCEPT 12 • Give the speed of sound in dry air at 20 °C. • Describe variation in the speed of sound in different media. • Explain the effect of temperature on the speed of sound. www.ck12.org Speed of Sound Has this ever happened to you? You see a ﬂash of lightning on the horizon, but several seconds pass before you hear the rumble of thunder. The reason? The speed of light is much faster than the speed of sound. What Is the Speed of Sound? The speed of sound is the distance that sound waves travel in a given amount of time. You’ll often see the speed of sound given as 343 meters per second. But that’s just the speed of sound under a certain set of conditions, speciﬁcally, through dry air at 20 °C
. The speed of sound may be very different through other matter or at other temperatures. Speed of Sound in Different Media Sound waves are mechanical waves, and mechanical waves can only travel through matter. The matter through which the waves travel is called the medium (plural, media). The Table 12.1 gives the speed of sound in several different media. Generally, sound waves travel most quickly through solids, followed by liquids, and then by gases. Particles of matter are closest together in solids and farthest apart in gases. When particles are closer together, they can more quickly pass the energy of vibrations to nearby particles. You can explore the speed of sound in different media at this URL: http://www.ltscotland.org.uk/resources/s/sound/speedofsound.asp?strReferringChannel=resources&strReferringPageI D=tcm:4-248291-64 42 www.ck12.org Concept 12. Speed of Sound Medium (20 °C) Dry Air Water Wood Glass Aluminum TABLE 12.1: speed of sound Speed of Sound Waves (m/s) 343 1437 3850 4540 6320 Q: The table gives the speed of sound in dry air. Do you think that sound travels more or less quickly through air that contains water vapor? (Hint: Compare the speed of sound in water and air in the table.) A: Sound travels at a higher speed through water than air, so it travels more quickly through air that contains water vapor than it does through dry air. Temperature and Speed of Sound The speed of sound also depends on the temperature of the medium. For a given medium, sound has a slower speed at lower temperatures. You can compare the speed of sound in dry air at different temperatures in the following Table 12.2. At a lower temperature, particles of the medium are moving more slowly, so it takes them longer to transfer the energy of the sound waves. Temperature of Air 0 °C 20 °C 100 °C TABLE 12.2: speed of sound Speed of Sound Waves (m/s) 331 343 386 Q: What do you think the speed of sound might be in dry air at a temperature of -20 °C? A: For each 1 degree Celsius that temperature decreases, the speed of sound decreases by 0.6 m/s. So sound travels through dry, -20 °C air at a speed of 319 m/s. Summary • The speed of sound is the distance that sound waves travel
in a given amount of time. The speed of sound in dry air at 20 °C is 343 meters per second. • Generally, sound waves travel most quickly through solids, followed by liquids, and then by gases. • For a given medium, sound waves travel more slowly at lower temperatures. Vocabulary • speed of sound : Speed at which sound waves travel, which is 343 m/s in dry air at 20 °C. Practice At the following URL, read about the speed of sound in different materials. Be sure to play the animation. Then answer the questions below. http://www.ndt-ed.org/EducationResources/HighSchool/Sound/speedinmaterials.htm 43 1. Describe what you hear when you play the animation. Explain your observations. 2. Name two properties of materials that affect the speed of sound waves. How do they affect the speed of sound? 3. Explain why sound waves moves more quickly through warmer air than cooler air. www.ck12.org Review 1. What is the speed of sound in dry air at 20 °C? 2. Describe variation in the speed of sound through various media. 3. Explain how temperature affects the speed of sound. 44 www.ck12.org Concept 13. Resonance with Sound Waves CONCEPT 13 Resonance with Sound Waves Objectives The student will: • Understand the conditions for resonance. • Solve problems with strings and pipes using the condition for resonance. Vocabulary • beat frequency • natural frequency: The frequency at which a system vibrates normally when given energy without outside interference. • resonance: Timing force to be the same as natural frequency. • sympathetic vibrations Introduction Many systems have a tendency to vibrate. When the forced vibration frequency is the same as the natural frequency, the amplitude of vibration can increase tremendously. A well-known example of this situation is pushing a person on a swing, Figure below. We know from study of simple pendulums that without being pushed, the person in the swing rocks back and forth with a frequency that depends on gravity and the length of the chain. f = 1 2p r g L This is one example of a natural frequency – the frequency at which a system vibrates normally when given energy without outside interference. Pushing on the person in the swing will affect the amplitude of the swinging. This is called forced vibration – when a periodic force from one object (the person pushing) affects the vibration of another object (the person swinging). To get the most effect
, the person pushing will start just at the very back of the swing. In other words, the frequency of how often they push is exactly the same as the frequency of the swing. Suppose they do not push at the right time, but instead push at some other frequency. That would mean that sometimes they are pushing forward when the swing is still going backward. In that case, the swing would slow down – i.e. the amplitude of the swing will be reduced. Timing the pushes to be the same as the natural frequency is called resonance. For this reason, the natural frequency is also known as the resonant frequency. If the pushes are timed just right, then even if each individual push is small, the vibration will get larger with each push. 45 www.ck12.org FIGURE 13.1 A classic example of an unfortunate consequence of a forced vibration at resonant frequency is what happened to the Tacoma Narrows Bridge, in 1940. See the link below. http://www.youtube.com/watch?v=3mclp9QmCGs In Figure below, the bridge is beginning to resonate, in part, due to the frequency of vibration of the wind gusts. In Figure below, we see that the bridge is no longer able to respond elastically to the tremendous amplitude of vibration from the forced vibration of wind energy (at its resonant frequency), and it is torn apart. FIGURE 13.2 Modern bridges are built to avoid this effect, but through history there are a number of documented situations where a forced vibration at resonance had dire results. The Broughton Suspension Bridge (1831) and the Angers Bridge (1850) are two examples of bridges believed to have collapsed due to the effect of soldiers marching at a regular pace that caused resonance. The Albert Bridge in West London, England has been nicknamed The Trembling Lady because it has been set into resonance so often by marching soldiers. Though soldiers no longer march across the bridge, there still remains a sign of concern as shown in Figure below. 46 www.ck12.org Concept 13. Resonance with Sound Waves FIGURE 13.3 FIGURE 13.4 Sympathetic Vibrations There is a typical classroom physics demonstration where one tuning fork is set into motion and an identical tuning fork, if placed closed enough, will also be set vibrating, though with smaller amplitude. The same effect occurs when tuning a guitar. One string is plucked and another, whose length is shortened by holding it
down some distance from the neck of the guitar, will also be set into vibration. When this condition is met, both strings are vibrating with the same frequency. We call this phenomenon sympathetic vibration. 47 www.ck12.org MEDIA Click image to the left for more content. FIGURE 13.5 https://www.youtube.com/watch?v=tnS0SYF4pYE In Figure above, a set of pendulums are ﬁxed to a horizontal bar that can be easily jostled. Pendulums A and E have the same length. If one of them is set swinging, the horizontal bar will be forced into moving with a period equal to that of the pendulum, which, in turn, will cause the other pendulum of the same length to begin swinging. Any pendulum that is close in length to pendulums A and E, for example, pendulum D, will also begin to swing. Pendulum D will swing with smaller amplitude than pendulums A and E since its resonant frequency is not quite the same as pendulums A and E. Pendulums with lengths dramatically different from pendulums A and E will hardly move at all. You can try a similar demonstration out yourself with the following simulation: http://phet.colorado.edu/en/simulation/resonance Resonance is a very common phenomenon, especially with sound. The length of any instrument is related to what note it plays. If you blow into the top of a bottle, for example, the note will vary depending on the height of air in the bottle. This plays an important role in human voice generation. The length of the human vocal tube is between 17 cm and 18 cm. The typical frequencies of human speech are in the range of 100 Hz to 5000 Hz. FIGURE 13.6 48 www.ck12.org Concept 13. Resonance with Sound Waves By using the muscles in their throat, singers change the note they sing. A dramatic example of this is breaking glass with the human voice. By singing at exactly the resonant frequency of a delicate wine glass, the glass will resonate with the note and shatter. http://www.youtube.com/watch?v=7YmuOD5X4L8 The resonance of sound is also a mechanical analogue to how a radio set receives a signal. The Figure below shows one of the earliest radio designs, called a crystal radio because the element which detected the radio waves was a
crystalline mineral such as galena. FIGURE 13.7 An old crystal radio set. In modern times, the air is ﬁlled with all manner of radio waves. In order to listen to your favorite radio station, you must tune your radio to resonate with only the frequency of the radio station. When you hear the tuning number of a radio station, such as “101.3 FM”, that is the measure in megahertz, 101:3 MHz = 1:013 108 Hz. The coiled wire (called an inductor) and the capacitor in Figure above act together to tune in a speciﬁc radio station. Effectively, the capacitor and inductor act analogously to a pendulum of a speciﬁc length that will only respond to vibrations of another pendulum of the same length. So when tuned, only a speciﬁc radio frequency will cause resonance in the radio antenna. Strings Fixed at Both Ends A case of natural frequency that you can observe plainly is when you pluck a string or stretched rubber band. Normally, the string will vibrate at a single widest point in the middle. This is called the fundamental or ﬁrst harmonic resonance of the string. This is the same as the natural frequency of a simple pendulum or mass on a spring. Because it vibrates all along its length, though, the string also lets us see further patterns of resonance. By vibrating the end of the string rather than just plucking it, we can force vibration at frequencies other than the ﬁrst harmonic. When the string is set into vibration, energy will travel down the string and reﬂect back toward the end where the waves are being generated. This steady pattern of vibration is called a standing wave. The points where the reﬂecting waves interfere destructively with the “generated’ waves are called nodes. The points where the reﬂecting waves interfere constructively with the generated waves are called anti-nodes. Figure below shows a string ﬁxed at both ends vibrating in its fundamental mode. There are two nodes shown and one antinode. The dashed segment represents the reﬂected wave. If you compare the wave shape of the ﬁrst harmonic to the wave shape of Figure above, it will be apparent that the ﬁrst harmonic contains one-half of a wavelength, l. Therefore
L, the length of the unstretched string, is equal to one-half the wavelength, which is 1 2 l1 = L! l1 = 2L: 49 www.ck12.org FIGURE 13.8 The second harmonic contains an entire wavelength 2 2 l2 = L! l2 = L as shown in Figure below. FIGURE 13.9 And the third harmonic contains one and one-half wavelengths 3 2 l3 = L! l3 = 2 3 L. FIGURE 13.10 If the pattern continues then the fourth harmonic will have a wavelength of 4 pressions for the length of the string in terms of the wavelength, a simple pattern emerges: L = 1. We can express the condition for standing waves (and of resonance, as well) as L = n the length of string and n = 1; 2; 3 : : :. 2 l4 = L! l4 = 1 2 L. Looking at the ex2 l2; 3 2 l3; 4 2 l4 : : : n L, where L is 2 l1; 2 2 ln or ln = 2 Check Your Understanding 1. How many nodes and anti-nodes are shown in Figure above.? Answer: There are three nodes and two anti-nodes. 2. If the length of the unstretched string is 20 cm, what is the wavelength for the 10th harmonic? Answer: ln = 2 5 (20 cm) = 4 cm n L! l10 = 2 10 L = 1 Strings Fixed at One End and Opened at One End A string ﬁxed at one only end displays a different standing wave pattern. In this case, the unbounded end of a string of length L is an antinode. The fundamental mode (the ﬁrst harmonic) for the length L of string contains only one- 50 www.ck12.org Concept 13. Resonance with Sound Waves fourth of a wavelength as shown in Figure below below. Therefore L, the length of the unstretched string, is equal to one-quarter the wavelength, which is 1 4 l1 = L! l1 = 4L. FIGURE 13.11 The second harmonic contains three-quarters of a wavelength 3 4 l2 = L! l2 = 4 3 L as shown in Figure above FIGURE 13.12 The third harmonic contains ﬁve-fourths of a wavelength 5 4 l2 = L! l2 = 4 5 L as
shown in Figure above The third harmonic contains ﬁve-fourths wavelengths as shown in Figure below. If the pattern continues, then the fourth harmonic will have a wavelength of 7 expressions for the length of the string in terms of the wavelength, a simple pattern emerges 1 We can express the condition for resonance as L = n. 7 L. Looking at the 4 l3; 7 4 l4 : : :. n L, where L is the length of string and n = 1; 3; 5 : : : 4 l4 = L! l4 = 4 4 ln or ln = 4 4 l2; 5 4 l1; 3 As long as the tension in the string remains ﬁxed, the velocity of the wave along the string remains constant. Does it seem reasonable that a sagging string will not support the same wave velocity as a taut string? Since v = l f product l f is constant as long as the wave velocity remains constant. Therefore, for a string vibrating in many different modes, we have v = l1 f1 = l2 f2 = l3 f3 : : :. 51 www.ck12.org FIGURE 13.13 Illustrative Example 1 1a. If the frequency of the ﬁrst harmonic for a string ﬁxed at both ends is f1, determine the frequency for successive harmonics in terms of f1. Answer: We know that ln = 2 v = l1 f1 = 2L f1 and substituted into fn = n fn = n n L and v = l f. Combining, we have v = 2 2L v. But v can be expressed n L f! fn = n 2L v, giving 2L (2L f1) = n f1! fn = n f1 1b. If the ﬁrst harmonic has frequency of 261 Hz, what frequencies do the second and third harmonics have? Answer: Since fn = n f1! 2(261 Hz) = 522 Hz fn = n f1! 3(261 Hz) = 783 Hz All whole number multiples of the ﬁrst harmonic (the fundamental) are called harmonics. String instruments, as well as non-string instruments, can actually vibrate with many different frequencies simultaneously (called modes). For example, a string may vibrate with frequencies 261 Hz, 522 Hz and 783 Hz simultaneously. One of attributes
of the “quality” or “timbre” of musical instruments depends upon the combination of the various overtones produced by the instrument. Check Your Understanding 1. A tuning fork has a frequency of 512 Hz stamped on it. When it is struck, a student claims she can hear higher frequencies from the tuning fork. Is this possible? Answer : Yes, it is. The tuning fork may be producing harmonics, in which case the student may be hearing frequencies in multiples of 512 Hz, such as 1,024 Hz and 1,536 Hz. 2. A string with a fundamental frequency of 220 Hz vibrates in its third harmonic with a wavelength of 60 cm. What is the wave velocity on the string? 52 www.ck12.org Concept 13. Resonance with Sound Waves Answer : v = l f but f = 3 f1 = 3(220 Hz) = 660 Hz, so l = 0:60 m v = (660 Hz)(0:60 m) = 396 m/s. Open and Closed Pipes and Tubes In our discussions of pipes, the length of the pipe will be assumed to be much greater than the diameter of the pipe. An open pipe, as the name implies, has both ends open. Though open pipes have antinodes at their ends, the resonant conditions for standing waves in an open pipe are the same as for a string ﬁxed at both ends. Thus for an open pipe we have: for n = 1; 2; 3:::; L = n 2 ln, or ln = 2 n L. There is a simple experiment your instructor may have you do in class that demonstrates resonance in an open tube. Roll two sheets of long paper into two separate tubes and use a small amount of tape to keep them rolled. Have the diameter of one tube just small enough to ﬁt inside the other tube so the inside tube can freely slide back and forth. Hold a struck tuning fork (your instructor will make sure the frequency is adequate) close to the end of the outer tube while the inside tube is moved slowly. When the total length of the tubes is the proper length to establish resonance, you’ll hear a noticeable increase in the volume of the sound. At this moment, there are standing waves present in the tubes. A closed pipe is closed at only one end. Closed pipes have the same standing wave patterns as a string ﬁxed at one end and unbound at the other end
. They therefore have the same resonant conditions as a string ﬁxed at only one end, for n = 1; 3 ln or ln = 4 n L. A closed pipe supporting the ﬁrst harmonic (the fundamental frequency) will ﬁt one-fourth of the wavelength, the second harmonic will ﬁt three-fourths, and so on, as shown in Figure below. Compare these pictures to those in Figure above for a string ﬁxed at only one end FIGURE 13.14 A standard physics laboratory experiment is to ﬁnd the velocity of sound by using a tuning fork that vibrates over a closed pipe as shown in Figure Figure above. The water level in a pipe is slowly changed until the ﬁrst harmonic is heard. http://demonstrations.wolfram.com/ResonanceInOpenAndClosedPipes/ 53 www.ck12.org FIGURE 13.15 Illustrative Example 2 Resonance is established in a hollow tube similar to that shown in Figure above with a tuning fork of 512 Hz. The distance from the tube opening to the water level is 16.8 cm. a. What is the velocity of sound according to this experiment? Answer: The wave velocity equation is v = l f. One-fourth of the wave occupies the length of the tube for the ﬁrst harmonic. So the wavelength of the resonant wave must be four times the length of the hollow tube. That is, l1 = 4L = 4 16:8 cm = 67:2 cm = 0:672 m v = (0:672 m)(512 Hz) = 344 m/s b. The velocity of sound changes with temperature as given by the formula v = 330 + 0:6T, where T is the temperature in degrees centigrade. Using the result of part A, determine the temperature at the location the experiment was conducted. Answer: We simply set the result of part A equal to the given equation: 344 = 330 + 0:6T! T = 23:3C (or about 74F ). References 1. Evan Long (Flickr: Clover_1). http://www.ﬂickr.com/photos/clover_1/4915240660/. CC-BY-NC 2.0 54 www.ck12.org Concept 13. Resonance with Sound Waves 2. Barney Elliot, Prelinger Archives. struction.
ogg. Public Domain http://commons.wikimedia.org/wiki/File:Tacoma_Narrows_Bridge_de 3.. http://commons.wikimedia.org/wiki/File:Tacoma-narrows-bridge-collapse.jpg. Public Domain 4. Russell James Smith (Flickr: russelljsmith). http://www.ﬂickr.com/photos/russelljsmith/2146210247/. CC-BY 2.0 5. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 6. CK-12 Foundation - Raymond Chou, using public domain image by Mariana Ruiz Villarreal (Wikimedia:. CC-BY- http://commons.wikimedia.org/wiki/File:Respiratory_system_complete_en.svg LadyofHats). NC-SA 3.0 7. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 8. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 9. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 10. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 11. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 12. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 13. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 14. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 15. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 55 CONCEPT 14 www.ck12.org Sound in a Tube Students will learn how to analyze and solve problems where standing waves (and hence sound) is produced in a tube. Students will learn how to analyze and solve problems where standing waves (and hence sound) is produced in a tube. Key Equations v = l f for a tube closed at one end f = nv=4L, where n is always odd for a tube open at both ends f = nv=
2L, where n is an integer Guidance In the case of a tube that is open at one end, a node is forced at the closed end (no air molecules can vibrate up and down) and an antinode occurs at the open end (here, air molecules are free to move). A different spectrum of standing waves is produced. For instance, the fundamental standing sound wave produced in a tube closed at one end is shown below. In this case, the amplitude of the standing wave is referring to the magnitude of the air pressure variations. This standing wave is the ﬁrst harmonic and one can see that the wavelength is l = 4L. Since v = l f, the frequency of oscillation is f = v=4L. In general, the frequency of oscillation is f = nv=4L, where n is always odd. Example 1 Question The objects A, B, and C below represent graduated cylinders of length 50 cm which are ﬁlled with water to the depths of 10, 20 and 30 cm, respectively as shown. 56 www.ck12.org Concept 14. Sound in a Tube a) If you blow in each of these tubes, which (A,B,C) will produce the highest frequency sound? b) What is the wavelength of the 1st harmonic (i.e. fundamental) of tube B? c) The speed of sound at room temperature is about 343 m/s. What is the frequency of the 1st harmonic for tube B? Solution a) The water forms the bottom of the tube and thus where the node of the wave will be. Thus the air column is where the sound wave can exist. The larger the air column, the larger the wavelength. Frequency is inversely proportional to wavelength, thus the tube with the smallest air column will have the highest frequency. So the answer is tube C. b) The air column is 50 cm - 20 cm = 30 cm. The ﬁrst harmonic has a quarter wavelength in the tube. Thus l = 4 L. Thus, l = 4 30cm = 120cm c) Using the wave equation for the ﬁrst harmonic (thus, n = 1) of a tube open at one end we get f = v 286Hz 4L = 343m/s 1:2m = Watch this Explanation MEDIA Click image to the left for more content. 57 www.ck12.org MEDIA Click image to the left for more content.
Time for Practice 1. 2. Aborigines, the native people of Australia, play an instrument called the Didgeridoo like the one shown above. The Didgeridoo produces a low pitch sound and is possibly the world’s oldest instrument. The one shown above is about 1.3 m long and open at both ends. a. Knowing that when a tube is open at both ends there must be an antinode at both ends, draw the ﬁrst 3 harmonics for this instrument. b. Calculate the frequency of the ﬁrst 3 harmonics assuming room temperature and thus a velocity of sound of 340 m/s. Then take a shot at deriving a generic formula for the frequency of the n th standing wave mode for the Didgeridoo, as was done for the string tied at both ends and for the tube open at one end. 3. Students are doing an experiment to determine the speed of sound in air. They hold a tuning fork above a large empty graduated cylinder and try to create resonance. The air column in the graduated cylinder can be adjusted by putting water in it. At a certain point for each tuning fork a clear resonance point is heard. The students adjust the water ﬁnely to get the peak resonance then carefully measure the air column from water to top of air column. (The assumption is that the tuning fork itself creates an anti-node and the water creates a node.) The following data were collected: TABLE 14.1: Wavelength (m) Speed of sound (m/s) Frequency fork (Hz) 184 328 384 512 1024 of tuning Length of air column (cm) 46 26 22 16 24 (a) Fill out the last two columns in the data table. (b) Explain major inconsistencies in the data or results. (c) The graduated cylinder is 50 cm high. Were there other resonance points that could have been heard? If so what would be the length of the wavelength? (d) What are the inherent errors in this experiment? 3. Peter is playing tones by blowing across the top of a glass bottle partially ﬁlled with water. He notices that if he blows softly he hears a lower note, but if he blows harder he hears higher frequencies. (a) In the 120 cm long tubes below draw three diagrams showing the ﬁrst three harmonics produced in the tube. Please draw the waves as transverse even though we know sound waves are longitudinal (reason for this
, obviously, is that it 58 www.ck12.org Concept 14. Sound in a Tube is much easier to draw transverse waves rather than longitudinal). Note that the tube is CLOSED at one end and OPEN at the other. (b) Calculate the frequencies of the ﬁrst three harmonics played in this tube, if the speed of sound in the tube is 340 m/s. (c) The speed of sound in carbon dioxide is lower than in air. If the bottle contained CO2 instead of air, would the frequencies found above be higher or lower? Knowing that the pitch of your voice gets higher when you inhale helium, what can we say about the speed of sound in He. Answers: 1. (b) 131 Hz, 262 Hz, 393 Hz; formula is same as closed at both ends 2. Discuss in class 3. (b) 70.8 Hz, 213 Hz, 354 Hz (c) voice gets lower pitch. Speed of sound in He must be faster by same logic. 59 Physics Unit 11: Electromagnetic Waves Patrick Marshall Jean Brainard, Ph.D. Ck12 Science Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable version of this book, as well as other interactive content, visit www.ck12.org AUTHORS Patrick Marshall Jean Brainard, Ph.D. Ck12 Science CK-12 Foundation is a non-proﬁt organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2014 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include
the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: April 6, 2014 iii Contents www.ck12.org 1 6 9 12 16 21 24 27 32 35 Contents 1 Electromagnetic Waves 2 Properties of Electromagnetic Waves 3 Electromagnetic Spectrum 4 Law of Reﬂection 5 Refraction of Light 6 Total Internal Reﬂection 7 Single Slit Interference 8 Double Slit Interference 9 Diffraction Gratings 10 Wave-Particle Theory iv www.ck12.org Concept 1. Electromagnetic Waves CONCEPT 1 Electromagnetic Waves • Deﬁne electromagnetic wave and electromagnetic radiation. • Describe the electric and magnetic ﬁelds of an electromagnetic wave. • Explain how electromagnetic waves begin and how they travel. • State how electromagnetic waves may interact with matter. • Identify sources of electromagnetic waves. Did you ever wonder how a microwave works? It directs invisible waves of radiation toward the food placed inside of it. The radiation transfers energy to the food, causing it to get warmer. The radiation is in the form of microwaves, which are a type of electromagnetic waves. What Are Electromagnetic Waves? Electromagnetic waves are waves that consist of vibrating electric and magnetic ﬁelds. Like other waves, electromagnetic waves transfer energy from one place to another. The transfer of energy by electromagnetic waves is called electromagnetic radiation. Electromagnetic waves can transfer energy through matter or across empty space. For an excellent video introduction to electromagnetic waves, go to this URL: http://www.youtube.com/watch?v=cfXz wh3KadE MEDIA Click image to the left for more content. Q: How do microwaves transfer energy inside a microwave oven? A: They transfer energy through the air inside the oven to the food. 1 May
the Force Be with You A familiar example may help you understand the vibrating electric and magnetic ﬁelds that make up electromagnetic waves. Consider a bar magnet, like the one in the Figure 1.1. The magnet exerts magnetic force over an area all around it. This area is called a magnetic ﬁeld. The ﬁeld lines in the diagram represent the direction and location of the magnetic force. Because of the ﬁeld surrounding a magnet, it can exert force on objects without touching them. They just have to be within its magnetic ﬁeld. www.ck12.org FIGURE 1.1 Q: How could you demonstrate that a magnet can exert force on objects without touching them? A: You could put small objects containing iron, such as paper clips, near a magnet and show that they move toward the magnet. An electric ﬁeld is similar to a magnetic ﬁeld. It is an area of electrical force surrounding a positively or negatively charged particle. You can see electric ﬁelds in the following Figure 1.2. Like a magnetic ﬁeld, an electric ﬁeld can exert force on objects over a distance without actually touching them. How an Electromagnetic Wave Begins An electromagnetic wave begins when an electrically charged particle vibrates. The Figure 1.3 shows how this happens. A vibrating charged particle causes the electric ﬁeld surrounding it to vibrate as well. A vibrating electric ﬁeld, in turn, creates a vibrating magnetic ﬁeld. The two types of vibrating ﬁelds combine to create an electromagnetic wave. You can see animations of electromagnetic waves at these URLs: http://www.youtube.com/ http://www.phys.hawaii.edu/~teb/java/ntnujava/emWave/emWave.htm watch?v=Qju7QnbrOhM&feature=related l 2 www.ck12.org Concept 1. Electromagnetic Waves FIGURE 1.2 FIGURE 1.3 How an Electromagnetic Wave Travels As you can see in the diagram above, the electric and magnetic ﬁelds that make up an electromagnetic wave are perpendicular (at right angles) to each other. Both ﬁelds are also perpendicular to the direction that the wave travels. Therefore, an electromagnetic wave is
a transverse wave. However, unlike a mechanical transverse wave, which can only travel through matter, an electromagnetic transverse wave can travel through empty space. When waves travel through matter, they lose some energy to the matter as they pass through it. But when waves travel through space, no energy is lost. Therefore, electromagnetic waves don’t get weaker as they travel. However, the energy is “diluted” as it travels farther from its source because it spreads out over an ever-larger area. 3 Electromagnetic Wave Interactions When electromagnetic waves strike matter, they may interact with it in the same ways that mechanical waves interact with matter. Electromagnetic waves may: www.ck12.org • reﬂect, or bounce back from a surface; • refract, or bend when entering a new medium; • diffract, or spread out around obstacles. Electromagnetic waves may also be absorbed by matter and converted to other forms of energy. Microwaves are a familiar example. When microwaves strike food in a microwave oven, they are absorbed and converted to thermal energy, which heats the food. Sources of Electromagnetic Waves The most important source of electromagnetic waves on Earth is the sun. Electromagnetic waves travel from the sun to Earth across space and provide virtually all the energy that supports life on our planet. Many other sources of electromagnetic waves depend on technology. Radio waves, microwaves, and X rays are examples. We use these electromagnetic waves for communications, cooking, medicine, and many other purposes. Summary • Electromagnetic waves are waves that consist of vibrating electric and magnetic ﬁelds. They transfer energy through matter or across space. The transfer of energy by electromagnetic waves is called electromagnetic radiation. • The electric and magnetic ﬁelds of an electromagnetic wave are areas of electric or magnetic force. The ﬁelds can exert force over objects at a distance. • An electromagnetic wave begins when an electrically charged particle vibrates. This causes a vibrating electric ﬁeld, which in turn creates a vibrating magnetic ﬁeld. The two vibrating ﬁelds together form an electromagnetic wave. • An electromagnetic wave is a transverse wave that can travel across space as well as through matter. When it travels through space, it doesn’t lose energy to a medium as a mechanical wave does. • When electromagnetic waves strike matter, they may be re�
��ected, refracted, or diffracted. Or they may be absorbed by matter and converted to other forms of energy. • The most important source of electromagnetic waves on Earth is the sun. Many other sources of electromag- netic waves depend on technology. Vocabulary • electromagnetic radiation : Transfer of energy by electromagnetic waves across space or through matter. • electromagnetic wave : Transverse wave consisting of vibrating electric and magnetic ﬁelds that can travel across space. Practice Watch the electromagnetic wave animation at the following URL, and then answer the questions below. http://www. youtube.com/watch?v=4CtnUETLIFs 4 www.ck12.org Concept 1. Electromagnetic Waves MEDIA Click image to the left for more content. 1. Identify the vibrating electric and magnetic ﬁelds of the wave. 2. Describe the direction in which the wave is traveling. Review 1. What is an electromagnetic wave? 2. Deﬁne electromagnetic radiation. 3. Describe the electric and magnetic ﬁelds of an electromagnetic wave. 4. How does an electromagnetic wave begin? How does it travel? 5. Compare and contrast electromagnetic and mechanical transverse waves. 6. List three sources of electromagnetic waves on Earth. References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Christopher Auyeung.. CC BY-NC 3.0 3. Christopher Auyeung.. CC BY-NC 3.0 5 CONCEPT 2 Properties of Electromagnetic Waves www.ck12.org • State the speed of light. • Describe wavelengths and frequencies of electromagnetic waves. • Relate wave frequency to wave energy. • Show how to calculate wavelength or wave frequency if the other value is known. What do these two photos have in common? They both represent electromagnetic waves. These are waves that consist of vibrating electric and magnetic ﬁelds. They transmit energy through matter or across space. Some electromagnetic waves are generally harmless. The light we use to see is a good example. Other electromagnetic waves can be very harmful and care must be taken to avoid too much exposure to them. X rays are a familiar example. Why do electromagnetic waves vary in these ways? It depends on their properties. Like other waves, electromagnetic waves have properties of speed, wavelength, and frequency. Speed of Electromagnetic Waves All electromagnetic waves travel at the same speed through empty
space. That speed, called the speed of light, is about 300 million meters per second (3.0 x 10 8 m/s). Nothing else in the universe is known to travel this fast. The sun is about 150 million kilometers (93 million miles) from Earth, but it takes electromagnetic radiation only 8 minutes to reach Earth from the sun. If you could move that fast, you would be able to travel around Earth 7.5 times in just 1 second! You can learn more about the speed of light at this URL: http://videos.howstuffworks.com/discove ry/29407-assignment-discovery-speed-of-light-video.htm MEDIA Click image to the left for more content. Wavelength and Frequency of Electromagnetic Waves Although all electromagnetic waves travel at the same speed across space, they may differ in their wavelengths, frequencies, and energy levels. 6 www.ck12.org Concept 2. Properties of Electromagnetic Waves • Wavelength is the distance between corresponding points of adjacent waves (see the Figure 2.1 ). Wavelengths of electromagnetic waves range from longer than a soccer ﬁeld to shorter than the diameter of an atom. • Wave frequency is the number of waves that pass a ﬁxed point in a given amount of time. Frequencies of electromagnetic waves range from thousands of waves per second to trillions of waves per second. • The energy of electromagnetic waves depends on their frequency. Low-frequency waves have little energy and are normally harmless. High-frequency waves have a lot of energy and are potentially very harmful. FIGURE 2.1 Q: Which electromagnetic waves do you think have higher frequencies: visible light or X rays? A: X rays are harmful but visible light is harmless, so you can infer that X rays have higher frequencies than visible light. Speed, Wavelength, and Frequency The speed of a wave is a product of its wavelength and frequency. Because all electromagnetic waves travel at the same speed through space, a wave with a shorter wavelength must have a higher frequency, and vice versa. This relationship is represented by the equation: Speed = Wavelength Frequency The equation for wave speed can be rewritten as: Frequency = Speed Wavelength or Wavelength = Speed Frequency Therefore, if either wavelength or frequency is known, the missing value can be calculated. Consider an electromagnetic wave that has a wavelength of 3 meters. Its speed, like the speed of all electromagnetic waves, is 3.0 10 8 meters per second.
Its frequency can be found by substituting these values into the frequency equation: Frequency = 3:0108 m/s 3:0 m = 1:0 108 waves/s, or 1.0 10 8 Hz Q: What is the wavelength of an electromagnetic wave that has a frequency of 3.0 10 8 hertz? A: Use the wavelength equation: Wavelength = 3:0108 m/s 3:0108 waves/s = 1:0 m You can learn more about calculating the frequency and wavelength of electromagnetic waves at these URLs: htt p://www.youtube.com/watch?v=GwZvtfZRNKk and http://www.youtube.com/watch?v=wjPk108Ua8k 7 www.ck12.org Summary • All electromagnetic waves travel across space at the speed of light, which is about 300 million meters per second (3.0 x 10 8 m/s). • Electromagnetic waves vary in wavelength and frequency. Longer wavelength electromagnetic waves have lower frequencies, and shorter wavelength waves have higher frequencies. Higher frequency waves have more energy. • The speed of a wave is a product of its wavelength and frequency. Because the speed of electromagnetic waves through space is constant, the wavelength or frequency of an electromagnetic wave can be calculated if the other value is known. Vocabulary • speed of light : Speed at which all electromagnetic waves travel through space, which is 3.0 10 8 m/s. Practice Use the calculator at the following URL to ﬁnd the frequency and energy of electromagnetic waves with different wavelengths. Use at least eight values for wavelength. Record and make a table of the results. http://www.1728.org /freqwave.htm Review 1. What is the speed of light across space? 2. Describe the range of wavelengths and frequencies of electromagnetic waves. 3. How is the energy of an electromagnetic wave related to its frequency? 4. If the frequency of an electromagnetic wave is 6.0 10 8 Hz, what is its wavelength? References 1. Christopher Auyeung.. CC BY-NC 3.0 8 www.ck12.org Concept 3. Electromagnetic Spectrum CONCEPT 3 Electromagnetic Spectrum • Describe electromagnetic radiation and its properties. • Give an overview of the electromagnetic spectrum. It’s a warm sunny Saturday, and Michael and Lavar have a big day planned. They’re going to ride across town to meet their friends
and then go to the zoo. The boys may not realize it, but they will be bombarded by electromagnetic radiation as they ride their bikes and walk around the zoo grounds. The only kinds of radiation they can detect are visible light, which allows them to see, and infrared light, which they feel as warmth on their skin. Q: Besides visible light and infrared light, what other kinds of electromagnetic radiation will the boys be exposed to in sunlight? A: Sunlight consists of all the different kinds of electromagnetic radiation, from harmless radio waves to deadly gamma rays. Fortunately, Earth’s atmosphere prevents most of the harmful radiation from reaching Earth’s surface. You can read about the different kinds of electromagnetic radiation in this article. Electromagnetic Radiation Electromagnetic radiation is energy that travels in waves across space as well as through matter. Most of the electromagnetic radiation on Earth comes from the sun. Like other waves, electromagnetic waves are characterized by certain wavelengths and wave frequencies. Wavelength is the distance between two corresponding points on adjacent waves. Wave frequency is the number of waves that pass a ﬁxed point in a given amount of time. Electromagnetic waves with shorter wavelengths have higher frequencies and more energy. A Spectrum of Electromagnetic Waves Visible light and infrared light are just a small part of the full range of electromagnetic radiation, which is called the electromagnetic spectrum. You can see the waves of the electromagnetic spectrum in the Figure 3.1. At the top 9 of the diagram, the wavelengths of the waves are given. Also included are objects that are about the same size as the corresponding wavelengths. The frequencies and energy levels of the waves are shown at the bottom of the diagram. Some sources of the waves are also given. For a video introduction to the electromagnetic spectrum, go to this URL: http://www.youtube.com/watch?NR=1&feature=endscreen&v=cfXzwh3KadE www.ck12.org FIGURE 3.1 • On the left side of the electromagnetic spectrum diagram are radio waves and microwaves. Radio waves have the longest wavelengths and lowest frequencies of all electromagnetic waves. They also have the least amount of energy. • On the right side of the diagram are X rays and gamma rays. They have the shortest wavelengths and highest frequencies of all electromagnetic waves. They also have the most energy. • Between these two extremes are waves that are commonly called light. Light includes infrared light, visible light, and ultraviolet light. The wavelengths, frequencies
, and energy levels of light fall in between those of radio waves on the left and X rays and gamma rays on the right. Q: Which type of light has the longest wavelengths? A: Infrared light has the longest wavelengths. Q: What sources of infrared light are shown in the diagram? A: The sources in the diagram are people and light bulbs, but all living things and most other objects give off infrared light. Summary • Electromagnetic radiation travels in waves through space or matter. Electromagnetic waves with shorter wavelengths have higher frequencies and more energy. • The full range of electromagnetic radiation is called the electromagnetic spectrum. From longest to shortest wavelengths, it includes radio waves, microwaves, infrared light, visible light, ultraviolet light, X rays, and gamma rays. Vocabulary • electromagnetic spectrum : Full range of wavelengths of electromagnetic waves, from radio waves to gamma rays. 10 www.ck12.org Practice Concept 3. Electromagnetic Spectrum At the ﬁrst URL below, read about electromagnetic waves with different frequencies. Then use the information to complete the table at the second URL. http://www.darvill.clara.net/emag/index.htm and http://www.darvill.clar a.net/emag/gcseemag.pdf Review 1. Describe the relationship between the wavelength and frequency of electromagnetic waves. 2. What is the electromagnetic spectrum? 3. Which electromagnetic waves have the longest wavelengths? 4. Identify a source of microwaves. 5. Which type of light has the highest frequencies? 6. Explain why gamma rays are the most dangerous of all electromagnetic waves. References 1. NASA.. public domain 11 www.ck12.org Law of Reﬂection CONCEPT 4 • Deﬁne reﬂection and image. • Compare and contrast regular and diffuse reﬂection. • State the law of reﬂection. These dancers are practicing in front of a mirror so they can see how they look as they performs. They’re watching their image in the mirror as they dance. What is an image, and how does it get “inside” a mirror? In this article, you’ll ﬁnd out. Reﬂected Light and Images Reﬂection is one of several ways that light can interact with matter. Light reﬂects off surfaces such as mirrors that do not transmit or absorb light. When
light is reﬂected from a smooth surface, it may form an image. An image is a copy of an object that is formed by reﬂected (or refracted) light. Q: Is an image an actual object? If not, what is it? A: No, an image isn’t an actual object. It is focused rays of light that make a copy of an object, like a picture projected on a screen. Regular and Diffuse Reﬂection If a surface is extremely smooth, as it is in a mirror, then the image formed by reﬂection is sharp and clear. This is called regular reﬂection (also called specular reﬂection). However, if the surface is even slightly rough or bumpy, an image may not form, or if there is an image, it is blurry or fuzzy. This is called diffuse reﬂection. Q: Look at the boats and their images in the Figure 4.1. Which one represents regular reﬂection, and which one represents diffuse reﬂection? 12 www.ck12.org Concept 4. Law of Reﬂection FIGURE 4.1 A: Reﬂection of the boat on the left is regular reﬂection. The water is smooth and the image is sharp and clear. Reﬂection of the boat on the right is diffuse reﬂection. The water has ripples and the image is blurry and wavy. In the Figure 4.2, you can see how both types of reﬂection occur. Waves of light are represented by arrows called rays. Rays that strike the surface are referred to as incident rays, and rays that reﬂect off the surface are known as reﬂected rays. In regular reﬂection, all the rays are reﬂected in the same direction. This explains why regular reﬂection forms a clear image. In diffuse reﬂection, the rays are reﬂected in many different directions. This is why diffuse reﬂection forms, at best, a blurry image. You can see animations of both types of reﬂection at this URL: htt p://toolboxes.ﬂexiblelearning.net.au/demosites/series5/508/Laboratory/StudyNotes/snRe�
��ectionMirrors.htm FIGURE 4.2 Law of Reﬂection One thing is true of both regular and diffuse reﬂection. The angle at which the reﬂected rays leave the surface is equal to the angle at which the incident rays strike the surface. This is known as the law of reﬂection. The law is illustrated in the Figure 4.3 and also in this animation: http://www.physicsclassroom.com/mmedia/optics/lr.cfm Summary • Reﬂection is one of several ways that light can interact with matter. When light is reﬂected from a smooth surface, it may form an image. An image is a copy of an object that is formed by reﬂected (or refracted) light. • Regular reﬂection occurs when light reﬂects off a very smooth surface and forms a clear image. Diffuse reﬂection occurs when light reﬂects off a rough surface and forms a blurry image or no image at all. • According to the law of reﬂection, the angle at which light rays reﬂect off a surface is equal to the angle at which the incident rays strike the surface. 13 www.ck12.org FIGURE 4.3 Vocabulary • image : Copy of an object that is formed by reﬂected or refracted light. • law of reﬂection : Law stating that the angle at which reﬂected rays of light bounce off a surface is equal to the angle at which the incident rays strike the surface. • reﬂection : Bouncing back of waves from a barrier they cannot pass through. Practice At the following URL, review the law of reﬂection and watch the animation. Then ﬁll in the blanks in the sentence below. http://www.physicsclassroom.com/mmedia/optics/lr.cfm 1. When a ray of light strikes a plane mirror, the light ray __________ off the mirror. 2. Reﬂection involves a change in __________ of a light ray. 3. The angle of incidence equals the angle between the incident ray and ________. 4. The angle of __________ equals the angle of incidence. 5. The normal line is __________ to the mirror. Review 1
. What is an image? 2. Identify the object and the image in the Figure 4.4. Which type of reﬂection formed the image: regular reﬂection or diffuse reﬂection? How do you know? FIGURE 4.4 3. What is the law of reﬂection? 4. Label the angle of incidence and the angle of reﬂection in the Figure 4.5. 14 www.ck12.org Concept 4. Law of Reﬂection FIGURE 4.5 References 1. Left: Kenneth Baruch; Right: Damian Gadal.. CC BY 2.0 2. Joy Sheng.. CC BY-NC 3.0 3. Christopher Auyeung.. CC BY-NC 3.0 4. Mike Baird.. CC BY 2.0 5. Zachary Wilson.. CC-BY-NC-SA 3.0 15 www.ck12.org CONCEPT 5 Refraction of Light • Deﬁne refraction. • Given data about the optical density of the media, predict whether the light will bend toward the normal or away from the normal. • State Snell’s Law and solve refraction problems using it. • Solve problems using the relationship between the index of refraction and the velocity of light in the media. • Explain effects caused by the refraction of light. When a light ray passes at an angle through the boundary between optically different media, the light does not travel in a straight line. The pencil in the glass of liquid shown above is a normal straight pencil. The light that travels from the pencil through the liquid, through the glass, and into the air is bent differently than light from the portion of the pencil that is not in the liquid. Your eye assumes the light from both portions of the pencil moved in a straight line, but the two portions of the pencil do not appear to be lined up. Your eye thinks the pencil is broken. Refraction of Light The speed of light is different in different media. If the speed of light is slower in a particular medium, that medium is said to be more optically dense. When a wave front enters a new medium at an angle, it will change directions. If the light is entering a more optically dense medium, the light bends toward the normal line. If the light is entering a less optically dense medium, the light will bend away from the normal line. Remember that the
normal line is the line perpendicular to the medium interface. 16 www.ck12.org Concept 5. Refraction of Light In the sketch below, light wave fronts are moving upward from the bottom of the page and encounter a boundary into a more optically dense medium. The light waves bend toward the normal line. Because the right end of the wave fronts enter the new medium ﬁrst, they slow down ﬁrst. When the right side of the wave front is moving more slowly that the left side, the wave front will change directions. When light is traveling from air into another medium, Snell’s Law states the relationship between the angle of incidence and angle of refraction is n = sin qi sin qr where qi is the angle of incidence, qr is the angle of refraction, and n is the ratio of the two sines and is called the index of refraction. Snell’s Law may be stated that a ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. The index of refraction is also related to the relative speeds of light in a vacuum and in the medium. n = speed of light in a vacuum speed of light in the medium When a ray of light is traveling from medium into another medium, Snell’s Law can be written as ni sin qi = nr sin qr. Table of Indices of Refraction Medium Vacuum Air Water Ethanol Crown Glass Quartz Flint Glass Diamond TABLE 5.1: n 1.00 1.0003 1.36 1.36 1.52 1.54 1.61 2.42 17 www.ck12.org Example Problem: A ray of light traveling through air is incident upon a slab of Flint glass at an angle of 40:0. What is the angle of refraction? Solution: n = sin qi sin qr so sin qr = sin qi 1:61 = 0:399 n = 0:643 The angle of refraction = sin1:399 = 23:5 Example Problem: What is the speed of light in a diamond? Solution: speed of light in diamond = speed of light in a vacuum n = 3:00108 m=s 2:42 speed of light in diamond = 1:24 108 m=s Effects of Refraction Bending the Sun’s Rays Because air is slightly more optically
dense than a vacuum, when sunlight passes from the vacuum of space into our atmosphere, it bends slightly towards the normal. When the sun is below the horizon and thus not visible on a direct line, the light path will bend slightly and thus make the sun visible by refraction. Observers can see the sun before it actually comes up over the horizon, or after it sets. Mirages In the Figure below, the sun shines on the road, heating the air just above the road. The difference in density between the hot air over the road and the surrounding air causes the hot air to refract light that passes through it. When you look at the road, you see a mirage. What appears to be water on the road is actually light coming from the sky that has been refracted as it passes through the hot air above the road. This phenomenon is common on hot roads and in the desert. Summary • The speed of light is different in different media. • When a wave front enters a new medium at an angle, it will change directions. If the light is entering a more optically dense medium, the light bends toward the normal line. If the light is entering a less optically dense medium, the light will bend away from the normal line. 18 www.ck12.org Concept 5. Refraction of Light FIGURE 5.1 • When light is traveling from air into another medium, Snell’s Law states that n = sin qi sin qr • The index of refraction is also related to the relative speeds of light in a vacuum and in the medium. n =. speed of light in a vacuum speed of light in the medium • When a ray of light is traveling from one medium into another medium, Snell’s Law can be written as ni sin qi = nr sin qr. Practice Use the video on refraction to answer the questions below. http://video.mit.edu/watch/mit-physics-demo-refraction-a-total-internal-reflection-12044/ MEDIA Click image to the left for more content. 1. What happens to the path of a light beam when it enters a new medium at an angle? 2. Light moving from air into water is bent ___________ the normal. 3. Light moving from water into air is bent ___________ the normal. Review 1. Light moving through air is incident on a piece of crown glass at an angle of 45. What is the
angle of refraction? 2. A ray of light passes from air into water at an incident angle of 60:0. Find the angle of refraction. 3. Light passes from water into a block of transparent plastic. The angle of incidence from the water is 31 and the angle of refraction in the block is 27. What is the index of refraction for the plastic? 4. The index of refraction of water is 1.36. What is the speed of light in water? 19 www.ck12.org 5. If the speed of light in a piece of plastic is 2:00 108 m=s, what is the index of refraction for the plastic? • refraction : The change of direction of a ray of light or sound in passing obliquely from one medium into another in which its wave velocity is different. • optically dense: Refers to the ability of a material to slow the light waves. The greater the optical density of a material, the greater the slowing effect. • Snell’s Law: For a light ray incident on the interface of two media, the sine of the angle of incidence times the index of refraction of the ﬁrst medium is equal to the sine of the angle of refraction times the index of refraction of the second medium. • index of refraction: The ratio of the speed of light in a vacuum to the speed of light in a medium under consideration. • mirage: An optical phenomenon that creates the illusion of water, often with inverted reﬂections of distant objects, and results from the refraction of light by alternate layers of hot and cool air. References 1. Image copyright leonello calvetti, 2013. http://www.shutterstock.com. Used under license from Shutter- stock.com 2. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 3. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 4. Michael Gil (Flickr: MSVG). http://www.ﬂickr.com/photos/msvg/5994891327/. CC-BY 2.0 20 www.ck12.org Concept 6. Total Internal Reﬂection CONCEPT 6 Total Internal Reﬂection • Describe total internal reﬂection. • Use the critical angle to determine when total internal reﬂection will
occur. Total internal reﬂection allows the light to travel down the optical ﬁber and not pass through the sides of the tube. The light continuously reﬂects from the inside of the tube and eventually comes out the end. Optical ﬁbers make interesting lamps but they are also used to transport telephone and television signals. Total Internal Reﬂection We already know that when light passes from one medium into a second medium where the index of refraction is smaller, the light refracts away from the normal. In the image below, the light rays are passing into an optically less dense medium; therefore, the rays bend away from the normal. As the angle of incidence increases, the light ray bends even further away from the normal. Eventually, the angle of incidence will become large enough that the angle of refraction equals 90, meaning the light ray will not enter the new medium at all. 21 www.ck12.org Consider a ray of light passing from water into air. The index of refraction for air is 1.00 and for water is 1.36. Using Snell’s Law, ni sin qi = nr sin qr, and allowing the angle of refraction to be 90, we can solve for the angle of incidence which would cause the light ray to stay in the old medium. ni sin qi = nr sin qr (1:36)(sin qi) = (1:00)(sin 90) sin qi = 0:735 and qi = 47 This result tells us that when light is passing from water into air, if the angle of incidence exceeds 47, the light ray will not enter the new medium. The light ray will be completely reﬂected back into the original medium. This is called total internal reﬂection. The minimum angle of incidence for total internal reﬂection to occur is called the critical angle. Total internal reﬂection is the principle behind ﬁber optics. A bundle of ﬁbers made out of glass or plastic only a few micrometers in diameter is called a light pipe since light can be transmitted along it with almost no loss. Light passing down the ﬁbers makes glancing collisions with the walls so that total internal reﬂection occurs. Summary • When light passes from one medium into a second medium with a smaller index of refraction, the light refracts away from the normal
. • If the angle of incidence becomes large enough that the angle of refraction equals 90, the light ray will not enter the new medium with the smaller angle of refraction. • Total internal reﬂection means the light ray will not enter the new medium but will be completely reﬂected back into the original medium. Practice MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=hBQ8fh_Fp04 In this video, a hole is drilled in the side of a plastic bottle ﬁlled with liquid. An arc of liquid shoots out through the hole. A laser pointer is aimed through the opposite side of the bottle so that the light also exits through the hole. The stream of liquid acts like an optical ﬁber and the light undergoes total internal reﬂection as it follows the stream of liquid. As the amount of liquid in the bottle decreases, the arc of the stream of liquid changes and the direction of the light follows the stream of liquid. Toward the end, the light beam is shining almost 90 from the direction of the laser pointer. The following video discusses total internal reﬂection. Use this resource to answer the questions that follow. http://www.khanacademy.org/science/physics/waves-and-optics/v/total-internal-reﬂection# 22 www.ck12.org Concept 6. Total Internal Reﬂection MEDIA Click image to the left for more content. 1. What phenomenon occurs when the light does not enter the new medium and remains in the old medium? 2. When does this phenomenon occur? Review 1. Find the critical angle for light passing from diamond into air, given ndiamond = 2:42 2. When two swimmers are under water in a swimming pool, it is possible for the interface between the water and the air to act as a mirror, allowing the swimmers to see images of each other if they look up at the underside of the surface. Explain this phenomenon. 3. Robert shines a laser beam through a slab of plastic and onto the interface between the slab of plastic and the air on the other side. The index of refraction for the plastic is 1.62. If the angle of incidence in the plastic is 54, will the laser beam pass out of the plastic into the air? • total internal reﬂection: When light
is passing from a medium of higher index of refraction into a medium of lower index of refraction is completely reﬂected by the boundary between the two media. • critical angle: The smallest angle of incidence at which a light ray passing from one medium to another less refractive medium will be totally reﬂected from the boundary between the two. • ﬁber optics: The science or technology of light transmission through very ﬁne, ﬂexible glass or plastic ﬁbers without energy loss making use of the principle of total internal reﬂection. References 1. Roshan Nikam (Flickr: roshan1286). http://www.ﬂickr.com/photos/31916678@N07/4753800195/. CC-BY 2.0 2. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 23 CONCEPT 7 Single Slit Interference • Explain how single slit diffraction patterns occur. • Use single slit diffraction patterns to calculate wavelength. www.ck12.org Though it looks like a double slit interference pattern, the pattern on the screen are actually the results of light diffracting through a single slit with the ensuing interference. Single Slit Interference Interference patterns are produced not only by double slits but also by single slits, otherwise known as single slit interference. In the case of a single slit, the particles of medium at both corners of the slit act as point sources, producing circular waves from both edges. These circular waves move across to the back wall and interfere in the same way that interference patterns were produced by double slits. In the sketch at below, the black lines intersect at the center of the pattern on the back wall. This center point is equidistanct from both edges of the slit. Therefore, the waves striking at this position will be in phase; that is, the waves will produce constructive interference. Also shown in the sketch, just above the central bright spot where the red lines intersect, is a position where destructive interference occurs. One of these red lines is one-half wavelength longer than the other, causing the two waves to hit the wall out of phase and undergo destructive interference. A dark bank appears at this position. 24 www.ck12.org Concept 7. Single Slit Interference Just as in double slit interference, a pair of similar triangles can be constructed in the interference pattern. The pertinent values
from these triangles are the width of the slit, w, the wavelength, l, the distance from the central bright spot to the ﬁrst dark band, x, and the distance from the center of the slit to back wall, L. The relationship of these four values is L or l = wx w = x L. l Example Problem: Monochromatic light of wavelength 605 nm falls on a slit of width 0.095 mm. The slit is located 85 cm from a screen. How far is the center of the central bright band to the ﬁrst dark band? Solution: x = lL w = (6:05107 m)(0:85 m) (9:5105 m) = 0:0054 m Summary • Interference patterns can also be produced by single slits. • In the case of a single slit, the particles of medium at both corners of the slit act as point sources, and produce circular waves from both edges. • The wavelength can be determined by this equation: l w = x L or l = wx L. Practice MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=_HO7LJDcqos Follow up questions. 1. The interference pattern appears as the slit becomes ___________ (wider, thinner). Review 1. The same set up is used for two different single slit diffraction experiments. In one of the experiments, yellow light is used, and in the other experiment, green light is used. Green light has a shorter wavelength than yellow light. Which of the following statements is true? 25 (a) The two experiments will have the same distance between the central bright band and the ﬁrst dark band. (b) The green light experiment will have a greater distance between the central bright band and the ﬁrst dark band. (c) The yellow light experiment will have a greater distance between the central bright band and the ﬁrst www.ck12.org dark band. 2. Why are the edges of shadows often fuzzy? (a) Interference occurs on the wall on which the shadow is falling. (b) Light diffracts around the edges of the object casting the shadow. (c) The edges of the object casting the shadow is fuzzy. (d) Light naturally spreads out. 3. Monochromatic, coherent light passing through a double slit will produce exactly the
same interference pattern as when it passes through a single slit. (a) True (b) False 4. If monochromatic light passes through a 0.050 mm slit and is projected onto a screen 0.70 m away with a distance of 8.00 mm between the central bright band and the ﬁrst dark band, what is the wavelength of the light? 5. A krypton ion laser with a wavelength of 524.5 nm illuminates a 0.0450 mm wide slit. If the screen is 1.10 m away, what is the distance between the central bright band and the ﬁrst dark band? 6. Light from a He-Ne laser (l = 632:8 nm) falls on a slit of unknown width. In the pattern formed on a screen 1.15 m away, the ﬁrst dark band is 7.50 mm from the center of the central bright band. How wide is the slit? • single slit interference: When monochromatic, coherent light falls upon a small single slit it will produce a pattern of bright and dark fringes. These fringes are due to light from one side of the slit interacting (interfering) with light from the other side. References 1. Luiz Sauerbronn. http://commons.wikimedia.org/wiki/File:Fresnel_Diffraction_experiment_DSC04573.JPG. Public Domain 2. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 3. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 26 www.ck12.org Concept 8. Double Slit Interference CONCEPT 8 Double Slit Interference • Deﬁne diffraction of light. • Deﬁne wave interference. • Describe double slit interference patterns. • Explain Thomas Young’s contibutions to physics. • Calculate the wavelength from a double slit interference pattern. FIGURE 8.1 When waves strike a small slit in a wall, they create circular wave patterns on the other side of the barrier. This is seen in the image above, where ocean waves create precise circular waves. The circular waves undergo constructive and destructive interference, which generates a regular interference pattern. Diffraction and Interference When a series of straight waves strike an impenetrable barrier, the waves stop at the barrier.
However, the last particle of the medium at the back corner of the barrier will create circular waves from that point, called the point source. This can be seen in the image below. This phenomenon is called diffraction, and it occurs in liquid, sound, and light waves. While the waves become circular waves at the point source, they continue as straight waves where the barrier does not interfere with the waves. Any two waves in the same medium undergo wave interference as they pass each other. At the location where the two waves collide, the result is essentially a summation of the two waves. In some places, a wave crest from one source will overlap a wave crest from the other source. Since both waves are lifting the medium, the combined wave crest will be twice as high as the original crests. Nearby, a wave trough will overlap another wave trough and the new 27 trough will be twice as deep as the original. This is called constructive interference because the resultant wave is larger than the original waves. Within the interference pattern, the amplitude will be twice the original amplitude. Once the waves pass through each other and are alone again, their amplitudes return to their original values. In other parts of the wave pattern, crests from one wave will overlap troughs from another wave. When the two waves have the same amplitude, this interaction causes them to cancel each other out. Instead of a crest or a trough, there is nothing. When this cancellation occurs, it is called destructive interference. www.ck12.org FIGURE 8.2 It is easy to see how waves emanating from multiple sources, such as drops of rainwater in still water, create interference patterns. But a single source of waves can create interference patterns with itself as a result of diffraction. The Double Slit Experiment A similar situation to the raindrops above occurs when straight waves strike a barrier containing two slits. These waves are cut off everywhere except for where the waves that pass through the two slits. The medium in the slits again acts as a point source to produce circular waves on the far side of the barrier. As long as these two circular waves have the same wavelength, they interfere constructively and destructively in a speciﬁc pattern. This pattern is called the wave interference pattern and is characterized by light and dark bands. The light bands are a result of constructive interference, and the dark bands occur because of destructive interference. 28 www.ck12.org Concept 8. Double Slit Interference In the early 1800�
�s, light was assumed to be a particle. There was a signiﬁcant amount of evidence to point to that conclusion, and famous scientist Isaac Newton’s calculations all support the particle theory. In 1803, however, Thomas Young performed his famous Double Slit Experiment to prove that light was a wave. Young shined a light onto the side of a sealed box with two slits in it, creating an interference pattern on the inside of the box opposite the slits. As seen above, interference patterns are characterized by alternating bright and dark lines. The bright lines are a result of constructive interference, while the dark lines are a result of destructive interference. By creating this interference pattern, Young proved light is a wave and changed the course of physics. Calculating Wavelength from Double Slit Pattern Using the characteristics of the double slit interference pattern, it is possible to calculate the wavelength of light used to produce the interference. To complete this calculation, it is only necessary to measure a few distances. As can be seen below, ﬁve distances are measured. In the sketch, L is the distance from the two slits to the back wall where the interference pattern can be seen. d is the distance between the two slits. To understand x, look again at the interference pattern shown above. The middle line, which is the brightest, is called the central line. The remaining lines are called fringes. The lines on either side of the central line are called the ﬁrst order fringes, the next lines are called the second order fringes, and so on. x is the distance from the central line to the ﬁrst order fringe. r 1 and r 2 are the distances from the slits to the ﬁrst order fringe. We know that the fringes are a result of constructive interference, and that the fringe is a result of the crest of two waves interfering. If we assume that r 2 is a whole number of wavelengths (conﬁrm for yourself that this is a logical assumption), then r 1 must be one more wavelength. This is because r 1 and r 2 are the distances to the ﬁrst order fringe. Mathematically, we can let r2 = nl and r1 = nl + l, where l is the wavelength and n is a constant. Using this relationship, we determine that r1 r2 = l. Looking again at the diagram, the red and blue triangles are similar,
which means that the ratios of corresponding sides are the same. The ratio of x to L in the red triangle is equal to the ratio of l to d in the blue triangle. For proof of this, visit http://www.physicsclassroom.com/class/light/u12l3c.cfm. From this, we can determine that the wavelength is dependent on x, d, and L: 29 l = xd L Example Problem: Monochromatic light falls on two narrow slits that are 0.0190 mm apart. A ﬁrst order fringe is 21.1 mm from the central line. The screen (back wall) is 0.600 m from the slits. What is the wavelength of the light? www.ck12.org Solution: l = xd L = (0:021 m)(0:000019 m) (0:600 m) = 6:68 107 m Summary • The last particle of medium at the back corner of an impenetrable barrier will act as a point source and produce circular waves. • Diffraction is the bending of waves around a corner. • Constructive interference occurs when two wave crests overlap, doubling the wave amplitude at that location. • Destructive interference occurs when a wave crest overlaps with a trough, causing them to cancel out. • Light is a wave, and creates an interference pattern in the double slit experiment. • An interference pattern consists of alternating bright and dark lines; the bright lines are called fringes. • In a double slit experiment, the wavelength can be calculated using this equation: l = xd L Practice http://www.youtube.com/watch?v=AMBcgVlamoU Follow up questions. 1. When the amplitude of waves add, it is called _________________ interference. 2. When the amplitude of waves subtract, it is called _________________ interference. 3. What do we call the phenomenon of light bending around a corner? Review 1. Destructive interference in waves occurs when (a) two troughs overlap. (b) crests and troughs align. (c) two crests overlap. (d) a crest and a trough overlap. 2. Bright bands in interference patterns result from (a) destructive diffraction. (b) destructive interference. (c) constructive diffraction. (d) constructive interference. 3. In a double slit experiment with slits 1:00 105 m apart, light casts the
ﬁrst bright band 3:00 102 m from the central bright spot. If the screen is 0.650 m away, what is the wavelength of this light? (a) 510 nm (b) 390 nm (c) 430 nm (d) 460 nm 4. Violet light falls on two slits separated by 1:90 105 m. A ﬁrst order bright line appears 13.2 mm from the central bright spot on a screen 0.600 m from the slits. What is the wavelength of the violet light? 5. Suppose in the previous problem, the light was changed to yellow light with a wavelength of 5:96 107 m while the slit separation and distance from screen to slits remained the same. What would be the distance from the central bright spot to the ﬁrst order line? 30 www.ck12.org Concept 8. Double Slit Interference 6. Light with a wavelength of 6:33 107 m is used in a double slit experiment. The screen is placed 1.00 m from the slits and the ﬁrst order line is found 65.5 mm from the central bright spot. What is the separation between the slits? • diffraction: Change in the directions and intensities of a group of waves after passing by an obstacle or through an aperture whose size is approximately the same as the wavelength of the waves. • monochromatic: Light having only one wavelength. • constructive interference: The interference of two or more waves of equal frequency and phase, resulting in their mutual reinforcement and producing a single amplitude equal to the sum of the amplitudes of the individual waves. • destructive interference: The interference of two waves of equal frequency and opposite phase, resulting in their cancellation where the negative displacement of one always coincides with the positive displacement of the other. References 1... CC BY-NC-SA 2.. Kathy Shield. CC BY-NC-SA 3. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 4. Pieter Kuiper. http://commons.wikimedia.org/wiki/File:SodiumD_two_double_slits.jpg. Public Domain 5. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 31 CONCEPT 9 www.ck12.org Diffraction Gratings • Understand the structure of a diffraction grating. • Explain the interference pattern formed
by diffraction gratings. • Use diffraction grating interference patterns to calculate the wavelength of light. Suppose we had a light bulb that emitted exactly four frequencies of light; one frequency in each of the colors red, yellow, green, and blue. To our eye, this bulb would appear white because the combination of those four colors produces white light. If viewed through a diffraction grating, however, each color of light would be visible. The original white light bulb is visible in the center of the image, and interference causes the light bulb to appear in each color to the left and the right. Diffraction Gratings Diffraction gratings are composed of a multitude of slits lined up side by side, not unlike a series of double slits next to each other. They can be made by scratching very ﬁne lines with a diamond point on glass, or by pressing plastic ﬁlm on glass gratings so that the scratches are replicated. The clear places between the scratches behave as slits similar to the slits in a double slit experiment and the gratings form interference patterns in the same general way that double slits do. With more slits, there are more light waves out of phase with each other, causing more destructive interference. Compared to the interference pattern of a double slit, the diffraction grating interference pattern’s colors are spread out further and the dark regions are broader. This allows for more precise wavelength determination than with double slits. The image below shows the diffraction pattern emanating from a white light. 32 www.ck12.org Concept 9. Diffraction Gratings Also in this image is the measurement for q, which can be used to calculate the wavelength of the original light source. The equation from the double slit experiment can be adjusted slightly to work with diffraction gratings. Where l is the wavelength of light, d is the distance between the slits on the grating, and q is the angle between the incident (original) light and the refracted light, l = xd L = d sin q (Note that x Looking at the equation, x = lL d, it should be apparent that as the distance between the lines on the grating become smaller and smaller, the distance between the images on the screen will become larger and larger. Diffraction gratings are often identiﬁed by the number of lines per centimeter; gratings with more lines per centimeter are usually more useful because the greater the number of lines
, the smaller the distance between the lines, and the greater the separation of images on the screen. L = sin q, using the small angle approximation theorem.) Example Problem: A good diffraction grating has 2500 lines/cm. What is the distance between two lines on the grating? Solution: d = 1 2500 cm1 = 0:00040 cm Example Problem: Using a diffraction grating with a spacing of 0.00040 cm, a red line appears 16.5 cm from the central line on the screen. The screen is 1.00 m from the grating. What is the wavelength of the light? Solution: l = xd L = (0:165 m)(4:0106 m) 1:00 m = 6:6 107 m Summary • Diffraction gratings can be made by blocking light from traveling through a translucent medium; the clear places behave as slits similar to the slits in a double slit experiment. • Diffraction gratings form interference patterns much like double slits, though brighter and with more space between the lines. • The equation used with double slit experiments to measure wavelength is adjusted slightly to work with diffraction gratings. l = xd L = d sin q 33 www.ck12.org Practice http://vimeo.com/39495562 Follow up questions. 1. How does a diffraction grating differ from single or double slit? 2. What happens when you increase the number of slits in a diffraction grating? Review 1. White light is directed toward a diffraction grating and that light passes through the grating, causing its monochromatic bands appear on the screen. Which color will be closest to the central white? 2. Three discrete spectral lines occur at angles of 10.1°, 13.7°, and 14.8° respectively in the ﬁrst order spectrum. If the grating has 3660 lines/cm, what are the wavelengths of these three colors of light? 3. A 20.0 mm section of diffraction grating has 6000 lines. At what angle will the maximum bright band appear if the wavelength is 589 nm? 4. Laser light is passed through a diffraction grating with 7000 lines/cm. The ﬁrst order maximum on the screen is 25° away from the central maximum. What is the wavelength of the light? • diffraction grating: A glass, plastic, or polished metal surface having a large number of very
ﬁne parallel grooves or slits and used to produce optical spectra by diffraction. References 1. CK-12 Foundation - Samantha Bacic, using light bulb images copyright Ruslan Klimovich, 2013. http:// www.shutterstock.com. Used under license from Shutterstock.com 2. Candace (Flickr: cosmiccandace). http://www.ﬂickr.com/photos/candace/315205005/. CC-BY 2.0 34 www.ck12.org Concept 10. Wave-Particle Theory CONCEPT 10 Wave-Particle Theory • State the wave-particle theory of electromagnetic radiation. • Describe a photon. • Identify evidence that electromagnetic radiation is both a particle and a wave. What a beautiful sunset! You probably know that sunlight travels in waves through space from the sun to Earth. But do you know what light really is? Is it just energy, or is it something else? In this article you’ll ﬁnd out that light may be more than it seems. The Question Electromagnetic radiation, commonly called light, is the transfer of energy by waves called electromagnetic waves. These waves consist of vibrating electric and magnetic ﬁelds. Where does electromagnetic energy come from? It is released when electrons return to lower energy levels in atoms. Electromagnetic radiation behaves like continuous waves of energy most of the time. Sometimes, however, electromagnetic radiation seems to behave like discrete, or separate, particles rather than waves. So does electromagnetic radiation consist of waves or particles? The Debate This question about the nature of electromagnetic radiation was debated by scientists for more than two centuries, starting in the 1600s. Some scientists argued that electromagnetic radiation consists of particles that shoot around like tiny bullets. Other scientists argued that electromagnetic radiation consists of waves, like sound waves or water waves. Until the early 1900s, most scientists thought that electromagnetic radiation is either one or the other and that scientists on the other side of the argument were simply wrong. Q: Do you think electromagnetic radiation is a wave or a particle? A: Here’s a hint: it may not be a question of either-or. Keep reading to learn more. 35 www.ck12.org A New Theory In 1905, the physicist Albert Einstein developed a new theory about electromagnetic radiation. The theory is often called the wave-particle theory. It explains how electromagnetic radiation can behave as both a wave and a particle. Einstein argued that when an electron
returns to a lower energy level and gives off electromagnetic energy, the energy is released as a discrete “packet” of energy. We now call such a packet of energy a photon. According to Einstein, a photon resembles a particle but moves like a wave. You can see this in the Figure 10.1. The theory posits that waves of photons traveling through space or matter make up electromagnetic radiation. FIGURE 10.1 Energy of a Photon A photon isn’t a ﬁxed amount of energy. Instead, the amount of energy in a photon depends on the frequency of the electromagnetic wave. The frequency of a wave is the number of waves that pass a ﬁxed point in a given amount of time, such as the number of waves per second. In waves with higher frequencies, photons have more energy. Evidence for the Wave-Particle Theory After Einstein proposed his theory, evidence was discovered to support it. For example, scientists shone laser light through two slits in a barrier made of a material that blocked light. You can see the setup of this type of experiment in the sketch below. Using a special camera that was very sensitive to light, they took photos of the light that passed through the slits. The photos revealed tiny pinpoints of light passing through the double slits. This seemed to show that light consists of particles. However, if the camera was exposed to the light for a long time, the pinpoints accumulated in bands that resembled interfering waves. Therefore, the experiment showed that light seems to consist of particles that act like waves. 36 www.ck12.org Concept 10. Wave-Particle Theory FIGURE 10.2 Summary • Electromagnetic radiation behaves like waves of energy most of the time, but sometimes it behaves like particles. From the 1600s until the early 1900s, most scientists thought that electromagnetic radiation consists either of particles or of waves but not both. • In 1905, Albert Einstein proposed the wave-particle theory of electromagnetic radiation. This theory states that electromagnetic energy is released in discrete packets of energy—now called photons—that act like waves. • After Einstein presented his theory, scientists found evidence to support it. For example, double-slit experi- ments showed that light consists of tiny particles that create patterns of interference just as waves do. Vocabulary • photon : Tiny “packet” of electromagnetic radiation that is released when an electron returns to a lower. • wave-particle theory : Theory
proposed by Albert Einstein that electromagnetic energy is released in discrete packets of energy (now called photons) that act like waves. Practice Watch the animation “Let There Be Light” at the following URL. Then create a timeline of ideas and discoveries about the nature of light. http://www.abc.net.au/science/explore/einstein/lightstory.htm Review 1. Why did scientists debate the nature of electromagnetic radiation for more than 200 years? 2. State Einstein’s wave-particle theory of electromagnetic radiation. 3. What is a photon? 4. After Einstein proposed his wave-particle theory, how did double-slit experiments provide evidence to support the theory? 37 References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Zachary Wilson.. CC-BY-NC-SA 3.0 www.ck12.org 38 Physics Unit 12: Static Electricity Patrick Marshall Jean Brainard, Ph.D. Ck12 Science Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable version of this book, as well as other interactive content, visit www.ck12.org AUTHORS Patrick Marshall Jean Brainard, Ph.D. Ck12 Science CK-12 Foundation is a non-proﬁt organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2014 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: April 6, 2014 iii Contents www.ck12.org Contents 1 Static Electricity and Static Discharge 2 Electric Charge and Electric Force 3 Transfer of Electric Charge 4 Forces on Charged Objects 5 Coulomb’s Law 6 The Electric Field 7 Electric Potential 1 4 7 12 19 22 25 iv www.ck12.org Concept 1. Static Electricity and Static Discharge CONCEPT 1 Static Electricity and Static Discharge • Describe static electricity. • Explain static discharge. • Outline how lightning occurs. You’re a thoughtful visitor, so you wipe your feet on the welcome mat before you reach out to touch the brass knocker on the door. Ouch! A spark suddenly jumps between your hand and the metal, and you feel an electric shock. Q: Why do you think an electric shock occurs? A: An electric shock occurs when there is a sudden discharge of static electricity. What Is Static Electricity? Static electricity is a buildup of electric charges on objects. Charges build up when negative electrons are transferred from one object to another. The object that gives up electrons becomes positively charged, and the object that accepts the electrons becomes negatively charged. This can happen in several ways. One way electric charges can build up is through friction between materials that differ in their ability to give up or accept electrons. When you wipe your rubber-soled shoes on the wool mat, for example, electrons rub off the mat onto your shoes. As a result of this transfer of electrons, positive charges build up on the mat and negative charges build up on you. Once an object becomes electrically charged, it is likely to remain charged until it touches another object or at least comes very close to another object. That’s because electric charges cannot travel easily through air, especially if the air is dry. 1 www.ck12.org Q: You’re more likely to get a shock in the winter when the air is very dry. Can you
explain why? A: When the air is very dry, electric charges are more likely to build up objects because they cannot travel easily through the dry air. This makes a shock more likely when you touch another object. Static Discharge What happens when you have become negatively charged and your hand approaches the metal doorknocker? Your negatively charged hand repels electrons in the metal, so the electrons move to the other side of the knocker. This makes the side of the knocker closest to your hand positively charged. As your negatively charged hand gets very close to the positively charged side of the metal, the air between your hand and the knocker also becomes electrically charged. This allows electrons to suddenly ﬂow from your hand to the knocker. The sudden ﬂow of electrons is static discharge. The discharge of electrons is the spark you see and the shock you feel. Watch the animation “John Travoltage” at the following URL to see an example of static electricity and static discharge. http://www.cabrillo.edu/~jmccullough/Physics/Electric_Forces_Fields.html How Lightning Occurs Another example of static discharge, but on a much larger scale, is lightning. You can see how it occurs in the following diagram and animation as you read about it below. http://micro.magnet.fsu.edu/electromag/java/lightning/index.html FIGURE 1.1 During a rainstorm, clouds develop regions of positive and negative charge due to the movement of air molecules, water drops, and ice particles. The negative charges are concentrated at the base of the clouds, and the positive charges are concentrated at the top. The negative charges repel electrons on the ground beneath them, so the ground below the clouds becomes positively charged. At ﬁrst, the atmosphere prevents electrons from ﬂowing away from areas of negative charge and toward areas of positive charge. As more charges build up, however, the air between the oppositely charged areas also becomes charged. When this happens, static electricity is discharged as bolts of lightning. At the URL below, you can watch an awesome slow-motion lightning strike. Be sure to wait for the real-time lightning strike at the end of the video. You’ll be amazed when you realize how much has occurred during that split-second discharge of static electricity. 2 www.ck12.org Concept 1. Static Electricity and Static Dis
charge http://www.youtube.com/watch?v=Y8oN0YFAXWQ&feature=related MEDIA Click image to the left for more content. Summary • Static electricity is a buildup of electric charges on objects. It occurs when electrons are transferred from one object to another. • A sudden ﬂow of electrons from one charged object to another is called static discharge. • Examples of static discharge include lightning and the shock you sometimes feel when you touch another object. Vocabulary • static discharge : Sudden ﬂow of electrons from an object that has a buildup of charges. • static electricity : Buildup of charges on an object that occurs through induction. Practice Watch the video at the following URL. Then answer the discussion questions. Read the background essay if you need help with any of the questions. http://www.teachersdomain.org/resource/phy03.sci.phys.mfe.zsnap/ Review 1. What is static electricity? 2. How does static discharge occur? 3. Explain why a bolt of lightning is like the spark you might see when you touch a metal object and get a shock. References 1. Zachary Wilson.. CC BY-NC 3.0 3 CONCEPT 2 Electric Charge and Electric Force www.ck12.org • Deﬁne electric charge. • Describe electric forces between charged particles. A lightning bolt is like the spark that gives you a shock when you touch a metal doorknob. Of course, the lightning bolt is on a much larger scale. But both the lightning bolt and spark are a sudden transfer of electric charge. Introducing Electric Charge Electric charge is a physical property of particles or objects that causes them to attract or repel each other without touching. All electric charge is based on the protons and electrons in atoms. A proton has a positive electric charge, and an electron has a negative electric charge. In the Figure 2.1, you can see that positively charged protons (+) are located in the nucleus of the atom, while negatively charged electrons (-) move around the nucleus. Electric Force When it comes to electric charges, opposites attract, so positive and negative particles attract each other. You can see this in the diagram below. This attraction explains why negative electrons keep moving around the positive nucleus of the atom. Like charges, on the other hand, repel each other, so two positive or two negative charges push apart. This is also shown in the diagram. The attraction
or repulsion between charged particles is called electric force. The strength of electric force depends on the amount of electric charge on the particles and the distance between 4 www.ck12.org Concept 2. Electric Charge and Electric Force FIGURE 2.1 them. Larger charges or shorter distances result in greater force. You can experiment with electric force with the animation at the following URL. http://www.colorado.edu/physics/2000/waves_particles/wavpart2.html FIGURE 2.2 Q: How do positive protons stay close together inside the nucleus of the atom if like charges repel each other? A: Other, stronger forces in the nucleus hold the protons together. Summary • Electric charge is a physical property of particles or objects that causes them to attract or repel each other without touching. • Particles that have opposite charges attract each other. Particles that have like charges repel each other. The force of attraction or repulsion is called electric force. 5 www.ck12.org Vocabulary • electric charge : Physical property of particles or objects that causes them to attract or repel each other without touching; may be positive or negative. • electric force : Force of attraction or repulsion between charged particles. Practice Read the ﬁrst four boxes of text at the following URL. Then write a concise paragraph explaining why direction E is the correct answer to the quick quiz. http://www.physics.wisc.edu/undergrads/courses/208-f07/Lectures/lect6.pdf Review 1. What is electric charge? 2. Make a simple table summarizing electric forces between charged particles. References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Zachary Wilson.. CC BY-NC 3.0 6 www.ck12.org Concept 3. Transfer of Electric Charge CONCEPT 3 Transfer of Electric Charge • Describe how the transfer of electrons changes the charge of matter. • Relate the transfer of electrons to the law of conservation of charge. • Compare and contrast three ways that electric charge can be transferred. Why is this girl’s hair standing straight up? She is touching a device called a van de Graaff generator. The dome on top of the device has a negative electric charge. When the girl places her hand on the dome, she becomes negatively charged as well—right down to the tip of each hair! You can see a video demonstrating a van
de Graff generator at this URL: http://www.youtube.com/watch?v=SREXQWAIDJk MEDIA Click image to the left for more content. Q: Why is the man’s hair standing on end? A: All of the hairs have all become negatively charged, and like charges repel each other. Therefore, the hairs are pushing away from each other, causing them to stand on end. 7 www.ck12.org Transferring Electrons The man pictured above became negatively charged because electrons ﬂowed from the van de Graaff generator to him. Whenever electrons are transferred between objects, neutral matter becomes charged. This occurs even with individual atoms. Atoms are neutral in electric charge because they have the same number of negative electrons as positive protons. However, if atoms lose or gain electrons, they become charged particles called ions. You can see how this happens in the Figure 3.1. When an atom loses electrons, it becomes a positively charged ion, or cation. When an atom gains electrons, it becomes a negative charged ion, or anion. FIGURE 3.1 Conservation of Charge Like the formation of ions, the formation of charged matter in general depends on the transfer of electrons, either between two materials or within a material. Three ways this can occur are referred to as conduction, polarization, and friction. All three ways are described below. However, regardless of how electrons are transferred, the total charge always remains the same. Electrons move, but they aren’t destroyed. This is the law of conservation of charge. Conduction The transfer of electrons from the van de Graaff generator to the man is an example of conduction. Conduction occurs when there is direct contact between materials that differ in their ability to give up or accept electrons. A van de Graff generator produces a negative charge on its dome, so it tends to give up electrons. Human hands are positively charged, so they tend to accept electrons. Therefore, electrons ﬂow from the dome to the man’s hand when they are in contact. You don’t need a van de Graaff generator for conduction to take place. It may occur when you walk across a wool carpet in rubber-soled shoes. Wool tends to give up electrons and rubber tends to accept them. Therefore, the carpet transfers electrons to your shoes each time you put down your foot. The transfer of electrons results in you becoming negatively charged and the carpet becoming positively
charged. 8 www.ck12.org Polarization Concept 3. Transfer of Electric Charge Assume that you have walked across a wool carpet in rubber-soled shoes and become negatively charged. If you then reach out to touch a metal doorknob, electrons in the neutral metal will be repelled and move away from your hand before you even touch the knob. In this way, one end of the doorknob becomes positively charged and the other end becomes negatively charged. This is called polarization. Polarization occurs whenever electrons within a neutral object move because of the electric ﬁeld of a nearby charged object. It occurs without direct contact between the two objects. The Figure 3.2 models how polarization occurs. FIGURE 3.2 Q: What happens when the negatively charged plastic rod in the diagram is placed close to the neutral metal plate? A: Electrons in the plate are repelled by the positive charges in the rod. The electrons move away from the rod, causing one side of the plate to become positively charged and the other side to become negatively charged. Friction Did you ever rub an inﬂated balloon against your hair? You can see what happens in the Figure 3.3. Friction between the balloon and hair cause electrons from the hair to “rub off” on the balloon. That’s because a balloon attracts electrons more strongly than hair does. After the transfer of electrons, the balloon becomes negatively charged and the hair becomes positively charged. The individual hairs push away from each other and stand on end because like charges repel each other. The balloon and the hair attract each other because opposite charges attract. Electrons are transferred in this way whenever there is friction between materials that differ in their ability to give up or accept electrons. Watch the animation “Balloons and Static Electricity” at the following URL to see how electrons are transferred by friction between a sweater and a balloon. http://www.cabrillo.edu/~jmccullough/Physics/Electr ic_Forces_Fields.html Q: If you rub a balloon against a wall, it may stick to the wall. Explain why. A: Electrons are transferred from the wall to the balloon, making the balloon negatively charged and the wall positively charged. The balloon sticks to the wall because opposite charges attract. Summary • Whenever electrons are transferred between objects, neutral matter becomes charged. For example, when atoms lose or gain electrons they become charged particles called ions. 9
www.ck12.org FIGURE 3.3 • Three ways electrons can be transferred are conduction, friction, and polarization. In each case, the total charge remains the same. This is the law of conservation of charge. • Conduction occurs when there is direct contact between materials that differ in their ability to give up or accept electrons. • Polarization is the movement of electrons within a neutral object due to the electric ﬁeld of a nearby charged object. It occurs without direct contact between the two objects. • Electrons are transferred whenever there is friction between materials that differ in their ability to give up or accept electrons. Vocabulary • law of conservation of charge : Law stating that charges are not destroyed when they are transferred between two materials or within a material, so the total charge remains the same. Practice At the following URL, review how charges are transferred through friction. Watch the animation and read the list of more-positive to less-positive materials. Then answer the questions below. http://www.regentsprep.org/regents/physics/phys03/atribo/default.htm 1. If you rub glass with a piece of plastic wrap, will the glass become positively or negatively charged? 2. Assume that after you pet your dog with very dry hands, you touch a metal doorknob and get a shock. Is electric charge transferred from your hand to the doorknob or the other way around? Review 1. How is charge transferred by a van de Graaff generator? 2. Compare and contrast the formation of cations and anions. 3. State the law of conservation of charge. 4. Explain how conduction and polarization occur, using the example of walking across a wool carpet in rubber- soled shoes and then reaching out to touch a metal doorknob. 10 www.ck12.org Concept 3. Transfer of Electric Charge 5. Predict what will happen to the charges of a plastic comb and a piece of tissue paper if you rub the tissue paper on the comb. ( Hint : Plastic tends to accept electrons and tissue paper tends to give up electrons.) References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Christopher Auyeung.. CC BY-NC 3.0 3. Flickr:olga.palma.. CC BY 2.0 11 CONCEPT 4 Forces on Charged Objects www.ck12.org • Describe the changes that occur in the sub-atomic arrangement in
matter when charged. • Describe how to charge an object. • Deﬁne conductors and insulators. • Understand the difference between conduction and induction. • Summarize the forces between charged objects. Lightning is the discharge of static electricity that has built up on clouds. Every year, the earth experiences an average of 25 million lightning strikes. Lightning bolts travel at speeds up to 60,000 miles per second, and can reach temperatures of 50,000°F, which is ﬁve times the temperature of the surface of the sun. The energy contained in a single lightning strike could light a 100 Watt light bulb 24 hours per day for 90 days. Forces on Charged Objects Electric charges exist within the atom. At the turn of the 20th century, J. J. Thomson and Ernest Rutherford determined that atoms contain very light-weight negatively charged particles called electrons and more massive, positively charged particles called protons. The protons are lodged in the nucleus of the atoms, along with the neutrally charged particles called neutrons, while the electrons surround the nucleus. When the number of electrons in the electron cloud and the number of protons in the nucleus are equal, the object is said to be neutral. Changes to the nucleus of an atom require tremendous amounts of energy, so protons are not easily gained or lost by atoms. Electrons, on the other hand, are held fairly loosely and can often be removed quite easily. When an object loses some electrons, the remaining object is now positively charged because it has an excess of protons. The electrons may either remain free or may attach to another object. In that case, the extra electrons cause that object to become negatively charged. Atoms that have lost electrons and become positively charged are called positive ions, and atoms that have gained electrons and become negatively charged are called negative ions. Electrons can be removed from some objects using friction, simply by rubbing one substance against another substance. There are many examples of objects becoming charged by friction, including a rubber comb through 12 www.ck12.org Concept 4. Forces on Charged Objects hair, and a balloon on a sweater. In both these instances, the electrons move from the second object to the ﬁrst, causing the ﬁrst object to become negatively charged and the second one positively charged. Friction between the tires on a moving car and the road cause the tires to become charged, and wind causes friction between clouds and air which causes clouds to become charged and can result
in tremendous bolts of lightning. A common method of producing charge in the lab is to rub cat or rabbit fur against stiff rubber, producing a negative charge on the rubber rod. If you hold a rubber rod on one end and rub only the tip of the other end with a fur, you will ﬁnd that only the tip becomes charged. The electrons you add to the tip of the rod remain where you put them instead of moving around on the rod. Rubber is an insulator. Insulators are substances that do not allow electrons to move through them. Glass, dry wood, most plastics, cloth, and dry air are common insulators. Materials that allow electrons to ﬂow freely are called conductors. Metals have at least one electron that can move around freely, and all metals are conductors. Forces are exerted on charged objects by other charged objects. You’ve probably heard the saying "opposites attract," which is true in regards to charged particles. Opposite charges attract each other, while like charges repulse each other. This can be seen in the image below. When two negatively charged objects are brought near each other, a repulsive force is produced. When two positively charged objects are brought near each other, a similar repulsive force is produced. When a negatively charged object is brought near a positively charged object, an attractive force is produced. Neutral objects have no inﬂuence on each other. A laboratory instrument used to analyze and test for static charge is called an electroscope. Seen below, an electroscope consists of a metal knob connected by a metal stem to two very lightweight pieces of metal called leaves, shown in yellow. The leaves are enclosed in a box to eliminate stray air currents. 13 www.ck12.org When a negatively charged object is brought near the knob of a neutral electroscope, the negative charge repels the electrons in the knob, and those electrons move down the stem into the leaves. Excess electrons ﬂow from the rod into the ball, and then downwards making both leaves negatively charged. Since both leaves are negatively charged, they repel each other. When the rod is removed, the electroscope will remain charged because of the extra electrons added to it. Conversely, if the rod is brought near the knob but doesn’t touch it, the electroscope will appear the same while the rod is near. That is, the negative charge in the rod repels the electrons in the ball, causing them to travel down to the
leaves. The leaves will separate while the rod is nearby. No extra electrons were added to the electroscope, meaning that the electrons in the electroscope will redistribute when the negatively charged rod is taken away. The leaves return to neutral, and they stop repelling each other. If the rod touches the knob, the electroscope leaves are permanently charged but if the rod is brought near but does not touch the knob, the electroscope leaves are only temporarily charged. If the leaves are permanently charged and the rod removed, the electroscope can then be used to determine the type of unknown charge on an object. If the electroscope has been permanently negatively charged, and a negatively charge object is brought near the knob, the leaves will separate even further, showing the new object has the same charge as the leaves. If a positively charged object is brought near a negatively charged electroscope, it will attract some of the excess electrons up the stem and out of the leaves, causing the leaves to come slightly together. Similar to the results of a negatively charged rod, if a positively charged rod is brought near the knob of a neutral electroscope, it will attract some electrons up from the leaves onto the knob. That process causes both of the leaves to 14 www.ck12.org Concept 4. Forces on Charged Objects be positively charged (excess protons), and the leaves will diverge. If the positively charged rob is actually touched to the knob, the rob will remove some electrons and then when the rob is removed, the electroscope will remain positively charged. This is a permanent positive charge. Charging an object by touching it with another charged object is called charging by conduction. By bringing a charged object into contact with an uncharged object, some electrons will migrate to even out the charge on both objects. Charging by conduction gives the previously uncharged object a permanent charge. An uncharged object can also be charged using a method called charging by induction. This process allows a change in charge without actually touching the charged and uncharged objects to each other. Imagine a negatively charged rod held near the knob, but not touching. If we place a ﬁnger on the knob, some of the electrons will escape into our body, instead of down the stem and into the leaves. When both our ﬁnger and the negatively charged rod are removed, the previously uncharged electroscope now has a slight positive charge. It was charged by induction. Notice that charging by induction causes the newly charged object to have the opposite
charge as the originally charged object, while charging by conduction gives them both the same charge. Summary • Electric charges exist with the atom. • Atoms contain light-weight, loosely held, negatively charged particles called electrons and heavier, tightly- held, positvely charged particles called protons. • When the number of electrons and the number of protons are equal, the object is neutral. • The loss of electrons gives an ion a positive charge, while the gain of electrons gives it a negative charge. • Materials that allow electrons to ﬂow freely are called conductors, while those that do not are called insulators. • Opposite charges attract, and like charges repel. • Charging an object by touching it with another charged object is called charging by conduction. Practice The following video shows a young woman placing her hands on a Van de Graf generator which then gives her a static charge. Use this resource to answer the two questions that follow. http://www.youtube.com/watch?v=87DqbdqBx8U 15 www.ck12.org MEDIA Click image to the left for more content. 1. What happens to her hair when she touches a ground? 2. What happens to her hair when she steps off the platform? This video shows the static charge from the Van de Graf generator. http://www.youtube.com/watch?v=prgu6AvauuI MEDIA Click image to the left for more content. This video demonstrates superconductivity that occurs at extremely low temperatures. http://www.youtube.com/watch?feature=player_embedded&v=nWTSzBWEsms MEDIA Click image to the left for more content. Additional Practice Questions: 1. When a glass rod is rubbed with a silk cloth and the rod becomes positively charged, (a) electrons are removed from the rod. (b) protons are added to the silk. (c) protons are removed from the silk. (d) the silk remains neutral. 2. Electric charge is (a) found only in a conductor. (b) found only in insulators. (c) conserved. (d) not conserved. 3. When two objects are rubbed together and they become oppositely charged, they are said to be charge by (a) conduction. (b) induction. 16 www.ck12.org Concept 4. Forces on Charged Objects (c) friction.
(d) grounding. 4. Two objects each carry a charge and they attract. What do you know about the charge of each object? (a) They are both charged positively. (b) They have opposite charged from each other. (c) They are both charged negatively. (d) Any of the above are possible. 5. A material that easily allows the ﬂow of electric charge through it is called a(n) (a) insulator. (b) conductor. (c) semiconductor. (d) heat sink. 6. What is the most common way of acquiring a positive static electrical charge? (a) by losing electrons (b) by gain protons (c) by losing protons (d) by gaining electrons (e) by switching positions of electrons and protons in the atom Review 1. How does friction generate static electricity? (a) Friction heats the materials, thus causing electricity. (b) Rubbing materials together displaces atoms, causing sparks to ﬂy. (c) Rubbing materials together can strip electrons off atoms, causing one material to become positive and the other to become negative. (d) Rubbing materials together causes neutrons and electrons to trade places. (e) None of the above. 2. What electrical charge does an electron have? (a) A negative charge. (b) A positive charge. (c) A neutral charge. (d) May be any of the above. (e) None of the above. 3. What happens when opposite charges get close to each other? (a) They repel each other. (b) They attract each other. (c) Nothing happens. (d) They attract surrounding objects. (e) They repel surrounding objects. 4. What is an electrical conductor? (a) A material that allows electrons to travel through it freely. (b) A material that doesn’t allow electrons to travel through it freely. (c) A material that melts at low temperature. (d) A material that creates free electrons. (e) None of the above. 5. Which of the following is a good insulator of electricity? 17 www.ck12.org (a) Copper (b) Iron (c) Rubber (d) Salt water (e) None of these. • electrons: A fundamental sub-atomic particle, meaning it cannot be broken into smaller particles. Electrons are found in the “electron cloud” surrounding an
atomic nucleus, or they may break free and exist as a free electron. • protons: A stable, positively charged, sub-atomic particle, found in atomic nuclei in numbers equal to the atomic number of the element. • neutral: A neutral particle, object, or system is one that has a net electric charge of zero. • conductors: Materials through which electric charge can pass. • insulator: Substances that block or retard the ﬂow of electrical current or charge. • positive ions: An atom or a group of atoms that has acquired a net positive charge by losing one or more electrons. • negative ions: An atom or a group of atoms that has acquired a net negative charge by gaining one or more electrons. • ions: An atom or a group of atoms that has acquired a net electric charge by gaining or losing one or more electrons. • electroscope: An instrument used to detect the presence and sign of an electric charge by the mutual attraction or repulsion of metal foils. • charging by conduction: Involves the contact of a charged object to a neutral object. • charging by induction: A method used to charge an object without actually touching the object to any other charged object. References 1. Courtesy of NOAA Photo Library, NOAA Central Library; OAR/ERL/National Severe Storms Laboratory (NSSL). http://www.photolib.noaa.gov/htmls/nssl0016.htm. Public Domain 2. Sweater: Image copyright Sibiryanka, 2013; Balloon: Image copyright simpleman, 2013. http://www.shut terstock.com. Used under licenses from Shutterstock.com 3. CK-12 Foundation - Christopher Auyeung.. CC-BY-NC-SA 3.0 4. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 5. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 6. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 18 www.ck12.org Concept 5. Coulomb’s Law CONCEPT 5 Coulomb’s Law • State Coulomb’s Law. • Describe how electric force varies with charge and separation of charge. • State the SI unit of charge. • Solve problems using Coulomb’s Law. Electric cars are becoming more popular. One large advantage for
electric cars is the low cost of operation, which may become an ever bigger advantage as gas prices climb. Energy costs for electric cars average about one-third of the cost for gasoline engine cars, but they can only travel about 200 miles per charge at this point. These cars run using the science of electrical charges and forces. Coulomb’s Law The questions regarding the relationship between the electrical force, the size of the charge, and the separation between the charges were solved by Charles Coulomb in 1785. He determined that electrical force between two charges is directly related to the size of the charges and inversely proportional to the distance between the charges. This is known as Coulomb’s Law. Fe = Kq1q2 d2 In this equation, q 1 and q 2 are the two charges, d is the distance between the two charges, and K is a constant of proportionality. F e is the electric force, which occurs as a result of interactions between two charged particles. For the purpose of calculating electric forces, we assume all charge is a point charge, in which the entire charge of the particle is located in a massless point. The SI unit of charge is the coulomb, C, which is the charge of 6:25 1018 electrons. The charge on a single electron is 1:60 1019 C. The charge on a single electron is known as the elementary charge. The charge on a proton is the same magnitude but opposite in sign. When the charges are measured in coulombs, the distance in meters, and the force in Newtons, the constant K is 9:0 109 N m2=C2. 19 www.ck12.org The electrical force, like all forces, is a vector quantity. If the two charges being considered are both positive or both negative, the sign of the electrical force is positive and this force is repulsive. If the two charges are opposite in sign, the force will have a negative sign and the force is attractive. Example Problem: Object A has a positive charge of 6:0 106 C. Object B has a positive charge of 3:0 106 C. If the distance between A and B is 0.030 m, what is the force on A? Solution: Fe = Kq1q2 d2 = (9:0109 Nm2=C2)(6:0106 C)(3:0106 C) = 180 N (0:030 m)2 The positive sign of the force indicates
the force is repulsive. This makes sense, because both objects have a positive charge. Example Problem: 6:0 106 C. Calculate the total force on q 2. In the sketch below, the charges are q1 = 10:0 106 C; q2 = 2:0 106 C, and q3 = Solution: Fe = Kq1q2 d2 = (9:0109 Nm2=C2)(10:0106 C)(2:0106 C) (2:0 m)2 = 0:045 N (towards q3) Fe = Kq2q3 d2 = (9:0109 Nm2=C2)(2:0106 C)(6:0106 C) (4:0 m)2 = 0:007 N (towards q3) Since the two forces act in the same direction, their absolute values can be added together; the total force on q 2 is 0.052 N towards q 3. Summary • Coulomb determined that electrical force between two charges is directly related to the size of the charges and inversely proportional to the distance between the charges: Fe = Kq1q2 d2 • The SI unit of charge is the coulomb, C, which is the charge of 6:25 1018 electrons. • The charge on a single electron is 1:60 1019 C and is known as the elementary charge. • The electrical force is a vector quantity that is positive in repulsion and negative in attraction. Practice The following video covers Coulomb’s Law. Use this resource to answer the questions that follow. http://www.youtube.com/watch?v=rYjo774UpHI MEDIA Click image to the left for more content. 1. What happens when like charges are placed near each other? 2. What happens when opposite charged are placed near each other? 3. What happens to the force of attraction if the charges are placed closer together? 20 www.ck12.org Concept 5. Coulomb’s Law Practice problems on Coulomb’s Law. http://physics.info/coulomb/problems.shtml Review 1. Suppose that two point charges, each with a charge of +1.00 C, are separated by a distance of 1.0 m: (a) Will the charges attract or repel? (b) What is the magnitude of the force between them? (c) If the distance
between them is doubled, what does the force become? 2. What is the electrical force between two balloons, each having 5.00 C of charge, that are 0.300 m apart? 3. Two spheres are charged with the same charge of -0.0025 C and are separated by a distance of 8.00 m. What is the electrical force between them? 4. A red foam ball and a blue foam ball are 4.00 m apart. The blue ball has a charge of 0.000337 C and is attracting the red ball with a force of 626 N. What is the charge on the red ball? • Coulomb’s Law: States the force of attraction or repulsion acting along a straight line between two electric charges is directly proportional to the product of the charges and inversely to the square of the distance between them. References 1. Image copyright testing, 2013. http://www.shutterstock.com. Used under license from Shutterstock.com 2. CK-12 Foundation - Samantha Bacic.. CC BY-NC-SA 3.0 21 CONCEPT 6 • Deﬁne an electric ﬁeld. • Solve problems relating to ﬁeld, force, and charge. www.ck12.org The Electric Field A plasma globe, such as the one pictured above, is ﬁlled with a mixture of noble gases and has a high-voltage electrode at the center. The swirling lines are electric discharge lines that connect from the inner electrode to the outer glass insulator. When a hand is placed on the surface of the globe, all the electric discharge travels directly to that hand. The Electric Field Coulomb’s Law gives us the formula to calculate the force exerted on a charge by another charge. On some occasions, however, a test charge suffers an electrical force with no apparent cause. That is, as observers, we cannot see or detect the original charge creating the electrical force. Michael Faraday dealt with this problem by developing the concept of an electric ﬁeld. According to Faraday, a charge creates an electric ﬁeld about it in all directions. If a second charge is placed at some point in the ﬁeld, the second charge interacts with the ﬁeld and experiences an electrical force. Thus, the interaction we observe is between the test charge and the ﬁeld and a second particle at some distance is no longer necessary. The
strength of the electric ﬁeld is determined point by point and can only be identiﬁed by the presence of test charge. When a positive test charge, q t, is placed in an electric ﬁeld, the ﬁeld exerts a force on the charge. The ﬁeld strength can be measured by dividing the force by the charge of the test charge. Electric ﬁeld strength is given the symbol E and its unit is Newtons/coulomb. 22 E = Fonqt qt www.ck12.org Concept 6. The Electric Field The test charge can be moved from location to location within the electric ﬁeld until the entire electric ﬁeld has been mapped in terms of electric ﬁeld intensity. Example Problem: A positive test charge of 2:0 105 C is placed in an electric ﬁeld. The force on the test charge is 0.60 N. What is the electric ﬁeld intensity at the location of the test charge? Solution: E = F q = 0:60 N 2:0105 C = 3:0 104 N/C Summary • An electric ﬁeld surrounds every charge and acts on other charges in the vicinity. • The strength of the electric ﬁeld is given by the symbol E, and has the unit of Newtons/coulomb. • The equation for electric ﬁeld intensity is E = F q. Practice The following video covers electric ﬁelds. Use this resource to answer the questions that follow. http://www.youtube.com/watch?v=lpb94QF0_mM MEDIA Click image to the left for more content. 1. What does it mean when a force is called a non-contact force? 2. What symbol is used to represent electric ﬁeld strength? 3. What is the relationship between the direction of the electric ﬁeld and the direction of the electric force? Review 1. The weight of a proton is 1:64 1026 N. The charge on a proton is +1:60 1019 C. If a proton is placed in a uniform electric ﬁeld so that the electric force on the proton just balances its weight, what is the magnitude and direction of the ﬁeld? 2. A negative charge of
2:0 108 C experiences a force of 0.060 N to the right in an electric ﬁeld. What is the magnitude and direction of the ﬁeld? 3. A positive charge of 5:0 104 C is in an electric ﬁeld that exerts a force of 2:5 104 N on it. What is the magnitude of the electric ﬁeld at the location of the charge? 4. If you determined the electric ﬁeld intensity in a ﬁeld using a test charge of 1:0 106 C and then repeated the process with a test charge of 2:0 106 C, would the forces on the charges be the same? Would you ﬁnd the value for E? 5. A 0.16 C charge and a 0.04 C charge are separated by a distance of 3.0 m. At what position between the two charges would a test charge experience an electric ﬁeld intensity of zero? • electric ﬁeld: A region of space characterized by the existence of a force that is generated by an electric charge. • electric ﬁeld intensity: Electric ﬁeld intensity or ﬁeld strength is described as the ratio of force to the amount of test charge. 23 References 1. User:Slimsdizz/Wikipedia. http://commons.wikimedia.org/wiki/File:Glass_plasma_globe.jpg. Public Domain www.ck12.org 24 www.ck12.org Concept 7. Electric Potential CONCEPT 7 Objectives The student will: Electric Potential • Understand how to solve problems using electric potential energy. • Understand how to solve problems using voltage differences. • Understand how to solve problems in a uniform electric ﬁeld. Vocabulary • electric potential: Energy per unit charge. • electric potential difference: The difference in electric potential between two points within an electric ﬁeld. • voltage: The amount of work done by the electric ﬁeld per unit charge in moving a charge between two points in the electric ﬁeld DV = W q, also known as a change in potential energy. Introduction In order to draw an analogy between gravitational potential energy and electrical potential energy, we liken the electric ﬁeld E to the gravitational acceleration g and the mass m of a particle to the charge q of a particle. Of course, g is assumed
constant (uniform) when we remain close to the surface of the Earth. As of yet, we have not encountered an example of a uniform electric ﬁeld E. But that won’t stop us from making a prediction! Since the gravitational potential energy of a mass m in a uniform gravitational ﬁeld is PEgravity = (mg)h, we predict the electric potential energy (PEelectric) of a charge (q) in a constant electric ﬁeld is PEelectric = (qE)h. Furthermore, the electric potential (the energy per unit charge) can be deﬁned as Ve = PEelectric will be dropped from now on.) We will discuss electric potential later. (The subscript “e” q It must be understood that, just as with gravity, the electric potential energy and the electric potential are measured at the same point. If a point charge q has electric potential energy PEx1 at point x1, the electric potential at x1 is Vx1 PEx1 = qVx1! Vx1 = PEx1 q q PEi q. Again, only differences in electric potential and electric potential energy are meaningful. That is, DPE or DV! Vf Vi = PE f The unit of electric potential is called the volt and from the deﬁnition above we see that the volt is equivalent to Coulomb! V = J Joules C. 25 Electric Potential Difference The electric potential difference is the difference in electric potential between two points within an electric ﬁeld. For example, a 1.5-volt battery has a potential difference of 1.5 volts (written 1.5 V ) between its positive and negative terminals. www.ck12.org Parallel Plate Conductors: A uniform Electric Field The equation E = k q r2 for the electric ﬁeld holds for point charges or for a charge distribution that effectively acts as a point charge. It turns out, however, that if opposite charges are placed on two parallel conducting plates, the electric ﬁeld between the plates is more or less uniform as long as the distance between the plates is much smaller than the dimensions of the plates. The plates can be charged by connecting them to the positive and negative terminals of a battery. A battery contains a substance (called an electrolyte) which causes two dissimilar metals to acquire opposite charges. The two dissimilar metals form the positive and negative terminals of the battery
. If a metal plate is connected to the positive terminal of the battery, and another metal plate is connected to the negative terminal of the battery, and the two plates brought closely together, a parallel plate arrangement (parallel-plate conductors) can be constructed with a uniform electric ﬁeld between the plates (seen edge on) in Figure below. We will see later that parallel plate conductors are also referred to as capacitors. FIGURE 7.1 Parallel plates. Just as in the case of the battery, one of the plates of the parallel-plate conductor will be at a higher potential than the other plate. Think, for example, of a standard AA battery with a voltage rating of 1.5V, Figure below. See the link below to learn more about how a battery works. http://phet.colorado.edu/en/simulation/battery-voltage 26 www.ck12.org Concept 7. Electric Potential FIGURE 7.2 volt battery Electrical Potential Energy In our gravitational analogy, the energy that a charge possesses at the plate with the higher potential is analogous to the energy a mass possesses above the ground. Additionally, now that we have found a way to create a uniform electric ﬁeld, we have an analog to a uniform gravitational ﬁeld. If a positive charge +q is placed at the positive plate in Figure below, it will be repelled by the positive charges on the plate and move toward the negative plate. (Think of +q as the object m falling toward the ground.) FIGURE 7.3 A positive charge moving toward the negative plate What is the force acting on the +q charge? Recall that the Coulomb force on a charge placed in an electrostatic ﬁeld is F = qE. The work that the electric ﬁeld does on the charge is equal to the negative change in the potential energy of the charge, just as in the gravitational case. We can ﬁnd an expression for the electric potential energy by ﬁnding the work that is done on the charge. Recall that W = FDx. We write 27 www.ck12.org Wf ield = FDx = (qE)Dx = DPE! qE(x f xi) = DPE! qEx f qExi = DPE The expression for the electric potential energy is thus: PEelectrical = qEx. Recall that the equation for the gravitational potential
energy is PEgravitational = mgh. We can compare the terms in the gravitational and electrical cases as follows Thus, we see that our prediction for the equation of electric potential energy stated in the introduction of the lesson, was correct! Check Your Understanding 1a. The electrical potential at the negative plate in Figure above is deﬁned as zero volts. What is the electrical potential energy of a charge +q = 15:0µC at the positive plate if the electric ﬁeld between the plates is 25:0 N The positive plate has position 6:00cm = 6:00 102 m according to Figure above. Answer : PEpositive plate = qEx = (15:0 106 C) 25:0 N C 1b. What is the change in the electrical potential energy DPE of the charge +q = 15:0µC if its potential changes from 1.5 V to 1.0 V? (6:00 102 m) = 3:75 104 J C? Answer : Just as in the case of a change in gravitational potential energy, the charge must lose potential energy, since it gains kinetic energy. The charge moves from the position xi = 6:00 102 m (1:5 V ) to the position xi = 4:00 102 m (1:0 V ). DPE = qEx f qExi = qE(x f xi) = N C (15:0 106 C) 25:0 (4:00 102 m 6:00 102 m) = 7:50 106 J: 1c. What is the work done on the charge by the electric ﬁeld? Answer: Wf ield = DPE = (7:50 106) = 7:50 106 J Notice that the electric ﬁeld does positive work on the charge, since the electric force and the displacement of the charge have the same direction. We should recall a very important point: It is only the change in potential energy that is meaningful, whether we are discussing the gravitational potential energy or the electrical potential energy. 28 www.ck12.org Concept 7. Electric Potential 2. An electron placed at the negative plate of a parallel-plate conductor will move toward the positive plate. The potential energy of the electron: A. Decreases B. Increases C. Remains the same. Answer : The correct answer is A. The electron is repelled by the negative charges of the conducting plate
and therefore gains kinetic energy. Just as an object that is dropped gains kinetic energy and loses potential energy, so does the electron. Recall our discussion of the conservation of energy. As long as the total energy remains conserved, the sum of the initial kinetic and potential energies must equal the sum of the ﬁnal kinetic and potential energies: KEi + PEi = KE f + PE f! DKE = DPE The gain in kinetic energy occurs due to the loss in potential energy. In order for the charges of the same sign to be brought together, as in the example above, positive work must be done by an external force against the electrostatic repulsion between the charges. The work increases the potential energy stored in the electric ﬁeld. When the charges are released, the potential energy of the ﬁeld is converted into the kinetic energies of the charges. The link below may be helpful in learning more about the work done upon charges in electric ﬁelds. http://www.youtube.com/watch?v=elJUghWSVh4 Electric Potential Difference in a Uniform Electric Field In working with the change in potential energy above, we wrote the equation DPE = qEx f qExi = qE(x f xi)! Let us call this Equation A. Recall that the electric potential was deﬁned at a speciﬁc point Vx1 = We therefore see that PEx1 = qVx1! Let us call this Equation B. Comparing Equation A and Equation B, we see that the electric potential can be expressed as Vx1 = Ex1. If the electric potential is deﬁned as V = 0 at x = 0, then the potential at any point in the electric ﬁeld is V = Ex. (Assuming that vector E is directed along the x axis).. PEx1 q Note: It is common to write V = Ed, where V is understood to mean the voltage (or potential difference) between the plates of a parallel-plate conductor, and d is the distance between the plates. Check Your Understanding Verify that the potential difference between the plates in Figure above is 1.5 V. Recall that the electric ﬁeld is E = 25:0 N Answer : V = Ed = 25 N C (6:00 102 m 0:00m) = 1:5 V C. Work We state
again: 1. The electric potential is deﬁned as the energy per unit charge! Vx1 = PEx1 q. 29 www.ck12.org 2. The electric potential difference (the voltage) is Vf Vi = PE f 3. An arbitrary reference level must be established for zero potential (just as in the case of gravitational potential q PEi q energy). 4. The units of electric potential and electric potential difference are J C since Vx1 = PEx1 q. It is often useful to express the voltage in terms of the work done on a charge. From Vf Vi = PE f But the work done on a charge by the ﬁeld is Wf ield = DPE. q, we have PE f PEi = q(Vf Vi)! DPE = q(Vf Vi). q PEi Combining DPE = q(Vf Vi) and Wf ield = DPE gives Wf ield = q(Vf Vi). An external force that does work on a charge in an electric ﬁeld exerts a force in the opposite direction to the ﬁeld (just as the external force acting on a spring acts opposite to the spring force). The work that an external force does is therefore Wexternal f orce = q(Vf Vi). The voltage can be thought of as the amount of work done by the electric ﬁeld per unit charge in moving a charge between two points in the electric ﬁeld DV = W q. We often refer to a change in potential as simply “the voltage.” In computing the work, it is often easier to ignore the sign in the equation and simply see if the force and displacement on the charge are in the same direction (positive work) or opposite to one another (negative work). Recall that the force and displacement need not be in the same direction or oppositely directed. In general, work is expressed as W = Fx cos q http://www.youtube.com/watch?v=F1p3fgbDnkY Other Units for the Electric Field! E = C, since! F q. But the electric ﬁeld has been also deﬁned using the scalar equation x. So the units of the electric ﬁeld can be also expressed as The electric ﬁeld has units N V = Ex
. Transposing terms, the electric ﬁeld is E = V volts per meter volts If we compare the units for the electric ﬁeld N m, we see that a (N m) is equivalent to a (C V ). A Joule can therefore be expressed as a Coulomb-Volt. Recall that work, measured in Joules, is the product of charge and voltage W = qDV. meter! V m. C and V Illustrative Example 16.2.1 All questions refer to Figure above. a. What is the potential at x = 2:0 cm in Figure below? The electric ﬁeld is E = 25:0 N C. Answer : The potential V varies directly with the position x between the plates (V = Ex). Thus, V = 25:0 N 0:50 V. C (2:0 102m) = b. Sketch a graph showing the relationship between the potential and the position. Answer: 30 www.ck12.org Concept 7. Electric Potential c. How much work is done by an external force F moving a 2:0 106 C charge from the positive plate to the negative plate? Answer: An external force must pull the charge away from the positive plate so the force will be in the same direction as the displacement. W = q(Vf Vi) = (2:0 106C)(0:00V 1:50V ) = 3:0 106 J d. What is the magnitude of the Coulomb force acting on the charge? Answer: W = F! 3:0 106 J = F(6:00 102 m) F = 3:0 106 J 6:00 102 m = 5:0 105 N Illustrative Example 16.2.2 a. A particle of mass m of 2:00 105 kg is has a charge q of +3:00 103 C. If the particle is released from the positive plate of a parallel-plate conductor with an electric ﬁeld E of 1:30 105 N C, determine the acceleration of the particle, see Figure below. Answer : If we ignore gravity, the only force acting on the particle is the electric force F = qE. Using Newton’s Second Law, the net force on the particle is equal to F = ma! qE = ma. The acceleration is a = Eq m = (1:30105 V m )(3:00103 C) 2:00
105 kg = 1:95 107 V C kgm. b. Show that the units V C kgm are equivalent to the units m s2. Answer : V C kg m = J kg m = N m kg m = N kg = kg m s2 kg = m s2 c. The plates have separation of 8.00 mm. Determine the velocity of the particle when it reaches the negative plate Answer : 31 www.ck12.org FIGURE 7.4 Illustrative Example 16.3.2 This is a kinematics problem, where the displacement and acceleration are known and the velocity is to be found. Recall the equation v2 f = v2 i + 2aDx.! v2 1:95 107 m s2 f = 0 + 2 v = 558:6! 5:59 102 m s : (8:00 103 m) = 312; 000 m2 s2 d. What is the potential difference between the plates? Answer : V = Ex = 1:30 105 V m (8:00 103 m) = 1:04 103 V e. How much work has the ﬁeld done on the particle as it moved from one plate to the other? W = qDV = (3:00 103 C)(1:04 103 V ) = 3:12 J Illustrative Example 16.2.3 An electron is accelerated from rest through a potential difference of 30,000 V. The mass of the electron is 9:11 1031 kg and the charge of the electron is 1:60 1019 C. Find its velocity. Answer : Recall that the Work-Energy Principle states that W = DKE. 32 www.ck12.org Concept 7. Electric Potential W = DKE W = qDV DKE = qDV 1 2 i = qDV mv2 1 2 mv2 f v f = Illustrative Example 16.2.4 vi = 0! v2 s f = 2qDV m! v f = r 2qDV m! 2(1:60 1019 C)(3:00 104 V ) 9:11 1031 kg = 1:026 108! 1:03 108 m s What magnitude of an electric ﬁeld is required to balance the gravitational force acting on an electron in Figure below? FIGURE 7.5 Illustrative Example 16.2.4-An electron suspended in an electric ﬁeld. Answer : Draw
a Free-Body-Diagram (FBD) of the situation. The electrostatic force that acts on the electron points upward and the gravitational force that acts upon on the electron points downward. The electron is suspended motionless (or moves with a constant velocity) when the net force on the electron is zero. 33 The net force on the electron must be zero, thus www.ck12.org F = 0! eF = mg! F = mg e mg e mg = eE! E = e2! but F = eE! (9:11 1031 kg) 9:81 m s2 (1:60 1019 C)2 E = = 3:49 108 V m http://www.youtube.com/watch?v=wT9AsY79f1k The Electron-Volt It is often convenient when dealing with small particles such as electrons, protons, and ions to express the energy of these particles with a smaller unit of measure. The electron-volt is deﬁned as the change in potential energy that an electron acquires when moving through a potential difference of 1 V, or equivalently, its change in kinetic energy after moving through a potential difference of 1 V. That is, PE = eV = (1:60 1019 C)(1:00 V ) = 1:60 1019 J. The energy 1:60 1019 J is deﬁned as one electron-volt. We write one-electron-volt as 1 eV = 1:60 1019 J. Check Your Understanding 1. What is the change in kinetic energy KE when an electron is released at the negative plate of a parallel plate conductor with a potential difference of 3,500 V? Express your answer in eV. Answer : The electron is repelled at the negative plate and therefore gains kinetic energy (and loses potential energy). The change in KE is positive and equal to = x eV 1 eV 1 V 3; 500 V DKE = 3; 500 eV! x = 3; 500 eV It is simplest to think that for every one volt of potential difference the particle experiences, it gains (or loses) 1 eV. 2. An alpha-particle (the nucleus of a helium atom) is ﬁred toward the positive plate of a parallel plate conductor and passes through a potential difference of 1,500 V. What is the change in its kinetic energy? Express your answer in eV.
Answer : Protons are the only charges inside the nucleus of an atom and so the alpha particle must be positively charged. A helium nucleus contains two protons (and two neutrons) with a total charge of 2(1:60 1019 C). The alpha particle must slow down due to the electrostatic repulsion from the positive plate. It must, therefore, lose kinetic energy and gain potential energy. Each proton loses 1,500 eV of kinetic energy. DKE = 3; 000 eV. 3. An electron and a proton both gain kinetic energy of 1 eV. True or False: Their speeds must be the same, since they both gained the same amount of energy. 34 www.ck12.org Concept 7. Electric Potential Answer : False. The mass of a proton is nearly 2000 times greater than the mass of an electron. Remember that kinetic energy depends on both the speed and mass of an object. Therefore, the ﬁnal speed of the electron will be much greater. Illustrative Example 16.2.5 a. An electron and a proton both gain kinetic energy of 1 eV. What is the ratio of the electron’s speed to the proton’s speed? Answer : As discussed above, though both particles gain the same kinetic energy, their speeds will not be the same, since they have different masses. The mass of the proton is nearly 2000 times as great as the electron’s so: = 1 2 mev2 e 1 2 mpv2 p = 1! mev2 e = mpv2 p! v2 e v2 p = mp me = 2000me me = 2; 000! = p 2; 000 = 44:7! 45 KEe KEp ve vp The electron will move about 45 times faster than the proton. b. What is the speed of a proton which has a kinetic energy of 37 MeV? The mass of a proton is 1:67 1027 kg. Answer : Because the electron-volts are a very small unit, they are typically expressed in KeV (1000 electron-volts) and MeV (one million electron-volts). The electron-volt is a convenient unit of measure but it is not an SI unit. In order to ﬁnd the velocity of a particle if its energy is given in units of eV, we must convert back into Joules. 37 MeV = (37 106)(1
:60 1019 J) = 5:92 1014 J 1 2 1 2 (1:67 1027 kg)v2 = 5:92 1014 J! mpv2 = 5:92 1014 J! v = 8:4 106 m s References 1. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 2. User:Asim18/Wikimedia Commons. http://commons.wikimedia.org/wiki/File:02_-_Single_Energizer_Ba ttery.jpg. CC-BY 3.0 3. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 4. CK-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 5. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 6. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 7. CK-12 Foundation - Raymond Chou.. CC-BY-NC-SA 3.0 35 Physics Unit 13: Circuits Patrick Marshall Jean Brainard, Ph.D. Ck12 Science James H Dann, Ph.D. Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) AUTHORS Patrick Marshall Jean Brainard, Ph.D. Ck12 Science James H Dann, Ph.D. CONTRIBUTORS Chris Addiego Antonio De Jesus López www.ck12.org To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-proﬁt organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2014 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK
-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: April 6, 2014 iii Contents www.ck12.org 1 4 7 10 12 15 19 22 25 28 32 35 Contents 1 Electric Circuits 2 Electric Current 3 Electric Resistance 4 Ohm’s Law 5 Resistance and Ohm’s Law 6 Energy Transfer in Electric Circuits 7 Ammeters and Voltmeters 8 Series Circuits 9 Resistors in Series 10 Parallel Circuits 11 Resistors in Parallel 12 Combined Series-Parallel Circuits iv www.ck12.org Concept 1. Electric Circuits CONCEPT 1 Electric Circuits • Deﬁne electric circuit. • Describe the parts of an electric circuit. • Show how to represent a simple electric circuit with a circuit diagram. Jose made this sketch of a battery and light bulb for science class. If this were a real set up, the light bulb wouldn’t work. The problem is the loose wire on the left. It must be connected to the positive terminal of the battery in order for the bulb to light up. Q: Why does the light bulb need to be connected to both battery terminals? A: Electric current can ﬂow through a wire only if it forms a closed loop. Charges must have an unbroken path to follow between the positively and negatively charged parts of the voltage source, in this case, the battery. Electric Circuit Basics A closed loop through which current can ﬂow is called an electric circuit. In homes in the U.S., most electric circuits have a voltage of 120 volts. The amount of current (amps) a
circuit carries depends on the number and power of electrical devices connected to the circuit. Home circuits generally have a safe upper limit of about 20 or 30 amps. 1 Parts of an Electric Circuit All electric circuits have at least two parts: a voltage source and a conductor. They may have other parts as well, such as light bulbs and switches, as in the simple circuit seen in the Figure 1.1. To see an animation of a circuit like this one, go to: http://www.rkm.com.au/animations/animation-electrical-circuit.html www.ck12.org FIGURE 1.1 • The voltage source of this simple circuit is a battery. In a home circuit, the source of voltage is an electric power plant, which may supply electric current to many homes and businesses in a community or even to many communities. • The conductor in most circuits consists of one or more wires. The conductor must form a closed loop from the source of voltage and back again. In the circuit above, the wires are connected to both terminals of the battery, so they form a closed loop. • Most circuits have devices such as light bulbs that convert electrical energy to other forms of energy. In the case of a light bulb, electrical energy is converted to light and thermal energy. • Many circuits have switches to control the ﬂow of current. When the switch is turned on, the circuit is closed and current can ﬂow through it. When the switch is turned off, the circuit is open and current cannot ﬂow through it. Circuit Diagrams When a contractor builds a new home, she uses a set of plans called blueprints that show her how to build the house. The blueprints include circuit diagrams. The diagrams show how the wiring and other electrical components are to be installed in order to supply current to appliances, lights, and other electric devices. You can see an example of a very simple circuit in the Figure 1.2. Different parts of the circuit are represented by standard circuit symbols. An ammeter measures the ﬂow of current through the circuit, and a voltmeter measures the voltage. A resistor is any device that converts some of the electricity to other forms of energy. For example, a resistor might be a light bulb or doorbell. The circuit diagram on the right represents the circuit drawing on the left. Below are some of the standard symbols used in circuit diagrams. 2 www.ck12.org Concept 1. Electric Circuits FIG
URE 1.2 Q: Only one of the circuit symbols above must be included in every circuit. Which symbol is it? A: The battery symbol (or a symbol for some other voltage source) must be included in every circuit. Without a source of voltage, there is no electric current. Summary • An electric circuit is a closed loop through which current can ﬂow. • All electric circuits must have a voltage source, such as a battery, and a conductor, which is usually wire. They may have one or more electric devices as well. • An electric circuit can be represented by a circuit diagram, which uses standard symbols to represent the parts of the circuit. Vocabulary • electric circuit : Closed loop through which current can ﬂow. Practice Take the electric circuit quiz at the following URL. Be sure to have your answers corrected. Try the quiz again if any of your answers are incorrect. http://www.myschoolhouse.com/courses/O/1/68.asp Review 1. What is an electric circuit? 2. Which two parts must all electric circuits contain? 3. Sketch a simple circuit that includes a battery, switch, and light bulb. Then make a circuit diagram to represent your circuit, using standard circuit symbols. References 1. Christopher Auyeung.. CC BY-NC 3.0 2. Christopher Auyeung.. CC BY-NC 3.0 3 www.ck12.org Electric Current CONCEPT 2 Objectives The student will: • Understand how electric current is deﬁned • Solve problems involving electric current Vocabulary • electric current: A ﬂow of charges under the inﬂuence of an electric ﬁeld, such as between the terminals of a battery. The rate I = DQ Dt at which charges ﬂow within a conducting wire past any point in the wire. Introduction The term electrical current is familiar to most people. Many electrical devices have electrical speciﬁcations printed on them. Figure below shows a typical AC adapter (“plug”) with its “specs.” Can you guess what the terms 5 VDC and 500 mA printed on the adapter mean? FIGURE 2.1 An electrical plug. Electric Current An electric current is a ﬂow of charges under the inﬂuence of an electric ﬁeld. A ﬂow of charges can be established, for instance
, between the terminals of a battery, as in Figure below. The rate I = DQ Dt at which charges ﬂow within 4 www.ck12.org Concept 2. Electric Current a conducting wire past any point in the wire is deﬁned as the electric current. The unit of current is coulombs Ampere, 1775-1836), Figure below. second which is called the ampere or amp (named for the French physicist Andre’-Marie The symbol A is used to represent the ampere. A rate of one coulomb per second is equivalent to one ampere: 1C 1s = 1A FIGURE 2.2 Figure above shows a ﬂow of electrons (e) from the positive terminal of a battery through a lightbulb to the negative terminal of a battery. FIGURE 2.3 Andre’-Marie Ampere One ampere is a very large current. The current of 1 A can easily kill a person. In fact, about 0.20 A can kill rather easily. Even relatively small voltage can produce these currents, which is why care must always be taken when dealing with all electrical appliances and any electrical device that is plugged into a wall outlet. A typical 12-V car battery can also be dangerous. Under the right circumstances, it does not take a huge voltage to cause deadly currents. It is common to express current in milliamperes 1 mA = 103 A, or microamperes 1 ¯A = 106 A. http://demonstrations.wolfram.com/ElectricCurrent/ 5 Illustrative Example 17.4.1 A total of 7:9 1012 electrons move past a point in a conducting wire every 1.45 s. What is the average current in the wire? Answer : www.ck12.org The total charge moving past the point is the product of the electric charge of an electron and the number of electrons moving past the point. The total charge is:! Q = 1:6 1019 (7:9 1012 electrons) = 12:6 107! 13 107 C The current is I = DQ 1:45 s = 8:69 107 A! 0:87 µA. Dt = 12:6107 C electron References 1. Ray Dehler (Flickr: raybdbomb). http://www.ﬂickr.com/photos/raybdbomb/2200741209/ 2. CK
-12 Foundation - Ira Nirenberg.. CC-BY-NC-SA 3.0 3.. http://commons.wikimedia.org/wiki/File:Andre-marie-ampere2.jpg. public domain. CC-BY 2.0 6 www.ck12.org Concept 3. Electric Resistance CONCEPT 3 Electric Resistance • Deﬁne resistance and identify the SI unit for resistance. • List factors that affect resistance. • Explain why resistance can be a help or a hindrance. These athletes are playing rugby, a game that is similar to American football. The players in red and blue are trying to stop the player in orange and black from running across the ﬁeld with the ball. They are resisting his forward motion. This example of resistance in rugby is a little like resistance in physics. What Is Resistance? In physics, resistance is opposition to the ﬂow of electric charges in an electric current as it travels through matter. The SI unit for resistance is the ohm. Resistance occurs because moving electrons in current bump into atoms of matter. Resistance reduces the amount of electrical energy that is transferred through matter. That’s because some of the electrical energy is absorbed by the atoms and changed to other forms of energy, such as heat. Q: In the rugby analogy to resistance in physics, what do the players on each team represent? A: Factors that Affect Resistance How much resistance a material has depends on several factors: the type of material, its width, its length, and its temperature. • All materials have some resistance, but certain materials resist the ﬂow of electric current more or less than other materials do. Materials such as plastics have high resistance to electric current. They are called electric insulators. Materials such as metals have low resistance to electric current. They are called electric conductors. 7 www.ck12.org • A wide wire has less resistance than a narrow wire of the same material. Electricity ﬂowing through a wire is like water ﬂowing through a hose. More water can ﬂow through a wide hose than a narrow hose. In a similar way, more current can ﬂow through a wide wire than a narrow wire. • A longer wire has more resistance than a shorter wire. Current must travel farther through a longer wire, so there are more chances for it to collide with particles of matter. • A cooler wire has less resistance than a warmer wire. Cooler
particles have less kinetic energy, so they move more slowly. Therefore, they are less likely to collide with moving electrons in current. Materials called superconductors have virtually no resistance when they are cooled to extremely low temperatures. Is Resistance Good or Bad? Resistance can be helpful or just a drain on electrical energy. If the aim is to transmit electric current through a wire from one place to another, then resistance is a drawback. It reduces the amount of electrical energy that is transmitted because some of the current is absorbed by particles of matter. On the other hand, if the aim is to use electricity to produce heat or light, then resistance is useful. When particles of matter absorb electrical energy, they change it to heat or light. For example, when electric current ﬂows through the tungsten wire inside an incandescent light bulb like the one in the Figure 3.1, the tungsten resists the ﬂow of electric charge. It absorbs electrical energy and converts some of it to light and heat. FIGURE 3.1 What’s wrong with this picture? (Hint: How does current get to the light bulb?) Q: The tungsten wire inside a light bulb is extremely thin. How does this help it do its job? A: Summary • In physics, resistance is opposition to the ﬂow of electric charges that occurs as electric current travels through matter. The SI unit for resistance is the ohm. • All materials have resistance. How much resistance a material has depends on the type of material, its width, its length, and its temperature. • Resistance is a hindrance when a material is being used to transmit electric current. Resistance is helpful when a material is being used to produce heat or light. 8 www.ck12.org Vocabulary Concept 3. Electric Resistance • resistance : Opposition to the ﬂow of electric charges that occurs when electric current travels through matter. Review 1. What is resistance? Name the SI unit for resistance. 2. Explain what causes resistance. 3. Describe properties of a metal wire that would minimize its resistance to electric current. 4. Extend the rugby analogy to explain why a longer wire has greater resistance to electric current. 5. Copper wires have about one-third the resistance of tungsten wires. Why would copper be less suitable than tungsten as a ﬁlament in an incandescent light bulb? References 1. lenetstan.. Used under license from Shutterstock.com 9 CONCEPT
4 • Explain Ohm’s law. • Use Ohm’s law to calculate current from voltage and resistance. www.ck12.org Ohm’s Law Look at the water spraying out of this garden hose. You have to be careful using water around power tools and electric outlets because water can conduct an electric current. But in some ways, water ﬂowing through a hose is like electric current ﬂowing through a wire. Introducing Ohm’s Law For electric current to ﬂow through a wire, there must be a source of voltage. Voltage is a difference in electric potential energy. As you might have guessed, greater voltage results in more current. As electric current ﬂows through matter, particles of matter resist the moving charges. This is called resistance, and greater resistance results in less current. These relationships between electric current, voltage, and resistance were ﬁrst demonstrated in the early 1800s by a German scientist named Georg Ohm, so they are referred to as Ohm’s law. Ohm’s law can be represented by the following equation. Current(amps) = Voltage(volts) Resistance(ohms) Understanding Ohm’s Law Ohm’s law may be easier to understand with an analogy. Current ﬂowing through a wire is like water ﬂowing through a hose. Increasing voltage with a higher-volt battery increases the current. This is like opening the tap wider so more water ﬂows through the hose. Increasing resistance reduces the current. This is like stepping on the hose so less water can ﬂow through it. If you still aren’t sure about the relationships among current, voltage, and resistance, watch the video at this URL: http://www.youtube.com/watch?v=KvVTh3ak5dQ 10 www.ck12.org Concept 4. Ohm’s Law Using Ohm’s Law to Calculate Current You can use the equation for current (above) to calculate the amount of current ﬂowing through a circuit when the voltage and resistance are known. Consider an electric wire that is connected to a 12-volt battery. If the wire has a resistance of 2 ohms, how much current is ﬂowing through the wire? Current = 12 volts 2 ohms = 6 amps Q: If a 120-volt voltage source is connected to a wire with
10 ohms of resistance, how much current is ﬂowing through the wire? A: Substitute these values into the equation for current: Current = 120 volts 20 ohms = 12 amps Summary • According to Ohm’s law, greater voltage results in more current and greater resistance results in less current. • Ohm’s law can be represented by the equation:’ Current(amps) = Voltage(volts) Resistance(ohms) • This equation can be used to calculate current when voltage and resistance are known. Vocabulary • Ohm’s law : Law stating that current increases as voltage increases or resistance decreases. Practice Review Ohm’s law and how to calculate current at the following URL. Then try to solve the two problems at the bottom of the Web page. Be sure to check your answers against the correct solutions. http://www.grc.nasa.gov /WWW/k-12/Sample_Projects/Ohms_Law/ohmslaw.html Review 1. State Ohm’s law. 2. An electric appliance is connected by wires to a 240-volt source of voltage. If the combined resistance of the appliance and wires is 12 ohms, how much current is ﬂowing through the circuit? 11 www.ck12.org CONCEPT 5 Resistance and Ohm’s Law • Deﬁne resistance. • Understand the unit for resistance: ohms. • Use Ohm’s Law to solve problems involving current, potential difference, and resistance. The bands of color on a resistor are a code that indicates the magnitude of the resistance of the resistor. There are four color bands identiﬁed by letter: A, B, C, and D, with a gap between the C and D bands so that you know which end is A. This particular resistor has a red A band, blue B band, green C band, and gold D band, but the bands can be different colors on different resistors. Based on the colors of the bands, it is possible to identify the type of resistor. the A and B bands represent signiﬁcant digits; red is 2 and blue is 6. The C band indicates the multiplier, and green indicates 10 5. These three together indicate that this particular resistor is a 26,000 Ohm resistor. Finally, the D band indicates the tolerance, in this case 5%, as shown by the gold band
. These terms will be explained over the course of this lesson. Resistance and Ohm’s Law When a potential difference is placed across a metal wire, a large current will ﬂow through the wire. If the same potential difference is placed across a glass rod, almost no current will ﬂow. The property that determines how much current will ﬂow is called the resistance. Resistance is measured by ﬁnding the ratio of potential difference, V, to current ﬂow, I. 12 R = V I www.ck12.org Concept 5. Resistance and Ohm’s Law When given in the form V = IR, this formula is known as Ohm’s Law, after the man that discovered the relationship. The units of resistance can be determined using the units of the other terms in the equation, namely that the potential difference is in volts (J/C) and current in amperes (C/s): R = volts amperes = joules/coulomb coulombs/second = joules seconds coulombs2 = ohms The units for resistance have been given the name ohms and the abbreviation is the Greek letter omega, W. 1.00 W is the resistance that will allow 1.00 ampere of current to ﬂow through the resistor when the potential difference is 1.00 volt. Most conductors have a constant resistance regardless of the potential difference; these are said to obey Ohm’s Law. There are two ways to control the current in a circuit. Since the current is directly proportional to the potential difference and inversely proportional to the resistance, you can increase the current in a circuit by increasing the potential or by decreasing the resistance. Example Problem: A 50.0 V battery maintains current through a 20.0 W resistor. What is the current through the resistor? Solution: I = V R = 50:0 V 20:0 W = 2:50 amps Summary • Resistance is the property that determines the amount of current ﬂow through a particular material. • V = IR is known as Ohm’s Law. • The unit for resistance is the ohm, and it has the abbreviation W. Practice The following video covers Ohm’s Law. Use this resource to answer the questions that follow. http://www.youtube.com/watch?v=uLU4LtG0_hc MEDIA Click image
to the left for more content. 1. What happens to current ﬂow when voltage is increased? 2. What happens to current ﬂow when resistance is increased? This website contains instruction and guided practice for Ohm’s Law. http://www.wisc-online.com/Objects/ViewObject.aspx?ID=DCE11904 Review 1. If the potential stays the same and the resistance decreases, what happens to the current? 13 www.ck12.org (a) increase (b) decrease (c) stay the same 2. If the resistance stays the same and the potential increases, what happens to the current? (a) increase (b) decrease (c) stay the same 3. How much current can be pushed through a 30.0 W resistor by a 12.0 V battery? 4. What voltage is required to push 4.00 A of current through a 32.0 W resistor? 5. If a 6.00 volt battery will produce 0.300 A of current in a circuit, what is the resistance in the circuit? • resistance: Opposition of a circuit to the ﬂow of electric current. • Ohm’s Law: V = IR. • Ohms: A resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere. References 1. Image copyright Robert Spriggs, 2013. http://www.shutterstock.com. Used under license from Shutter- stock.com 14 www.ck12.org Concept 6. Energy Transfer in Electric Circuits CONCEPT 6 Energy Transfer in Electric Circuits • Explain how devices convert electrical energy to other forms. • Use P = I2R and E = I2Rt for calculations involves energy transfer in electrical circuits. • Describe the reason for the use of high voltage lines for transmitting electrical energy. • Deﬁne the kilowatt-hour. Part of the electrical grid, an electrical transmission sub-station receives extremely high current levels, then passes the electrical energy on to as many as 200,000 homes. Approximately 5000 megawatt-hours of energy passes through this particular substation each year. Energy Transfer in Electric Circuits Electric power is the energy per unit time converted by an electric circuit into another form of energy. We already know that power through a circuit is equal to the voltage multiplied by the current in a circuit: P = V
I. It is possible to determine the power dissipated in a single resistor if we combine this expression with Ohm’s Law, V = IR. This becomes particularly useful in circuits with more than one resistor, to determine the power dissipated in each one. Combining these two equations, we get an expression for electric power that involves only the current and resistance in a circuit. P = I2R The power dissipated in a resistor is proportional to the square of the current that passes through it and to its resistance. 15 www.ck12.org Electrical energy itself can be expressed as the electrical power multiplied by time: E = Pt We can incorporate this equation to obtain an equation for electrical energy based on current, resistance, and time. The electrical energy across a resistor is determined to be the current squared multiplied by the resistance and the time. E = I2Rt This equation holds true in ideal situations. However, devices used to convert electrical energy into other forms of energy are never 100% efﬁcient. An electric motor is used to convert electrical energy into kinetic energy, but some of the electrical energy in this process is lost to thermal energy. When a lamp converts electrical energy into light energy, some electrical energy is lost to thermal energy. Example Problem: A heater has a resistance of 25.0 W and operates on 120.0 V. a. How much current is supplied to the resistance? b. How many joules of energy is provided by the heater in 10.0 s? Solution: a. I = V b. E = I2Rt = (4:8 A)2(25:0 W)(10:0 s) = 5760 joules 25:0 W = 4:8 A R = 120:0 V Think again about the power grid. When electricity is transmitted over long distances, some amount of energy is lost in overcoming the resistance in the transmission lines. We know the equation for the power dissipated is given by P = I2R. The energy loss can be minimized by choosing the material with the least resistance for power lines, but changing the current also has signiﬁcant effects. Consider a reduction of the current by a power of ten: How much power is dissipated when a current of 10.0 A passes through a power line whose resistance is 1.00 W? P = I2R = (10:0 A)2(1:00 W) = 100: Watts How much power
is dissipated when a current of 1.00 A passes through a power line whose resistance is 1.00 W? P = I2R = (1:00 A)2(1:00 W) = 1:00 Watts The power loss is reduced tremendously by reducing the magnitude of the current through the resistance. Power companies must transmit the same amount of energy over the power lines but keep the power loss minimal. They do this by reducing the current. From the equation P = V I, we know that the voltage must be increased to keep the same power level. The Kilowatt-Hour Even though the companies that supply electrical energy are often called “power” companies, they are actually selling energy. Your electricity bill is based on energy, not power. The amount of energy provided by electric current can be calculated by multiplying the watts (J/s) by seconds to yield joules. The joule, however, is a very small unit of energy and using the joule to state the amount of energy used by a household would require a very large number. For that reason, electric companies measure their energy sales in a large number of joules called a kilowatt hour (kWh). A kilowatt hour is exactly as it sounds - the number of kilowatts (1,000 W) transferred per hour. 1:00 kilowatt hour = (1000 J=s)(3600 s) = 3:6 106 J Example Problem: A color television uses about 2.0 A when operated on 120 V. a. How much power does the set use? b. If the TV is operated for 8.00 hours per day, how much energy in kWh does it use per day? c. At $0.15 per kWh, what does it cost to run the TV for 30 days? 16 www.ck12.org Solution: Concept 6. Energy Transfer in Electric Circuits a. P = V I = (120 V )(2:0 A) = 240 W b. E = (240 J=s)(8 h)(3600 s=h) c. Cost = (1:92 kW h)(30)($0:15) = $8:64 = 1:92 kW h 3:6106 J=kW h Summary • Electric power is the energy per unit time converted by an electric circuit into another form of energy. • The formula for electric power is P = I2R. • The electric energy transferred
to a resistor in a time period is equal to the electric power multiplied by time, E = Pt, and can also be calculated using E = I2Rt. • Electric companies measure their energy sales in a large number of joules called a kilowatt hour (kWh) which is equivalent to 3:6 106 J. Practice The following video is on electrical energy and power. Use this resource to answer the questions that follow. http://youtu.be/NWcYBvHOiWw MEDIA Click image to the left for more content. 1. What is this video about? 2. What is the deﬁnition of electrical power? 3. What happens to the electrical energy that is not converted into work? Instruction and practice problems related to the energy delivered by an electric circuit: http://www.physicsclassroom.com/Class/circuits/u9l2d.cfm Review 1. A 2-way light bulb for a 110. V lamp has ﬁlament that uses power at a rate of 50.0 W and another ﬁlament that uses power at a rate of 100. W. Find the resistance of these two ﬁlaments. 2. Find the power dissipation of a 1.5 A lamp operating on a 12 V battery. 3. A high voltage (4:0 105 V ) power transmission line delivers electrical energy from a generating station to a substation at a rate of 1:5 109 W. What is the current in the lines? 4. A toaster oven indicates that it operates at 1500 W on a 110 V circuit. What is the resistance of the oven? • electrical energy: Energy is the ability to do work, so electrical energy is the work done by an electrical circuit. • kilowatt hour: An amount of energy equal to 3:6 106 Joules. 17 References 1. User:Cutajarc/Wikipedia. http://en.wikipedia.org/wiki/File:Melbourne_Terminal_Station.JPG. Public Do- main www.ck12.org 18 www.ck12.org Concept 7. Ammeters and Voltmeters CONCEPT 7 Ammeters and Voltmeters • Describe the primary difference between the construction of ammeters and voltmeters. • Describe whether ammeters should be placed in circuits in series or parallel and explain why. • Describe whether voltmeters should be placed in
circuits in series or parallel and explain why. This photo is of the interior of the control room for a nuclear power plant. Many of the meters are reading information about the water temperature and the nuclear reaction that is occurring, but the majority of the meters are reading data about the electric energy being generated. Ammeters and Voltmeters Ammeters and voltmeters are cleverly designed for the way they are used. Ammeters measure the current of a circuit, and voltmeters measure the voltage drop across a resistor. It is important in the design and use of these meters that they don’t change the circuit in such a way as to inﬂuence the readings. While both types of meters are technically resistors, they are speciﬁcally designed to make their readings without changing the circuit itself. 19 www.ck12.org Ammeter An ammeter measures the current traveling through the circuit. They are designed to be connected to the circuit in series, and have an extremely low resistance. If an ammeter were connected in parallel, all of the current would go through the ammeter and very little through any other resistor. As such, it is necessary for the ammeter to be connected in series with the resistors. This allows the ammeter to accurately measure the current ﬂow without causing any disruptions. In the circuit sketched above, the ammeter is m 2. Voltmeter In contrast, a voltmeter is designed to be connected to a circuit in parallel, and has a very high resistance. A voltmeter measures the voltage drop across a resistor, and does not need to have the current travel through it to do so. When a voltmeter is placed in parallel with a resistor, all the current continues to travel through the resistor, avoiding the very high resistance of the voltmeter. However, we know that the voltage drop across all resistors in parallel is the same, so connecting a voltmeter in parallel allows it to accurately measure the voltage drop. In the sketch, the voltmeter is m 1. Summary • Ammeters measure the current through a resistor. • Ammeters have low resistances and are placed in the circuit in series. • Voltmeters measure the voltage drop across a resistor. • Voltmeters have high resistances and are placed in the circuit in parallel. Practice MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=liwan6-w-Pw
In this video, a circuit is established with a power supply, which also has an attached voltmeter, and a lamp (resistor). After the circuit is established, a voltmeter and an ammeter are alternately placed in the circuit. Follow up questions: 1. What happens when the ammeter is connected in parallel with the lamp? 2. Why do the problems occur when the narrator in the video places the ammeter in parallel with the lamp? 20 www.ck12.org Review Concept 7. Ammeters and Voltmeters 1. In the sketch at above, there are four positions available for the placement of meters. Which position(s) would be appropriate for placement of an ammeter? a. 1 b. 3 c. 4 d. All of them. e. None of them. 2. Which position(s) would be appropriate for placement of a voltmeter? a. 1 b. 2 c. 3 d. All of them. e. None of them. 3. Which position could hold an ammeter that would read the total current through the circuit? a. 1 b. 2 c. 3 d. 4 e. None of them. 4. Which position could hold a voltmeter that would read the total voltage drop through the circuit? a. 1 b. 2 c. 3 or 4 d. All of them. e. None of them. • ammeter: A measuring instrument used to measure the electric current in a circuit. • voltmeter: An instrument used for measuring electrical potential difference between two points in an electric circuit. References 1. Image copyright rtem, 2013. http://www.shutterstock.com. Used under license from Shutterstock.com 2. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 3. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 21 CONCEPT 8 www.ck12.org Series Circuits • Describe a series circuit. • Understand current as it passes through a series circuit. • Understand voltage drops in a series circuit. • Understand resistance in a series circuit with multiple resistors. • Calculate current, voltage drops, and equivalent resistances for devices connected in a series circuit. Resistors, including electrical appliances, have a particular current at which they operate most effectively and safely, and excessive current can cause irreparable damage. Therefore, it is important to limit the amount of current that may pass through a particular electrical circuit.
There are a number of safety devices used in electrical circuits to limit the current; fuses, circuit breakers, and surge suppressors. When fuses, such as those shown above, are placed in an electrical circuit, all the current must pass through the wire in the fuse. Series Circuits Electrical circuits are often modeled by using water in a river. The potential energy of the water is the highest at the source of the river and decreases as the water ﬂows down the river toward the end. When the water reaches the ocean, its potential energy has become zero. The circuit shown above has a similar situation. The current in this circuit is drawn in the direction of the electron ﬂow. It starts at the battery on the left, where electrons leave the 22 www.ck12.org Concept 8. Series Circuits negative terminal and travel around the circuit. Since all of the current travels across each resistor, these resistors are said to be in series. A series circuit is one in which all of the current must pass through every resistor in the circuit. Returning to the water analogy, there is only one riverbed from the top of the mountain to the ocean. Consider the series circuit sketched above. This circuit has a voltage drop for the entire circuit of 120 V and has three resistors connected in series. The current in this circuit is drawn in terms of electron ﬂow. The electrons leave the potential difference source at the negative terminal and ﬂow through the three resistors, starting with R 3. Though they have a small amount of resistance, the resistance of the connecting wires is so small in relation to the resistors that we ignore it. Therefore, we say that there is no voltage drop when the current passes through the connecting wires. The voltage drops occur when the current passes through each of the resistors and the total voltage drop for the entire circuit is equal to the sum of the voltage drops through the three resistors. VT = V1 +V2 +V3 The current through each of the resistors must be exactly the same because the current in a series circuit is the same everywhere. The current is moving in the entire circuit at the same time. IT = I1 = I2 = I3 Since the current passes through each resistor, the total resistance in the circuit is equal to the sum of the resistors. In the circuit above, the total resistance is: RT = R1 + R2 + R3 = 30 W + 15 W + 15 W =
60 W Therefore, the total current and the current through each resistor is I = V R = 120 V 60 W = 2:0 A: The individual voltage drops can be calculated using the current through each resistor and each resistor’s individual resistance. V1 = I1R1 = (2:0 A)(30 W) = 60 V V2 = I2R2 = (2:0 A)(15 W) = 30 V V3 = I3R3 = (2:0 A)(15 W) = 30 V Example Problem: Four 15 W resistors are connected in series with a 45 V battery. What is the current in the circuit? Solution: RT = 15 W + 15 W + 15 W + 15 W = 60 W I = V R = 45 V 60 W = 0:75 A 23 www.ck12.org Summary • A series circuit is one in which all of the current must pass through every resistor in the circuit. • VT = V1 +V2 +V3 • IT = I1 = I2 = I3 • RT = R1 + R2 + R3 Practice The following video is on series circuits. Use this resource to answer the questions that follow. MEDIA Click image to the left for more content. http://www.youtube.com/watch?v=qO391knBRjE 1. How do the voltage drops across the two light bulbs in the video related to the total voltage drop for the entire circuit? 2. In the video, what was the assumed voltage drop for the connecting wires and the switch? 3. What was the current through the second light bulb as compared to the current through the ﬁrst light bulb. Review 1. There are three 20.0 W resistors connected in series across a 120 V generator. (a) What is the total resistance of the circuit? (b) What is the current in the circuit? (c) What is the voltage drop across one of the resistors? 2. A 5.00W, a 10.0W, and a 15.0W resistor are connected in a series across a 90.0 V battery. (a) What is the equivalent resistance of the circuit? (b) What is the current in the circuit? (c) What is the voltage drop across the 5.00W resistor? 3. A 5.00W and a 10.0W resistor are connected in series across an unknown voltage. The total current in
the circuit is 3.00 A. (a) What is the equivalent resistance of the circuit? (b) What is the current through the 5.00W resistor? (c) What is the total voltage drop for the entire circuit? • series circuit: One in which all of the current must pass through every resistor in the circuit. References 1. Image copyright sevenke, 2013. http://www.shutterstock.com. Used under license from Shutterstock.com 2. CK-12 Foundation - Samantha Bacic.. CC-BY-NC-SA 3.0 24 www.ck12.org Concept 9. Resistors in Series CONCEPT 9 Resistors in Series Students will learn how to analyze and solve problems involving circuits with resistors in series. Students will learn how to analyze and solve problems involving circuits with resistors in series. Key Equations Guidance Rtotal = R1 + R2 + R3 + : : : Resistors in Series: All resistors are connected end to end. There is only one river, so they all receive the same current. But since there is a voltage drop across each resistor, they may all have different voltages across them. The more resistors in series the more rocks in the river, so the less current that ﬂows. Example 1 A circuit is wired up with two resistors in series. Both resistors are in the same ’river’, so both have the same current ﬂowing through them. Neither resistor has a direct connection to the power supply so neither has 20V across it. But the combined voltages across the individual resistors add up to 20V. Question: What is the total resistance of the circuit? Answer: The total resistance is Rtotal = R1 + R2 = 90 W + 10 W = 100 W Question: What is the total current coming out of the power supply? Answer: Use Ohm’s Law (V = IR) but solve for current (I = V =R). Itotal = Vtotal Rtotal = 20V 100 W = 0:20 A Question: How much power does the power supply dissipate? Answer: P = IV, so the total power equals the total voltage multiplied by the total current. Thus, Ptotal = ItotalVtotal = (0:20 A)(20V ) = 4:0 W. So the Power Supply is outputting 4W (i.e. 4 Joules of energy per second). Question:
How much power does each resistor dissipate? Answer: Each resistor has different voltage across it, but the same current. So, using Ohm’s law, convert the power formula into a form that does not depend on voltage. 25 www.ck12.org P = IV = I(IR) = I2R: P90 W = I2 P10 W = I2 90 WR90 W = (0:2 A)2(90 W) = 3:6W 10 WR10 W = (0:2 A)2(10 W) = 0:4W Note: If you add up the power dissipated by each resistor, it equals the total power outputted, as it should–Energy is always conserved. Question: How much voltage is there across each resistor? Answer: In order to calculate voltage across a resistor, use Ohm’s law. V90 W = I90 WR90 W = (0:2 A)(90 W) = 18V V10 W = I10 WR10 W = (0:2 A)(10 W) = 2V Note: If you add up the voltages across the individual resistors you will obtain the total voltage of the circuit, as you should. Further note that with the voltages we can use the original form of the Power equation (P = IV ), and we should get the same results as above. P90 W = I90 WV90 W = (18V )(0:2 A) = 3:6W P10 W = I10 WV10 W = (2:0V )(0:2 A) = 0:4W MEDIA Click image to the left for more content. Watch this Explanation 26 www.ck12.org Simulation Concept 9. Resistors in Series • http://simulations.ck12.org/Resistor/ Time for Practice 1. Regarding the circuit below. a. If the ammeter reads 2 A, what is the voltage? b. How many watts is the power supply supplying? c. How many watts are dissipated in each resistor? 2. Five resistors are wired in series. Their values are 10W, 56W, 82W, 120W and 180W. a. If these resistors are connected to a 6 V battery, what is the current ﬂowing out of the battery? b. If these resistors are connected to a 120 V power suppluy, what is the current
ﬂowing out of the battery? c. In order to increase current in your circuit, which two resistors would you remove? 3. Given the resistors above and a 12 V battery, how could you make a circuit that draws 0.0594 A? Answers to Selected Problems 1. a. 224 V b. 448 W c. 400 W by 100 W and 48 W by 12 W 2. a. 0.013 A b. 0.27 A c. 120W and 180W 3. need about 202W of total resistance. So if you wire up the 120W and the 82W in series, you’ll have it. 27 CONCEPT 10 www.ck12.org Parallel Circuits • Describe a parallel circuit. • Understand current as it passes through a parallel circuit. • Understand voltage drops in a parallel circuit. • Understand resistance in a parallel circuit with multiple resistors. • Calculate voltage drops, currents, and equivalent resistances when devices are connected in a parallel circuit. Electrical circuits are everywhere: skyscrapers, jumbo jets, arcade games, lights, heating, and security... very few complex things work without electrical circuits. Since the late 1970s, electrical circuits have primarily looked like this. The circuits are formed by a thing layer of conducting material deposited on the surface of an insulating board. Individual components are soldered to the interconnecting circuits. Circuit boards are vastly more complicated than the series circuits previously discussed, but operate on many similiar principles. Parallel Circuits Parallel circuits are circuits in which the charges leaving the potential source have different paths they can follow to get back to the source. In the sketch below, the current leaves the battery, passes through the orange switch, and then has three different paths available to complete the circuit. Each individual electron in this circuit passes through only one of the light bulbs. After the current passes through the switch, it divides into three pieces and each piece passes through one of the bulbs. The three pieces of current rejoin after the light bulbs and continue in the circuit to the potential source. 28 www.ck12.org Concept 10. Parallel Circuits In the design of this parallel circuit, each resistor (light bulb) is connected across the battery as if the other two resistors were not present. Remember that the current going through each resistor goes through only the one resistor. Therefore, the voltage drop across each resistor must be equal to the total voltage drop though the circuit. VT = V1 = V2

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