11
heinemann
Physics
n o
t t s
u n i
1
2 &
i t i
d d
e
n
2
Carmel Fry Rob Chapman Keith Burrows Doug Bail Geoff Millar Henry Gersh
Contents INTRODUCTION
V
WAVE-LIKE PROPERTIES OF LIGHT
C H A P T E R 1 T h e N a t u r e o f Wa v e s
2
1.1 Introducing waves 1.2 Representing wave features 1.3 Waves and wave interactions
3 10 18
Chapter Review
25
CHAPTER 2
2. 1 2. 2 2. 3 2. 4
Two Models for Light
27
Modelling simple light properties Refraction of light Applications of refraction L ight described as an electromagnetic wave Li
28 36 48 56
2.5 Di Dispersion and polarisation of light waves Chapter Review CHAPTER 3
Mirrors, Lenses and Optical Systems
3.1 Geometrical optics and plane mirrors 3.2 Applications of curved mirrors: concave mirrors 3.3 Convex mirrors
65 70 71
72 76 84
3.4 Refraction and lenses 3.5 Concave lenses 3.6 Optical systems Chapter Review
91 97 10 2 112
Area of Study Review—Wave-like Review—Wave-like Properties of Light
113
NUCLEAR AND RADIOACTIVITY PHYSICS
CHAPTER 4
Nuclear and Radioactivity Physics
118
4.1 Atoms, isotopes and radioisotopes 4.2 Ra Radioactivity and how it is detected 4.3 Properties of alpha, beta and gamma radiation
11 9 124 13 1
4.4 Ha Half-life and activity of radioisotopes 4.5 Ra Radiation dose and its effect on humans Chapter Review Area of Study Review — Nuclear and Radioactivity Radioactivity Physics
1 36 14 2 1 47 149
MOVEMENT
CHAPTER 5
Aspects of Motion
152
5. 1 D De escribing motion in a straight line 5.2 Graphing motion: position, velocity and acceleration 5.3 Equations of motion
15 3 16 4 1 72
5.4 Vertical motion under gravity Chapter Review
17 6 1 81
Contents INTRODUCTION
V
WAVE-LIKE PROPERTIES OF LIGHT
C H A P T E R 1 T h e N a t u r e o f Wa v e s
2
1.1 Introducing waves 1.2 Representing wave features 1.3 Waves and wave interactions
3 10 18
Chapter Review
25
CHAPTER 2
2. 1 2. 2 2. 3 2. 4
Two Models for Light
27
Modelling simple light properties Refraction of light Applications of refraction L ight described as an electromagnetic wave Li
28 36 48 56
2.5 Di Dispersion and polarisation of light waves Chapter Review CHAPTER 3
Mirrors, Lenses and Optical Systems
3.1 Geometrical optics and plane mirrors 3.2 Applications of curved mirrors: concave mirrors 3.3 Convex mirrors
65 70 71
72 76 84
3.4 Refraction and lenses 3.5 Concave lenses 3.6 Optical systems Chapter Review
91 97 10 2 112
Area of Study Review—Wave-like Review—Wave-like Properties of Light
113
NUCLEAR AND RADIOACTIVITY PHYSICS
CHAPTER 4
Nuclear and Radioactivity Physics
118
4.1 Atoms, isotopes and radioisotopes 4.2 Ra Radioactivity and how it is detected 4.3 Properties of alpha, beta and gamma radiation
11 9 124 13 1
4.4 Ha Half-life and activity of radioisotopes 4.5 Ra Radiation dose and its effect on humans Chapter Review Area of Study Review — Nuclear and Radioactivity Radioactivity Physics
1 36 14 2 1 47 149
MOVEMENT
CHAPTER 5
Aspects of Motion
152
5. 1 D De escribing motion in a straight line 5.2 Graphing motion: position, velocity and acceleration 5.3 Equations of motion
15 3 16 4 1 72
5.4 Vertical motion under gravity Chapter Review
17 6 1 81
CHAPTER 6
Newton’s Laws of Motion
183
6.1 Force as a vector 6.2 Newton’s first law of motion
184 19 1
6.3 Newton’s second law of motion 6.4 Newton’s third law of motion Chapter Review
19 6 20 4 21 3
CHAPTER 7
Work, Energy, Power and Momentum
215
7.1 Work
21 6
7.2 7.3 7.4 7.5
22 2 231 23 9 2 48
Mechanical energy Energy transformation and power The relationship between momentum and force Conservation of momentum
Chapter Review Area of Study Review—Movement Review—Movement
252 253
ELECTRICITY
CHAPTER 8
Concepts in Electricity Co
258
8.1 Electric charge
259
8.2 8.3 8.4 8.5
2 66 2 72 2 80 28 8
Electric forces and fields Electric current, EMF and electrical potential Resistance, ohmic and non-ohmic conductors Electrical energy and power
Chapter Review CHAPTER 9
9.1 9.2 9.3 9.4
Electric Circuits
Simple electric circuits Circuit elements in parallel C ells, batteries and other sources of EMF Ce Household electricity
Chapter Review Area of Study Review—Electricity
29 5 297
2 98 304 3 09 31 7 32 2 324
APPENDIX A
328
APPENDIX B
330
SOLUTIONS
331
GLOSSARY
343
INDEX
348
Introduction Heinemann Physics 11 is the first book in the exciting series, Heinemann Physics, written specifically for the VCE syllabus at years 11 and 12.
First and foremost in the minds of the authors has been a desire to write a text that will support students’ learning in physics while making the subject interesting, enjoyable and meaningful. The book has been written using clear and concise language throughout, and all concepts have been fully explored first in general and then illustrated in context. Much care has been taken to use illustrative material that is fresh, varied and appealing to a wide range of students of both sexes. The book boasts many features that will help students and teachers find it easy to use. Each of the book’s nine chapters has been divided into a number of selfcontained sections. At the end of each section is a set homework-style questions that are designed to reinforce the main points. Further, more demanding questions are included at the end of the chapter. These could be used for assignment or tutorial work. A further set of exam-style questions is included to cover each Area of Study. Study. These could be used for revision. There are over 1000 questions in the text and all answers are supplied. Within each section, the concept c oncept development and worked examples occupy the main 2/3rd column. The remaining 1/3rd column has been set aside for some of the 600 photographs and diagrams, as well as small snippets of interesting ‘Physics File’ information. The longer pieces of high-interest and context material are contained in the full-page-width ‘Physics in Action’ sections. Both Physics in Action and Physics Files are clearly distinguishable f rom other material. HEINEMANN PHYSICS 11 SECOND EDITION
The second edition of Heinemann Physics 11 has been fully revised and upgraded to match the content and focus of Units 1 and 2 of the 2004 VCE Physics Study Design. Any section which contains background or extension material outside the syllabus is contained under a section heading in a non-shaded banner.
The new edition is presented as a student pack consisting of textbook and ePhysics 11 student CD. Successful features of the first edition have been retained while improvements in design and presentation will make the book even easier and more stimulating to use. Heinemann ePhysics 11 features a complete electronic copy of the textbook plus all Detailed Studies. Each Detailed Study is a selfcontained unit of work, structured to ensure efficient and effective coverage of the chosen topic. A major innovation is the inclusion of Interactive Tutorials Tutorials that model mo del and simulate key physics concepts. Cross references in the textbook to these and Practical Activities suggest when it is most appropriate to undertake these new activities.
Detailed Studies
See Heinemann ePhysics CD for an Interactive Tutorial
PRACTICAL ACTIVITY 9 Reflection in a plane mirror
Introduction
v
Heinemann Physics 11 second edition Support Material:
Heinemann Physics 11 Teacher’ eacher’ss Resource and a nd Assessment Disk makes planning, structuring and implementing the new syllabus easy. It contains PhysicsBank 11, an electronic database of questions and worked solutions, a wide range of innovative and accessible practical activities and Sample Assessment Tasks, Tasks, which teachers can use directly di rectly or modify. A detailed course outline and work program is also included.
vi
PHYSIC PHY SICS S 11
About the Authors DOUG BAIL
Is an experienced physics educator and writer with a particular interest in the development and integration of new technologies into science teaching. He has been Head of Science and Agriculture at Tintern Anglican Girls Grammar School and maintains a passion for making physics relevant, stimulating and accessible to all students. He led the development of the new Practical Activities which form part of the teacher support material. These activities were extensively trialed throughout Australia and include a range of activities from teacher demonstration to discovery-based investigation suiting a range of learning styles and needs. This includes many short activities for when time is limited! KEITH BURROWS
Has been teaching senior physics in Victorian schools for many years. He is a member of the Australian Institute of Physics Victorian Education Committee and was actively involved with the VCAA in the design of the new course. Keith was a VCAA representative involved in introducing the new VCE course to physics teachers in Victoria and running workshop sessions for teachers. He is particularly keen to portray ‘The Big Picture’ of physics to students. Keith would like to acknowledge Maurizio Toscano Toscano of the Melbourne University Astrophysics Group who has provided invaluable help and advice in the preparation of the Astronomy and Astrophysics detailed studies. ROB CHAPMAN
Has taught physics for over twenty years and has been keen to explore the possibilities presented by changing technologies over the years. He has been Science Coordinator at St Columba’ Columba’s College in Essendon where he was instrumental in introducing the use of datalogging technology to junior science and senior physics classes. Rob is currently teaching Senior Physics at PEGS (Penleigh and Essendon Grammar School). He has written a wide variety of curriculum support material including physics units for the CSFII. Rob has also produced physics trial examination papers and is the author of the acclaimed Physics 12—A student guide . CARMEL FRY
Has 14 years involvement in development of text, CD and on-line curriculum materials for VCE physics. She is Senior Teacher Teacher and co-coordinator co-coo rdinator of Senior Physics at Eltham College of Education where she is currently managing the integration of IT into physics education. Carmel is the author of numerous texts and multimedia resources. She was a VCAA representative involved in introducing the new VCE course to physics teachers in Victoria and running workshop sessions for teachers. Carmel has led the development of the Interactive Tutorials which are an exciting innovation on the student CD.
Newton’s laws of motion
vii
HENRY GERSH
Has taught physics and mathematics in a wide variety of courses at TAFE colleges and universities. He has led the development of the PhysicsBank utilising his vast experience in question creation and the writing of clear but detailed worked solutions. GEOFFREY MILLAR
Has had 26 years experience teaching science and senior physics in two states. He has been very involved in the development of good curriculum, methodology and practice in physics, and has a continuing interest in the subject. He was Director of Curriculum at Geelong College for seven years. REVIEW PANEL
Principal reviewer/consultant Dr Colin Gauld has taught mathematics and physics at secondary schools and science method to prospective science teachers at the University of New South Wales. Wales. At present he is a Visiting Fellow in the School of Education at the t he University of New South Wales. Wales. The teacher review panel consisted of experienced VCE physics teachers and physics educators. The authors and publisher would like to thank the following people for their input: Dr Barbara Moss, David Jagger Jagger,, Martin Mahy Mahy,, Dr Maurizio Toscano, Toscano, Peter Kolsch, Vincent Vignuoli, Lyndon Webb and Catie Morrison. Acknowledgments Acknowledgments The publishers wish to thank the following organisations who kindly gave permission to reproduce copyright material in this book:
AAP, pp. 159, 192 (left) ANT Photo Library, p. 312 Australian Picture Library, Library, pp. 66, 139, 160(all), 168 (right), 184 (a), 201, 257 Keith Burrows, pp. 280 Coo-ee Picture Library, pp. 45 (right), 184 (c) Malcolm Cross, pp. 23 (both right), 63 (both), 76 (right), 88, 213 (all), 249 Dale Mann, p. 268 (all) Mark Fergus, pp. 36 (both), 77, 97, 102, 184 (bottom), 282 (both) NASA, pp. 71, 192 (right) Newspix, p. 155 PASCO Scientific p. 161(right) Photodisc, pp. 1, 2, 4 (left), 8 (both), 15, 20, 27, 29, 31, 50, 133, 140, 158, 173, 184 (d), 196, 222 (top), 229, 232 (top, centre and bottom right), 300 Photolibrary.com, Photolibrary .com, pp. 3, 4 (right), (right), 22, 45 (left), 72, 109, 118, 122, 124 (both), 129 (bottom), 142, 183 Rainbow, p. 146 RMIT/Craig Mills, pp. 37 (both), 233 Sport:The Library, Library, pp. 151, 153 (both), 168 (left), (left), 184 (b), 185, 209, 225 (both), 236, 241 The Age, p. 169 The Picture Source/Terry Source/Terry Oakley, pp. 23 (left) Tom Taylor, Tmatio Tmation, n, p. 293 Victoria Police Traffic Traffic Office, Offic e, p. 174 Every effort has been made to trace and acknowledge copyright material. The author and publisher would welcome any information from people who believe they own copyright material in this book. viii
PHYSIC PHY SICS S 11
WAVE-LIKE PROPERTIES OF LIGHT
W
hat benefit could there be in knowing more about light? Well, everyone knows about the common benefits that come from our ability to manipulate the path of light. For hundreds of years we have been able to magnify small objects with microscopes and look into the distance with telescopes. Spectacles have helped people with eye defects to see their world in sharp focus. Today our ability to deal with the wave-like properties of light means that enormous reflecting telescopes collect weak light signals from space and optical fibres carry our communications across the world. In this area of study you will learn just how these devi ces work and develop an insight into the nature of light itself. Our knowledge of light in this technological age means that humans now produce more light
than at any other time in history. Since we rely largely on incandescent and fluorescent lighting, every bit of light energy produced means that considerable heat is simultaneously produced, and the implications of heating our world are well accepted. Fortunately quantum physics tells us that there is a better way to produce light. The electrons in an atom give out a flash of light whenever they jump from a high energy level to a low one. Small electronic devices called light emitting diodes (LEDs) use this effect, but for a long time they couldn’t be used to make light that was similar to sunlight. But the technology is approaching. LEDs in development may mean that besides being about 20 times more efficient than current light globes, they give off no heat and last a life time. You never need to burn your fingers changing a light bulb again!
CHAPTER 1
The nature of waves
W
ould you know what was coming if you were sitting on a picturesque beach in Hawaii and suddenly the coastal water in front of you seemed to retract before your eyes, leaving the tethered fishing boats sitting on beds of sand? You may have enough knowledge of coastal waves to realise that the water at the shore was being dragged back to help to build a giant tsunami way out to sea. Tsunami is the Japanese term for the phenomenon that used to be called a tidal wave. Since it’s nothing to do with the tide the name tidal wave is being dropped. Regardless of what you’d call it, you would be advised to get to high ground, and quickly! The ability to forewarn the affected coastal people of the occurrence of a tsunami anywhere in the world would undoubtedly save lives. Appropriately, it may well be our knowledge of a different category of waves—gravity waves —that someday allows us to do this. Gravity waves are not just any ordinary type of wave. Albert Einstein in his general theory of relativity predicted their existence early last century. General relativity treats the Universe as a four-dimensional surface called space–time. Gravitational waves are the curvature of space–time caused by the motion of matter. If a gravity wave arrived at Earth it would cyclically shrink and stretch the dimensions of everything around us, bu t by such miniscule amounts that even the strongest gravity waves are nearly impossible to detect. Einstein’s general theory of relativity was the same theory that successfully predicted the ‘bending’ of the path of light by the gravitational fields of massive objects. It was not until the 1970s that strong experimental evidence for the existence of gravitational waves in space was found, though they haven’t been detected here on Earth yet. Aside from showing us where the black holes, supernovae, etc. are located throughout the Universe, the detection of gravity waves should tell us all about the big bang and break down our limits regarding how far into space we can ‘see’. Along the way Australian gravity physicists have invented a device that can accurately monitor coastal ocean waves and provide warnings of potentially life-threatening swells. We too will focus our attention on the water as we begin our own study of the nature of waves.
BY THE END OF THIS CHAPTER
you will have covered material from the study of the wave-like properties of light including: • an explanation of the role of conceptual modelling used by scientists to explain phenomena • how waves involve the transfer of energy without the transfer of matter • the differences between transverse waves and longitudinal waves • how to represent waves • how to define waves by their amplitude, wavelength, period and frequency • the speed of travel of waves • the relationship between the velocity, frequency and wavelength of a wave.
1.1
Introducing waves
Why combine the study of waves and light? The first part of your course involves a study of the wave nature of light. As you embark on this study you will be walking in the footsteps of many famous physicists from the past, who were devoted to the quest of revealing the true nature of light. In the following chapters the question as to whether light has a wave nature is addressed. Before such a discussion can begin, we must have an understanding of the nature of waves themselves! Like the physicists that preceded us we will study the waves that can be seen on the surface of water and the waves that can be made to travel along springs and strings. Through this examination we will be able to describe how waves behave and collate a listing of the properties of waves. In particular, we will be looking for the rules of behaviour that seem to be true for waves alone, and not for other mechanisms of motion. Once we have put together the rules describing the behaviour of waves the question as to whether light has a wave nature can be addressed. You may have recognised that our quest is really just a quest to find a satisfactory model for the behaviour of light. Scientists rely heavily on models when they attempt to explain all kinds of phenomena. If an unknown or mysterious entity or observation can be linked with something with which we are familiar, then we can get closer to understanding it. For example, in the early 1900s physicists described the unknown structure of the atom by modelling it on the familiar structure of the Solar System. They depicted the orbits of electrons around the nucleus as comparable to the orbits of the planets around the Sun. This was a most useful model at the time and although not completely accurate, it set the scene for future progress regarding our knowledge of the atom. If waves are to be our chosen model for light then they must appear to behave largely in the same manner as light does. That is, if a wave model for light is to be accepted then it will need to be able to explain the known behaviours of light. A very successful model would illustrate all of the behaviours of light. Perfect modelling is rare in science. Rather it is more likely that we make use of the insight that a particular model provides and, as was the case with our early models of the atom, use it as a stepping-stone to furthering our understanding.
Figure 1.1 If we can learn enough about the properties of waves we can address the question ‘Does light have a wave n ature?’.
Physics file Australia has joined the quest to detect gravity waves with the commencement of construction of the Australian International Gravitational Observatory (AIGO) just north of Perth, Western Australia. This facility will use tiny changes in the path of laser light to detect the elusive gravity waves.
Check out the Interactive Tutorial, The Wave Equations.
Waves transfer energy Sometimes it is really obvious that energy is being transferred. A golf club hits a golf ball and the ball flies through the air, or the water stored in a dam is released, making a turbine spin, or a volcano erupts suddenly spurting out hot lava and heating the surrounding region. In all of these cases energy is transferred from one location to another. Later in the course we look at the concept of energy in detail and study its various forms. For the moment, simply appreciate that energy is an abstract idea. An understanding that it allows work to be done and items to be moved around is sufficient. There is another manner in which energy can be transferred from one location to another. This mechanism does not involve a single body carrying the energy with it from its origin to its final location, but rather the energy is
The nature of waves
3
carried through the particles of a substance. A dramatic example of this is a tsunami—a huge ocean wave created when there is a movement in the Earth’s crust under the sea. The energy created at the location of the shift in the crust is passed along by the particles of the ocean water at speeds of around 800 km h−1, and can reach the coastline in the form of a towering water wave that causes devastation. None of the water particles that flow onto the shore will have been originally located near the source of the tsunami. Only the energy has been passed along. (a)
(b)
Figure 1.2 (a) ‘Particles’ carry energy as they move, this energy can be transferred to another item as it collides with it. (b) Waves carry energy through a medium without the need for an item to have travelled from th e source to the receiver.
Any time we observe energy to have been transferred from one location to another by the passing of the energy from one particle to the next within a substance we say that a mechanical wave has been created. The substance carrying the wave is called the medium. Note that in order to pass on the wave the particles within the medium each temporarily possess some (kinetic) energy and pass it along to the adjacent particle through physically vibrating against it. As the wave energy passes through, each individual particle of the medium will not have any overall change in its position. This is why a floating piece of driftwood will be observed to merely bob up and down as waves pass by. All WAVES involve the transfer of energy without a net transfer of matter.
PRACTICAL ACTIVITY 1 Disturbance and propagation of a disturbance
4
WAVE-LIKE PROPERTIES OF LIGHT
Later we will see that mechanical waves are not the only category of waves that exist. Radio waves and microwaves, for example, also transfer energy from one place to another without a net transfer of matter. There are many waves that can carry energy without requiring a medium. Some of these are visited later in the course. Remember that our objective is to gain an understanding of the general properties of waves. We shall focus our attention on the tangible and readily observed mechanical waves that can be seen to travel in water, springs and strings.
Mechanical waves A mechanical wave involves the passing of a vibration through an elastic medium. Energy must be present at the source of the wave and this energy is described as being carried by the wave . Overall the medium itself is not displaced. Examples of mechanical waves include the vibrations in the earth that we call an earthquake, the sound waves emitted by a speaker, and the disturbance that travels along a guitar string when it is plucked. A model of an elastic medium is shown in Figure 1.3. Balls joined together by springs represent the particles of an elastic medium. Each ‘particle’ occupies its own mean (average) position. An initial disturbance of the first particle to the right will result in energy being passed along from particle to particle. The particles are not all disturbed at the same time; rather the disturbance gradually passes from one particle to the next. Also note that, for example, as particle 2 pushes against particle 3, particle 3 will push back on particle 2. Hence particle 2 is returned to its mean position after it has played its role in passing on the energy. Ideally all of the energy that was present initially will be passed right through the medium. In practice, the temperature of a medium will increase ever so slightly due to their movement.
1
2
3
4
5
Figure 1.3
This model of an elastic medium helps us to envisage the passage of mechanical waves through a medium.
Wave pulses and continuous waves When a single disturbance is passed through a medium in the manner discussed we say that a wave pulse has occurred. Each particle involved in carrying the energy is displaced once as the pulse passes through, and then the particles gradually oscillate back to their mean positions. Many examples of wave motion, however, involve more than one initial disturbance or pulse at the origin. Continuous waves are created when there is a repetitive motion or oscillation at the wave source. Energy is carried away from the source in the form of a continuous wave. A vibrating speaker producing sound waves in
(a)
one initial disturbance
wave pulse
PRACTICAL ACTIVITY 2 Waves in a slinky (b)
continuous vibration at source
Figure 1.4
(a) A single wave pulse can be sent along a slinky spring. (b) A continuously vibrating source can establish a periodic wave. The nature of waves
5
air forms a continuous wave, for example. When a medium is carrying a continuous wave the particles of the medium will vibrate about their mean position in a regular, repetitive manner. These are also called periodic waves as the motion of the particles repeats itself after a particular period of time.
Transverse and longitudinal waves Since all waves carry energy, for any wave the direction of travel of energy can be considered. There are two clearly different categories of mechanical waves. Longitudinal waves involve particles of the medium vibrating parallel to the direction of travel of the energy. An example of this is shown in Figure 1.5a. As the operator vibrates his hand in a line parallel to the axis of the spring, a longitudinal pulse is created. The particles of the medium (or the windings of the spring in this case) will vibrate in the direction shown. The vibrations are parallel to the direction of travel of the wave. Sound waves are a common example of longitudinal waves. When a speaker cone vibrates it causes nearby air molecules to vibrate as shown in Figure 1.5b and this is parallel to the direction in which the sound energy is sent.
Physics file Water waves are often classified as transverse waves but this is an approximation. If you looked carefully at a cork bobbing about in gentle water waves you would notice that it doesn’t move straight up and down but that it has a more elliptical motion. It moves up and down, and very slightly forward and backward as each wave passes. However, since this second aspect of the motion is so subtle, in most circumstances it is adequate to treat water waves as if they were purely transverse waves.
Transverse waves are created when the direction of the vibration of the particle of the medium is 90 ° (perpendicular) to the direction of travel of the wave energy itself. Figure 1.4a shows an example of how this could be achieved. As the operator shakes her hand in a direction perpendicular to the axis of the spring, a transverse disturbance is created. Each particle of the medium will be moved as a pulse passes through. The particles each vibrate around their mean position, but this vibration is perpendicular to the direction that the energy is travelling in.
PRACTICAL ACTIVITY 3 Waves in a rope
(a)
vibration of source
1. vibration of medium 2. next pulse created 3.
4. wave energy
vibration of source (b) wave energy
vibration of air molecule speaker
Figure 1.5
(a) When the vibratory motion and the direction of travel of the wave energy are parallel to one another, a longitudinal wave has been created. (b) Sound waves are longitudinal waves since the molecules of the medium (air molecules) vibrate in the direction of travel of energy
6
WAVE-LIKE PROPERTIES OF LIGHT
Sources of one-, two- and three-dimensional waves Another convenient classification system for waves considers the number of dimensions that the wave energy travels in. One-dimensional waves occur when longitudinal or transverse waves are sent along a spring or rope. The energy travels along the length of the conducting medium. Two-dimensional waves allow energy to be spread in two dimensions. Waves travelling across surfaces are two-dimensional. Ripples travelling outward across the water’s surface when a stone is dropped into a pond is a
PHYSICS IN ACTION
Modelling a longitudinal wave position
e m i t
If you don’t have a slinky spring handy you can still get the idea of a longitudinal wave using the handy model provided by Figure 1.6. Use two A5 pieces of paper. Place one sheet so that it covers all except the top few millimetres of the diagram. Place the other sheet so that there is a 2 mm slot created between the sheets at the top of the diagram. Now maintaining the 2 mm slot between the pages, slide the pages down the diagram, taking about 4 seconds to reach the bottom of the diagram. As you watch the slot you should be able to see ‘longitudinal waves’ travelling to the right. Try varying your sliding speed. Then figure out how it works!
Figure 1.6 Looking at these wavy lines through a slit gives the impression of longitudinal waves moving to the right.
The nature of waves
7
Physics file Seismic wave detectors don’t just pick up the vibrations from earth tremors. The demise of the space shuttle Columbia, the sinking of the Russian Kursk submarine and the collapse of the World Trade Center towers in New York all registered on different seismographs around the world.
Figure 1.7
(a) The ripples on the surface of this pond are described as two-dimensional waves since energy travels outwards in two dimensions. (b) Energy travelling outward in all directions, as in this bomb blast, forms a three-dimensional wave.
8
WAVE-LIKE PROPERTIES OF LIGHT
familiar example of these (see Figure 1.7a). Earthquakes, amongst other effects, produce two-dimensional seismic waves that are mechanical waves travelling across the surface of the Earth. The Sun has a version of these too. Solar flares have been found to be the cause of solar quakes . These twodimensional waves travel across the surface of the Sun and although they travel across distances equal to ten Earth diameters, they look just like ripples in a pond. When you speak you create three-dimensional waves since the sound wave energy spreads out in all three dimensions, though obviously the majority of the energy travels directly outward from the source. Designers of particular speaker-systems attempt to ensure that sound waves are spread out equally in all directions. Figure 1.7b shows a three-dimensional pressure wave emitted by a bomb blast.
1.1 SUMMARY
INTRODUCING WAVES
• Scientists use models to link an unknown entity or observation to something that we are familiar with, in order to gain a better understanding of it. • Knowledge of general wave properties will allow the possible wave nature of light to be assessed. • Energy must be present at the source of any wave. • All waves involve the transfer of energy without a net transfer of matter. • A substance carrying a wave is called a medium. • A mechanical wave is the passing of energy from one particle to the next within an elastic medium.
• A wave pulse occurs when a single disturbance is passed through a medium. • Continuous waves are created when there is a repetitive motion or oscillation at the wave source. Energy is carried away from the source in the form of a continuous or periodic wave. • Longitudinal waves occur when particles of the medium vibrate in the same direction as the d irection of travel of the energy. • Transverse waves are created when the direction of the vibration of the particle of the medium is perpendicular to the direction of travel of the wave energy itself.
1.1 QUESTIONS 1
Describe two ways in which energy can be transmitted.
2
What is the difference between a continuous wave and a pulse?
3
Classify each of the items below as a continuous wave, a pulse or neither: an opera singer holding a note for a long time an explosion c a flag flapping in the wind d dominos standing up in a row and the first one is knocked onto the second, etc. e a tsunami that is caused by a single upward shift in a section of a seabed. One end of a long spring is tied to a hook in a wall and the spring is pulled tight. The free end is then shaken up and down.
7
Mechanical waves are made up of a series of pulses. B Mechanical waves must have a vibrating item at their source C All waves transmit energy but don’t transmit materials. D All waves travel at right angles to the vibration of the particles in the medium. A spring was initially at rest and under slight tension when a series of compressions were sent along it as shown. A
a
b
4
Which of the following statements is incorrect ?
8
Z
Is the resultant wave transverse or longitudinal? b Describe the motion of a particle that is part of a longitudinal wave compared with one that is part of a transverse wave. A slinky spring runs from east to west across the floor of a room and is held at each end. At one end a person gives one quick shake by moving her hand in a northerly and then a southerly direction.
Y
X
a
5
Is the wave in the spring longitudinal or transverse? Is the wave in the spring continuous or a pulse? c Draw an example of how the spring might look at one moment in time. A slinky spring runs from east to west across the floor of a room and is held at each end. At one end a person oscillates her hand periodically in an easterly and then a westerly direction. a
undisturbed spring
9
b
6
a b c
How many oscillations had the hand completed at the moment shown? b In what direction are the following points about to move? i X ii Y iii Z Using apparatus like that shown in Figure 1.3, draw a sequence of five or six diagrams showing the passage of a transverse wave pulse along the entire length of the spring. a
10
Explain the following observation: Although transverse waves cannot travel through the middle or lower sections of a body of water, they can travel along its surface.
Is the wave in the spring longitudinal or transverse? Is the wave in the spring continuous or a pulse? Daw an example of how the spring might look at one moment in time.
The nature of waves
9
1.2
Representing wave features Displacement–distance graphs If a continuous wave was travelling across the surface of water, and we were able to freeze it instantaneously, a cross-section would look something like Figure 1.8a. If the wave then continued, a brief moment later it will have moved slightly to the right and the water particles will have taken up new positions as shown in Figure 1.8b and then Figure 1.8c. The floating cork, like the particles of the medium itself, demonstrates a vertical vibratory motion. It is displaced up then down, then up, then down. Instead, a continuous transverse wave could be sent along a piece of rope or a spring, and the particles of the medium would display a similar behaviour to the up and down motion of the water particles.
wave source
wave travels right
(a)
(b)
(c)
original water level
cork now lower
crest
trough
Figure 1.8
As the wave moves to the right the displacement of the particles of the medium can be tracked using a cork. (a) The cork is on the crest of a wave. (b) The cork has moved lower as the wave moves to the right. (c) The cork is now in the trough of a wave.
A more convenient way of representing waves is to draw a graph of particle displacement against distance from the source. Keep in mind that the mean position of each water particle is the undisturbed level (flat surface) of the water. On the vertical axis we plot the displacement of each particle from its original level at a particular moment in time. The horizontal axis is used to represent the various locations across the water’s surface. Therefore the graph shows the displacement of all particles along the path of the wave, at a particular instant. In this case the chosen instant is the wave position shown in Figure 1.8c.
The shape of the graph in Figure 1.9 relates directly to what we see on the surface of the water. However, these types of graphs can also be used to represent waves that are not so readily visible. Sound waves in air are often 10
WAVE-LIKE PROPERTIES OF LIGHT
t n e e l m c e i t r c a a l P p s i d
Figure 1.9
Distance from the source
The graph of displacement versus distance from the source of a wave is effectively freezing the wave at a moment in time, in other words taking a snapshot.
represented via displacement–distance graphs but in this case the vertical axis is used to show the forward and backward displacement of the air molecules as the sound wave passes through. Re-visit Figure 1.5a. If a longitudinal pulse was sent down a spring (by giving a quick push along its axis), then the vertical axis could be used to represent the forward and backward displacement of the particles of the medium.
The speed of waves Rather than just examining one snapshot, a sequence of graphs can be used to represent a wave that is moving across to the right (see Figure 1.10). By tracking the progress of one crest as it moves to the right, the speed at which the wave is moving can be determined. The use of a dashed line in Figure 1.10 is just to help you keep track of the initial trough and crest that were created. Note that points P and Q and all particles of the medium simply oscillate vertically, whilst the crests and troughs ‘move’ steadily to the right.
Figure 1.10 As each disturbance is created it will be carried away from the source by the medium. The nature of waves
11
Worked example 1.2A . Use the series of graphs shown in Figure 1.10 to determine: a
the average speed of the wave
b
the horizontal speed of par ticle P
c
the average vertical speed of particle P between t = 0 s and t = 0.025 s.
Solution a
Since speed is the measurement of the distance an item travels in a certain time, examining the progress of the first crest it travels from d = 0.01 m to d = 0.05 m in a time period of 0.100 seconds. Average speed
=
=
= =
PRACTICAL ACTIVITY 4 The speed of sound by clap and echo
b
Particle P is vibrating vertically. It has zero horizontal speed.
c
Particle P covers a vertical distance of 5 period. distance travelled Average speed = time taken 0.005 = 0.025 = 0.20 m s−1
Infrared (heat) waves travel away from the source at the same speed as light in a vacuum or in air, 3 × 108 m s−1. Different temperatures show up as different colours in an infrared photograph.
12
WAVE-LIKE PROPERTIES OF LIGHT
×
10−3 m (or 5 mm) in this time
In mechanical waves the speed of the wave is largely determined by the properties of the medium and, of course, by the type of disturbance that is being carried by the medium. (Sometimes the speed of a wave can also be affected by the frequency of the source; this is discussed later.) You may have observed a common example of how the properties of a medium can be altered in order to change the speed of a wave. Try sending a pulse along a slinky spring and make a mental note of how quickly it is carried away. Now stretch the spring across a greater distance, increasing the tension in the spring, and send a similar pulse along it. You should have been able to noticeably increase the speed at which the wave travels. Tension is one example of a property of an elastic medium that affects wave speed. Table 1.1 shows some common waves and typical speeds at which they are carried by their medium. Table 1.1
Figure 1.11
distance travelled time taken (0.05 − 0.01) 0.100 0.04 0.100 0.40 m s−1 or 40 cm s−1
Typical speeds of waves in some common mediums.
Source of wave
Medium
Mechanical pulse Guitar plucking Sound source
slinky spring guitar string air at 20°C water rock vacuum (no medium)
Infrared waves
Typical speed (m s–1)
200 300 344 1450 1500–3500 3 × 108
The frequency and period of a wave Every mechanical wave must have a vibrating source. The rate at which the source vibrates directly affects the nature of the wave formed. The frequency of a source is the number of full vibrations or cycles that are completed per second. For example, a dipping rod in a ripple tank may move up and down 30 times each second. It will therefore create 30 crests and 30 troughs on the water’s surface every second . If any given point on the water’s surface were selected, then 30 complete waves would travel past this point per second. Frequency is a measurement of cycles per second (s−1), and this unit has been appropriately named after Heinrich Hertz (1857–1894) who did important work with radio waves. Hence 1 cycle per second (s−1) equals 1 hertz (Hz). The FREQUENCY of a wave source, f , in hertz (Hz), is the number of vibrations or cycles that are completed per second. Or the frequency of a wave travelling in a medium is the number of complete waves that pass a given point per second.
The time interval for one vibration or cycle to be completed is called the period, T , which is measured in seconds (s). This will also be the time between successive wave crests arriving at a given point. Since a decrease in the frequency of a wave will result in a longer period between waves, the relationship between frequency and period is an example of inverse variation. For example, if ten crests pass a given point in 1 second, then the frequency of the wave must be 10 Hz and the period of the wave would be one-tenth of a second or 0.1 s.
FREQUENCY , f =
1 T
where f = frequency of the wave in hertz (Hz) T = period of the wave in seconds (s)
These ideas about waves can be investigated in the Interactive Tutorial entitled The Wave Equations.
Worked example 1.2B . A student lays a long heavy rope in a straight line across a smooth floor. She holds one end of the rope and shakes it sideways, to and fro, with a regular rhythm. This sends a transverse wave along the rope. Another student standing halfway along the rope notices that two crests and troughs travel past him each second. a
What is the frequency of the wave in the rope?
b
What is the frequency of vibration of the source of the wave?
c
How long does it take for the student to produce each complete wave in the rope?
Solution a
Since frequency is defined as the number of complete waves that pass a given point per second, f = 2 Hz.
To produce a wave with a frequency of 2 Hz, the source must have the same frequency of vibration; that is, 2 Hz. 1 c f = T 1 1 ∴ T = = = 0.5 s T 2 It takes 0.5 s for each cycle to be completed. b
The nature of waves
13
Physics file Keep in mind that displacement–time graphs are looking at the motion of a particular particle . Recall our original definition of a wave as involving energy moving in a medium and realise that these graphs are not showing energy travelling. Therefore these diagrams are not actually graphing a ‘wave’. The familiar shape of this graph occurs because the motion of the particle is periodic; that is, a repeating cycle.
Displacement–time graphs The effects of mechanical waves can be investigated using displacement–time graphs . In these graphs the movement of one particle of the medium is monitored as a continuous wave passes through. As with the previous graphs we studied, the vertical axis may be used to represent displacements perpendicular to the wave’s direction (as in transverse waves) or parallel to the wave’s direction (as in longitudinal waves). Regardless, the displacement is measured relative to the mean position of the particle. Since the horizontal axis indicates time values, the period of the continuous wave can be directly read from the graph. Figure 1.12 shows the displacement–time graph that would apply to the situation described in Worked example 1.2B. Note that the graph covers two complete cycles; that is, two complete waves have passed by.
Figure 1.12 When determining the period of a wave directly from a displacement– time graph it does not matter at which part of the cycle you begin the period measurement.
Wavelength and amplitude PRACTICAL ACTIVITY 5 Waves in a ripple tank
Physics file In many of the waves examined in this chapter there is no decrease in amplitude shown as the wave travels through its medium. This is an idealisation. You will have noticed that pulses sent along springs will die out eventually. Internal resistance within real springs turns some of the wave’s energy to heat. The energy of a circular wave is spread over a larger and larger wavefront as the circumference of the circular wavefront grows. As it moves outward, each section decreases in amplitude because it carries a smaller portion of the wave’s total energy.
14
WAVE-LIKE PROPERTIES OF LIGHT
Recall that earlier we examined graphs that show the displacement of all particles along the path of a continuous wave, at a particular instant. Graphs of particle displacement versus distance from the source can be used to determine the wavelength of a continuous wave. Examine Figure 1.13. Clearly there are particles within the medium that have identical displacements at the same time, such as points A and B. The wavelength of a continuous wave is the distance between successive points with the same displacement and moving in the same direction . These points are said to be in phase with one another. The symbol used for wavelength is the Greek letter lamda, λ . Like all length measurements in physics, the standard unit used is the metre (m). In Figure 1.13 the points X and Y have the same displacement and direction of movement and so they can also be described as being one wavelength apart from each other. Note that although points P and Q have the same dis1 placement, they will not be moving in the same direction. They are only 2λ apart from one another. In the next section we will examine how the frequency of the wave source, and the velocity that the medium allows the wave, combine to determine the wavelength of the wave that is produced. The amplitude, A, of a wave is the value of the maximum displacement of a particle from its mean position. The displacement of particles in a continuous wave will vary between a value of A and − A, as shown in Figure 1.13. The more energy provided by the source of the wave, the larger the amplitude of the wave. For example, in water waves the amplitude obviously corresponds directly to the height of the wave. In sound waves the amplitude determines the loudness of the sound.
Figure 1.13 When determining the wavelength of a wave directly from a displacement–distance graph it does not matter at which part of the cycle you begin the wavelength measurement.
PHYSICS IN ACTION
Huygens’s principle All sorts of waves, such as the circular water waves seen in Figure 1.14a, can also be represented in diagrams like that shown in Figure 1.14b. Lines are used to represent a certain part of the wave, such as the crests. If the diagram were drawn to scale the distance between the lines would represent the wavelength, λ . These diagrams are particularly useful should you want to indicate the region over which the wave energy has spread.
(a)
In 1678 Christiaan Huygens suggested a model that provides an explanation for how waves are carried through a medium. His model coincides with what we see in situations like that shown in Figure 1.14a. Huygens’s principle is a method that uses geometry to predict the new position of a wavefront, if the original position of the
(b)
(c)
Figure 1.14 (a) Circular water waves. (b) Evenly spaced lines can represent the crests of a wave travelling outward, according to Huygens’s principle.(c) Every point on a wavefront is a source of secondary circular wavelets, according to the principle.
The nature of waves
15
wavefront is known. The principle states that every point on a wavefront may be considered the source of small secondary circular wavelets. These wavelets spread out with exactly the same speed as the original wavefront. The new wavefront is then found by drawing a tangent to all of the secondary wavelets. This is called the envelope of the wavelets and is shown in Figure 1.14c. Figure 1.14c shows the points on a wavefront that are
1.2 SUMMARY
REPRESENTING WAVE FEATURES
• A mechanical wave can be represented at a particular instant by a graph of particle displacement against distance from the source. • The frequency of a wave, f , is the number of vibrations or cycles that are completed per second, or t he number of complete waves that pass a given point per second. Frequency is measured in hertz (Hz). • The period, T , is the time interval for one vibration or cycle to be completed. • Frequency, f =
sources of secondary circular wavelets. These wavelets move at speed v and so during time interval t cover a distance of vt . The speed, v , has been assumed to be the same for all wavelets. Although we have only examined the spread of a circular wave, Huygens was renowned for the use of his principle in explaining the reflection and refraction of waves at boundaries (which is discussed later in this text).
1 where f is the frequency of the wave T
in hertz (Hz), and T is the period of the wave in seconds (s).
• A graph of particle displacement versus time can be drawn for the particles of a medium that is carrying a continuous wave. The period of the wave can be directly read from this graph. • Graphs of particle displacement versus distance from the source can be used to determine the wavelength of a continuous wave. • The wavelength, λ , of a continuous wave is the distance between successive points having the same displacement and moving in the same direction; that is, the distance between points that are in phase. • The amplitude, A , of a wave is the value of the maximum displacement of a particle from its mean position.
1.2 QUESTIONS 1
2
Calculate the frequency and period of: a
a spring that undergoes 40 vibrations in 50 seconds
b
a pendulum that completes 250 full swings in one and a half minutes.
In a ripple tank the trough of a water wave travels 70 cm in 2.5 seconds. Calculate the speed of the wave in metres per second.
3
What usually happens to the amplitude of the vibration of a circular water wave as it spreads out? Why?
4
A pebble is dropped into a pool and after 3.00 seconds 24 wave crests have been created and travelled out from where the pebble entered the water. What is the frequency and period of the water wave that was created?
5
A piston in a car engine completes 250 complete up-anddown movements every half a minute. a
16
What is the frequency of vibration of the piston?
WAVE-LIKE PROPERTIES OF LIGHT
b
What is its period?
c
Assuming that the piston started from a central position and moved up. Where will it be after: i
6
7
1 period?
ii
1
14 periods?
iii
1
12 periods?
Which of the following statements is correct? A
Period is the measurement of the length of a wave.
B
The amplitude of a wave is dependent upon the frequency.
C
The more energy put into a wave the greater the wavelength.
D
The more energy put into a wave the greater the amplitude.
Examine the wave represented in Figure 1.10. What is the wavelength of the wave?
8
A longitudinal wave enters a medium and causes its particles to vibrate periodically. Draw a displacement–time graph that could demonstrate the motion of the first affected particle of the medium for the first two cycles. Begin with a positive displacement; i.e. in the direction of travel of the wave.
9
The displacement–distance graph shows a snapshot of a transverse wave as it travels along a spring towards the right.
a
Use the graph to determine the wavelength and the amplitude of this wave.
b
At the moment shown, state the direction in which the following particles are moving: Q, S.
c
Assuming that the wave is travelling at 12 m s −1 to the right, and no energy is lost, draw the displacement–distance graph for this wave 0.05 seconds after the moment shown. Label the points P, Q, R and S.
10
The displacement–time graph shows the motion of a single air molecule, P, as a sound wave passes by travelling to the right.
a
Use the graph to determine the amplitude, period and frequency of this sound wave.
b
State the displacement of the particle P at: i t = 1 ms ii t = 2.5 ms iii t = 5.5 ms
c
Draw the displacement–time graph for the particle, Q, which is positioned half a wavelength to the right of particle P. Show the same 4 ms time interval.
d
If sound is actually a longitudinal wave, why does this graph look more like a transverse wave?
The nature of waves
17
1.3
Waves and wave interactions The wave equation The frequency of a source of a mechanical wave and the velocity of that wave in the medium together determine the resulting wavelength of the wave. For example, a horizontal bar vibrating at frequency, f , may be used as the dipping element in a laboratory ripple-tank as shown in Figure 1.15. Once one crest is created, assume that it travels away from the source at a known speed, v . Since the definition of speed is: speed =
Figure 1.15 The medium carrying the wave and the frequency of its source together determine the wavelength of a wave.
distance travelled time taken
This can be rearranged to: distance travelled = speed × time taken. Consider the first period, T , of the wave’s existence. The distance that the first wave will be able to cover before the next wave is created behind it is determined by the speed at which the medium allows the wave to travel. The distance that the first wave travels during one period—by definition—is the wavelength of the wave, λ . Therefore we acknowledge that the ‘distance travelled’ = λ when the ‘time taken’ = T . Substituting into: distance travelled = speed × time taken λ = v × T
The frequency and period of a wave are inversely related: T =
1 f
Hence, the above relationship can also be expressed as: λ=
v f
The WAVE EQUATION links the speed, frequency and wavelength of a wave:
v = f λ where v = speed of the wave in metres per second (m s −1)
f = frequency of the wave in hertz (Hz), and λ=
wavelength of the wave in metres (m).
Note that for a medium of a given speed, the use of a higher frequency source would result in waves that are closer together; that is, waves of a shorter wavelength. A low frequency source would produce longer wavelength waves (see Figure 1.16). For a given wave speed: λ∝
18
WAVE-LIKE PROPERTIES OF LIGHT
1 f
Figure 1.16 (a) For a medium of a given speed, the use of a low frequency source produces waves with a long wavelength. (b) With less time between the creation of successive waves, a high frequency source produces waves with a shorter wavelength.
An implication of the wave equation that is worth noting is that a source that has a specific frequency of vibration is able to produce waves of different wavelengths, depending upon the medium that carries the wave. Consider a submarine that puts out a high frequency tone of 20 000 Hz. If this same frequency tone were sent both into the water and into the air, the waves produced in the water would have a much longer wavelength that the waves produced in the air. This is because sound waves travel about four times faster in water than in air (see Figure 1.17). For a source of a given frequency: λ ∝ v
Figure 1.17 Since sound waves travel much faster in water than in air, the waves produced by a tone of a given frequency have a much longer wavelength when they travel through water than when they travel through air.
Worked example 1.3A . A person standing on a pier notices that every 4.0 seconds the crest of a wave travels past a certain pole that sticks out of the water. The crests are 12 metres apart from one another. Calculate: a
the frequency of the waves
b
the speed of the waves.
Solution a
The period of the wave is 4.0 s. 1 Since f = T f = =
b
1 4.0 0.25 Hz
Since the crests are 12 m apart the wavelength is 12 m. v = f λ =
0.25
=
3.0 m s−1
×
12
Waves meeting barriers Mechanical waves travel through a medium. Commonly a situation will occur in which the wave travels right through to a point where the medium physically ends. An example of this is the wave created as a child leaps into a pool; it travels until it reaches the pool wall. At the boundary of the medium the energy that was being carried by the wave may undergo different processes. Some of the energy may be absorbed by or transmitted into a new medium, and some energy may be reflected .
See the Medical Physics Detailed Study for a study of ultrasound waves in the body.
PRACTICAL ACTIVITY 6 Reflection of waves in a ripple tank
The nature of waves
19
Physics file The phase change of a wave on reflection from a fixed end can be explained in terms of Newton’s third law of motion. When the pulse arrives at the fixture the rope exerts a force on the fixture. The fixture exerts an equal and opposite force on the rope. This produces a pulse that is in the opposite direction to the original pulse; that is, a change in phase has occurred.
The extent to which these processes occur depends on the properties of the boundary. We shall examine the case of a transverse wave pulse travelling in a heavy rope that has one end tied to a wall. As shown in Figure 1.18a the wave travels to the boundary and we can see that it is reflected with almost no energy loss since the original amplitude is maintained. The wave, however, has been inverted; this can also be described as a reversal in phase . (The definition of phase was discussed in the previous section.) Since a crest would reflect as a trough and a trough would reflect as a crest, we can say that the phase of the 1 wave has been shifted by 2λ . A WAVE REFLECTING FROM THE FIXED END of a string will undergo a phase reversal; that is, a phase shift of
λ
2
.
Now consider the situation where the end of the rope is instead free to move. As shown in Figure 1.18b, the wave travels to the end of the rope and we can see that it is reflected with no reversal in phase. Since a crest would reflect as a crest and a trough would reflect as a trough, we can say that there was no change of phase. A WAVE REFLECTING FROM THE FREE END of a string will not undergo a phase reversal.
Figure 1.18 (a) The reflection of a wave at an unyielding boundary produces a phase 1
shift of 2 λ . Note that otherwise the shape of the wave is unaltered. (b) The reflection of a wave at a free-end boundary does not produce a phase shift.
PHYSICS IN ACTION
Reflections NOT wanted! The stealth bomber is an aircraft that is designed so that its body is as poor a reflector as possible. The main way in which a passing aircraft is detected by others is with the use of radar. A radar transmitter sends out pulses of radio waves or microwaves and a receiver checks for any reflections from passing aircraft. By analysing the reflections, radar systems can work out the position, speed and perhaps even the identity of the passing aircraft.
Stealth aircraft are supposed to create as little reflection of these waves as possible. The shape of the stealth aircraft is the most important factor. It does not have any large vertical panels on the fuselage that would act like mirrors, nor a large vertical tail. It has no externally mounted devices such as missiles or bombs. It does not include any surfaces that meet at right angles. These would act like the corners in a billiard table and bounce the waves r ight back to their source. Instead curved surfaces on the stealth bomber are designed to reflect waves sideways or upward wherever possible. A thick coat of special paint that
20
WAVE-LIKE PROPERTIES OF LIGHT
absorbs radio waves is used on its surface. Although not completely undetectable, with the right shape and coating a large stealth plane can produce the same amount of wave reflection as an average sized marble!
Figure 1.19 The reflection of radar waves from aircraft usually reveals their position but the stealth plane is designed for minimal reflection.
Superposition: waves interfering with waves In the case of a continuous wave being sent toward a boundary, a situation can be created where two waves may be travelling in the one medium, but in different directions . The incident waves will meet the waves that have already reflected from the boundary. When two waves meet they interact according to the principle of superposition. The principle of SUPERPOSITION states that when two or more waves travel in a medium the resulting wave, at any moment and at any point, is the sum of the displacements associated with the individual waves.
Physics file When a note is played on a musical instrument sound waves with many different wavelengths are produced simultaneously. The richness of a tone is largely determined by how many different wavelengths make up the sound wave. The tone with the longest wavelength determines the overall perceived pitch of the note but the number of overtones (other wavelengths present) will add to its timbre .
Consider a spring where a transverse pulse has been sent from each end, as shown by the sequence of events in Figure 1.20. When the pulses reach the same point in the spring the resulting wave will be the sum of the displacement produced by the individual pulses. The principle of superposition is therefore the same as the ‘addition of ordinates’ process that is carried out on graphs. Simply sum the y -values of each of the pulses to see the resulting wave. PRACTICAL ACTIVITY 7 Diffraction of continuous water waves
PRACTICAL ACTIVITY 8 Interference of water waves
Figure 1.20 Superposition of two pulses of the same amplitude travelling toward one another.
In Figure 1.20 the initial pulses have particle displacements in the same direction and therefore constructive interference occurs. Notice that after interacting with each other, the two pulses have continued on unaffected. This is an observed property of waves. They are able to pass through one another, momentarily interact according to the superposition principle, and then continue on as if nothing had happened.
The nature of waves
21
In Figure 1.21 destructive interference occurs since the initial pulses have particle displacements in opposite directions. If the crest of one pulse has exactly the same dimensions as the trough in the approaching pulse then the two pulses will momentarily completely cancel each other out, as shown in Figure 1.21. If the amplitude of one of the waves is larger than the other then only partial cancellation will occur.
Figure 1.22 The superposition of continuous waves that are in phase and travelling in the same direction will result in constructive interference.
Physics file We have looked at how waves can reflect back along a string from a fixed end. Essentially this is what happens to the waves sent along a bowed violin string or a plucked guitar string. The numerous reflected waves add together according to the principle of superposition with some important effects. For each mode of vibration shown in Figure 1.24, at some spots on the string constructive interference will occur. In other spots destructive interference occurs. Since each particular mode of vibration has set locations for these spots, the wave is called a standing wave .
Figure 1.21 Superposition of two pulses of equal but opposite amplitudes travelling toward one another.
In the case of interference between continuous waves, the principle of superposition is still applicable. If two waves are exactly in phase and are travelling in the same direction , then constructive interference will occur along the entire length of the wave. The two waves need not have the same amplitude. In Figure 1.22 one wave is twice the amplitude of the other wave and the resultant wave is shown. Interesting effects are observed when two waves of different wavelengths are travelling in the same direction and interfere with one another. Figure 1.23 shows the addition of two waves, where one wavelength is exactly three times longer than the other. This is a relatively simple example. Imagine the complexity of the sound-wave patterns produced when instruments in an orchestra are played simultaneously. Or of the wave patterns that are produced on the surface of water in a busy harbour.
Figure 1.24 One mode in which a string can vibrate involves destructive interference happening right at the centre point of the string.
22
WAVE-LIKE PROPERTIES OF LIGHT
Figure 1.23 The addition of waves of different wavelengths results in complex wave patterns.
PHYSICS IN ACTION
Diffraction and interference effects We have seen how a wave can spread out from a point source, but waves are also capable of bending around obstacles or spreading out after they pass through a narrow gap. This bending of the direction of travel of a wave is called diffraction. Figure 1.25 shows the diffraction of water waves as they pass by an obstacle.
the crests that arrive at these locations, and the surface of the water remains relatively undisturbed. The regions of destructive interference appear grey and flat in the photograph. These regions of destructive interference appear to radiate from a point between the sources.
Both diffraction and interference effects are only observed when energy is being carried by waves, not when energy is being carried by particles.
Figure 1.25
Rather than only travelling directly forward, notice how the wavefronts spread out to fill the region behind the obstacle.
Diffraction effects can be seen with two-dimensional waves, such as on the surface of water, and also with threedimensional sound waves. This explains why we can hear sounds that were originally made around the corner of a building. The sound waves bend their direction of travel— that is, diffract—around the corner of the building to reach the listener’s ears.
Figure 1.26 Significant diffraction occurs when the wavelength is at least as large as the aperture.
The extent to which diffraction occurs depends on the relative dimensions of the aperture or obstacle that the wave passes, and the wavelength of the wave. Most noticeable diffraction occurs if the wavelength is at least as large as the aperture is wide. When waves of the same wavelength are sent through large and narrow apertures as in Figure 1.26, only waves passing through the narrow aperture produce noticeable diffraction. Spreading water waves produce interference patterns that are characteristic of waves. Consider two sources of spherical waves (Figure 1.27). In some locations constructive interference occurs and waves of relatively large amplitude are seen. These regions have lots of contrast in the photograph; that is, alternating bright and dark bands are seen. In other regions destructive interference occurs. Troughs arriving from the other source always cancel out
Figure 1.27 The interference pattern produced by two point sources in phase.
The nature of waves
23
Now we can look at light! Now that we have put together the rules describing the characteristics of waves, the question as to whether light has a wave nature can be addressed. Waves have numerous characteristics and they have been worth examining in their own right. We have been able to conclude that: • waves involve the transfer of energy without an overall transfer of matter • mechanical waves require a vibrating item at their source and a medium to carry them • waves can be categorised as longitudinal or transverse • the wave equation, v = f λ , describes the relationship between the speed, frequency and wavelength of a wave • waves can reflect at boundaries and this will sometimes produce a change of phase • waves can be added according to the principle of superposition and this can result in constructive or destructive interference. In Chapter 2 we will go on to discuss whether it is appropriate to use waves as our chosen model for light. For this to be fitting, light must appear to behave largely in the same manner as waves do. That is, if a wave model for light is to be accepted, then it will need to explain the known behaviours of light. A very successful model would illustrate all of the behaviours of light. This is not likely. It is more likely that we will be able to make use of the insight that waves provide, and use this insight to further our understanding of the nature of light. 1.3 SUMMARY
WAVES AND WAVE INTERACTIONS
• The frequency of the source and the speed of the wave in the medium determine the wavelength of a mechanical wave. • The wave equation states: v = f λ , where v = speed of the wave in metres per second (m s−1), f = frequency of the wave in hertz (Hz), and λ = wavelength of the wave in metres (m). 1 • For a wave of a given speed: λ ∝ . f • For a source of a given frequency: λ ∝ v . • A wave reflecting from a fixed end of a string will undergo a phase reversal; that is, a phase shift of λ . 2
• A wave reflecting from a free end of a string will not undergo a phase reversal. • The principle of superposition states that when two or more waves travel in a medium the resulting wave, at any moment, is the sum of the displacements associated with the individual waves. • Constructive interference occurs when two waves meet that have particle displacements in the same direction. • Destructive interference occurs when two waves meet that have particle displacements in opposite directions.
1.3 QUESTIONS 1
2
3
24
a
What happens to the wavelengths of the waves in a ripple tank if the frequency of the wave source is doubled?
4
A submarine’s sonar equipment sends out a signal with a frequency of 35 kHz. If the wave travels at 1400 m s −1, what is the wavelength of the wave produced?
b
What happens to the speed of the waves in a ripple tank if the frequency of the wave source is halved?
5
Which of the following statements is incorrect?
The source of waves in a ripple tank vibrates at a frequency of 15.0 Hz. If the wave crests are 40.0 mm apart, what is the speed of the waves in the tank? A wave travels a distance of 50 times its wavelength in 10 seconds. What is its frequency?
WAVE-LIKE PROPERTIES OF LIGHT
A
When two pulses interact the resulting wave, at any moment, is the sum of the displacements associated with the individual waves.
B
After two waves interact with each other they will continue on through the medium unaffected.
6
7
8
C
For two pulses to interfere destructively they must have opposite amplitudes.
D
For two continuous waves to interfere constructively they must have identical amplitudes.
Will a transverse wave reaching the fixed end of a string undergo a phase reversal? Two waves are travelling in the same direction in a medium. They undergo constructive interference along the entire length of the wave. What two statements can be made about the two waves?
9
10
Draw the resultant displacement versus distance graph for two superimposed continuous waves that are in phase and travelling in the same direction. Each wave has a wavelength of 4 cm and amplitude of 1 cm. Show two complete cycles. Draw the resultant displacement versus distance graph for two superimposed continuous waves travelling in the same direction. Each wave has a wavelength of 4 cm and amplitude of 1 cm, but one wave is one-quarter of a wavelength behind the other.
Assuming the following diagram shows the displacement– distance graphs of two waves at a particular instant, show the addition of the two waves according to the principle of superposition.
CHAPTER REVIEW The following information applies to questions 1 to 3. A pulse is travelling along a light spring. The diagram below shows the position of the pulse at
t = 0 s. The pulse is moving at a speed of 40 cm s−1 to the right. 1
Use a set of scaled axes to draw the displacement–distance graph for the pulse at the moment shown.
2
Draw the displacement–distance graph for the pulse 0.5 seconds later. Clearly show the location of point P.
3
Draw the displacement–time graph for the point Q for a time
interval of 2.0 seconds, beginning at t = 0. 4
List an example of a one-, two- and three-dimensional wave.
5
A guitar string is plucked near one end. A wave moves along the string and another wave is produced in the air. State whether each wave is transverse or longitudinal.
The following information applies to questions 6 to 9. The diagram shows two successive amplitude–distance graphs for a periodic transverse wave travelling in a string. The time interval that passed between the tracings of the two graphs is 0.20 s. The graphs are drawn exactly to scale. 6
State the amplitude of the wave.
7
State the wavelength of the wave.
8
Calculate the velocity of the wave.
9
Calculate the frequency and period of the wave.
10
Which of the following statement(s) is/are incorrect (one or more answers)? A
All mechanical waves require a medium to carry the wave.
B
All mechanical waves transfer energy.
C
In wave motion some of the material is carried along with the wave.
D
Mechanical waves permanently affect the transmitting medium.
The nature of waves
25
CHAPTER REVIEW 11
What is the period of the wave that: a
b
12
13
involves 5.0 crests of water lapping against a breakwater each 20 seconds? is produced by a flute playing the note middle C (512 Hz)?
A transverse wave travels along a string towards an end that is free to move. Which of the following statements is true? A
The wave will reflect with no phase change.
B
The wave will reflect with a
Find the frequency of the waves that have periods of: a
0.35 s
b
4.0 × 103 s
c
10−2 s.
A wave pulse is sent simultaneously from both ends of a spring. When the pulses meet they momentarily completely cancel out one another. a
b
26
14
What is the term that describes this occurrence? Make statements about three features of the wave pulses.
WAVE-LIKE PROPERTIES OF LIGHT
phase change of
C D
15
16
λ
The following information applies to questions 17 to 20. Wave A has a wavelength of 4.0 cm, a period of 2.0 seconds and an amplitude of 1.5 cm. Wave B has a wavelength of 2.0 cm, a period of 1.0 second and an amplitude of 1.5 cm. 17
Draw a scaled displacement– distance graph for wave A. Show two full waves.
18
Draw a scaled displacement– distance graph for wave B. Show four full waves.
19
If wave A and wave B were sent into the same medium and they are travelling in the same direction, draw the resultant displacement–distance graph. Show two full waves.
20
Draw a displacement–time graph for a particle in the medium that carries wave A only. Show two complete cycles.
.
2 The wave will not be reflected. The wave will reflect faster than the incident wave.
Waves travelling in a ripple tank have a wavelength of 7.0 mm and travel at 60 cm s−1. What is the frequency and period of the waves? One end of a long spring is firmly connected to a wall fitting. Briefly explain how a transverse wave can be created and carried by the spring.
CHAPTER 2
Two models for light
I
t is a common trait of humans that when we seek to understand something we will intuitively attempt to link the unknown with the known. In your earlier schooling a physical representation or model was probably used to teach you about nature’s water cycle, or multiplication, or the properties of gases. Young students benefit from the use of tangible items such as models; things that can be seen and touched. As we grow, our knowledge and understanding can still benefit from the use of a modelling approach, but our models can be more sophisticated. When computer-generated pictures were used to model the complex equations of fractal geometry they had an amazing similarity to some structures found in nature. Fractal images model things such as coastlines and snowflakes and they have become popular works of art. A model is a system of some type that is well understood and that is used to build a mental picture or analogy for an observed phenomenon, in our case the behaviour of light. A good model will appear to behave in the same manner as the entity being investigated. A model for light needs to be able to explain the observations of light that have already been made and ideally it would predict new behaviours. Therefore, throughout this chapter, when deciding upon a model for light we must examine each of its known behaviours in turn and assess the effectiveness of the chosen model.
BY THE END OF THIS CHAPTER
you will have covered material from the study of the wave-like properties of light including: • wave and ray models for light • modelling reflection and refraction • refractive index and Snell’s law • total internal reflection • optical fibres, and material and modal dispersion • electromagnetic radiation • colour components of white light and dispersion • polarisation of light waves.
2.1 A
Modelling simple light properties B
Now that we have a thorough appreciation of the properties of waves, the question can be asked: Is light a wave? If a wave is defined as the sum of its properties, does light exhibit all of the properties that are known to belong to waves?
C
Figure 2.1
Light from the lamp can only be seen if the pinholes lie in a straight line. This means that light must travel from the lamp to the eye along a straight line. (a)
Curiosity about the nature of light has occupied the minds of physicists for centuries. The beginning of human interest in the nature of light dates back to the ancient Greek, Arabian and Chinese philosophers. In the early 19th century, evidence suggested that light could be modelled as a wave since it exhibited the same set of properties as other things that had already been defined as waves: water waves, sound waves, vibrations in springs and strings. If light exhibits sufficient properties in common with these known waves, then surely it too could be assumed to ‘be’ a wave? The story of the development of a scientific model for light is not straightforward. The discussion of light as a wave did not exist in isolation. The giants of physics became embroiled in a famous ongoing scientific debate that posed the question: Is light made up of particles or waves? In this section we look at how the very simplest behaviours of light can be readily modelled as either particles or waves.
(b)
Modelling straight-line propagation
diverging rays
converging rays
parallel rays
(c)
Light streaming through trees on a misty morning, the projector’s beam in a dusty cinema, our limited view when peeping through a keyhole and the distinct shape of shadows are all evidence for the straight-line or rectilinear path of light. These examples provide evidence that light—transmitted in a uniform medium (i.e. a substance which is unchanging in its constitution)— travels in straight lines. Our awareness of the rectilinear propagation of light allows us to judge the distance to objects. The mechanism by which our eyes and brain interpret a three-dimensional world is complex, but it relies on the assumption that light in a uniform medium travels in straight lines. The following simple experiment can be performed to demonstrate that light travels a straight path in a uniform medium. Make a pinhole in each of three identical pieces of card. Place card A close to a light source, and position card B a little further away, as shown in Figure 2.1. Then, holding card C in front of your eye so that you can always see through the hole, adjust its position so that you can see the light from the lamp. This will only be possible when all three pinholes lie in the same line; that is, when the pinholes are co-linear . The conclusion that can be drawn is that light must travel in straight lines.
(d)
Figure 2.2
(a) A beam of light is made up of a bundle of rays. (b) Rays can be diverging, converging or parallel to one another. (c) An idealised point source of light emits rays of light in all directions. (d) Very distant sources of light are considered to be sources of parallel rays. 28
WAVE-LIKE PROPERTIES OF LIGHT
This property of light was first modelled by considering that light was particle-like in nature. Consider a beam of light shining from a powerful torch. If light is assumed to be corpuscular or particle-like in nature then the direction of travel of the light energy can be represented by rays (Figure 2.2a). The idea of a light ray is a useful concept as it can successfully model the behaviour of light in the situations illustrated. A beam of light can be thought of as a bundle of rays. A strong light source, such as the Sun, could therefore be thought of as producing a very large number of light particles or rays. Light sources, in conjunction with other optical elements, such as lenses or mirrors, can produce rays of light that diverge, converge or travel parallel to each other (Figure 2.2b). In each case the rays are an indication of the direction
of travel of the light, essentially light is being modelled as a stream of particles. The incandescent (filament) light globes and fluorescent tubes in your home emit light in all directions. A point source of light is an idealised light source that emits light equally in all directions from a single point (Figure 2.2c). No single point source of light exists in reality, but a small filament lamp can be considered a good approximation. Lasers and special arrangements of light sources with mirrors or lenses can produce parallel rays of light in a beam. Very distant point sources of light can also be considered to be sources of parallel light rays. For example, on the Earth we treat the light rays that reach us from the Sun as though they were parallel to each other. This is because at such a large distance from the source, the angle between adjacent rays would be so tiny as to be considered negligible (Figure 2.2d). Later in this study we will also see how the ray model of light conveniently allows us to represent and understand the behaviour of light as it interacts with mirrors and lenses to form images. Although a particle description for light and the accompanying ray model are convenient for representing the behaviour of light in all of these cases, it has long been understood that light is not made up of ordinary particles. Light involves the transfer of energy from a source, but there are no tangible particles carrying this energy. With developing technology over the last two centuries physicists have been able to make more and more sophisticated observations of light. Later in the chapter we will see that a more refined model of light incorporates the wave-like properties of light. The ray approach is still useful. If light is considered to be a wave emanating from its source, then rays may simply be used to represent the direction of travel of the wavefronts (see Figure 2.3). The point source of light discussed above may be considered to be a point source of spherical wavefronts, like the ripples that travel out from a stone dropped into a pond (Figure 2.4).
Modelling reflection The reflection of waves was discussed in Chapter 1. Light has been observed to obey the same laws of reflection that apply to waves and so evidence is provided for the argument that light is a wave. Using a wave model, the reflection of light would be represented as a series of wavefronts striking a surface and reflecting as shown in Figure 2.5. However it is far more common to model the reflection of light using ray diagrams and the conventions associated with them.
wavefronts travel outward from torch rays travel outward from torch
Figure 2.3
Rays can be used to represent the direction of travel of light waves leaving the torch.
spherical wavefronts travel outwards
point source
Figure 2.4
A point source of light may be considered to be a point source of spherical waves. Both the particle and the wave model are consistent with the observation that the intensity reduces with the square of the distance from the source.
PRACTICAL ACTIVITY 9 Reflection in a plane mirror
Consider the plane mirror drawn in Figure 2.6. We define a normal to the surface of the mirror as the line perpendicular (at 90°) to the mirror’s surface (a)
(b)
(c)
Figure 2.5
When studying reflection, ray diagrams are the most convenient way of representing the path of light. Two models for light
29
plane (flat) mirror i n c i d e n t r a y
angle of incidence normal
angle of reflection
Experiment shows that whenever reflection occurs, the angle of incidence always equals the angle of reflection. In addition, the light reflects in such a way that the incident ray, the normal and the reflected ray all lie in the same plane. The law of reflection can then be re-stated using a ray model for light.
i r
a y d r e t l e c r e f
Figure 2.6
When light reflects from a plane mirror, the angle of incidence equals the angle of reflection: i = r . incident ray
at the point where an incoming or incident ray strikes the mirror surface. The angle made between an incident ray and the normal is the angle of incidence, denoted i . The ray strikes the mirror and reflects with an angle of reflection, r , which is the angle between the reflected ray and the normal.
~4% ~96%
glass layer metal layer paint layer
Figure 2.7
Most of the incident light on a mirror is reflected from the silvered surface at the back of the mirror. The glass on the front and the paint on the back serve to protect the reflective surface from damage. (a)
The LAW OF REFLECTION states that the angle of incidence, i , is equal to the angle of reflection, r (i = r ). The incident ray, normal and reflected ray will all lie in the same plane.
A normal household mirror is constructed with three separate layers: a layer of transparent glass, a thin coating of aluminium or silver deposited onto the glass to reflect the light and a backing layer of protective paint (Figure 2.7). When a beam of light strikes the surface of the mirror a tiny amount of the light energy (about 4%) is reflected from the front surface of the glass, but most of the light continues to travel through the glass and is reflected from the metal surface at the back. These reflected rays produce the image that is seen in the mirror.
Regular and diffuse reflection To some extent at least, light will reflect from all surfaces, but only some surfaces will produce a clearly defined image. If parallel rays of light are incident on a plane mirror or a flat polished metal surface, they will remain parallel to each other on reflection (Figure 2.8a). This is regular reflection (sometimes called specular reflection ) and, as a result, a clear image can be produced. Common examples of regular reflection include the reflection of light from plane mirrors, glossy painted surfaces and still water such as in a lake. When light is reflected from a roughened or uneven surface, it is scattered in all directions as shown in Figure 2.8b. This is diffuse reflection. Parallel rays of incident light will be reflected in what seem to be unpredictable directions. Each ray obeys the law of reflection, but the surface is irregular so that normals drawn at adjacent points have completely different directions. Thus, light is reflected in many different directions. Most materials produce diffuse reflection. For example, when looking at this page, you can see the printing because the lighting in the room is reflected in all directions due to diffuse reflection. If the page behaved as a regular reflector, you would also see the (reflected) images of other objects in the room.
(b)
Figure 2.8
(a) Regular reflection from a smooth surface occurs when parallel rays of incident light are reflected parallel to each other. (b) Diffuse reflection occurs at an irregular surface. Here, the incoming parallel rays are reflected at all angles.
Diffuse and regular reflection are the two extreme cases of how light can be reflected. In reality most surfaces display an intermediate behaviour. For example, the pages of a glossy magazine may allow a blurry image of the reader’s face to be formed, but the printing can still be seen. The surface produces reflection that lies somewhere between pure diffuse reflection and pure regular reflection. Can you think of any other surfaces that do this? What do they all have in common? To predict the extent to which diffuse and regular reflection occurs at a surface, one must examine the surface on a microscopic scale. If the irregularities in the surface are small compared with the wavelength of the incident light, then regular reflection occurs. If the irregularities are comparable in size to the wavelength of light, then more diffuse reflection occurs. The wavelength of light is discussed more fully later in the chapter.
30
WAVE-LIKE PROPERTIES OF LIGHT
PHYSICS IN ACTION
Eclipses From our everyday experience, we know that a shadow is formed when a solid body obstructs the path of light from the source. This can occur on a large scale when the Moon and Sun are aligned with the Earth to produce different types of eclipses . Figure 2.9 shows how the Earth, Moon and Sun must be aligned to produce a lunar eclipse . In a lunar eclipse, the Earth’s shadow falls across the face of the Moon. However, the Earth’s shadow consists of two distinct parts. The complete shadow, the umbra (the Latin for shadow or shade), is the darkest region, and when the Moon passes through this, a total eclipse of the Moon occurs. There are also regions known as penumbra (literally ‘almost an umbra’) where the Moon is neither fully illuminated nor fully in shadow. When the Moon only passes through the penumbra of the Earth’s shadow, the Moon appears dimmer than usual. Even during a total eclipse, the Moon does not completely disappear. Rather, it appears very dim and red in colour. This is because it is still illuminated by a small amount of light which has travelled through the Earth’s atmosphere. The atmosphere acts like a prism, splitting the white sunlight into its component colours through the process of refraction , and the distances and angles involved are such that most of the light that reaches the Moon is red. Looking at the relative positions of the Earth, Moon and Sun in Figure 2.9, one might expect a lunar eclipse to occur once during each revolution of the Moon around the Earth, i.e. once per month. This does not occur because the plane of the Moon’s orbit is slightly tilted with respect to the Earth’s (a) total eclipse
Sun
Earth
Earth’s orbit
Moon’s orbit
Earth’s main shadow
Figure 2.9
During a lunar eclipse the Moon travels into the Earth’s shadow.
orbit around the Sun. This means that the Moon often travels only partially into the shadow region and so only a portion of its surface is obstructed. The Moon travels at a speed of about 1 km s−1 through the Earth’s shadow, which means that the longest time a total eclipse of the Moon can last is 1 hour 42 minutes. Between 9 and 12 total eclipses of the Moon can be seen from the Earth every decade. A solar eclipse occurs when the Moon comes between the Earth and the Sun, casting its shadow onto the Earth. Figure 2.10 illustrates two types of solar eclipse. The Moon orbits the Earth in an elliptical path, so its distance from the Earth varies. The relative distances between the Earth, Moon and Sun determine whether an eclipse is total or annular (from the Latin word annulus , meaning ‘ring’). When the Moon is relatively close to the
Moon’s orbit
Sun
(b) annular eclipse
Sun
(c) Earth
Moon
Moon’s orbit
Earth
Moon
Figure 2.10 (a) During a total solar eclipse the Sun disappears behind the disc of the Moon. Depending on the relative positions of the Earth, Moon and Sun, this can last for as long as 7 minutes. (b) and (c) When the Moon is at its farthest from the Earth, its disc is no longer large enough to cover the Sun, and an annular eclipse occurs, in which a thin ring (or annulus) of the Sun’s disc remains visible and the Moon blocks out only the central region. Two models for light
31
Earth, and the Sun, Moon and Earth are aligned, the Moon’s umbra reaches the Earth (Figure 2.10a). Observers in the region of the Earth’s surface covered by the umbra see a total eclipse of the Sun. Observers only just outside the main shadow but still within the penumbra see a partial eclipse of the Sun, in which only a portion of the disc of the Sun is obscured. A total eclipse of the Sun will never last more than 7 minutes at any location on the Earth and most last only 2 or 3 minutes.
When the Moon is further from the Earth, the umbra does not quite reach the Earth’s surface. Viewed from the Earth, the Moon does not completely block out the Sun. Its angular size is too small to cover the whole disc of the Sun, and at mid-eclipse a thin ring of the Sun’s disc can still be seen around a dark Moon. This explains the term annular eclipse . Annular eclipses occur slightly more frequently than do total eclipses.
Reflection, absorption and transmission
Physics file When light travels past our eyes it cannot be seen. Light is invisible unless some of it is reflected by tiny particles in the air into our eyes. The particles might be dust, fog or smoke. An effective demonstration of this is to mark the path of a laser beam in a darkened room with chalk dust.
After a beam of light strikes an object, there are three processes that can occur: some of the light may be reflected from the surface, some may be transmitted through the material, and some may be absorbed into the surface. This behaviour of light recommends a wave model, as viewing light as a particle would make it difficult to explain the light energy undergoing three different processes. Most materials are opaque to visible light; that is, they do not allow any light to pass through them. For example, brick, plaster and cardboard are impervious to light. Opaque materials will reflect some light and absorb the rest. Other materials are transparent . A transparent material will allow a significant amount of light to pass through it. It may absorb some, and some may even be reflected from the surface of the material. Clear glass, Perspex, water and plastic food wrap are common examples of transparent materials.
incident light
reflected light
boundary between two surfaces absorbed light transmitted light
Figure 2.11 Light incident on the surface of a transparent material is partly reflected, partly transmitted and partly absorbed by the material. The relative amounts of the light experiencing these processes will depend on the nature of the material in question.
32
WAVE-LIKE PROPERTIES OF LIGHT
Some materials classified as transparent allow some of the incoming light to pass through but distort the path of this light so that no clear image can be seen through the material. Although the rays of light have passed through the material, the relationship between them has been altered. Such materials are called translucent , and examples include frosted glass, tissue paper and fine porcelain. Translucent materials are particularly useful if an area needs to be illuminated but privacy is required. Frosted or mottled glass is often used for bathroom windows. In other situations, a translucent material is used to deliberately scatter light. For example, the cover around a fluorescent lamp or the ‘pearl’ finish of an incandescent globe can soften household lighting by diffusing it, thereby producing less harsh shadows. Light is a form of energy, and when light is absorbed by a material, the energy it carries is converted directly into heat, warming the material up. Some of the light energy will also be reflected, bouncing directly from the surface. Experience tells us that a shiny, smooth surface tends to reflect a greater proportion of an incoming light beam than a roughened surface. Figure 2.11 illustrates the three possibilities for the behaviour of light falling, or incident , on a transparent material. Many transparent materials will only absorb tiny amounts of the light energy falling on them. For this reason, we will choose to ignore absorption in our discussions. However, it is important to note that no material is able to allow 100% of the incident light to pass through. There are no perfectly transparent materials; some reflection and absorption of the incident light will always occur.
Physics file
Figure 2.12 Multiple images are formed by the window, which is simultaneously
Euclid, a philosopher and mathematician (330–260 BC) described the law of reflection in his book Catoptrics . However, Euclid also upheld Plato and Ptolemy in their misguided belief in extramission . Euclid claimed that vision was possible because rays from our eyes spread out in all directions and fell on the objects that we see. He proposed that more rays fell on closer objects and so they were seen more clearly. Very small or distant objects were supposed to be difficult to see because they would lie between adjacent rays. Echoes of this idea continued well into the 14th century when vision was still described in terms of ‘extra-mitted’ visual rays emanating from the eye. Roger Bacon in the late 16th century, however, proposed that light actually travelled from the object to the observer’s eyes.
reflecting and transmitting light from outside and inside the shop respectively.
Figure 2.12 shows the effect of the transmission and reflection of light occurring simultaneously. If you look into a shop window you can often see an image of your own face and the streetscape behind you as well as the items on display in the window. The image of your face and the streetscape are the result of reflection: the window is acting like a mirror. However, you also see the items inside the shop as a consequence of the transmission of the light reflected from objects inside the shop through the glass.
The Medical Physics Detailed Study looks at the reflection, absorption and transmission of ultrasound waves.
PHYSICS IN ACTION
The pinhole camera The operation of a pinhole camera provides further evidence that light travels in straight lines. The operation of a camera is more easily explained using a ray model for light. Rays are used to represent the direction of travel of light through the camera. A pinhole camera consists of a sealed box with a small pinhole in the centre of one side. When open, the pinhole allows a limited amount of light into the camera and forms an image on the opposite inside wall of the box. If the opposite wall of the box is lined with photographic film, a permanent image can be developed. The amount of light that can enter the camera is determined by the size of the pinhole, and because this is small, the object to be photographed must either be well illuminated or be a luminous object itself. Typical exposure times
reach several minutes, so it is only practical to use a pinhole camera to photograph stationary objects. Ray tracing can be used to determine the size and nature of the image that will be produced on the film. First, consider only the uppermost tip of the object being photographed. An infinite number of rays can be considered to emanate from this point. Only a few of these rays will pass into the camera because of its tiny aperture. These rays continue on to strike the photographic film lining the back wall of the camera. These rays will not all fall on exactly the same point on the photographic plate, but if the pinhole is small, they lie sufficiently close together for an image to be formed. The geometry of the camera dictates that rays leaving the top of the object strike the bottom of the film.
Two models for light
33
Similarly, the rays from the bottom of the object strike the top of the film. This means that the image on the film is inverted relative to the object (Figure 2.13). screen, film or photographic plate
pinhole distant object
real inverted image
Figure 2.13 The pinhole camera. If the object is 10 times as far from the pinhole as the film is, then the size of the image will be one-tenth of the size of t he object.
The image in a pinhole camera is faint because only a little light has been allowed to reach the photographic film. To make a brighter image, a larger diameter pinhole might be used, but this will not produce satisfactory results. The image will be brighter but it will be blurred. This is because the larger hole will allow rays from one point on the object to strike different points on the film. We say that the rays are not focused .
2.1 SUMMARY
WAVE-LIKE PROPERTIES OF LIGHT
You can build your own pinhole camera using any container (card or metal) that can be sufficiently sealed to block out all light except that falling on the pinhole. It may help to paint the inside of the box matt black to prevent scattered light from reflecting off the walls and back on to the film. To make the pinhole, punch a nail hole in one wall and cover it with aluminium foil. A pinhole in the foil of about 1 mm diameter will produce good results. You will need to load the film and seal the box in darkness; it is a good idea to practise this a few times first. Alternatively, you can replace the wall opposite the pinhole with tracing paper or another translucent material to act as a viewing screen. This also needs to be shielded from exterior light so that the image is not flooded out. This can be done by surrounding this end of the camera with a cardboard tube. If a photograph is to be taken, mounting the camera on a stand is a good idea. The camera will produce best results with bright, distant objects. Outdoor scenery works well.
MODELLING SIMPLE LIGHT PROPERTIES
• Light travels in a straight path in a uniform medium. • A straight ray model of light implies its particle nature, but rays can also be used to represent the direction of travel of light waves. • When describing the reflection of light, light can be readily modelled as either a particle or a wave. • The law of reflection states that the angle of incidence is equal to the angle of reflection ( i = r ), and the incident ray, normal and reflected ray lie in the same plane.
34
On examining the geometry in the ray diagram for a pinhole camera, it is clear that a relationship exists between the distances from the pinhole to the object and image, and the relative heights of the object and image. It can be seen by using similar triangles that if an object is at a distance equal to 10 times the distance from the pinhole to the film, the height of the image will be one-tenth of that of the object.
• Smooth reflective surfaces produce regular (specular) reflection, whereas rough surfaces produce diffuse reflection. • Light can be reflected, transmitted and/or absorbed at the surface of a material. • Materials can be classified as transparent, translucent or opaque to the passage of light.
2.1 QUESTIONS 1
Describe three situations or phenomena that provide evidence for the statement ‘light travels in straight lines’.
2
On a particular day in Melbourne at noon, the Sun was at an angle of elevation of 70° above the horizon. Find the length of the shadow of:
3
4
5
a
a 10 m flagpole
b
a 1.8 m person
c
a 50 m building.
reflect light
B
emit light
i ii iii v vi b
C
transmit light
D
absorb light
a
Describe the construction of an ordinary plane mirror.
b
Under certain conditions, a double image can be seen in a mirror. Why?
8
the duco of a new car the surface of calm water a pane of glass aluminium foil matt paint on a wall frosted glass.
Why is it impossible to see an image of yourself in a sheet of paper?
An observer stands at position P, near a plane mirror as shown. Which of the objects A, B, C and D can be seen in the mirror?
B
P
A D C
Use the law of reflection to trace the path of the rays of light shown in the diagram. Calculate the angle of incidence and the angle of reflection at each surface. 9
a
b
40°
90°
6
Classify the following surfaces as producing diffuse reflection or regular reflection:
iv
A child has glow-in-the-dark stars on her bedroom ceiling. The reason they can be seen in a darkened room at night is because they: A
7a
60°
90°
Describe a situation where both the partial reflection and partial transmission of light occur. How can you tell that both phenomena are occurring at the same time?
10
Oceanographers refer to the region in the oceans in which some light from the Sun is able to penetrate as the ‘photic’ zone. If seawater is transparent, why doesn’t the photic zone extend to the ocean floor? Two mirrors are placed at right angles as shown in the diagram, and a small object is viewed in the mirrors. Draw the path for rays travelling from the object to the observer as they reflect from the mirrors. (Hint: there are three possible paths.)
eye
Two models for light
35
2.2
Refraction of light Refraction
See Heinemann ePhysics CD for an Interactive Tutorial on Refraction.
Light travels in a straight path if it is travelling in a uniform medium , but as soon as light enters a different medium its path may be bent. Evidence of the bending of light is shown in Figure 2.14 where a person’s face can be seen through a glass of water. Some of the person’s face can be seen directly. Light must be travelling along a straight path from the person’s face to the observer’s eyes. However, notice that parts of the person’s face can also be seen through the glass of water. The light rays from the person’s face passing through the glass of water have been re-directed or bent by the water towards the observer’s eyes. The bending or change of direction of light as it passes from one medium to another is called refraction .
Figure 2.14 Refraction occurs because the light changes speed as it enters a medium of different optic al density. In this case the light reflected fr om the person’s face is bent as it enters and leaves the glass of water. As a result the face is seen ‘inside’ the glass of water.
Various common phenomena are caused by refraction. Examples include the bend which appears in a straw that is standing in a glass of water (Figure 2.15), the strangely shortened appearance of your legs as you stand in a waistdeep swimming pool and the ‘puddles of water’ which you see on the road ahead on a warm day.
REFRACTION is the bending of the path of light due to a change in speed as it enters a medium of different optical density.
Figure 2.15 The refraction of light makes the straw appear to have a bend in it. The appearance of the straw is explained in Figure 2.20.
36
WAVE-LIKE PROPERTIES OF LIGHT
To fully understand these phenomena, the refraction of light can be investigated by using a block of glass and a narrow beam of light. Figure 2.16 shows a light beam travelling through air and entering a semicircular glass block. When light strikes the surface of a material some of the incident light is reflected, some is transmitted and some is absorbed by the material. The transmitted ray deviates from its original direction of travel. This change in
direction occurs at the boundary between the air and the glass, and the ray is said to have been refracted. Refraction occurs because the light changes speed as it enters a medium of different optical density . Later discussion will examine the speed of light in different media and how greater changes in speed cause more significant deviation of the beam.
Refraction and a ray/particle approach The refraction of light can be represented using a ray/particle approach. The ray of light which strikes the boundary between two media is called the incident ray . A normal to the boundary is drawn at the point where the incident ray strikes. The angle between the normal and the incident ray is called the angle of incidence , i . The angle between the normal and the transmitted or refracted ray is called the angle of refraction , r . The incident ray, normal and refracted ray all lie in the same plane (Figure 2.17). The angle of deviation , D , is the angle through which the ray has been deviated; hence D = (i − r ). Refraction is only noticeable if the angle of incidence is other than 0 °. If the incident ray is perpendicular to the boundary, i.e. i = 0°, the direction of travel of the transmitted ray will not deviate even though the speed of light has altered. An example of this can be seen in Figure 2.16 as the ray leaves the prism and continues in a straight path. When light travelling through air enters a more optically dense medium such as glass (in which it must travel more slowly), it will be refracted so that the angle of refraction is smaller than the angle of incidence. We say that the path of light has been deviated ‘towards the normal’. When light passes from glass to air it speeds up, as it has entered a less optically dense medium. The angle of refraction will be larger than the angle of incidence. The path of light is described as being refracted ‘away from the normal’. The behaviour of light undergoing refraction can be summarised by two statements. • When a light ray passes into a medium in which it travels slower (a more optically dense medium) it is refracted towards the normal.
Figure 2.16 When a light ray strikes the surface of a glass prism the transmitted ray is refracted because of a change in speed of the light. The bending occurs at the boundary of the two media.
angle of incidence i D r
angle of deviation angle of refraction
Figure 2.17 The angles of incidence, refraction and deviation are defined as shown. If an incident ray is perpendicular to the boundary between two media, i.e. i = 0°, the direction of travel of the ray does not deviate. An example of this can be seen as light leaves the glass prism.
• When a light ray passes into a medium in which it travels faster (a less optically dense medium) it is refracted away from the normal.
The path of refracted light is ‘reversible’. Figure 2.18 shows a ray of light incident on the left-hand side of a rectangular prism. It undergoes refraction towards the normal at the air–glass boundary. It then continues in a straight path through the glass until it strikes the glass–air boundary where it is refracted away from the normal. At each boundary the ray’s path deviates through the same sized angle ; hence, the ray which finally emerges from the prism is parallel to the original incident ray. If the light ray was sent in the opposite direction through the prism, i.e. if the starting and finishing points of the light ray were swapped, the light ray would trace out this same path— but in reverse.
Refraction and the wave approach Although the way in which light is reflected is modelled equally effectively using either a particle or wave approach, this is not the case with respect to
Figure 2.18 The path of light through any optical element is reversible since the amount of deviation at any boundary is determined by the change in speed of the light.
Two models for light
37
refraction . The wave model is better able to explain the change in direction that is observed as light enters a medium in which its speed is altered.
PRACTICAL ACTIVITY 10 Refraction of continuous water waves
(a)
Consider a light wave to be travelling at an angle towards a boundary between two media as shown in Figure 2.19a. For example, the light may be travelling from air into water. As soon as the light wave enters the water it will slow down . At the moment shown in the diagram the section of wavefront AB that first enters the water will be travelling at a slower speed than the section of the same wavefront BC that has not yet entered the water. This first section of the wavefront then effectively lags behind the position that it would have held had it been able to continue at its initial faster speed. The overall result of this delay is that the direction of travel of the overall wavefront is altered. Figure 2.19b shows how, once a number of wavefronts have passed into the second medium, the direction of travel of the overall wave has been deviated from its original course. A similar arrangement can be constructed for the refraction of light as it speeds up . S
original direction of travel
(b)
C
S
air
wavefront air (fast medium) R
C
R water
B A
B
water (slow medium)
A
new direction of travel
Figure 2.19 The change in the direction of light that is associated with a change of speed is called refraction. Refraction can be modelled by treating light as a wave.
The bent straw
O
Figure 2.20 The immersed portion of the straw is apparently shifted upwards due to refraction. This is because the rays appear to have come from a raised position in the glass of water.
38
WAVE-LIKE PROPERTIES OF LIGHT
Objects partially immersed in water will be distorted because of refraction; that is, they will appear to have a kink in them. The photograph of the ‘bent’ straw illustrates this. Figure 2.20 shows the path of the rays which produce this illusion. As each ray of light emitted from the base of the straw encounters the water–air boundary it is refracted away from the normal, since the ray enters a less optically dense medium. The observer perceives these rays from the base of the straw to have come from a position higher up in the glass of water. Ray tracing can be carried out for each point along the straw, resulting in the image shown. The straw appears to have a bend in it because the part of the straw which is submerged appears closer to the air–water boundary.
Apparent depth Just as the position of the submerged portion of the straw is apparently shifted, the actual depth of a body of water, or any transparent substance, cannot be judged accurately by an external observer because of refraction effects. Young children often jump into a pool believing it to be much shallower than it really is. Have you ever reached for an object at the bottom of a body of water and been surprised to find that you can’t reach it?
Consider an object O at the bottom of a pool as shown in Figure 2.21. Rays from the object are refracted away from the normal at the water–air boundary. If the observer looks down into the water from directly above, he or she will perceive these rays to have come from a closer position as shown. Since the floor of the pool will appear to be much closer than it really is, the water seems safely shallow. The extent to which the depth of the water is altered is affected by the angle from which it is viewed.
observer
Physics file Although ray diagrams are commonly used to represent the refraction of light, keep in mind that these describe only the direction of travel of the refracted light and do not tell the whole story. Although wave diagrams of refraction are more complicated, they are more convincing since they reveal the reasons for the change in direction of travel of the light.
air water apparent depth of pool
O
O
Figure 2.21 The apparent depth of a body of water appears less than its actual depth because of refraction. Light rays from a point of the bottom of the pool are refracted at the water’s surface. The observer perceives these rays to have come from an elevated location and interprets this as an i ndication of shallow w ater.
Worked example 2.2A . Predict the approximate path of light through each of the following prisms. In each case identify the normal, angle of incidence and angle of refraction. a
b
air
air
glass
Perspex
Two models for light
39
c
d
air glass
air glass water
Solution a
b
r
i r
i
air
glass
Perspex c
d
i
air
air
glass
glass i
water
r
r
At each boundary between media the light must refract either towards or away from the normal to the boundary. The light ray is refracted towards the normal on entering each prism and away from the normal when leaving the prism. When an incident ray meets a boundary with an incident angle of 0°, no deviation occurs.
The law of refraction Physics file Willebrord Snell (1591–1626) is commonly accredited with the discovery of the law of refraction around 1621. He did not immediately publish his findings and meanwhile the French scientist René Descartes (1596–1650) published his own derivation of the law of refraction. This caused a dispute within the scientific community at the time, with some claiming that Descartes had seen Snell’s work. It is worth noting that in France the law of refraction known elsewhere as Snell’s law is called Descartes’ law.
40
WAVE-LIKE PROPERTIES OF LIGHT
Scientists spent many years trying to find the relationship between the angle of incidence and the angle of refraction produced at the boundary between a given pair of media. At very small angles, doubling the angle of incidence appeared to double the angle of refraction, but this relationship does not hold for larger angles of incidence. Around 1621 Willebrord Snell, a Dutch scientist, found that for a given pair of media, the sine of the angle of incidence was directly proportional to the sine of the angle of refraction, i.e. sin i ∝ sin r . This relationship is now known as Snell’s law: sin i . . . . . . . . . . . . . (i) = constant sin r Each combination of a pair of materials has a different constant. For example the air–water interface has a different constant than the glass–water interface. The constant is called the relative refractive index , denoted n *. Literally, an index (or listing ) of numbers was created to describe the amount
of refraction or bending occurring at the boundary of numerous pairs of transparent media. A higher relative refractive index for a given pair of media indicated that more bending occurred. This was a cumbersome system which was later refined, as will be explained later in this chapter. sin i sin r Each pair of media will have a specific relative refractive index. RELATIVE REFRACTIVE INDEX: n * =
Worked example 2.2B . A student shone a thin beam of light on to the side of a glass block. She noted that when the angle of incidence was 40 °, the light passed into the block with an angle of refraction of 25°. Furthermore, when the angle of incidence was 70° the angle of refraction was 39°. Determine the relative refractive index, n *, of the air–glass boundary.
Solution The first pair of data gives: sin i sin 40° n * = = = 1.5 sin r sin 25° The second pair of data gives: sin i sin 70° n * = = = 1.5 sin r sin 39° i.e. the same relative refractive index, as expected.
Optical density and the speed of light Different transparent media allow light to travel at different speeds. Light travels fastest in a vacuum, more slowly in water and even more slowly in glass. We say that glass is more optically dense than water. Table 2.1 shows the speed of light in various media. The amount of refraction occurring at any boundary depends upon the extent to which the speed of light has been altered, i.e. the ratio of the two speeds of light in the two different media. Figure 2.22 represents light travelling in air and meeting three substances of different optical density. Light bends most when its speed is most significantly altered. The medium carrying the incident ray is identified as medium 1 and the medium carrying the refracted ray is medium 2. The angle of refraction depends on the speed of light in the two media and the angle of incidence. sin i v 1 . . . . . . . . . . . . . (ii) = sin r v 2
air water
air diamond
air glass
least refraction
most refraction
Figure 2.22 In each case light is entering a medium of greater optical density. The bending of the path of the light depends on the ratio of the speeds in the two different media. The path of light deviates most when the change in speed is greatest.
where v 1 is the speed of light in medium 1 and v 2 is the speed of light in medium 2.
The index of refraction Since it is only the ratio of the speeds of light in the two different media which determines the degree of refraction, each medium can be allocated an absolute refractive index , n . This is obtained by comparing the speed of light in the medium in question with the speed of light in a vacuum:
Table 2.1 The speed of light in different media (quoted for yellow light Medium
Vacuum Air Water Glass Diamond
λ=
589 nm). Speed of light (m s–1)
3.00 × 3.00 × 2.25 × 2.00 × 1.24 ×
108 108 108 108 108
Two models for light
41
Table 2.2
Absolute index of refraction, n (quoted for yellow light λ = 589 nm).
n =
Medium
c = speed of light in a vacuum = 3.0 × 108 m s−1
Index (n )
Vacuum Air Ice Water Quartz Light crown glass Heavy flint glass Diamond
1.0000 1.00029 1.31 1.33 1.46 1.51 1.65 2.42
PRACTICAL ACTIVITY 11 Investigating refraction: Snell’s law
speed of light in a vacuum speed of light in the medium
c n = v medium
By definition the refractive index of a vacuum would be exactly 1. Light travels only marginally slower in air and so the refractive index of air is 1.0003, but in most cases a value of 1.00 is sufficiently accurate. Materials in which light travels slowest will have the highest indices of refraction. For example, if a particular medium allowed light to travel at half the speed it does in a vacuum, then the refractive index of the medium would be 2. The refractive index can therefore be considered an indication of the ‘bending power’ of a material. Table 2.2 lists the absolute refractive indices for various media. By definition of the absolute refractive index: v 1 c n 2 c × v 2 = n 1 Hence, Snell’s law can now be expressed as: sin i n 2 = sin r n 1
. . . . . . . . . . . . (iii)
Living nearly 200 years before scientists were able to measure the speed of light in air with any degree of accuracy, Snell was not aware that the refractive index was linked to the speed of light in a particular medium. We can now combine the equations (i), (ii) and (iii) developed above so that Snell’s law is fully expressed. The Heinemann ePhysics CD Interactive Tutorial Refraction allows you to check out Snell’s law.
SNELL’S LAW:
sin i v 1 n 2 = = = n * sin r v 2 n 1
Worked example 2.2C . A ray of light passes from air into quartz whose absolute refractive index is 1.46. If the angle of incidence of the light is 40°, calculate: a
the angle of refraction
b
the angle of deviation of the ray
c
the speed of light in the quartz.
Assume the index of refraction of air is 1.00 and the speed of light in air is 3.0 × 108 m s−1.
Solution Draw a diagram to model the situation. As the light is slowed down the rays should bend towards the normal .
N i = 40°
a
medium 1 = air medium 2 = quartz
boundary
r
angle of deviation
42
WAVE-LIKE PROPERTIES OF LIGHT
List the data: i = 40° n 1 = 1.00 n 2 = 1.46 r = ? sin i n 2 Then: = sin r n 1 sin 40° 1.46 = sin r 1.00
Hence ∴
1.00 × sin 40° 1.46 r = 26°
Physics file
sin r =
The refractive index is also dependent on the colour of the light travelling through the medium. Since white light is made up of light of different colours, the refractive index must be quoted for a specific wavelength of light. Typically yellow light of wavelength 589 nm is used since this can be considered an average wavelength of white light. If the refractive index for a particular sample of crystal quartz was quoted as 1.55, red light would have a slightly higher refractive index of 1.57 and violet light would have a slightly lower refractive index of 1.54. This observation produces dispersion, discussed later in the chapter.
Note that r is smaller than i , as expected. b
The D = = =
c
n 1 = n 2 = v 1 = v 1 = v 2
angle of deviation is equal to the difference between i and r . (i − r ) 40° − 26° 14° 1.00 1.46 3.0 × 108 m s−1 n 2 n 1
1.46 3.0 × 108 = 1.00 v 2 v 2 = 2.1 × 108 m s−1 PHYSICS IN ACTION
Huygens’s wavelets and refraction In 1678 the Dutch mathematician Christiaan Huygens published his ideas on the nature and propagation of light. (At about the same time Newton developed his corpuscular theory.) Huygens’s idea was that light acted like a wave. In his model he suggested that each point along a wavefront of light could be considered to be a point source for small, secondary wavelets. Each wavelet was spherical, and the wavelets radiated from their point source in the general direction of the wave propagation (i.e. the light beam). The envelope or common tangent of the wavelets then became the new wavefront as shown in Figure 2.23.
incident light
λ 1
λ 1 A λ 2
θ1
boundary
θ2
B
λ 2
refracted light
Figure 2.24 Huygens’s approach allowed the refraction of light—as quantified by Snell’s law—to be acc urately modelled.
ray
source
ray
initial wavefront ray new wavefront
Figure 2.23 The envelope of the wavelets caused the for mation of the new wavefront.
Figure 2.24 shows how Huygens’s principle can be used to explain refraction . In the initial medium the spacing between wavefronts is λ 1. In the second (slower) medium the wavelength, λ 2, will be reduced in correspondence with the ratio of the velocity of light in each medium such that: λ 1 v 1 λ 2 = v 2 Consider the wavefront that is just approaching the boundary labelled AB and the wavefront just leaving it. Using the right-angled triangle drawn above the boundary (with hypotenuse AB), we can state: λ 1 sin θ1 = AB
Two models for light
43
Using the right-angled triangle below the boundary with hypotenuse AB, we can state: λ 2 sin θ2 = AB
Hence Snell’s law, which states that: sin θ1 λ 1 v 1 sin θ2 = λ 2 = v 2 can be derived from a wave model of light.
Therefore: sin θ1 λ 1 AB λ 1 sin θ2 = AB × λ 2 = λ 2
Refraction in the atmosphere apparent position of star
light from star is refracted as it travels through the atmosphere
real position of star
layers of increasing optical density
Figure 2.25 The refractive index of the atmosphere is not uniform; hence, the path of light from a star is refracted, apparently altering its position.
Although air has previously been considered to be a uniform medium, there are circumstances when in fact it is not uniform. Consider the envelope of air surrounding the Earth. The atmosphere is not uniform: the optical density of air increases closer to the Earth’s surface. Although this variation occurs gradually, the situation can be represented by a series of horizontal layers of increasing refractive index as shown in Figure 2.25. As light from an object such as a star travels through each boundary between layers it is refracted towards the normal. The observer therefore believes the light to have come from a higher position in the sky. In reality the bending must occur gradually rather than at distinct intervals, and the amount of bending has been greatly exaggerated in the diagram. A maximum amount of refraction occurs when objects are quite low in the sky. This is because the angle of incidence on the atmosphere is greatest and the light must travel through a wider atmospheric band. The amount of refraction is not noticeable to the human eye as when a maximum amount of refraction occurs stars would only be shifted in position by less than 1°. Refraction by the atmosphere also extends the length of the day. Whenever you watch a sunset you see the Sun for a few minutes after it has actually passed below the horizon. This is because light from the Sun is refracted as it enters and travels through the Earth’s atmosphere as shown in Figure 2.26. This effect is greatest at sunset and sunrise when the angle of incidence of the Sun’s rays on the atmosphere is greatest. The atmosphere consists of moving layers of air and so the optical density of the layers is continually changing. Since light rays must travel through this varying medium, light from the one object will follow slightly different paths at different times. This is one reason for the twinkling of stars and the apparent wriggling of distant objects on a warm day. Sun appears to be on the horizon atmosphere position of observer
actual path of sunlight
Figure 2.26 The Sun is still visible even though it is actually below the horizon.
44
WAVE-LIKE PROPERTIES OF LIGHT
PHYSICS IN ACTION
Mirages Figure 2.27 Desert mirage. When air is heated from below, an inferior mirage can occur where an image is displaced downwards. This diagram is simplified. The air layers of different temperature will not be uniform nor parallel to the plane of the ground. Light from the sky is gradually refracted by the air layers so that it is travelling slightly upwards when it enters the eye. The diagram exaggerates this bending. The observer sees an image of the sky on the distant road ahead and interprets this as a body of water.
Mirages are often believed to be the insane illusions of thirsty desert wanderers, but a mirage is the image of a real object and can be explained in terms of the refraction of light. A mirage is a displaced and often distorted image occurring when layers of air of different temperature cause the path of light to bend. The severity and consistency of this temperature gradient determines many features of the observed apparitions. An inferior mirage refers to the downward displacement of an image. A superior mirage means the image is displaced upwards . The following discussion examines just a few of the many different types of mirages that occur.
Inferior mirages An inferior mirage occurs when the air at ground level is warmer than the air immediately above, i.e. the air is being heated from below. This situation often arises in the afternoon of a hot, sunny day above a black bitumen road or above the sands of the desert. Air of higher temperature is less optically dense and hence has a lower refractive index. Light from the sky is refracted as it travels through the layers of air of different optical density as shown in
light from sky
cool air hot air mirage
Figure 2.27. As a result the light ray is travelling upwards as it enters the observer’s eyes and the image is then seen on the ground ahead. Driving on a warm day, you often see an image of the sky on the road ahead; this is interpreted as a body of water.
Floating on water A fascinating inferior mirage often occurs above shallow bodies of water in the early morning. The water retains its heat overnight but the surrounding land does not. Cool air from above the land flows over the warmer water and is heated from below. Thus air temperature decreases with height. However, the temperature gradient is not uniform. The temperature drops quickly near the water’s surface, but at greater heights the decrease in temperature is more gradual. If an observer examines a person in the distance the image of the person is displaced downwards but the bottom of the object is displaced more than the top of the object since these lower rays travel through a stronger temperature gradient. Thus the person is irregularly enlarged. This phenomenon is called towering .
Figure 2.28 The picture on the left shows a mirage on a road. In the picture on the right, the horizon shows the effect of towering.
Two models for light
45
2.2 SUMMARY
REFRACTION OF LIGHT
• Refraction is the bending of light due to a change in speed as it enters a medium of different optical density. The greater the change in speed of light the greater the bending. • Adopting a wave model for light allows the direction of bending to be explained. For convenience, rays are used to represent the direction of travel of the light waves. • The angle between the normal and the incident ray is called the angle of incidence. The angle between the normal and the transmitted or refracted ray is called the angle of refraction. • When a light ray passes into a medium in which it travels more slowly (a more optically dense medium) it is refracted towards the normal. When a light ray
passes into a medium in which it travels faster (less optically dense) it is refracted away from the normal. • Each transparent medium is allocated an absolute refractive index, n , determined by the speed at which light can travel in the medium compared with the speed of light in a vacuum, c : c n = v medium • Snell’s law describes the relationship between the angle of incidence, i , and the angle of refraction, r , for a given pair of media. Medium 1 carries the incident ray. Medium 2 carries the refracted ray. v is the velocity of light in the medium and n is the absolute refractive index of the medium: sin i v 1 n 2 sin r = v 2 = n 1 = n *
2.2 QUESTIONS 1
a
The figure below represents a situation involving the refraction of light. Which of the lines labelled A–E is: i ii iv
b
A
B i
the boundary between two media? the normal?
iii
the refracted ray?
v
the incident ray?
i
glass medium A
the reflected ray?
glass medium B
Explain what happens to the speed of light as it crosses the boundary between medium 1 and medium 2. How do you know? A
B C
C
D i
i
glass medium C
glass medium D
medium 1 D medium 2
3
Using Figure 2.18 as your reference, show how a wave model can be used to explain the refraction of light as it passes through the boundary into a medium in which its speed is increased.
4
Explain the following observations.
E 2
The following diagrams show light passing from glass into different media labelled A, B, C and D. a
Which media are more optically dense than the glass?
b
Which medium has a refractive index less than the refractive index of glass?
c
Which medium has the highest refractive index?
d
46
Which medium has a refractive index very close to the refractive index of this sample of glass?
WAVE-LIKE PROPERTIES OF LIGHT
5
a
When you are standing in a shallow pool you appear shorter than usual.
b
On a warm day a person sees a ‘puddle’ on the road ahead.
When light passes through a plane of glass it is refracted. This does not cause the distortion of an image seen through the glass because:
A
the emerging rays are perpendicular to the incident rays
B
the index of refraction of glass is too small to cause distortion
C
D 6
a
a
What is the relative refractive index for light passing from water into diamond if an incident angle of 30° produces an angle of refraction of 16°?
b
Light travels from water (n = 1.33) into glass (n = 1.60). The incident angle is 44°. Calculate the angle of refraction.
the displacement of light rays is too small to be noticed unless the glass is very thick most of the light is reflected, not refracted
9
The speed of light in a particular transparent plastic is 2.00 × 108 m s−1. Calculate the refractive index of the plastic. The speed of light in a vacuum is 3.00 × 108 m s−1.
A ray of light is incident on the surface of water in a fish tank. The incident ray makes an angle of 32.0° with the surface of the water. The light that is transmitted makes an angle of 50.4° with the surface. Calculate: a
the angle of incidence
What is the speed of light in water (n = 1.33)?
b
the angle of refraction of the transmitted light
A student wishes to determine the refractive index of a particular sample of glass by experiment. By passing a narrow beam of light from air into the glass, she measures the angles of refraction, r , produced using a range of incident angles, i . Her results are shown.
c
the angle of reflection
d
the angle of deviation of the transmitted ray.
b 7
8
i (degrees)
25 30 35 40 45 50 55 60 65
10
A ray of light travels from air, through a layer of glass and then into water as shown. Calculate angles a, b and c. air (n = 1.00)
glass (n = 1.50)
water (n = 1.33)
r (degrees) 16 19 22 25 28 31 33 35 37 a
Plot a graph of sin i versus sin r .
b
Determine the gradient of the graph, i.e. the relative refractive index of light passing from air into glass.
c
d
Assuming the refractive index of air is 1.00, what is the refractive index of the glass sample used? Calculate the velocity of light in the glass.
40° a b c
Two models for light
47
2.3
Applications of refraction Physics file
When light refracts at a surface the transmitted transm itted ray ray becomes becomes less intense intense as the angle angle of of incidence incidence increase increases. s. Place Place any small flat piece of glass on this page; a microscope slide will do. Look onto the slide from directly above and you will easily observe the writing below. Now move your head so that you are looking through the slide from a gradually decreasing angle of elevation. The page beneath the glass should become gradually darker. The amount of light transmitted transm itted through through the glass glass and and towards toward s your eyes is becom becoming ing less. less.
= 45° 45° i = water air = 70° 70° r =
= 49° 49° i = water air
90°° r < ∼ 90
Critical angle and total internal reflection When light is incident upon the boundary between two media reflection, transmission and absorption may occur. occur. As the angle of incidence increases, the intensity of the reflected beam increases and less light is transmitted. Consider the case where light is travelling from water into air. Since the transmitted light enters a less optically dense medium it travels faster and is refracted away from the normal . The series of diagrams in Figure 2.29 shows the effect that increasing the angle of incidence has on the transmitted light. As the angle of incidence is increased, the angle angle of refraction also increases. increases. This continues until, at a certain angle of incidence called the critical angle, i c , the angle of refraction will be almost 90° and the transmitted ray travels just along the surface of the water. The The angle of refraction can increase no further. If the angle of incidence is then increased beyond the critical angle no ray is transmitted and total internal reflection will occur occur.. This is appropriately named, as none of the incident light energy is able to be transmitted into the next medium; it is totally reflected into the medium carrying the incident ray. In effect, as the angle of incidence increases, the intensity of the reflected beam gradually becomes stronger, stronger, until at an angle of refraction of more than 90° all of the light is reflected and no light is transmitted at all. The critical angle can be found for any boundary between two media by using Snell’s law. If the refractive indices of the two media are known, a presumption of an angle of refraction of 90° allows the critical (incident) angle to be calculated: calculated: sin i n 2 = sin r n 1 If the incident angle is equal to the critical angle, i.e. i = i = i c , then r = 90°. The above equation becomes: n 2 sin i c = n sin 90° 1 Now sin 90° = 1; hence the critical incident angle is given by: n 2 sin i c = n 1
50°° i = 50 water air
Figure 2.29 The critical angle for light travelling from water into air is approximately 49°. If the incident angle is greater than 49° total internal reflection occurs.
PRACTICAL ACTIVITY 12 Total internal reflection in prisms
48
WAVE-LIKE PROPERTIES OF LIGHT
The CRITICAL ANGLE , i c, is the angle of incidence which produces an angle of refraction of 90° 90° as light is transmitted into a medium in which it travels at a higher speed. n 2 sin i c = n 1
In a situation where light is travelling from a slower medium into air the above equation can be simplified, since n 2 (air) = 1.00. 1 Therefore, sin i c = n 1
Worked example 2.3A . An underwater light shines upwards from the centre of a swimming pool which is 1.50 m deep. Determine the radius of the circle of light which is seen from above. (n (n air = 1.00, n water = 1.33)
Solution
Physics file
Step 1. 1. Determine the critical angle for the water–air boundary. n air sin i c = n water sin i c =
1.00 1.33
i c = sin−1 =
1.00 1.33
i c = 48.8 48.8°° Step 2 . Draw a diagram to represent the situation. Considering one-half of the cone of light produced, since the critical angle is 48.8° 48.8 °, the angle QOP is 48.8° 48.8°.
R
air
P
Q
water 48.8°° 48.8 1.50 m 48.8°° 48.8
Step 3 . Using trigonometry trigonometry,, R 1.50 hence R = 1.50 tan48.8 tan48.8°° = 1.71 m
tan48.8°° = tan48.8
light source
Since the refractive index of any given medium depends on the colour of the light travelling through the medium, each colour of light will have a slightly different critical angle. For example, if the critical angle for red light travelling from glass to air was 40.5° 40.5°, then yellow light will have a slightly smaller critical angle of 40.2° 40.2°. Violet light would have an even smaller smaller critical angle of 39.6° 39.6°. Although these values are very similar, they are sufficiently different to cause the dispersion of white light into its component colours—the colours of the rainbow!
(a)
The radius of the circle of light is 1.71 m.
Optical instruments Total internal reflection is used in many optical instruments, including cameras, periscopes and binoculars. When light reflects from a mirror there is always some loss in intensity of the incident light and reflection can occur from both the front and rear surfaces of the mirror, causing problems. By contrast, almost no loss of intensity occurs with total internal reflection. Since the refractive index for glass is around 1.5, the critical angle for light travelling from glass to air is approximately 42 °, and a glass prism with internal angles of 45° can be used as a mirror in the applications discussed below. below.
(b)
eyepiece
Figure 2.30a shows the construction of a simple periscope. Light enters the top glass prism perpendicular to the glass surface and so no refraction (deviation) occurs at this stage. The light passes through the prism and strikes the back surface at an angle of 45°. The angle of incidence is greater than the critical angle and so the light can only be totally internally reflected. The light travels down the tube of the periscope, enters the lower prism and is again reflected. The surfaces of the prism do not need to be silvered for reflection to occur. Binoculars use a compound prism which is constructed of four 45 ° prisms (Figure 2.30b). The light actually undergoes four reflections on its passage from the objective lens to the eyepiece. This lengthens the path that the light must travel and hence more compact binoculars can be made for a given magnification. Today’s binoculars are as effective as the telescopes of the past, which had to be be many times longer. longer.
Optical fibres
objective
Figure 2.30 Good quality periscopes and binoculars use 45° glass prisms for the total internal reflection of light rather than mirrors. This means reduced loss in the intensity of light and eradicates the problems caused by reflection occurring at both the front and rear surfaces of mirrors.
An optical fibre uses total internal reflection to carry light with very little energy loss. Since the 1950s fibre optics has been used in the flexible fibrescope, a device that allows doctors to see inside the human body. body. In 1968 optical fibres were proposed as carriers of information for communication. Two models for light
49
However, at the time there were numerous technological problems, the main However, one being the loss of considerable light energy as the signal travelled along the fibre. By 1970 the production of very pure fibres reduced energy losses, making fibre optics feasible. Since then their efficiency has improved even more and this technology has made it possible to transfer large amounts of data at remarkable speeds, enabling the dream of a worldwide computer network to be realised. With rapidly increasing computer usage and the growth of the Internet we have come to rely heavily on optical communication systems. Concurrently a whole group of opto-electronic devices have been created that will create data suitable for use in these systems. There has been continuing pressure to continue to improve the capacity and data transfer speeds of these communication systems, and the technology in this field is literally changing day by day.
The transmission process Essentially optical fibres carry information in a digital format; that is, light signals turn on and off at very fast rates. Optical communication systems take information from a device, convert it into a digital form if needed, and impress this digital signal onto a carrier frequency that has been produced by a laser or LED. This This is called modulation. The signal is then fed into an optical fibre for transmission (coupling). A critical factor is the range of frequencies of light that the optical fibre is able to carry efficiently. Each fibre has a limiting bandwidth (frequency range) and a set number of different signalwavelengths that are allowable in this bandwidth. For example, the different signal wavelengths employed may be not allowed allowed to be any closer than than about 0.3 nm in spacing. (See later discussion.) During transmission attenuation (energy losses along the way) will occur. Therefore repeater stations are used to receive the weak incoming signal and boost it before sending it further along the fibre. Regenerators may also be used to remove noise and distortion in the signal at this stage. Depending on the quality of the fibre, repeater stations are inserted about every 100 km along a long-distance optical cable, but with improvements in optical fibres and the power of signals, they will be needed less frequently in future.
A magnified optical fibre torch.
When a signal reaches its destination it must be de-modulated (removed from its carrier wave) and the signal processed as required by the end user. The signal must arrive with only as much distortion as can be compensated for at this end. As we will see later, it is this ‘end state’ state’ of the signal which limits how fast (that is how densely) data can be sent along the fibre.
Different types of fibre Optical fibres can be divided into two categories depending on the manner in which information is carried: single-mode (thin core) fibres and multimode (large core) fibres. The term ‘mode’ is synonymous with pathway. A single-mode fibre allows only one path on which light can travel. It contains a central core of glass of a relatively high refractive index surrounded by a layer of glass of lower refractive refractive index called the cladding . A coating made of a spongelike, shock-absorbing plastic designed to cushion the fibre from any impacts protects the glass fibre (Figure 2.31a). Note the tiny dimensions of the fibre. A micron is one-millionth of a millimetre, and a human hair is typically about 70 microns in diameter diameter.. 50
WAVE-LIKE PROPERTIES OF LIGHT
Single-mode fibres are used in high-speed, long-distance telecommunications. For example, Melbourne and Sydney are linked by optical cable containing numerous single-mode fibres. Although single-mode fibres are more difficult, and therefore more expensive, to produce, they allow only one path for light thereby avoiding a complication called modal dispersion . Also single-mode fibres allow a larger bandwidth (i.e. range of signal frequencies that can be transmitted within the distortion limits), but due to their dimensions the coupling (feeding the signal in) to these fibres is more difficult. A multimode fibre can have two different forms: the step-index multimode fibre or the graded-index multimode fibre. A step-index multimode fibre has the same structure as the single-mode fibre described above, but it has a much larger core made of uniform glass (Figure 2.31b). A graded index fibre also has a large core, but its refractive index gradually decreases from the centre to the outer diameter of the fibre. (a)
(b) core 8.3 micron
cladding 125 micron
core 50 100 micron
coating 250, 500 or 900 micron
cladding 125 or 140 micron
coating 250 900 micron
Figure 2.31 Compare the dimensions and structure of the different fibres. (a) Single-mode fibres, such as those used in highspeed telecommunications, result in much less distortion of the optical signal. (b) Multimode fibres, such as those used in local area networks, are cheaper to produce but have more distortion problems. These are adequate for use over shorter distances.
It is impossible to distinguish between single-mode and multimode fibres with the the naked eye; there there is no difference difference in their their outward outward appearance, only in the diameter of the core fibre. Typically Typically between 2 and 36 of these coated glass fibres are grouped together inside a tube with a gel-type substance between them. Each tube of fibres is called an optical unit . A cable is formed by placing optical units around a strength member, which bears the tension in the cable. The entire cable is covered with a further protective coating as shown in Figure 2.32. This outer plastic coating is designed to protect the fibre from the environment in which it is placed and so the material from which it is made depends on where the fibre is located. Optical cables can be designed to be placed under water, alongside overhead electrical wires, beneath the ground, inside walls or between computers in a room.
The path of light in i n a step-index multimode fibre Light rays are sent down the central core fibre. If the fibre is straight, most of the rays will travel along the axis of the fibre. Some light will strike the boundary between the core and the cladding, particularly if the fibre is bent. Any ray striking the boundary at an angle greater than the critical angle is totally internally reflected. The size of the critical angle is determined by the refractive indices of the core and cladding (see Figure 2.33).
Two models for light
51
(a)
(b)
strength-bearing member gel
Figure 2.32 (a) A tube containing a group of fibres is called an optical unit. Tubes typically contain between 2 and 36 fibres. An even number of fibres is always used used since each of the fibres of a pair carry light in opposite directions. (b) An optical cable is formed by placing optical units around a tension-bearing member. The strength of this member will depend on the distance which the cabler must span. The entire cable is covered by a plastic coating to protect it from the weather. weather.
Worked example 2.3B . A particular step-index multimode fibre has a core of refractive index 1.460 and cladding of refractive index 1.440. Calculate the critical angle of the core–cladding boundary of this optical fibre.
Solution List the data: n core = 1.460, n cladding = 1.440 n sin i c = 2 n 1 i c = sin−1 1.440 1.460 = 80.51 80.51°° cladding (n = 1.440)
81°° 81
rays striking here are lost
86° 86°
9°
cone of acceptance
light source
13°° 13 core (n = 1.460) 13°° 13 9°
rays striking here are lost
81°° 81
86°° 86
cladding (n = 1.440)
Figure 2.33 Outer rays from the light source which enter the core at an angle of incidence greater than 13° will continue on to strike the core–cladding boundary at an angle of incidence less than the critical angle of 81 °. These rays will therefore be transmitted into the cladding and be lost. Rays originally within the cone of of acceptance strike the cladding at an angle greater than 81 ° and are therefore totally internally reflected. These reflected rays carry the signal along the fibre.
52
WAVE-LIKE PROPERTIES OF LIGHT
The core and cladding are designed so that the critical angle for an optical fibre is typically above 80 °. Hence, only light rays undergoing glancing collisions with the core–cladding boundary are totally internally reflected. Although this causes more light energy to be lost at coupling, it means that all of the light rays emerging from the end of the fibre have travelled a path of only approximately the same length .
Modal dispersion In any multimode fibre the light rays from the one original pulse will arrive over a small time interval. The fact that the different paths (modes) that the light signals will take do differ in length, and therefore time, is called modal dispersion. Any spreading out in the time of arrival of a single pulse directly limits the number of pulses per second that can be sent along the fibre; that is, modal dispersion in multimode fibres is a limiting factor in the data transfer rates that can be achieved. Since the extent of modal dispersion will be greatly affected by the length of the cable, multimode fibres are adequate for use in short-distance communications such as between computers in adjacent buildings or throughout a multistorey building. The recent improvements in multimode fibres have been driven by the demand for increased data transfer capacity. One such improvement is in the structure of the graded-index multimode fibre. The central core of this fibre has the highest refractive index, but the refractive index (i.e. density) reduces radially outward to the cladding. Instead of following the zig-zagged path discussed earlier the rays will spiral more smoothly along the fibre. More importantly, importantly, rays spending more time near the central axis have a shorter path but their speed is slower due to the higher n value. Rays travelling a longer path by spending more time nearer to the cladding are able to travel more swiftly. This can significantly reduce the modal dispersion of a signal.
Material dispersion Multimode fibres often use an LED as their source of light but as LEDs do not emit light of a single, pure frequency they are problematic. In our discussion the refractive indices of materials have been stated as absolute values, but in fact we can do no more than state average n values—the refractive index of light in the middle of the visible spectrum, for example. The refractive index of a material is actually different for different frequenci frequencies es of light. This means that the degree of refraction of a given light signal is dependent upon the frequency of the light. Within a multimode fibre different frequencies of light will follow different paths and arrive at slightly different times, this is called material dispersion. This problem can be overcome by using single-frequency signals from lasers, instead of LEDs, as the light source. Alternatively signals of wavelengths that do not diverge much from a value of 1.3 µm can be used, since minimal dispersion occurs in silica glass with light around this wavelength. This is called minimising the bandwidth .
Energy losses in fibres The main area in which light energy is lost is at the source of light—where the light enters the fibre. Rays diverging markedly from the axis of the fibre will strike the core–cladding boundary at an angle smaller than the critical angle Two models for light
53
and so pass into the cladding. A ‘cone of acceptance’ occurs as shown in Figure 2.33. The fibre will not transmit rays outside this cone. Light energy is also lost at bends in the fibre since bending increases the probability that light will strike the core–cladding boundary at an angle less than the critical angle and so be transmitted into the cladding. Manufacturers quote a minimum bending radius for each fibre. When installed, fibres must not be bent beyond this value. Although fibres fibres cannot be bent sharply around around corners like electrical conducting wire, besides their immense data-carrying capacities, they do have other considerable advantages over electrical conductors. Optical fibres have a small diameter and high tensile strength; therefore smaller, lighter cables and connectors are used. Since glass is made from sand, an abundant resource, fibres are relatively cheap to produce. Fibres are immune to electrical interference and corrosion and do not produce an electromagnetic field. They can therefore be used in a range of environments in which electrical conductors cannot be used. 2.3 SUMMARY
APPLICATIONS OF REFRACTION
• As the angle angle of incidence incidence of light light onto a transparen transparentt surface is increased, proportionally more light is reflected and less light is refracted. • When light light enters enters a less opticall optically y dense medium medium it is refracted away from the normal. At the critical incident angle, i c, the angle of refraction is 90°. • If the incident incident angle angle is greater greater than the critical critical angle, angle, i c, total internal reflection occurs. • The critica criticall angle angle is given by: n 2 sin i c = n 1 • Optic Optical al fibres fibres carry information information in in a digital format format on a carrier frequency frequency.. • Sing Single-mod le-mode e fibres are used used for long distances; distances; step-
index multimode fibres and graded-index multimode fibres are used for shorter distances. • Optic Optical al fibres fibres have a cladding cladding of a lower lower refractive refractive index and light is carried along the fibre via total internal reflection. • Mod Modal al dispersion dispersion is the spreadi spreading ng out of the arrival arrival time of a single pulse of light due to the different path lengths taken by the light. This dispersion is reduced in graded-index fibres and does not apply to single-mode fibres. • Materi Material al dispersion dispersion is caused by the different different refractive refractive indices of different frequencies of light. Material dispersion is the limiting factor in determining the bandwidth of optical fibres.
2.3 QUESTIONS 1
2
54
Can total internal reflection occur as light strikes the boundary from: a
= 1.00) to glass (n = 1.55)? air (n (n = (n =
b
glass to air?
c
= 1.33)? glass to water (n (n =
d
= 1.55) to glass (n = 1.58)? glass (n (n = (n =
D
normal air glass
The ray is reflected only.
B
The ray is refracted only.
C
The ray is reflected and refracted.
WAVE-LIKE PROPERTIES OF LIGHT
i c
1
The diagram shows four rays incident on the boundary between glass and air. Ray 2 meets the boundary at the critical incident angle. For each of the rays 1–4 choose the option option which which best best describe describes s what what happens happens as it strikes the boundary. A
The ray is reflected and transmitted.
2 3 3
4
Determine the critical angle for light travelling from: a
= 2.42) into air diamond (n (n =
b
flint glass ((n n = = 1.60) into air
4
5
6
7
c
water (n = 1.33) into air
d
glass (n = 1.50) into water (n = 1.33).
8
The critical angle for light passing from oleic acid into air is 43.2°. Calculate the index of refraction of oleic acid. The speed of light in a particular sample of clear plastic is 1.80 × 108 m s−1. Determine the critical angle for light passing from this plastic into air. a
Explain the term modal dispersion and how it impacts upon data transfer rates in optical fibres.
9
10
c
Find out how optical fibres are used in medicine.
a
Explain how the effects of material dispersion in single-mode fibres can be minimised.
b
Explain the terms modulation, bandwidth, attenuation and cone of acceptance.
Use a diagram to explain why the over-bending of an optical fibre can result in energy losses. a
A particular step-index optical fibre has a core of refractive index 1.547 and cladding of refractive index 1.532. Determine the critical angle of incidence for light striking the core–cladding boundary.
b
Describe two approaches regarding optical fibres that can be adopted in order to reduce modal dispersion.
a
How are the reflecting prisms inside binoculars arranged to decrease the required length of each barrel?
b
If the core and cladding are more widely different in refractive index values, does this result in a larger or smaller critical angle?
b
In precise optical instruments prisms are used for reflection rather than mirrors. Why?
c
What are the advantages and disadvantages of a large critical angle for the core–cladding boundary?
Two models for light
55
2.4
Light described as an electromagnetic wave What is light? In the late 1600s it was known to involve the transfer of energy from one place to another. In Isaac Newton’s time a corpuscular model and wave model for light had seemed equally valid. We have discussed these two proposed models of light along with their respective explanations of the reflection and refraction of light. In spite of considerable endeavour on behalf of scientists it was not until the early 1800s that one model prevailed. Thomas Young discovered that sources of light were able to interfere with each other just like sound waves and water waves do. This finding led to a universally accepted wave theory for light. Furthermore, the speed of light could be measured for the first time in history, and the wave model of refraction that was discussed in Section 2.2 was validated. Meanwhile another area of physics had been developing. By the 1860s investigations being carried out on different forms of electromagnetic radiation led to the finding that visible light itself is just one of the many forms of electromagnetic radiation (EMR).
Electromagnetic waves Electricity and magnetism were once considered to be separate subjects. However, moving charges create magnetic fields. Similarly a changing magnetic field can be used to create electricity. In 1864 James Clerk Maxwell used mathematical equations to describe how charges moving periodically in a conductor would set up alternating electric fields and magnetic fields in the nearby region. Maxwell knew that the magnetic and electric fields travelled through space. He calculated their speed and found it to be 300 000 km s−1, exactly the same as the speed of light! Also, he devised mathematical expressions to describe the magnetic and electric fields. The solution to these expressions was found to be the equation of a wave . Maxwell has shown that light is an electromagnetic wave. Today we know that the electromagnetic spectrum includes a wide range of frequencies (or wavelengths). All electromagnetic waves are created by accelerating charges which result in a rapidly changing magnetic field and electric field travelling out from the source at the speed of light, as shown in Figure 2.34. Note that the electric field component and the magnetic field component are at right angles to each other and to their direction of travel. Electromagnetic radiation meets the description of a transverse wave as discussed in Chapter 1.
Figure 2.34 Since all electromagnetic waves travel with the same velocity the only thing that differentiates one form of EMR from another is the frequency (and, therefore, the wavelength). 56
WAVE-LIKE PROPERTIES OF LIGHT
The many forms of EMR are essentially the same, differing only in their frequency and, therefore, their wavelength. The electromagnetic spectrum is roughly divided into seven categories depending on how the radiation is produced and the frequency. The energy carried by the electromagnetic radiation is proportional to the frequency. High-frequency short-wavelength gamma rays are at the high-energy end of the spectrum. Low-frequency long wavelength radio waves carry the least energy. Humans have cells in their eyes which can respond to EMR of frequencies between approximately 400 THz and 800 THz; these frequencies make up the visible light section of the electromagnetic spectrum. Recall from Chapter 1 that for any wave the relationship between its frequency and its wavelength is given by: v = f λ. All electromagnetic radiation travels at a speed of 3.00 × 108 m s−1 in space and so this significant speed has been allocated the symbol c.
For all ELECTROMAGNETIC RADIATION f = where f is the frequency of the EMR (Hz) c is the speed of the EMR = 3 × 108 m s−1 λ is the wavelength of the EMR (m)
c λ
Figure 2.35 shows the different categories of EMR. Note the range of frequencies and wavelengths is enormous. The range of frequencies (or wavelengths) constituting visible light occurs near the middle of the spectrum. Our eyes cannot perceive any wavelengths of EMR outside of this range.
Gamma rays
Frequency
Wavelength
1022 Hz
10–14 m
1016 Hz
10–8 m
X-rays
Ultraviolet
Visible spectrum
700–400 THz
400–700 nm
Infrared
1012 Hz
10–4 m
Microwaves
1010 Hz
10–2 m
TV
108 Hz
10 m
Radio
106 Hz
102 m
The Medical Physics Detailed Study looks at the uses of X-rays and gamma rays in medicine. X-rays are used in CT scans. Gamma rays are used to image the body and treat tumours.
Figure 2.35 The electromagnetic spectrum. Two models for light
57
Worked example 2.4A . The EMR given off by a sample of sodium as it is burned has a wavelength of 589 nm. What is the frequency of this radiation? How would we detect the radiation?
Solution f = =
c λ 3 × 108 589 × 10−9
= 5.09 × 1014 Hz = 509 THz This frequency of EMR lies in the visible light section of the electromagnetic spectrum, therefore we would see it! It is actually yellow light. PHYSICS IN ACTION
Other forms of EMR Radio waves Accelerating a positive or negative charge can produce EMR. Electrons oscillating in a conducting wire, such as an antenna, produce the radio waves that bring music to your home. The long-wavelength low-energy electromagnetic waves blanket the surrounding region, and aerials can receive the signal many kilometres from the source. As a result of the radio waves, electrons in the receiving aerial wire will oscillate, producing a current that can be amplified. Radio waves can be transmitted over very long distances, including around the Earth’s surface, by reflection from layers in the atmosphere.
satellites create an image by sensing infrared radiation and converting it into a visible picture.
Ultraviolet waves Ultraviolet waves have wavelengths shorter than violet light—so our eyes cannot detect them—but no greater than about 10 nm. Many insects can detect the ultraviolet light that is commonly reflected from flowers. Although ultraviolet light is less energetic than gamma- or X-rays, it is known to cause skin cancer particularly with increased exposure.
Microwaves
Silicon atoms are able to absorb some frequencies in the ultraviolet region of the spectrum, reducing your chances of getting sunburnt through glass.
Microwaves are EMR of wavelengths ranging from about 1 mm to about 10 cm.
X-ray waves
The microwaves that cook your dinner are produced by the spin of an electron or nucleus. Microwave links are used to allow computer systems to communicate remotely and radar equipment uses microwave frequencies of centimetre wavelengths.
Infrared waves Infrared or heat radiation includes the wavelengths that our skin responds to. When you feel the warmth from the Sun or an electric bar heater you are actually detecting infrared radiation. All objects that are not at a temperature of absolute zero radiate EMR. The hotter the object the more radiation is emitted, and the further along the spectrum the radiation is. Night scopes and infrared spy
58
WAVE-LIKE PROPERTIES OF LIGHT
X-rays are produced when fast-moving electrons are fired into an atom. The name is a result of scientists not knowing what they were when they were first detected, hence the letter ‘X’. X-rays can pass through body tissue and be detected by photographic film, and so are used in medical diagnosis. They have extensive safety testing, security and quality control applications in industry.
Gamma-ray waves The highest energy, smallest wavelength radiation is the gamma ray, which is produced within the nucleus of an atom. Gamma rays are one of the three types of emissions that come from radioactive (unstable) atoms. Gamma rays are extremely penetrating and require dense material to absorb them.
Coloured light, different wavelengths Our eyes are responsive to many different colours of light from the deepest red through to the brightest violet, the visible spectrum (Figure 2.36). Each variation in colour or shade is caused by light of a different wavelength. Traditionally the colours quoted as making up the visible spectrum are red, orange, yellow, green, blue and violet. However, as shown in Figure 2.36, the actual allocation of separate names for the colours is difficult since they merge into one another. The wavelengths associated with visible light are very small: they range from approximately 390 nanometres (or 3.9 × 10−7 m) for violet light to around 780 nanometres for red light.
red wavelength ≈ 780 nm
violet wavelength ≈ 390 nm
Figure 2.36 Visible light is one category of EMR. The spectrum of visible light contains a myriad of colours. Each different colour or hue is light of a dif ferent wavelength.
The colour of an object that we see is actually a physiological response to the particular wavelength(s) of light entering our eyes. Our colour-sensing system, consisting of the eye, nerve conductors and the brain, can discriminate between hundreds of thousands of different colours. However, different combinations of wavelengths of light can evoke the same response from our brain. In other words, there are a number of different ways in which to make an object appear a particular shade of yellow, for example. These methods are explained below.
Viewing objects under white light Black, white and grey objects We perceive light as white light if it contains roughly equal amounts of each of the colours of the visible spectrum. The page of this book appears white because it is reflecting all of the colours (wavelengths) of visible light in roughly equal proportions. Sunlight, incandescent light and fluorescent light all produce the same general sensation of white light. Figure 2.37 shows their component colours. The light from incandescent and fluorescent globes does not appear to be quite as white as sunlight. This is because sunlight is very evenly distributed across the spectrum but an incandescent source radiates considerably more red light than blue light and a fluorescent source favours blue wavelengths of light.
Physics file You wouldn’t expect a person renowned as a great scientist to be superstitious. For many years the spectrum of colour was listed as being made up of seven separate colours rather than the six colours listed today. Isaac Newton carried out famous experiments producing the spectrum of colour and recombining it into white light. In his writings indigo (a very dark blue) was stated as lying between blue and violet. The separate identification of indigo light is strange as it really does not appear as prominently as the other six main colours. Newton was rather mystical in his religious beliefs and seven was considered to be a ‘perfect’ number somehow related to the natural laws governing the Universe, and so he deliberately identified seven colours in the visible spectrum.
y t i s n e t n i e v i t a l e R
Sunlight
y t i s n e t n i e v i t a l e R
Incandescent light
y t i s n e t n i e v i t a l e R
Fluorescent light
Figure 2.37 The colour components of sunlight, incandescent light and fluorescent light. All are referred to as sources of white light, but their spectral compositions vary, affecting the colour of an illuminated object.
Two models for light
59
When incident light strikes the surface of an object, it may be absorbed, transmitted and/or reflected. If all of the white light falling on a surface is absorbed , the object will appear black as no light is reflected, and the object will be warmed in the process. If a surface reflects only a small proportion of the incident white light, and this reflected light is evenly distributed throughout the spectrum, the object will appear grey.
Coloured objects When white light is incident on the surface of a coloured object, some colours (wavelengths) will be absorbed and some colours will be reflected. Pigments are responsible for giving an object its colour. A yellow object is seen if the pigments in the surface of the object reflect yellow light and absorb red, orange, green, blue and violet light. However, the situation is rarely this clear cut. Orange and green light have similar wavelengths to yellow light as they appear either side of yellow light in the spectrum. So, to some degree, the same pigments will reflect some orange and some green light. Other colours may also be reflected but to a much lesser degree. An object’s colour is therefore often determined by the predominantly reflected colour of light. Coloured paints are produced by the addition of pigments to white paint. White paint reflects light evenly across the visible spectrum. If blue pigment was added the pigment would absorb the majority of red, orange and yellow light, reflecting only blue light and a limited amount of green and violet since these are the adjacent spectral colours (Figure 2.38). If yellow pigment was then added to the same blue paint, the paint would now appear green. Your experience of painting in art classes tells you this. This can be explained because the yellow pigment absorbs red, blue and violet light and reflects only yellow light and a limited amount of the adjacent spectral colours orange and green. The only colour which has been reflected by both pigments is green. All other colours have been absorbed by one or the other of the pigments. Hence, green light dominates the reflected spectrum and the paint appears green as shown in Figure 2.38.
Figure 2.38 Combining blue and yellow pigments results in the reflection of only green light since all other colours of the visible spectrum have been absorbed by either the blue or the yellow pigment.
green
white light
white light
white light
blue
yellow
green
violet
orange
green blue + yellow paint = green paint
An examination of the way in which pigment colours combine shows that all colours of the visible spectrum may be produced by mixing various combinations of the three primary pigment colours red, yellow and blue. These primary colours themselves cannot be produced from any other pigment combinations. Recall from your childhood art experience: red + yellow = orange red + blue = purple yellow + blue = green red + yellow + blue = brown/black
60
WAVE-LIKE PROPERTIES OF LIGHT
Worked example 2.4B . Pigment A reflects mostly violet light, with some red and blue, but absorbs all green, yellow and orange light. Pigment B reflects mostly green light, with some yellow and blue light, but absorbs all violet, red and orange light. A person dyes a white shirt by placing it in a solution of the two pigments. What colour will the shirt become?
Solution Blue is the only colour not absorbed by a pigment. Green, yellow and orange are absorbed by pigment A. Violet, red and orange are absorbed by pigment B. The shirt will appear blue.
Viewing objects under coloured light
Physics file The cones in the retina are responsible for colour vision. There are three different types of cones, each having maximum sensitivity to different colours. These are somewhat loosely referred to as red, green and blue photoreceptors. Any visible light excites all three receptors to varying degrees. The relative proportions of these three signals are thought to determine the perceived colour. For example, if all three receptors are equally stimulated we see an object as white.
When an object is illuminated by white light it is said to be demonstrating its true colour. However, the colour of an object can be dramatically changed when seen under differently coloured light. Consider the situation where a green object is illuminated by pure red light or pure violet light. Since the object is only capable of reflecting green light, all of the incident light is absorbed and the object appears black. If the same green object was then illuminated by cyan light (made up of blue and green light) the object would again appear green since it reflects the green component of the incident light.
‘Impure’ coloured objects The physiology of the eye results in another fascinating observation. An object may appear yellow for example, not because the pigments reflect yellow light of wavelength approximately 590 nm, but because red (∼700 nm) and green (∼540 nm) light are being reflected. It seems as if our colour-sensing system ‘averages’ the detected wavelengths of light and interprets the colour as yellow even if there is no light of approximately 590 nm entering the eye. Objects creating their colour in this manner are said to have an impure colour. If such an impure yellow object was viewed in red light it would appear red, since it is only capable of reflecting red and green light. Similarly if it was illuminated by green light it would appear green.
Figure 2.39 The flag appears to be different colours under different coloured light. Two models for light
61
Worked example 2.4C . A wall paint contains pigment dyes capable of reflecting light of two different wavelengths: 790 nm and 600 nm, corresponding to red and yellow light respectively. What colour would the wall appear if illuminated by: a
white light?
b
yellow light?
c
blue light?
Solution a
White light contains all colours of the visible spectrum. Only red and yellow light would be reflected. The eye/brain ‘averages’ the colour of the reflected light and perceives the wall to be orange.
b
If illuminated by yellow light only, the yellow pigments can reflect any light and so the wall appears yellow.
c
If only blue light is incident on the wall, none of this light is able to be reflected and so the wall would appear black.
Colour addition or mixing light sources In 1807 Thomas Young discovered that combining red, green and blue light on a screen produced white light. In fact various combinations of these three colours of light could create all of the other colours of the spectrum. Red, green and blue are therefore called the primary colours of light. None of the primary colours can be produced by a combination of the other primary colours. Note that the primary colours of light are not the same as the primary pigment colours discussed earlier.
PRACTICAL ACTIVITY 13 Colour addition and subtraction
Figure 2.40 Pairs of the primary colours of light overlap to produce the secondary colours yellow, cyan and magenta. When all three primary colours of light overlap white light is produced.
62
WAVE-LIKE PROPERTIES OF LIGHT
Figure 2.40 shows the three primary colours of light overlapping to produce other colours. The particular colours formed by the overlapping of pairs of primary colours are called cyan, magenta and yellow. Any group of colours which combine to form white light are called complementary colours. All three primary colours when combined form white light. Combining any two primary colours forms the complementary colour of the remaining primary colour. So, for example, when red and green are combined they form the complement of blue, which is yellow. Yellow is the complement of blue. Cyan is the complement of red, and magenta is the complement of green.
Colour television Television screens produce coloured pictures yet only utilise three different colours: red, green and blue. These colours are produced when electron beams strike the tiny phosphor dots lining the screen. Figure 2.41a shows a greatly magnified picture of a screen. It is actually made up of thousands of tiny coloured dots called pixels. Different parts of the television screen appear to be different colours because of the relative abundance of the three primary colours. Figure 2.41b shows how the different colours are created. If white is required all three colours will be produced. Because these dots are so close, our eye interprets this as a uniform area of white light. If magenta light is required only the blue and red dots will be stimulated. In fact all colours are produced by altering the proportions of red, green and blue dots stimulated in any particular area of the television screen.
(a)
(b)
Colour subtraction or filtering When a beam of light strikes the surface of a transparent medium such as glass, some of the light is reflected, some is absorbed and some is transmitted. Coloured filters are chosen specifically for the colours of light that they will absorb and transmit. If white light is shining through a blue filter it will allow mostly blue light to pass through, with a small amount of green and violet light as well, since they are the adjacent spectral colours. The other colours are absorbed by the filter. This is why the filter appears blue. Hence, filters can be used to produce any colour of light from a white light source. This process is called colour subtraction since many colours have been subtracted from white light. Astronomers use narrow band filters when observing the Sun. These filters allow only a small range of wavelengths of light to be transmitted, highlighting features of the Sun’s surface.
(c) (b)
Blue
Green
Red
Mixture white cyan yellow
Worked example 2.4D . A burglar is wearing a white shirt, yellow trousers and blue cap. He is observed by two witnesses. Anna is wearing red-tinted glasses, Brad is wearing yellow-tinted glasses. How would each witness describe the burglar?
Solution Anna: Red shirt, black trousers and cap. As a red filter allows only red light to pass, the white shirt would appear red since all other colours are absorbed. The yellow light reflected from the burglar’s trousers and the blue light from the cap cannot pass through the red filter and so the trousers and cap appear black.
magenta
Figure 2.41 The yellow seen on the television screen is, like all other colours, composed of only red, green and blue dots. The tiny dots on the screen are so close together that they cannot be recognised as separate dots, but blend together to form a continuous picture. Different areas of the television screen are made to produce red, green and blue dots in varying proportions, thus creating the various colours seen on the television screen.
Brad: Yellow shirt, yellow trousers and black cap. As a yellow filter allows only yellow light to pass, the white shirt would appear yellow since all other colours are absorbed. The yellow light reflected from the burglar’s trousers passes through the filter and so the trousers are seen as yellow. The blue light reflected from the cap cannot pass through the yellow filter and so the cap appears black.
Two models for light
63
LIGHT DESCRIBED AS AN ELECTROMAGNETIC WAVE
2.4 SUMMARY
• Visible light is only a small part of the electromagnetic spectrum. The many forms of EMR are essentially the same, differing only in their frequency and, therefore, their wavelength. • All EMR travels at a speed of 3.0 × 108 m s−1 in a vacuum. • White light contains approximately equal proportions of red, orange, yellow, green, blue and violet light. • The colours of light that an object reflects and absorbs
determine its colour. For example, red objects mostly reflect red light and absorb all other colours. • The primary pigment colours are red, blue and yellow. • The primary colours of light are red, green and blue. Combining the three primary colours of light produces white light. • A filter allows some colour(s) to be transmitted but all other colours are absorbed. A filter can change the colour of light through colour subtraction.
2.4 QUESTIONS 1
2
3
4
Calculate the wavelength of: a
microwaves of frequency 3
b
ultraviolet radiation of frequency 1015 Hz.
a
List three different types of electromagnetic radiation and describe a use for each.
b
List two properties common to all forms of electromagnetic radiation.
64
Hz
A painting of a person wearing a yellow jumper is observed under white light. The yellow pigment in the painting:
D 6
red, green and blue yellow, cyan and magenta.
Students are experimenting with the lighting for their school play. They want to produce some dramatic lighting effects. Determine the colour formed from a mixture of: a
red and blue light
b
red, blue and green light
c
blue and yellow light
d
green and magenta light.
7
A spinning top is decorated with all the colours of the rainbow, yet when spun it appears almost white. Why?
8
What colour light emerges when:
A
absorbs yellow, orange and green light
B
reflects red, blue and violet light
C
absorbs red, blue and violet light
a
a red filter is placed in the path of white light?
D
absorbs only yellow light.
b
a red filter and then a yellow filter is placed in the path of white light?
c
a red filter and then a yellow filter is placed in the path of red light?
The current Australian flag has blue, red and white sections. Assuming each colour was produced by a single pigment, what colour would each section appear if illuminated by: a
pure blue light
Explain how you could use pigments and/or filtered lighting to make:
b
pure red light
a
a white object appear violet
c
pure yellow light
b
a red object appear black
c
a yellow object appear green.
d
5
C × 1010
yellow light which is made by combining red and green light.
The primary colours for light are: A
red, green and yellow
B
red, blue and yellow
WAVE-LIKE PROPERTIES OF LIGHT
9
10
Explain how the spectrum of light reflected from a sky blue T-shirt is different from the spectrum of light reflected by a pastel blue T-shirt.
2.5
Dispersion and polarisation of light waves
Dispersion We have examined how the recognition of the wave nature of light allowed the development of a full explanation of the refraction of light as it changes speed. For example, the change in direction of travel of the light wave as it entered an optically denser medium occurred because a section of the wavefront entered a slower medium. The slowing down of this section of the wavefront, but not the section still travelling in the original medium, causes the overall wave to veer from its original direction of travel. Recall that white light is made up of many different frequencies (colours) of light. For some materials the speed at which light is transmitted is actually slightly different for different frequencies (colours) of light. This means that on refraction different colours of light will take slightly different paths. This results in the spreading out of the white light into its component colours. This is called the dispersion of white light. Prisms split white light into its component colours. It took scientists many years to be able to explain this phenomenon. Prior to Isaac Newton it was thought that glass prisms altered the incoming white light by varying degrees to produce the spectrum of colour. Newton carried out his investigations into dispersion and was the first to conclude that white light is actually made up of the colours of the spectrum and therefore recombining these colours would produce white light. Figure 2.42 shows the dispersion of white light as it passes through a triangular prism. The light is dispersed both on e ntering and leaving the prism, so that as the light emerges the range of colours spreads over quite a wide angle. There are no distinct boundaries where one colour finishes and another begins.
Figure 2.42 Dispersion of white light by a triangular glass prism. On entering and leaving the prism, the violet light is most significantly altered in speed and so it is refracted through the greatest angle. Red light is slowed less and so is refracted the least.
PRACTICAL ACTIVITY 14 Light and a continuous spectrum
As light enters a prism, it refracts due to a change in speed . Why does light slow down when it enters a more optically dense medium? The light energy is being momentarily absorbed and then re-radiated by the atoms which make up the medium. Different colours of light interact differently with these atoms. As a result they travel at different speeds within the medium and so are refracted through different angles. Of the colours which constitute the visible spectrum, violet light is slowed down the most and so is refracted through the greatest angle. Red light is slowed least and so is refracted the least. A similar situation occurs when light speeds up on entering a new medium. Different colours are refracted through different angles. Effectively, a particular medium, glass for example, has a different refractive index for each colour of light. Light flint glass has a refractive index of 1.62 for red light and 1.67 for violet light. Quartz has a refractive index of 1.45 for red light and 1.47 for violet light. In a vacuum, however, all colours of light travel at the same speed of 3.000 × 108 m s−1.
Two models for light
65
Physics file Diamond has a relatively high refractive index of 2.42. Hence, the critical angle for diamond is a relatively small 24°. White light is slightly dispersed on entering a diamond. Because of the shape of a diamond, once light has entered the diamond any ray striking the diamond–air boundary is likely to have an incident angle greater than 24° and it is therefore totally internally reflected. The special shape of a diamond means that the dispersed beam of light is likely to undergo a number of internal reflections before it meets a boundary at an incident angle of less than 24°, each reflection spreading the beam a little wider. After a number of internal reflections the light leaves the diamond. If the diamond is appropriately shaped, single colours of light are seen to be scattered by the diamond.
Worked example 2.5A . A narrow beam of white light enters a crystal quartz prism with an angle of incidence of 35°. In air, the white light travels at a speed of 3.00 × 108 m s−1. In the prism the different colours of light are slowed to varying degrees. The refractive index for red light in crystal quartz is 1.54 and for violet light the refractive index is 1.57. Calculate: a
the angle of refraction for the red light
b
the angle of refraction for the violet light
c
the angle through which the spectrum is dispersed
d
the speed of the red light in the crystal quartz.
Solution a
n 2 sin i = sin r n 1
1.54 sin 35° = 1.00 sin r
b
r = 21.9° n 2 sin i = sin r n 1 1.57 sin 35° = 1.00 sin r r = 21.4°
c d
Angle of dispersion = 21.9° − 21.4° = 0.5° n 2 v 1 = n 1 v 2 1.54 3.00 × 108 = v 2 1.00 v 2 = 1.95 × 108 m s−1
Figure 2.43 Flashes of coloured light can be seen emerging from a diamond as it is viewed from different angles.
PHYSICS IN ACTION
Rainbows Water droplets in the air disperse white light into colours to produce a rainbow just as a glass prism does. Rainbows are seen only when you have your back to the Sun, and many water droplets form a cloud in front of you. White light enters the water droplet, reflects from the back of the droplet (due to total internal reflection) and then leaves the droplet. On both entering and leaving the droplet the white light is slightly dispersed since water has a slightly different refractive index for each colour of light (Figure 2.44). To see what will happen, we will examine the path of the two extremes of the spectrum, red and violet light. The paths of all of the other colours of light will lie between these. 66
WAVE-LIKE PROPERTIES OF LIGHT
Because of dispersion, the red and violet rays leave the drop in different directions. Therefore an observer cannot see both the red and violet light emitted from the one droplet. If the violet light from a particular raindrop is entering your eye then all of the other colours reflected by that droplet must miss your eye. The different colours observed in a rainbow must come from different raindrops. Those raindrops sending the red light to your eye must be higher in the sky since the red light emerges more downward than the violet light. In fact all of the droplets which send red light to your eye lie on an arc of about 42° as measured from the original direction of travel of the
Sun’s rays (Figure 2.45). The droplets reflecting violet light lie on an arc of about 40 °, causing the shape of the rainbow! No two people can actually see the same rainbow because to view a rainbow the light reflected and dispersed by a particular set of raindrops is directed towards your dispersion white light from the Sun
total internal reflection
red violet
eye. Those same drops cannot send the same colours of light to your neighbour. A portion of the particular drops producing red light for you may be forming the green section of your neighbour’s rainbow.
direction of Sun’s rays all drops on this arc appear red to viewer’s eye
from Sun
t
l e d o r e v i
t l e i o v
dispersion violet
42° d 40°
all drops on this arc appear violet to viewer’s eye
r e
red
Figure 2.44 The rainbow is commonly seen because of the dispersion and reflection of light by water droplets. Less intense rainbows sometimes accompany a bright er rainbow. These are due to light reflecting inside the droplet more than once before emerging.
Figure 2.45 All drops lying on the outer arc reflect red light in the direction of the observer. All droplets on the lower arc reflect violet light to the observer’s eye.
Polarisation Further evidence for the wave nature of light is the finding that light can be polarised. Consider that light is an electromagnetic wave with associated electric and magnetic fields that vary. Each of these fields varies at a right angle to the direction in which the wave travels. For this discussion we need only think about the varying electric field associated with light, as it is this that largely determines how light interacts with materials. We can therefore represent light as shown in Figure 2.46. Think of how light is produced by a normal light globe inside a torch. Light is emitted from many different atoms in the filament and lots of light waves may be sent in a particular direction. However, the electric fields of these light waves will not be aligned. This is shown in Figure 2.46a. This is called unpolarised light. Most light sources, including our Sun, produce unpolarised light. If the light waves did have their electric fields aligned with one another as shown in Figure 2.46b, we would call this polarised light . The fact that unpolarised light can be converted into polarised light provides strong evidence that light is actually a wave.
Techniques for polarising light The most familiar way in which unpolarised (non-aligned) light can be converted to polarised (aligned) light is by using polarising filters. These filters have molecules that will block all electric-field-wave components except those whose plane is aligned in a particular direction. Figure 2.47 demonstrates this process using a ‘slit’ to represent the filter. Keep in mind that filters are actually solid materials, usually special plastics.
Figure 2.46 (a) Unpolarised light, such as that emitted by the Sun or a globe, has electric field variations that are not in alignment with one another. (b) Polarised light waves, such as laser light waves, have the electric fields that vary in the same plane as one another.
Physics file The fact that light can be polarised provides strong evidence that light is actually a transverse wave since longitudinal waves cannot be polarised.
Two models for light
67
Figure 2.47 (a) This filter allows vertically polarised light to pass through. (b) All horizontally polarised light is blocked. (c) The horizontal component of the light is suppressed, resulting in vertically polarised light of reduced amplitude. The emerging light will be less bright than the incident light.
PRACTICAL ACTIVITY 15 Polarisation effects with light
A light wave that has its electric field varying in a plane aligned with the filter will pass straight through the filter, maintaining its original amplitude. All of the light energy passes through. Figure 2.47a shows a vertical polarising filter. It allows light with vertically oriented electric field variation to pass through. A light wave that has its electric-field plane completely out of alignment with the filter will be blocked. That is, the light will be absorbed by the filter, as shown in Figure 2.47b.
Figure 2.48 Polarising materials crossing over at right angles to one another will prevent any light from passing through.
If a light wave has only a component of its electric-field plane corresponding to that of the filter, then only this component of the wave will be transmitted. The emerging wave has significantly reduced amplitude. Therefore a portion of the light energy does not pass through the filter, as indicated in Figure 2.47c. The emerging light is described as ‘vertically polarised’. Should unpolarised light be incident on the filter, only vertically polarised light would emerge. A pair of polarising filters can therefore be placed at right angles to one another to prevent all light from passing through. One filter may block all of the horizontal electric field components and the other filter may block the vertical components as shown in Figure 2.48.
Why Polaroid sunglasses work When outdoors on a bright, sunny day, the smooth, highly reflective, horizontal surfaces around you are a significant contributor to the amount of light entering your eyes. Bring to mind the glare that can occur from the surface of water or snow. Fortunately light that is reflected from smooth, horizontal surfaces tends to be polarised (aligned) in a horizontal direction. An appropriately oriented polarising filter can be employed. Lenses in a pair of Polaroid sunglasses are polarising filters oriented to block the horizontal wave components, allowing only vertical components through. Hence the intensity of light—that is, the glare—is markedly reduced.
68
WAVE-LIKE PROPERTIES OF LIGHT
2.5 SUMMARY
DISPERSION AND POLARISATION OF LIGHT WAVES
• Dispersion is the spreading of white light into its component colours in a spectrum. • Dispersion occurs as white light enters or leaves a prism because different colours of light are refracted by different amounts. • Violet light has the greatest refractive index and is refracted through the greatest angle. Red light has the smallest refractive index and is refracted through the smallest angle.
• Unpolarised light waves have electric-field variations that are not in alignment with one another. Polarised light waves have electric fields that vary in the same plane as one another. • Polarising filters are able to convert unpolarised light into polarised light, providing strong evidence that light is actually a wave.
2.5 QUESTIONS Assume the index of refraction for air is n = 1.00. 1
a
Which colour of light travels fastest in Perspex: red, green or blue?
b
If red, green and blue light passed from air into a Perspex block, which colour of light would be slowed down the most?
c
7
Which colour of light—red, green or blue—would be refracted the most as it passed into the Perspex block?
A particular prism of glass has a refractive index of 1.55 for violet light and 1.50 for red light. A beam of white light is incident on the prism at an angle of 40.0 ° and is dispersed. a
Which colour of light will have the slowest speed in glass: red or violet?
b
Which colour is refracted most: red or violet?
c
What is the angle of refraction for the red light at the air–glass boundary?
2
How does polarisation support a wave model for light?
d
3
With the use of a diagram, show how a narrow beam of white light will be dispersed on both entering and leaving a triangular glass prism.
e
4
You have two identical pairs of sunglasses. How could you find out whether the sunglasses were polarising or not polarising?
5
A piece of glass and a diamond are cut to exactly the same size and shape. They are illuminated by the same white light. Why does the diamond appear to sparkle more colourfully than the glass?
6
Explain why the use of a very narrow bandwidth (range of frequencies) reduces the amount of material dispersion in optical fibres.
8
9
10
What is the angle of refraction for the violet light at the air–glass boundary? Over what angle will the spectrum be spread?
A particular plastic has a refractive index of 1.455 for red light and 1.650 for violet light. For light passing from this plastic into air, which colour of light would have the greatest critical angle? Explain why dispersion provides evidence for a wave nature of light. A polarising filter is positioned so that it produces vertically polarised light. If another filter is oriented at an angle of 30° to this filter, what happens to the intensity of the light?
Two models for light
69
CHAPTER REVIEW CHAPTER REVIEW 1
As light travels from quartz to water does it: a b
2
3
4
5
List seven different categories of electromagnetic radiation and give an application of each.
b
List two features common to all types of electromagnetic radiation.
9
A
green
B
yellow
C
bright red
D
black or dark brown
11
Explain the following observations. Use a diagram where appropriate. a
b
12
b
c
d
13
Which medium has the highest index of refraction?
A ray of light exits a glass block. On striking the inside wall of the glass block, the ray makes an angle of 58.0° with the glass–air boundary. The index of refraction of the glass is 1.52.
14
What is the angle of incidence? Assume n air = 1.00. What is the angle of refraction? What is the angle of deviation of the ray? What is the speed of light in the glass?
a
Explain how the use of a laser as a monochromatic light source reduces material dispersion in optical fibres.
b
Describe and explain two ways in which modal dispersion in multimode optical fibres can be reduced.
WAVE-LIKE PROPERTIES OF LIGHT
A particular step-index optical fibre has a core of refractive index 1.557 and cladding of refractive index 1.542. Determine the critical angle of incidence for light striking the core–cladding boundary. Explain why snowboarders and fishermen are likely to wear polarising sunglasses.
At the boundary between medium 1 and medium 2, the angle of refraction is smaller than the angle of incidence. Which medium allows light to travel faster?
Scuba divers notice that when they are only submerged a few metres the ocean waters appear blue. Why?
10
If a dark red jumper is viewed under yellow light, it would appear:
a
70
refract towards or away from the normal?
a
b
7
speed up or slow down?
Describe the practical work which you would carry out if you were given a glass prism and asked to determine its refractive index and critical angle.
a
6
8
16
17
A stone at the bottom of a shallow pond seems closer to you than it really is.
Explain how you could use pigments and/or filtered lighting to make: a
a white object appear orange
b
a blue object appear black
c
a yellow object appear green.
Explain how the spectrum of colour reflected from a black object is different from the spectrum of light reflected by a grey object.
A
higher than it really is
B
lower than it really is the same as viewed from above the water
A narrow beam of white light enters a crown glass prism with an angle of incidence of 30°. In air, the white light travels at a speed of 3.00 × 108 m s−1. In the prism the different colours of light are slowed to varying degrees. The refractive index for red light in crown glass is 1.50 and for violet light the refractive index is 1.53.
a
the angle of refraction for the red light
b
the angle of refraction for the violet light
c
the angle through which the spectrum is dispersed
d
the speed of the violet light in the crystal quartz.
The speed of light in air is 3.00 × 108 m s−1. As light strikes an ai r–Perspex boundary, the angle of incidence is 43.0° and the angle of refraction is 28.5 °. Calculate the speed of light in the Perspex. What colour of light emerges when: a
a green filter is placed in the path of white light?
b
a green filter and then a yellow filter are placed in the path of white light?
c
a green filter and then a yellow filter are placed in the path of green light?
18
What is the relative refractive index for light passing from Perspex into water if an incident angle of 17.0° produces an angle of refraction of 14.5°?
19
a
What is the critical angle for light passing from: i
In mid-afternoon a diver looks up from below the surface of the water. His judgement of the position of the Sun will be:
C
15
A star can still be seen even though it is actually positioned below the horizon.
Calculate:
b
20
diamond (n = 2.42) to air?
ii
glass (n = 1.50) to air?
iii
water (n = 1.33) to air?
What effect do the relative sizes of the refractive indices have on the size of the critical angle?
Why does a ray of light which passes through plate glass emerge parallel to its original direction of travel? Has the ray of light been refracted?
CHAPTER 3
Mirrors, lenses and optical systems
W
ithout question the most useful optical device in existence is the lens. Our entire view of the world relies on a pair of lenses situated at the front of our eyes. Many everyday and scientific devices utilise lenses. Magnifying glasses, movie projectors, telescopes, binoculars and microscopes rely on specially designed systems of lenses. All of these devices manipulate the path of light to produce useful images that would otherwise be more difficult, or indeed impossible, to see. Telescopes were first pointed at the night sky in the early 17th century. Since then technology has greatly advanced the capabilities of these devices. The clearest optical images of our night skies are obtained by enormous reflecting telescopes. These are placed on mountain tops to reduce the degrading effects of the shimmering atmosphere. The Hubble Space telescope, launched in 1990, orbits the Ear th above the atmosphere so that clearer images can be produced. It has become infamous because once launched it was discovered that the images did not provide the dramatic improvement in clarity that was expected. An error in design meant that the clarity of the images was hardly better than that produced by the largest telescopes on the ground! This was an error that proved very expensive to fix. Whole new technologies had to be developed to be able to send a repair team into orbit! In this chapter we use a ray model of light to examine how mirrors and lenses alter the path of light to produce images. We look at the design of optical systems and the function of the eye.
BY THE END OF THIS CHAPTER
you will have covered material from the study of the wave-like properties of light including: • describing the ray model of light as derived from the wave model • application of a ray model of light to behaviours of light, including reflection and refraction • optical devices, including lenses and mirrors • colour dispersion in lenses • application of mathematical modelling to light phenomena.
3.1
Geometrical optics and plane mirrors The wave-model of light acknowledged In Chapter 2 we have examined how the wave model of light was paramount in explaining numerous properties of light. We have seen that by considering light to be a transverse electromagnetic wave an understanding of light has been reached in relation to: • the linear propagation of light waves in a uniform medium • the regular reflection of light waves from smooth surfaces and diffuse reflection from irregular surfaces • the refraction of light waves as they change speed at the boundary between two media • the dispersion of white light into its component colours (wavelengths) • the existence of light as a part of the continuous electromagnetic spectrum • the polarisation of light waves. In addition to these wave ideas that we have already studied, the diffraction and interference properties of light were discovered in 1803 by Thomas Young. Although not part of this study, these discoveries were crucial in demonstrating the wave-like nature of light. Young showed that when light passes through a narrow slit, bright bands are formed in regions on a screen that a particle model would predict to be in shadow. Light was observed to bend its path as it passed through the slit; that is, diffract . When a pair of slits is used, alternating bright and dark bands form, imitating the interference that occurs between two sets of water waves. These important wave-like behaviours of light are covered in Heinemann Physics 12 .
Figure 3.1
In 1803, Thomas Young discovered the diffraction and interference properties of light.
In the early 19th century scientists were satisfied that light truly had a wavelike nature. In 1819 Augustin Fresnel presented a wave theory of light that explained diffraction and interference effects and it appeared that the matter was settled upon. It was not until the next century that observations were made that once again questioned the pure wave-like nature of light. As your later studies may illustrate, there are some behaviours of light that a pure wave model simply cannot explain.
Geometrical optics When studying the path of light through many common optical devices, maintaining a wave model for light is unnecessarily complex. In this chapter, rays are used extensively to describe the path of light in optical systems. Note that this approach is justified only in the absence of diffraction. Diffraction is the bending of the direction of travel of light as it passes through an aperture; light will ‘flare out’ as it goes through a narrow gap and rays cannot adequately represent its path. However, significant diffraction only occurs if the size of the aperture is approximately as small as the wavelength of the light itself. In this chapter the dimensions of the mirrors, prisms and len ses discussed are much greater than the wavelength of light and hence diffraction effects can be ignored. In these circumstances light travels in straight lines according to the laws of reflection and refraction, and so pathways are accurately represented with rays. These conditions are called the conditions of geometrical optics.
72
WAVE-LIKE PROPERTIES OF LIGHT
Figure 3.2
The path of light through many optical systems involving (a) lenses and (b) mirrors can be accurately represented using rays. The modelling of image formation in simple optical systems is far simpler using geometrical optics rather than a wave-optics approach.
Ray tracing to locate and describe images
PRACTICAL ACTIVITY 9 Reflection in a plane mirror
To investigate the images produced by mirrors and lenses ray tracing can be carried out. Using rays, known pathways of light are modelled on a scaled, twodimensional diagram and the characteristics of the resulting image can be identified. The image can then be fully described by its: • nature —Is the image real or virtual? (discussed later) • orientation —Is the image upright or inverted? • position —Where is the image in relation to the mirror? • size (including magnification) —What is the height of the image? By what factor has the size of the image changed?
Images in a plane mirror
(a)
To understand how an image forms in a plane mirror , it is helpful to distinguish between how we see an object when we are looking at it directly and the image of the object in a plane mirror. To begin, consider an observer looking directly at another person’s feet. The observer will see a foot because light from some external source is reflected from a point on the foot and this enters the observer’s eye. Specifically, the path of two slightly diverging rays can be traced from a point on the foot to the observer’s eyes (Figure 3.3a). The pupil of the eye is not just a point aperture. It collects light over an area and the eye will focus diverging rays onto the retina at the back of the eye. This is the mechanism by which we see and hence the foot is judged to be at A. When a person stands in front of a plane mirror, light from a point on his or her foot will travel to the mirror and be reflected. All rays obey the law of reflection, and a pair of slightly diverging rays can be traced as they leave the foot, reflect from the mirror and enter the observer’s eyes (Figure 3.3b). To form an image, the observer’s eyes focus the diverging light rays just as they did in the previous situation. Knowing that light travels in straight lines, the eye/ brain system interprets the rays as having come from a single point behind the mirror. This is where the image is seen in the mirror. Figure 3.3b illustrates geometrically how the eye/brain subconsciously extrapolates diverging rays of light until they meet and are interpreted as an image. It is important to remember that the rays behind the mirror are not actually there. They are
A you
your friend 2m
(b) real rays
virtual rays
you
your image
mirror 1m
1m
Figure 3.3
(a) Diverging rays from your friend’s foot are focused by the eye in order that you see them. (b) Diverging reflected rays enter the eye in the same way and appear to have come from the same point. Mirrors, lenses and optical systems
73
virtual rays , and by convention they are shown as dashed lines. When an image forms behind an optical element like a mirror, it is classified as a virtual image. Other optical systems form a real image where the rays converge to meet in reality. A real image can be shown on a screen—like the image from a slide projector. Virtual images cannot be projected on to a screen.
reflecting surface of plane mirror 50 cm
This ray-tracing exercise can be repeated for all parts of the body to locate the complete image of the whole person. Every point on the object is matched by a corresponding image point, which is the same distance behind the mirror surface that the object is in front. The object and image are exactly the same size. All images formed by plane mirrors have the following characteristics: • the image is always upright • the image is the same distance behind the mirror as the object is in front • the image is the same size as the object.
reflecting surface of plane mirror 50 cm
50 cm
Worked example 3.1A . Use ray tracing to locate the image of a pom-pom on the top of this girl’s hat, which is 50 cm from the mirror as shown.
i r
Solution To locate an image, the path of at least two rays coming from a point on the object must be followed. Each ray obeys the law of reflection, i = r , and the image will lie at the point where they meet after extrapolating the rays back. This point lies 50 cm behind the mirror. Physics file
Multiple mirrors
When looking into a mirror, you do not appear as others see you. For example, a careful inspection of the image of your left hand in a mirror reveals that it appears as a right hand. This is an example of the back-to-front effect in a plane mirror which results in a change of ‘handedness’. The image is neither vertically nor laterally inverted (though this is often stated). Something at the top left of the object is also at the top left of the image as we see it. However, the image of your left eye in a mirror will be directly in front of your left eye as you face the mirror, but this eye would appear to be the right eye of the ‘person’ standing in the mirror.
90° − α α
α
90° − α
α
90° − α
Figure 3.4
Two mirrors are placed at right angles to one another. Light is reflected from each mirror and emerges parallel to its original direction of travel. 74
WAVE-LIKE PROPERTIES OF LIGHT
If two plane mirrors are arranged at 90° to one another, incident light from an object will reflect from one mirror and then the other as shown in Figure 3.3. The ray will finally emerge parallel to the incoming light ray. If the angle of incidence for the incoming light ray is α, then the angle of reflection is also α. This means the angle of incidence as it strikes the second mirror is now 90° − α, and it leaves with an angle of reflection 90° − α. The beam therefore has been deflected through a total angle of α + α + (90° − α) + (90° − α) = 180°. Being deflected through 180° means that it reflects from the mirrors along a path that is parallel to its incoming path. If three mirrors are set up at 90 ° to each other—like the corner of a cube— then again any incident ray will reflect and emerge parallel to the incident ray in three dimensions. This situation is exploited in reflectors used on cars and bicycles and seen on roadside posts. If you look carefully at the surface of a bicycle reflector it appears to be made up of rows and rows of tiny little box corners. Light from the headlights of an approaching car is reflected back towards the car so that a driver can see the cyclist. There is a three-dimensional corner reflector placed on the Moon by astronauts. Laser light from Earth is reflected directly back to its source. A measurement of the time this journey takes allows the distance to the Moon to be calculated to within centimetres. Finally, if two mirrors are placed at an angle θ to each other, light from an object will reflect to produce a number of images. Mirrors at 90° to each other will produce three images. As the angle is reduced, the number of images increases. The number of images seen in a pair of mirrors placed at an angle θ to each other is given by the following relationship: 360° −1 number of images = θ Use a pair of mirrors to verify this.
3.1 SUMMARY
GEOMETRICAL OPTICS AND PLANE MIRRORS
• In the 19th century the wave nature of light was accepted. • Geometrical optics is the branch of optics that models the path of light using straight rays. This is accurate as long as diffraction (spreading) effects are negligible. • Ray-tracing procedures can be used to determine the image formed by a mirror or lens. An image of a point exists when rays from the point converge (or appear to converge) at another point. • When using a ray diagram at least two rays must be traced in order to locate an image.
• The complete description of an image requires that the position, nature, orientation and size of the image be stated. The nature of an image is either ‘real’ or ‘virtual’. The orientation of an image is either ‘upright’ or ‘inverted’ relative to the object. • Diverging rays of light entering our eyes are interpreted as having come from a single point behind a plane mirror, where a virtual image is seen. • A plane mirror produces an image which is always virtual, upright and unchanged in size.
3.1 QUESTIONS 1
What are two behaviours of light that can be modelled effectively using both a particle and a wave model for light?
2
In what circumstances is it acceptable to model the path of light with rays rather than waves?
3
When light passes through tiny apertures (such as a pinhole) the image produced has blurred edges. Why does this suggest that the particle model of light is inadequate?
4
Is the image formed in a plane mirror always:
5
6
7
8
a
upright or inverted?
b
enlarged, diminished or the same size as the object?
c
real or virtual?
9
Complete the ray paths in the following diagrams to locate the image of the object in the plane mirror.
a
observer’s eye object
mirror
b
object
In order to fully describe an image what four characteristics should be commented upon? a
List the features common to all images produced by plane mirrors.
b
What is a virtual image? Sketch a ray diagram to illustrate your answer.
observer’s eye
A girl stands 7 m away from a full-length vertical plane mirror in a boutique. a
How far must she walk to appear to be 3 m from her image?
b
She walks towards the mirror at 1.5 m s −1. With what speed does her image appear to walk towards her?
A 1.6 m tall person stands in front of a vertical dressing mirror. Her eyes are at a height of 1.5 m above the ground. a
What length mirror is needed for the person to be able to see a full-length image of herself?
b
How high above the ground should the bottom of the mirror be positioned?
mirror 10
a
At the football, smaller people are often stuck behind others who are up to 20 cm taller. Design yourself a periscope that could assist in such a situation.
b
Your idea for a ‘footy-scope’ is taken up by the AFL and manufactured. However, the bottom mirror falls out of a poorly constructed periscope at the football. If the user were to look directly up the tube, how would the image differ from that seen in the twomirror periscope?
Mirrors, lenses and optical systems
75
3.2
Applications of curved mirrors: concave mirrors The applications of plane mirrors in everyday life are limited due to the features of the image that can be produced. For example, a plane mirror will be useless where magnification is required—the image in a plane mirror is always the same size as the reflected object. Curved mirrors provide a greater variety of options depending on the extent and sense of the curvature of the mirror and the placement of the object (Figure 3.5). It’s simply a matter of choosing the right mirror for the task to be undertaken. Shop-security mirrors, shaving and make-up mirrors and dentist’s mirrors all use a curved mirror to produce an image that is appropriate for the situation in which it is used.
(a)
(b)
Figure 3.5
In one mirror the image is upright and significantly reduced in size. The image in the other mirror has been magnified many times.
All curved mirrors are either concave or convex . A concave mirror is curved like the inside of the bowl of a spoon, whereas a convex mirror is shaped like the back of a spoon. (One way to remember which is which is to recall that a con-cave mirror forms a small cave or may cave in.)
Figure 3.6
Each ray obeys the law of reflection, resulting in (a) converging rays or (b) diverging rays. In each case, a focal point can be defined. A concave mirror has a real focus, and the focal length is positive. The focus for the convex mirror is virtual, since its position is determined by extrapolating the reflected rays behind the mir ror.
Physics file In reality, parallel rays of light are not brought to a perfect focus by a spherical mirror. The rays directed near to the centre of the mirror will meet in a region near an approximate focus, but rays farther from the centre miss this region completely. As a consequence, any image formed in a spherical mirror will be distorted. This is called spherical aberration , and, literally, an aberration is a distortion. The farther from the centre of a mirror one looks, the more distorted the image is.
Curved mirrors are usually spherical in shape—as if the mirror has been made from a portion of a sphere—because these are cheap and easy to manufacture. More specialised applications require a mirror that is parabolic in shape. As you will see, the image of an object seen in each mirror type can be analysed in the same way. To understand how a curved mirror produces an image, it is important to understand how the curved mirrors reflect light. Any curved mirror can be considered to be made up of a number of tiny plane mirrors. A twodimensional representation of this is shown in Figure 3.6, although a real mirror will of course be three-dimensional. If parallel rays of light shine directly on the surface of each mirror, the rays reflect, thus obeying the law of reflection for the position of the (plane) mirror at which it strikes. Parallel rays will converge to a single point on reflection from a concave mirror. This point is called the focal point or focus of the mirror. Parallel rays striking a convex mirror will diverge from an imaginary focal point located behind the mirror. Unfortunately, most spherical concave mirrors are only capable of bringing parallel light rays to an approximate focus, as the reflected rays do not intersect at precisely the one single point (Figure 3.7a). This causes blurring of an image near the edges of the mirror. This distortion is called spherical aberration, and is particularly noticeable if the mirror is large compared to its radius of curvature. If an undistorted image, free of spherical aberration, is required, a parabolic mirror is needed. A parallel beam of light will reflect from a parabolic mirror to a sharp focus (Figure 3.7b). Spherical mirrors are often used in inexpensive torches and in security and dental mirrors where the accuracy of the focal point is not a great concern.
76
WAVE-LIKE PROPERTIES OF LIGHT
Parabolic mirrors are more difficult to produce and therefore are more expensive. Parabolic mirrors are usually reserved for situations where a precise focus and undistorted images are required, such as in reflecting telescopes.
(a)
spherical mirror
The focus of a concave mirror can be used in reverse. It is possible to place a point light source at the focus, have the light reflect from the mirror, and create a beam of parallel rays. Car headlights, halogen spotlights and the torches used by security guards all employ this design. Because the rays of light emitted from the light source strike the mirror and emerge parallel to one another, the beam is strong even at a large distance (Figure 3.8). parabolic mirror
(b)
parabolic mirror
parallel beam of light F
i r
globe placed at focal point produces a parallel beam
Figure 3.8
A bright light globe placed at the focal point of a parabolic mirror produces a parallel beam of light. This arrangement is used in spotlights where one wants a penetrating beam, e.g. car headlights, rescue searchlights.
A dentist uses a concave mirror placed close to the back of a tooth to see a magnified upright image. Shaving and make-up mirrors also produce magnified upright images but, again, you have to stand close to the mirror. Concave mirrors are also capable of forming small, inverted images (Figure 3.9). The type of image produced by a particular concave mirror depends on the placement of the object relative to the mirror.
Image formation in a concave mirror
F , focal point
Figure 3.7
(a) Spherical mirrors do not have a precisely defined principal focus, and the further a ray of light is from the principal axis of the mirror, the greater the inaccuracy. The effect is most noticeable if the mirror is large compared to its radius of curvature. (b) A parabolic concave mirror will bring parallel rays to a perfect focus.
PRACTICAL ACTIVITY 16 Concave mirrors
Before explaining how the image is found, there are some terms used to describe a spherical mirror which need to be understood (Figure 3.10). The centre of the mirror is called the pole , denoted P . The principal axis is a straight line perpendicular to the surface of the mirror at the pole. Rays parallel to the principal axis will meet at the principal focus , denoted F . The distance between the pole and the principal focus is the focal length of the mirror, denoted f . The radius of the sphere from which the mirror is assumed to have been made is the radius of curvature of the mirror, R . Finally, the centre of curvature of the mirror is the centre of the sphere of which the mirror forms a part and is denoted C . Just as for the plane mirror, the path of some rays must be determined as they are reflected from the mirror. First, it is assumed that one of the rays leaves the base of the object and travels along the principal axis to the pole of the mirror. This ray reflects along itself and will form the base of the image. The position of the top of the image is determined by constructing at least two rays emanating from the top of the object. The image of this point forms
Figure 3.9
A concave mirror forms an enlarged image of a close object.
Mirrors, lenses and optical systems
77
(a) Ray 1 principal axis object
Concave mirror is a portion of a sphere.
F
(b) Ray 2
principal axis
R , radius of curvature P , pole
F , principal focus
C
object
C , centre of curvature
f , focal length
(c) Ray 3 i r
object
P
where these rays intersect. There are a number of specific rays which strike the surface of the mirror whose path can be easily predicted. Naturally, these rays obey the law of reflection.
(d) Ray 4 object
F
(e)
object image
Figure 3.11 The path of four rays leaving the top of an object reflect from the mirror to produce an image. (a) This ray travels parallel to the principal axis and reflects through F . (b) Ray 2 passes through C and is reflected back along its incident path. (c) The ray striking the pole will reflect as if it has struck a small plane mirror, i r . (d) The ray passing through F reflects parallel to the principal axis. (e) Intersecting rays locate the image. =
Physics file The quickest method to measure the focal length of a concave mirror is to stand near a window, holding the mirror so that an image of a distant object forms on a small white card or screen that you hold in your other hand. Light rays from a distant source are considered to be parallel to one another, so you are effectively focusing parallel rays of light. The distance from the mirror to the screen approximates the focal length of the mirror.
78
Figure 3.10 Terminology used to describe spherical mirrors.
WAVE-LIKE PROPERTIES OF LIGHT
Graphical ray tracing: concave mirrors To determine the image of an object in a curved mirror, the object is positioned on the principal axis to the left of the mirror. Between two and four rays are traced from the top of the object: the top of the image is found at the intersection of these rays. The rest of the image fills in the space from this point to the principal axis, and is perpendicular to the principal axis. The four rays whose path can be determined with ease in a concave mirror are described below and shown in Figure 3.11. Ray 1: A ray of light which travels parallel to the principal axis will be reflected through the principal focus of the mirror. See Figure 3.11a. Ray 2: A ray of light emanating from the object and travelling through the centre of curvature of the mirror will strike the surface at an angle of 90 °, i.e. along a normal. This ray will therefore be reflected back along the path from whence it came. See Figure 3.11b. Ray 3: The ray which strikes the pole of the mirror will reflect at an angle equal to the angle of incidence. At the pole, the mirror acts like a plane mirror and the principal axis becomes a normal. See Figure 3.11c. Ray 4: The ray emanating from the object and travelling through the principal focus before striking the mirror will be reflected from the mirror parallel to the principal axis. See Figure 3.11d.
If any two of these four rays are traced, the point at which they intersect determines the location of the image. See Figure 3.11e. Once the position of the image is found, we know that any ray emanating from the top of the object and striking the mirror must reflect through this point, and a clear image is produced.
Ray diagrams Particular conventions apply to the construction of a ray diagram (Figure 3.12). • A vertical line called the optical axis represents the reflecting surface, i.e. the back of the mirror. Although the mirror surface is curved, the optical axis is
a straight vertical line. A ray diagram tends to use a much larger vertical scale than horizontal scale, thus allowing the mirror to be represented by a straight line with little loss of accuracy. • The optical axis is perpendicular to the principal axis, and the pole, P , is placed at the intersection between the optical axis and the principal axis. A small curved mirror symbol is placed here to indicate the type of mirror being used. • The principal focus of the mirror, F , and its centre of curvature, C , are located on the principal axis to scale. • Traditionally, the object is on the left of the optical axis, along with the eye that will view the image. The object is usually represented as a small vertical arrow. optical axis
Physics file Any ray of light, parallel and close to the principal axis, reflecting from a spherical mirror, will travel to the focal point, F . The distance from the focus to the pole of the mirror is called the focal length of the mirror, f , and this is half the distance from the pole to the centre of curvature for the mirror, R . Concave mirrors have a real focal point as the rays intersect at this point. Convex mirrors have a virtual focal point as the rays only appear to have come from this point: R f = 2
u
principal axis object
C
image
P
F v
f
F
C
Figure 3.12 Layout for a ray diagram to find the image of an object in a concave mirror.
When constructing a ray diagram, the vertical scale for the diagram does not have to be the same as the horizontal scale. This is particularly useful if a tiny object such as a small whisker is to be viewed in a mirror with a focal length of 20 cm. The distance from the object to the mirror (or optical axis) is called the object distance , denoted u , and the distance from the optical axis to the image is the image distance , v . As already stated, at least two rays from the top of the object must be drawn to locate the position of the image. It is a good idea to use a third ray to check that sufficient care has been taken. All three rays should meet at a single image point.
R
Remember that four characteristics of the resulting image can be commented on: the nature, orientation, size and position of the image. Figure 3.13 shows the images produced when the same object is placed various distances from a concave spherical mirror. When an object is placed at a relatively large distance (u > R ) from a concave mirror, the image formed is real, diminished and inverted (Figure 3.13a). Figure 3.13b shows that when the object is placed at the centre of curvature of the mirror ( u = R ), the image formed is also real and inverted but is exactly the same size and distance from the mirror as is the object. When the object is placed between the centre of curvature and the focal point ( f < u < R ), as depicted in Figure 3.13c, the image is again real and inverted, but now it is magnified. When an object is placed at the principal focus of a concave mirror (u = f ) the reflected rays will emerge parallel to one another. As discussed earlier, this is the arrangement employed in car headlamps, torches and the like, where a globe is placed at the focal point of a concave mirror. The rays leaving the top of the object strike the mirror and emerge parallel to one another so the image can be considered to be at infinity (see Figure 3.13d). Mirrors, lenses and optical systems
79
(a)
(b)
object
C image
object
Image real diminished inverted
F
C
F
Image real same size inverted
image object
(c)
Image real magnified inverted
F
C
image
object
(d)
C
F
object
(e)
Image formed at infinity
C
F
Image virtual magnified upright
Figure 3.13 (a)–(c) A concave mirror will produce a real image when the object is placed farther than the focal point, and (d), (e) a virtual image when at or closer than the focal length.
Figure 3.13e illustrates what happens when an object is placed inside the focal length of the mirror ( u < f ). On reflection, the rays diverge like those from a plane mirror. This means that the rays cannot intersect to create a real image. Rather, our eyes interpret the diverging rays as coming from an image behind the mirror—and a virtual, upright and magnified image results. The location of the image is found by extrapolating the rays back to their intersection point as shown. Since it is a virtual image, it cannot be projected on to a screen. If one is available, mount a concave mirror on a wall and check that the predictions from the ray diagrams in Figure 3.13 are valid. Stand a few metres back from the mirror and then slowly walk towards the mirror. As you walk towards the mirror you should notice that your image alters. A concave mirror can produce different images depending on how close the object is to the mirror. As mentioned earlier, the complete description of an image requires the nature, orientation, position and magnification to be stated. Magnification refers to the size of an image relative to the object. If the image is larger than the object, it can be described as enlarged or magnified . A smaller image is diminished or reduced . Sometimes the actual height of an image has to be found—a carefully drawn scale diagram is required here. The magnification, M , of an image is the factor by which the size of the object has been multiplied; this is the ratio of the height of the image to the height of the object. A positive magnification value is used to indicate that the
80
WAVE-LIKE PROPERTIES OF LIGHT
image is upright relative to the object. A negative magnification value is used to indicate that the image is inverted relative to the object. If the magnification factor is greater than one, |M | > 1, this indicates a magnified image. A diminished image has a magnification factor of less than one, |M | < 1. If |M | = 1, the image and object are the same size. For example, a mirror producing a magnification of −1.2 means that the image is inverted and 20% larger than the object. The magnification of an image is also given by the ratio of the image v distance to the object distance (M = − ). Notice from Figure 3.12 that the u ratio of the heights of the image and object is exactly the same as the ratio of the image and object distances.
The MAGNIFICATION FACTOR , M , in optics is given as the ratio of the image height to the object height. image height image distance M = = object height object distance
M =
H i v = − H o u
Summary of images formed by concave spherical mirrors Table 3.1 summarises the images formed by concave mirrors. Note that all real images are inverted and all virtual images are upright. Table 3.1
Summary of images formed by concave spherical mirrors.
Position of object
Beyond C At C Between C and F At F Between F and P
Position of image
between C and F at C beyond C image at infinity behind mirror
Description of image Real or virtual
Upright or inverted
Enlarged or diminished
real real real — virtual
inverted inverted inverted — upright
diminished same size enlarged — enlarged
optical axis
object
Worked example 3.2A . A man stands in front of his shaving mirror and is disappointed with the image he sees. The focal length of the concave mirror is 50 cm, and he is standing 1.5 m from its pole. Use a ray diagram to explain what he sees in the mirror.
(1.5 m)
Solution Draw a scale diagram representing the situation. Choose two rays to trace. In this case rays 1 and 4 have been drawn. The intersection between the reflected rays lies below the principal axis. The image is therefore real and inverted, but half the size of the original whisker—not much use when shaving!
F (0.5 m)
0.75 m object
image
F
0.25 m
0.5 m optical axis
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81
Worked example 3.2B .
optical axis
A dentist wishing to view a cavity in a tooth holds a concave mirror of focal length 12 mm at a distance of 8 mm from the tooth. Describe the image produced.
object F (12 mm) (8 mm)
Solution Draw a graphical scale diagram representing the situation. Choose two rays to trace. In this case rays 1 and 3 have been drawn. The reflected rays do not intersect, and so they must be traced back to locate the virtual image. The dentist sees a virtual, upright image of the tooth that is magnified approximately three times. optical axis
optical axis
virtual image
object
F
F
object
These two worked examples have illustrated problems in which the characteristics of the mirror and image are given. It is the image that has to be found. The following worked example shows a more difficult problem in which the position of the object and image are given, and the focal length has to be found.
Worked example 3.2C . A person stands 3.0 m from a concave mirror that produces an upright image magnified by a factor of 1.5. Determine the focal length of the mirror.
Solution optical axis ray 3 principal object axis
3 cm
position of image
Draw a graphical scale diagram representing the situation. Because the height of the person is not given, make the object 2 cm high. Choose ray 3 to draw. If the image is upright, it must also be virtual. Being magnified by 1.5, the image must be 3 cm high at some point behind the mirror. Now we know the position of the top of the image, so we work from ray 1 to find the focal length. A ruler is placed aligning the top of the image to the point where the ray from the top of the object meets the optical axis. This ray will reflect through the focus. Measuring from the following diagram, and using the scale, f = 9 m. optical axis ray 1
Horizontal scale: 1 cm = 1 metre
principal axis object
F
9m 82
WAVE-LIKE PROPERTIES OF LIGHT
3m
image
3.2 SUMMARY
APPLICATIONS OF CURVED MIRRORS: CONCAVE MIRRORS
• A concave mirror is curved like the inside of the bowl of a spoon. A convex mirror is curved like the back of a spoon. • A curved mirror is usually spherical or parabolic in shape. Parabolic mirrors create more precise images but are more difficult and costly to produce. Spherical mirrors are used in everyday situations. • A concave mirror can produce different images depending on the position of the object. • There are four rays, incident on a concave mirror, whose path can be predicted to determine the
position, nature, orientation, size and magnification of the image. At least two rays must be traced to locate an image when using a ray diagram. • The nature of an image is either ‘real’ or ‘virtual’. The orientation of an image is either ‘upright’ or ‘inverted’ relative to the object. image height image distance • Magnification = = object height object distance M =
H i v =− H o u
3.2 QUESTIONS 1
2
3
4
a
Distinguish between a concave and a convex mirror. Give one application for each.
b
What is meant by the term ‘radius of curvature’ for a spherical mirror?
c
The radius of curvature of a spherical concave mirror is 50 cm. What is its focal length?
d
Use a diagram to define the following terms associated with a spherical concave mirror: centre of curvature, principal axis, principal focus, pole.
a
A small light bulb is placed at the focus of a parabolic concave mirror. Use a diagram to show the path of the light after it reflects from the mirror.
b
State one use for such an arrangement of a globe and mirror.
c
What is spherical aberration?
5
A person looks into the bowl of a soup spoon and sees an inverted image of herself. Use a ray diagram to model this situation, estimating the focal length of the spoon and any other distances you need.
6
A concave mirror of focal length 20 cm is used to produce an image of an object. How far from the mirror must the object be placed in order to form an image that is:
What is the nature of the image?
b
How far will the image of each whisker be from the mirror?
c
What will the size of the image of each whisker be?
A pen is held 60 cm from a concave mirror of focal length 20 cm. Describe the nature of its image.
inverted and the same size as the object
b
upright and twice the size of the object.
7
A cavity in a tooth is actually 2 mm long. A dentist uses a concave mirror of focal length 15 mm to view the tooth. The mirror is held 8 mm from the tooth. Approximately how big will the cavity appear? Describe the nature of the image formed.
8
A person wants to use a make-up mirror with a focal length 40 cm to insert contact lenses. She stands too far back and sees an image of her face that is true to size but inverted. How far is her face from the mirror? Where can she stand to see an upright, magnified image of herself?
9
A student wants to use a concave mirror to magnify a specimen by a factor of 3. A concave mirror with a focal length of 40 cm is chosen. Use a ray diagram to determine the position of the specimen and the characteristics of the image.
While shaving, a man stands 24 cm from a concave mirror whose focal length is 36 cm. His aim is to shave off his 2 mm long whiskers. Use a ray diagram to answer the following. Different horizontal and vertical scales will aid accuracy. a
a
10
The image of an object in a concave mirror is 20 cm behind the mirror when the object is placed 5 cm in front of the mirror. What is the focal length of the mirror?
Mirrors, lenses and optical systems
83
3.3
Convex mirrors Convex mirrors are very common in everyday life, often seen as large mirrors used for security in the ceiling corners of shops or for traffic safety at ‘blind corners’, and as rear-view mirrors for particularly long vehicles. Convex mirrors can be spherical, parabolic, ellipsoidal or hyperbolic in shape, but the last three are only used in telescopes and other specialised equipment.
parallel incident rays
Figure 3.14 Convex mirrors are used on the road to help drivers view traffic around C F
principal axis
virtual focal point
Figure 3.15 The parallel rays reflected from a convex mirror will diverge in such a way that they appear to come from a single point—a virtual focus. 84
WAVE-LIKE PROPERTIES OF LIGHT
sharp corners.
The reflecting surface of a spherical convex mirror is again a portion of a sphere, but in this case the reflective surface is on the outside of the sphere. Incident light rays parallel to the principal axis of a convex mirror will diverge on reflection. If the rays are drawn back behind the mirror, they appear to come from one single point—the virtual principal focus of the mirror (Figure 3.15). The distance from the pole of the mirror to this point is the focal length
of the mirror and, as with a concave mirror, the focal length is half the radius of curvature. Convex mirrors (like plane mirrors) can only produce a virtual image. A convex mirror always provides a wide field of view and an image that is always upright. Examine Figure 3.16. Two observers are looking into mirrors of the same size. The observer looking into the convex mirror is able to see rays coming from a much wider field of view than the observer looking into the plane mirror.
w e i v f o d l e i f e d i w
i r
convex mirror C
r i
w e i v f o d l e i f w o r r a n
i r
plane mirror (a) Ray 1
r i
object
Figure 3.16 Convex mirrors allow a very wide field of view, and so are used when the image of a large region is required.
F
C
F
C
F
C
F
C
principal axis
(b) Ray 2
Graphical ray tracing To understand how an image is formed in a convex mirror, we must again trace the path of rays from a common point on the object as they reflect from the mirror. This is exactly the same process that was followed with plane and concave mirrors.
object
(c) Ray 3
Again, the object is placed to the left of the mirror with its base on the principal axis. Any two of four rays can be drawn from the top of the object to find the image. These rays are chosen because we are able to predict exactly the path that they follow. Either of the other two rays may be used as a ‘check’. Ray 1: This ray travels parallel to the principal axis and reflects as if it came from the virtual principal focus of the mirror. See Figure 3.17a.
object
(d) Ray 4
Ray 2: This ray travels towards the centre of curvature of the mirror and strikes the mirror surface along the normal at the point, i.e. the angle of incidence is 0°, and so it will reflect along the path from whence it came. See Figure 3.17b. Ray 3: This ray strikes the pole of the mirror and reflects at an angle equal to the angle of incidence. The principal axis acts as the normal at this point and so the ray passes below the principal axis at the distance of the object through a point which is the same height as the object. See Figure 3.17c. Ray 4: The ray of light directed towards the virtual principal focal point reflects parallel to the principal axis. See Figure 3.17d.
The path of the light described in these four rays emphasises that light reflecting from a convex mirror will always diverge in front of the mirror. As these rays never meet, a real image cannot be formed and so no image from a convex mirror may be projected onto a screen. Rather, the image will be virtual, forming where the rays appear to ‘meet’ behind the mirror. Once the tip of the image has been located, the rest of the image will lie between this point and the principal axis.
P
object
(e)
image object
F
C
Figure 3.17 Four rays reflected by a convex mirror are chosen to locate and describe the position of the image. The position of the image can be found by using only two of the rays, and any of the others can be a check. These rays are chosen because their path can be predicted with accuracy. Mirrors, lenses and optical systems
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When constructing a ray diagram for a convex mirror, the same conventions apply as for concave mirrors. A vertical line, the optical axis, representing the reflecting surface, is drawn perpendicular to the principal axis for the mirror. A small convex mirror symbol, whose pole lies at the origin for the axes, indicates the type of mirror used. The object is always placed to the left of the mirror and the principal focal point will lie to the right of the mirror (behind it). The horizontal and vertical scales need not be the same, and, in many situations, should be different in order to aid accuracy. Figure 3.18 shows the effect of placing the same object at various distances from a convex mirror. All the images lie behind the mirror between the pole and the focal point and are all upright and diminished. Each image, however, has a different magnification which depends on the position of the object from the mirror. As the object is moved closer to the mirror, the image size increases. Because of the curvature of the mirror the image can never be quite as large as the object; hence the magnification is always less than one ( | M | < 1). (a)
object
image F
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image F
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(b)
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(c)
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Figure 3.18 Ray tracing for a convex mirror. All images are upright, virtual and diminished. As the object is brought closer to the mi rror, the image increases in size, but it will never be the same size as the object.
Worked example 3.3A . A shop uses a convex mirror of focal length 2.0 m for security purposes. If a 1.5 m tall person is standing 4.0 m from the mirror, describe the nature of the image seen. What is the magnification of the image?
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WAVE-LIKE PROPERTIES OF LIGHT
