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1IZTJDT AS Exclusively endorsed by OCR for GCE Physics A Roger Hackett and Robert Hutchings 01865 888080 In Exclusive Partnership 934 physics.prelims.indd 3 15/11/07 8:06:06 am Heinemann is an imprint of Pearson Education Limited, a company incorporated in England and Wales, having its registered office at Edinburgh Gate, Harlow, Essex CM20 2JE. Registered company number: 872828 www.heinemann.co.uk Heinemann is a registered trademark of Pearson Education Limited Text © Roger Hackett, Robert Hutchings 2007 First published 2007 12 11 10 09 08 07 10 9 8 7 6 5 4 3 2 1 British Library Cataloguing in Publication Data is available from the British Library on request. ISBN 978 0 435691 82 0 Copyright notice All rights reserved. No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS (www.cla. co.uk). Applications for the copyright owner’s written permission should be addressed to the publisher. Edited by Anne Russell, Melissa Wesley Index compiled by Indexing Specialists Glossary and module summaries compiled by Graham Bone Designed by Kamae Design Project managed and typeset by Wearset Ltd, Boldon, Tyne and Wear Original illustrations © Pearson Education Limited 2007 Illustrated by Wearset Ltd, Boldon, Tyne and Wear Picture research by Q2AMedia Cover photo © Science Photo Library Printed in the UK by Scotprint Ltd Acknowledgements We would like to thank the following for their invaluable help in the development and trialling of this course: Graham �������������������� Bone, Amanda Hawkins, Dave Keble, Maggie Perry, Gareth Price and Simon Smith. The authors and publisher would like to thank the following for permission to reproduce photographs: p3 Professor Harold Edgerton/Science Photo Library; p6 Maksim Samasiuk/Shutterstock; p13 Photo Researchers, Inc./ Photolibrary.com; p15 Steve Allen/Science Photo Library; p19 L Photo Researchers, Inc./Photolibrary.com; p19 R Choon Hwa Yeo/ Istockphoto; p20 Helen Shorey/Shutterstock; p21 Photo Researchers, Inc./Photolibrary.com; p22 Professor Harold Edgerton/ Science Photo Library; p29 NASA/Science Photo Library; p32 Harcourt Index/Getty Images/Brand X Pictures; p32 T Alan Smillie/ Shutterstock; p32 TM Transtock Inc./Alamy; p32 B Mercedes-Benz USA; p48 L Mercedes-Benz USA; p48 R Trip/Alamy; p49 Detlev Van Ravenswaay/Science Photo Library; p55 Gilbert Iundt/TempSport/ Corbis; p57 Nasa; p60 NASA/Science Photo Library; p64 Eliza Snow/Istockphoto; p67 T Skyscan Photolibrary/Alamy; p67 M Skyscan/Science Photo Library; p67 B Flashon/Shutterstock; p69 Mercedes-Benz USA; p70 Bethan Collins/Shutterstock; p81 Harcourt Index/Corbis; p84 T Andrew Lambert Photography/Science Photo Library; p84 B Andrew Lambert Photography/Science Photo 934 physics.prelims.indd 2 Library; p87 Alfred Pasieka/Science Photo Library; p93 Astrid & Hans Frieder Michler/Science Photo Library; p95 Andrew Lambert Photography/Science Photo Library; p102 Pascal Goetgheluck/ Science Photo Library; p104 Mauro Fermariello/Science Photo Library; p108 Sheila Terry/Science Photo Library; p117 David Woods/Shutterstock; p127 Andrew Lambert Photography/Science Photo Library; p135 Gary Hincks/Science Photo Library; p136 TL Sam Ogden/Science Photo Library; p136 TM Richard R. 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Any omissions will be rectified in subsequent printings if notice is given to the publisher. Websites There are links to websites relevant to this book. In order to ensure that the links are up-to-date, that the links work, and that the sites are not inadvertently linked to sites that could be considered offensive, we have made the links available on the Heinemann website at www.heinemann.co.uk/hotlinks. When you access the site, the express code is 1813P. Exam Café student CD-ROM © Pearson Education Limited 2007 The material in this publication is copyright. It may be edited and printed for one-time use as instructional material in a classroom by a teacher, but may not be copied in unlimited quantities, kept on behalf of others, passed on or sold to third parties, or stored for future use in a retrieval system. If you wish to use the material in any way other than that specified you must apply in writing to the publisher. Original illustrations, screen designs and animation by Michael Heald Photographs © iStock Ltd Developed by Elektra Media Ltd Technical problems If you encounter technical problems whilst running this software, please contact the Customer Support team on 01865 888108 or email [email protected] 15/11/07 8:06:06 am 1 Contents Introduction vi UNIT 1 Mechanics (G481) Module 1 Motion 1 2 3 4 5 6 7 8 9 10 Physical quantities and units Estimated physical quantities Scalar and vector quantities Vector calculations Vector resolution Definitions in kinematics Graphs of motion Constant acceleration equations Free fall Measurement of g 2 4 6 8 10 12 14 16 18 20 22 Summaryandpracticequestions End-of-moduleexaminationquestions 24 26 Module 2 Forces in action 1 Forceandthenewton 2 Motionwithnon-constantacceleration 3 Equilibrium 4 Centreofgravity 5 Turningforces FurtherquestionsA 6 Density 7 Pressure 8 Carstoppingdistances 9 Carsafety 28 30 32 34 36 38 40 42 44 46 48 Summaryandpracticequestions End-of-moduleexaminationquestions 50 52 Module 3 Work and energy 1 Workandthejoule 2 Theconservationofenergy 3 Potentialandkineticenergies FurtherquestionsB 4 Powerandthewatt 5 Efficiency 6 Deformationofmaterials 7 Hooke’slaw 8 TheYoungmodulus 9 Categoriesofmaterials 54 56 58 60 62 64 66 68 70 72 74 Summaryandpracticequestions End-of-moduleexaminationquestions 76 78 UNIT 2 Electrons, waves and photons (G482) Module 1 Electric current 1 Electriccurrentandcharge 2 Kirchhoff’sfirstlaw 3 Electrondriftvelocity 80 82 84 86 Summaryandpracticequestions End-of-moduleexaminationquestions 88 90 Module 2 Resistance 1 Electromotiveforce 2 Potentialdifference 3 ResistanceandOhm’slaw 4 Resistanceofcircuitcomponents 5 Resistivity 6 Theeffectoftemperatureonresistivity 7 Electricalpower 8 Domesticelectricalsupply 9 Chargingforelectricalenergy 92 94 96 98 100 102 104 106 108 110 Summaryandpracticequestions End-of-moduleexaminationquestions 112 114 Module 3 DC circuits 1 Seriescircuits 2 Parallelcircuits 3 Circuitanalysis1 4 Circuitanalysis2 5 Thepotentialdivider FurtherquestionsC 116 118 120 122 124 126 128 Summaryandpracticequestions End-of-moduleexaminationquestions 130 132 Module 4 Waves 134 136 138 140 142 144 146 148 150 152 1 Wavemotion 2 Waveterminology 3 Wavespeed 4 Waveproperties FurtherquestionsD 5 Electromagneticwaves 6 Polarisation 7 Interference 8 TheYoungdouble-slitexperiment iv 934 physics.prelims.indd 4 15/11/07 8:06:08 am Contents 9 The diffraction grating Further questions E 10 Stationary waves 11 Stationary wave experiments 12 Stationary longitudinal waves 154 156 158 160 162 Summary and practice questions End-of-module examination questions 1 End-of-module examination questions 2 164 166 168 Module 5 Quantum physics 170 172 174 176 1 The energy of a photon 2 The photoelectric effect 1 3 The photoelectric effect 2 4 Wave–particle duality 5 Energy levels in atoms Further questions F 6 Spectra 178 180 182 184 Summary and practice questions End-of-module examination questions 186 188 Appendix: Accuracy and significant figures 190 Answers 192 Glossary 204 Index 210 934 physics.prelims.indd 5 15/11/07 8:06:10 am Introduction Howtousethisbook In this book you will find a number of features planned to help you. • Module opener pages – these carry an introductory paragraph that should set the context for the topics covered in the module. They also have a short set of questions that you should already be able to answer from your previous science courses. • Double-page spreads filled with information about each topic. • End-of-module summary pages to help you link all the topics within each module together. • End-of-module examination questions. These have been selected to show you the types of question that may appear in your examination. 1.1 Module contents UNIT 2 Module 2 1 2 Resistance 3 Introduction This module is about more than just resistance. It is about the meaning of the word ‘resistance’ and the resistance, and hence the behaviour, of the many different components that are used in electrical and electronic circuits. Some electrical components are very low-tech. For example, the heating element in a convector heater is nothing more than a piece of resistance wire. A fuse is often a piece of copper wire in a ceramic tube. Light bulbs are now manufactured in great quantities, but the first commercial light bulbs, which contained a carbon filament in a vacuum, did not go on sale until 1897, 100 years after Volta made his first battery. The problem was to get the filament hot enough without it melting. The filament in standard bulbs is a finely coiled coil of tungsten wire heated to about 2200 °C in an argon atmosphere. These bulbs radiate only about 10% of their power as light; the rest of the power is wasted in heating the surroundings. At present there is pressure on people and governments to fit low-energy fluorescent light bulbs rather than these conventional tungsten filament bulbs. Module 1 Free fall 9 Motion acceleration. When answering questions on free fall you need to deal with horizontal movement and vertical movement entirely separately. You must also be careful with + and – signs. The worked example shows how this information can be used. By the end of this spread, you should be able to . . . 1 Apply the equations for constant acceleration to situations with uniform velocity in one Electromotive force direction and constant acceleration in a perpendicular direction, including to motion of bodies falling in the Earth’s gravitational field. Potential difference Free fall A B C 1 Explain how experiments carried out by Galileo overturned Aristotle’s ideas of motion. Resistance and Ohm’s law Resistance of circuit components Resistivity 6 The effect of temperature on resistivity 7 Electrical power 8 Domestic electrical supply 9 Charging for electrical energy Figure 4 Constant horizontal velocity and downwards acceleration Free fall 4 5 An object undergoing free fall on the Earth has an acceleration g = 9.8118 m s–2. This figure is not quite a constant. It depends to a certain extent on where it is measured. At the North Pole g = 9.8322 m s–2 while in Singapore, near the Equator, it is 9.7803 m s–2. It also decreases slightly with altitude (i.e. the distance from the centre of the Earth). In this spread, however, we shall assume that it is constant, and that air resistance has a negligible effect. Velocity 36 m s1 39n Level of throwing Figure 6 Throwing a cricket ball Ground Figure 1 The Leaning Tower of Pisa, where Galileo is said to have demonstrated the effect of gravity Velocity upwards/m s1 13.2 10 0 1 2 3 4 6.4 10 In this module you will learn about: • the symbols used for components in electrical circuits • the distinction between electromotive force (e.m.f.) and potential difference (p.d.) • current – voltage characteristics for different components • resistance and Ohm’s law • resistivity • electrical power and energy. This problem is illustrated in Figure 6. The first step towards solving it is to resolve the velocity into horizontal and vertical components (explained on spread 1.1.5). Horizontal component of velocity at start = 36 cos 39° = 28.0 m s–1 Vertical component of velocity at start = 36 sin 39° = 22.7 m s–1 Answer (a) Vertical movement from throwing to the top of the ball’s arc (upwards regarded as positive) Use v2 = u2 + 2as where s is the maximum height at which v = 0. This gives 0 = 22.72 + 2 r (– 9.81)s 19.6s = 515.3 515.3 s = ______ = 26.3 m 19.6 The acceleration of free fall is vertically down towards the centre of the Earth. Galileo is said to have measured the acceleration of free fall in his famous experiment by dropping balls from the top of the Leaning Tower of Pisa. (There are strict instructions on the Tower of Pisa at present to stop anyone repeating his experiment. No one is now allowed to drop anything from the top of the tower!) Galileo was actually more concerned with the discovery that the acceleration of free fall is the same for all objects, whatever their mass. This contrasts with ancient Greek ideas: Aristotle assumed, without experimenting, that heavier objects would fall faster than lighter ones. 23.0 20 3.4 0 A cricket ball is thrown with a velocity of 36 m s–1 at an angle of 39° to the horizontal. (a) What height above the throwing point does it reach? (b) How long will it take to fall back to the level at which it was thrown? (c) What horizontal distance will it travel during this time? Why might a higher than expected value also be interesting? 30 The photograph is a view of the surface of a microprocessor (a silicon chip) of the sort used in all computers. This is very much a high-tech device. The magnification in the photograph is ×480. Most of what you see are connecting links between various tiny components. In most computers a fan is needed to keep the chips cool while they carry out millions of calculations per second. Developers are aiming to make microprocessors such as this one even smaller and to lower their power requirements so that no cooling fan is necessary. 5 Time/s 16.2 20 26.0 30 Figure 2 Object thrown vertically upwards Test yourself (b) Use s = ut + ½at2 for the entire vertical movement. The vertical displacement is therefore zero (it is back to the level at which it started). This gives 0 = 22.7t + ½ (–9.8t2) and dividing through by t 2 r 22.7 = 9.8t 45.4 t = _____ = 4.63 s 9.8 If an object is thrown rather than dropped, the situation is different. An object thrown vertically upwards still has the constant acceleration g downwards. The graph illustrating this is shown in Figure 2. Here, the object starts with an upward velocity of 23.0 m s–1. There is a moment at the top of its motion where the object has zero velocity, but still has an acceleration of 9.81 m s–2. The graph must be a straight line of gradient 9.81 because the gradient of a velocity–time graph gives the acceleration, 9.81 m s–2. There is no need to consider the upward and the downward motion separately. Each second the velocity changes by 9.81 m s–1, as shown by the figures on the velocity axis on the graph. One final point: do not keep rounding numbers when going through a lengthy calculation, or your final answer could be some way out. Any rounding of numbers should be done when quoting answers rather than in the course of your calculations. A good rule of thumb is to quote all the figures you are certain about and one about which you are uncertain – but no more. In the above example, you were given two significant figures in the question so you can be reasonably sure of two significant figures in your answer. The third figure is doubtful so quote the three answers as: (a) 26.3 m (b) 4.63 s (c) 130 m When in doubt about the number of significant figures to use, use three. You are most unlikely to be wrong by more than one significant figure. Questions 1 Copy the path of the ball in Figure 6 above. Include the velocity vector and its components at the throwing point. (a) (i) Add to your sketch the velocity vector at the highest point reached by the ball. What is its magnitude? (ii) Add to your sketch the velocity vector at 3.0 s into the flight of the ball. (b) (i) Calculate the horizontal and vertical components of the velocity at 3.0 s. (ii) Calculate the velocity at 3.0 s. This effect is illustrated by Figure 4. All the horizontal arrows are the same length, indicating constant horizontal velocity. The vertical arrows show a downward Figure 3 One mediaeval theory of a cannon ball’s trajectory Examiner tip (c) During all of this time the horizontal velocity has remained constant, so: the horizontal distance travelled = 28.0 m s–1 r 4.63 s = 129.6 m When a batsman hits a cricket ball for four, the ball has both vertical and horizontal motion. Soldiers and archers in the middle ages attempted to address this problem, working out theories for the flight of cannon balls and arrows. An example is shown in Figure 3. In fact, in the absence of air resistance, the horizontal velocity of any projectile remains constant while it is accelerating downwards. 1 What is this symbol 7 used for? 2 What equation defines resistance? 3 Name (i) a material which is a good electrical conductor; (ii) a material used for electrical insulation. Figure 5 Multiple-flash photography showing travel in two dimensions (a ball bouncing) Worked example Drilling for oil One method oil companies use when prospecting for suitable sites to drill oil wells is to measure g very accurately. To do this they use a gravimeter, which accurately times a ball falling in a vacuum. Attraction to the Earth is more dependent on the ground just beneath your feet than on matter further away. So, if at a certain location the value of g is expected to be 9.8167 m s–2, but turns out to be 9.8162 m s–2, then the implication is that there is something less dense underground than rock. It might be oil (or water). 20 1.3 Module 3 Work and energy summary Practice questions 1 The table shows how the braking distance (see spread Stress Stress Stress 1.2.8) for a car of mass 800 kg varies with the initial speed of the car when a constant braking force F is applied. Stress = Strain Strain Strain Ductile Polymeric Brittle Efficiency = Useful output energy 100 Total input energy F A Strain = Conservation of energy Speed (m s–1) 0 10 20 30 40 Distance (m) 0 6 24 x 96 (a) Calculate the kinetic energy in J of the car when it is travelling at 20 m s–1. (b) How much work in J is done by the braking force to bring the car to rest from 20 m s–1? Extension Length (c) Calculate the value of the braking force in N. (d) Calculate the braking distance x in m of the car from an initial speed of 30 m s–1. Elastic (e) Write down a general equation which will allow you to calculate the braking distance of the car from any initial speed. Materials (f) One simplistic method of measuring the severity of a car crash is by the amount of kinetic energy which must be dissipated. Using this method, determine whether a car hitting a wall at 20 m s–1 is a worse crash than two identical cars each travelling at 10 m s–1 in opposite directions making a head-on collision. Hooke’s law Efficiency Elastic limit Create and destroy F = kx 2 A model car Transfer and transform Force Work and energy Gradient = k Extension Forms of energy Power Chemical energy Electrical potential energy Electromagnetic wave energy Gravitational potential energy Internal energy Nuclear energy Sound energy Kinetic energy Area under line = Work done W = 12 Fx Figure 1 runs on a narrow flexible track. It has been formed into a vertical circular loop as shown in Figure 1. C v r u u A B (a) The car approaches the loop at speed u. Explain why the speed v at C must be less than u. (b) Using the law of conservation of energy show that u and v are related by the equation W = Fx cos U P= W t u2 = v2 + 4gr where g is the acceleration due to gravity. (c) There is a minimum value of v necessary ____ for the car to reach C. Unless u is greater or equal to 5gr the car will fall off the track before reaching C. Write down an expression for the minimum value of v. 1W = 1Js –1 Joule W = F Distance moved in the direction of the force (e) The boy playing with this toy only has enough track to raise the end 40 cm above the floor. Calculate the minimum speed at which he must release the car if it is to perform the stunt. 1 (d) For a loop of radius 20 cm, find the minimum height to which the end of the track, beyond the left-hand side of the diagram, must be raised for the car to complete the circle, when released from rest at the end of the track. Module 2 Examination questions Forces in action Examination questions Young modulus of steel. He stretches a fine wire of length 1.5 m and cross-sectional area 5.2 r 10–8 m2 using a force of 20 N. He measures the extension to be 2.8 mm. Velocity Cable (a) Calculate (i) the stress in the wire, (ii) the strain caused and (iii) the value of the Young modulus of steel from these data. 0 Sea (b) A very tall building requires a series of lifts to reach all floors of the building. The reasons for this arrangement include convenience and logistics as well as physics. The following calculation suggests one of the physical problems. The cables of length 70 m supporting a lift consist of two steel ropes each of 100 strands giving a total cross-sectional area of 1.0 r 10–4 m2. Consider a full lift to carry 8 passengers of average mass 75 kg. Calculate by how much an empty lift moves downwards when it is entered by 8 passengers. A girl of mass 55 kg is rescued by a man of mass 75 kg. The two are attached to the cable and are lifted from the sea to the helicopter. The lifting process consists of an initial acceleration followed by a period of constant velocity and completed by a final deceleration. (a) Name the two main forces acting on the two people being lifted. [2] (b) Calculate the combined weight in N of the man and girl. [1] (c) Calculate the tension in N in the cable during (i) the initial acceleration of 0.55 m s–2 and (ii) the period of constant velocity. [4] (d) Calculate the final deceleration in m s–2 if the tension in the cable is 1240 N. [2] (e) Sketch a graph of velocity v against time t for the complete lifting process. Numerical values are not required. [3] (OCR 2821 Jun04) 2.0 r 1020 N. To try to visualise the magnitude of this force imagine that the force holding the Earth and Moon together is provided instead by a steel cable. (b) Your answer to (a) should be about 400 km. Show that the mass of this cable would be greater than the mass of the Moon which is 7 r 1022 kg. The distance between the Earth and the Moon is 4 r 108 m and the density of steel is 7.9 r 103 kg m–3. 2 5 The ultimate tensile stress is not the value looked at by engineers when considering safety in their design of a building or aircraft or whatever. It is the yield stress at which permanent or plastic deformation starts. The basic rule is that the maximum stresses in the design should be no more than one quarter of the yield stress. A rocket-propelled model car accelerates from rest along a horizontal track as shown in Figure 2. Thrust 3.0 N Figure 2 (a) The mass of the car is 0.80 kg and the forward thrust provided by the rocket is 3.0 N. Calculate the initial acceleration in m s–2 of the car. [2] (b) The speed of the car increases at a decreasing rate to a maximum speed. The car then continues along the track at this constant speed. Throughout the motion the forward thrust on the car remains constant at 3.0 N. Explain, in terms of the forces acting on the car, why (i) the acceleration of the car decreases as the speed increases and (ii) the car reaches a constant speed. [4] (c) When the car has travelled further down the track, a parachute, attached to the rear of the car, is opened. The forward thrust remains unchanged at 3.0 N. (a) The yield stress of steel is 3 r 108 N m–2 and the ultimate breaking stress is about 109 N m–2. How much smaller is the maximum stress used in design than the ultimate breaking stress for steel? (b) A tug boat assists a tanker to dock at a quay using a steel cable in which the maximum safe tension is 6 r 105 N. Calculate the diameter of this steel cable. (c) Explain why it is better to use a long cable to tow a boat or a car, rather than a short one. Hint: First consider the extension and increase in strain when there are sudden changes in motion, e.g. a jerk. (d) Calculate the strain energy stored in 100 m of cable at the maximum safe tension. Take the Young modulus of steel to be 2.1 r 1011 N m–2. Time t Figure 3 Figure 1 4 The gravitational force between the Earth and the Moon is 0 (i) Use the graph to describe how the motion of the car changes from the instant the parachute is opened. [2] (ii) Explain why the motion of the car changes in the way you have described. [2] (OCR 2861 Jan07) 3 4 Figure 4 shows the two forces acting on a raindrop falling at velocity v through still air. Drag force F Cross-sectional area A Velocity v Raindrop of mass m Weight Figure 4 (a) The drag force F is related to the velocity v by the expression F = KAv2 where K is a constant depending on the shape of the raindrop is the density of air A is the cross-sectional area of the raindrop and v is the velocity of the raindrop. (i) Show that the unit of force N can also be written as kg m s–2. [1] (ii) Show that K has no units (is a dimensionless quantity). [1] (iii) What can be said about the two forces acting on the raindrop when it is falling at constant velocity (terminal velocity) vT? Explain your reasoning. [2] (iv) Show that the terminal velocity vT of a raindrop of mass m is given by ____ mg ____ vT = KA where g is the gravitational field strength. [2] 5 77 radius cross-sectional area weight r A mg 2r 4A 8mg Explain why, as the size of the drop increases from r to 2r (i) the cross-sectional area is increased by a factor of 4 and (ii) the weight is increased by a factor of 8. [2] (c) Use the equation in (a)(iv), and the information in the table, to show that a raindrop of radius 2r will fall with a terminal velocity about 1.4 times greater than a raindrop of radius r. [2] (OCR 2821 Jan06) A child sits on a swing Chain and is pulled by a 35n horizontal force P so that Tension in chains the chains make an angle with the vertical of 35°. Horizontal force P Figure 5 shows the forces Vertical acting in this position. Swing seat The combined mass of the child and swing seat Weight Figure 5 is 28 kg. (a) Calculate the combined weight in N of the child and swing seat. [2] (b) Use a labelled vector triangle to determine the force P in N required to hold the swing stationary in the position shown in Figure 5. [4] (c) State and explain what happens to the tension in the chains if the swing is pulled so that the chains make a larger angle with the vertical. A numerical answer is not required. [2] (OCR 2821 Jan04) (a) (i) Define centre of gravity. [1] (ii) Using a sketch, define the moment of a force. [2] (b) Figure 6 shows a computer resting on a tabletop that is hinged at A. 0.80 m 6 Figure 7 shows a stationary oil drum floating in water. Cross sectional area 0.25 m2 0.75 m Water Figure 7 The oil drum is 0.75 m long and has a cross-sectional area of 0.25 m2. The air pressure above the oil drum is 1.0 r 105 Pa. (a) Calculate the force in N acting on the top surface of the oil drum due to the external air pressure. [2] (b) The average density of the oil drum and contents is 800 kg m–3. Calculate the total weight in N of the oil drum and contents. [3] (c) Calculate the force in N acting upwards on the base of the drum. [1] (OCR 2821 Jun05) Computer Tabletop B A F 0.25 m 200 N mgh = 12 mv 2 76 The tabletop has a mass of 5.0 kg and its centre of gravity is 0.40 m from the axis of the hinge at A. The computer has a weight of 200 N acting through a point 0.25 m from the hinge at A. The tabletop is supported to maintain it in a horizontal position by a force F acting vertically at B. The distance AB is 0.80 m. (i) Calculate the weight in N of the tabletop. [1] (ii) Draw the tabletop and forces shown in Figure 6 acting on it. Add and label an arrow to represent the weight W of the tabletop. [1] (iii) Apply the principle of moments about the hinge at A to determine the vertical force F in N applied at B that is required to maintain the tabletop in equilibrium. [3] (iv) The tabletop must experience a resultant force of zero in order to be in equilibrium. Explain how the forces acting on the tabletop fulfil this condition. [2] (OCR 2821 Jan03) (b) The table below shows information about two different sized spherical raindrops. Figure 3 is a sketch graph showing how the velocity of the car changes from the moment the parachute opens at time t = 0. Figure 1 shows a helicopter that has a cable hanging from it to the sea below. 3 A student takes a single measurement to measure the (a) Calculate the minimum diameter of such a cable. Use the ultimate tensile stress of steel, 1.5 r 109 N m–2. Assume that there are no frictional forces. The car is not driven by any motive force. It just freewheels. Work done Watt Gravitational potential energy = mgh Kinetic energy = 12 mv 2 Stress Strain Young modulus = Plastic Sankey diagrams 1.2 Work and energy Pract ice quest ions 21 Figure 6 52 53 Within each double-page spread you will find other features to highlight important points. 2.5 Learning objectives 5 Energy levels in atoms By the end of this spread, you should be able to . . . 1 Explain the information provided by spectral lines. 1 Describe the origin of emission spectra. 1 Use the relationships hf or hc/λ = E1 – E2. Spectra People have always been fascinated by light – think of all the myths and stories surrounding rainbows. It was not until the seventeenth century, however, that a serious scientific study of light was undertaken. In 1666 Isaac Newton established, by using prisms, that white light could be split into a variety of colours and that this colour spectrum could then be combined back into white light. You can repeat Newton’s experiment by looking at a television screen through a strong magnifying glass. Colour TV screens can only produce the three primary light colours – red, green and blue. They do not produce white light. A white-coloured area represents a region of the screen where these three primary colours are of equal brightness. The production of other colours just requires the three primary colours to be combined in the appropriate proportions. How Science Works Colour spectra Visible light consists of electromagnetic waves that the eye can detect. When all the different wavelengths of visible light fall on the eye’s retina at the same time, white light is seen. However, visible light can be split into its different wavelengths, i.e. the visible light spectrum. Each colour band represents a very small range of wavelengths. During the eighteenth century, improvements in optical instruments enabled a closer study of spectra. In 1814 the German physicist Joseph von Fraunhofer discovered dark lines on the Sun’s spectrum where certain frequencies are missing. A photo of the Sun’s spectrum is shown in Figure 1. These lines were not explained until later in the century, by which time the spectra from gas discharge lamps containing hot gases, such as hydrogen, sodium, and mercury vapour, had been studied. A series of line spectra from different gases is shown in Figure 2. You can see that only certain frequencies are present and each element produces a unique pattern, like an elemental fingerprint! Line spectra are produced by hot gases, where the atoms are far apart. Term in bold Module 5 Quantum physics Energy levels in atoms Energy levels in atoms In 1913, Niels Bohr was the first physicist to apply quantum theory to radiation. He suggested that for atoms, the classical mechanics of Newton had to be modified. He proposed that within an atom’s structure there existed specific energy levels. As electrons moved closer to the nucleus, they emitted radiation – the electrons were moving from a higher energy level to a lower one (i.e. returning to their ground state). Bohr’s theory was a mixture of Newtonian ideas and quantum theory. Today, atomic theory is now based solely on quantum mechanics. However, the relationship between the frequency of radiation emitted and the energy levels is still upheld. The energy of a quantum of radiation, hf, is given by the equation: Energy 0 Transitions producing visible light –3.41 eV Transitions producing ultraviolet radiation Questions hf = E1 – E2 where E1 is the energy associated with the energy level the electron has left, and E2 is the energy associated with the level the electron moves to. Transitions producing infrared radiation –0.85 eV –1.51 eV –13.6 eV Ground state Figure 3 An energy-level diagram for the hydrogen atom The energy levels in the hydrogen atom are shown in Figure 3. The zero on this diagram is taken to be when the electron is a long way from the atom’s nucleus. The electron’s energy falls as it moves nearer to the nucleus. Worked example Some street lamps have a very intense orange light: they are sodium lamps. Two lines in the spectrum of a sodium lamp have wavelengths of 589.0 and 589.6 nm. (a) What are the frequencies of these wavelengths? (b) What energy changes do these transitions correspond to in electronvolts? Answer (a) Since: c = f&, c f = __. The two frequencies are: & 2.998 r 108 ___________ = 5.090 r 1014 Hz; and 589.0 r 10–9 2.998 r 108 ___________ = 5.085 r 1014 Hz 589.6 r 10–9 (b) For the 589.0 nm wave: E1 – E2 = hf = 6.626 r 10–34 r 5.090 r 1014 Hz = 3.373 r 10–19 J 3.373 r 10–19 J = __________________ = 2.105 eV 1.602 r 10–19 J eV–1 For the 589.6 nm wave: E1 – E2 = hf = 6.626 r 10–34 r 5.085 r 1014 Hz = 3.369 r 10–19 J Figure 1 The Sun’s spectrum showing the dark lines crossing it at certain wavelengths 180 Figure 2 Spectra from a series of different hot gases 3.369 r 10–19 J = __________________ = 2.103 eV 1.602 r 10–19 J eV–1 1 Explain how you can find that the ionisation energy of a hydrogen atom is 13.6 eV from Figure 3. What is the minimum speed that a free electron must have to ionise a hydrogen atom by direct collision with the bound electron? 2 Suppose that an atom has two energy levels of value E1 and E2 above the ground-state level, taken as zero energy. Photons are emitted from this atom at wavelengths 620 nm (red), 540 nm (green) and 290 nm (UV). (a) Sketch the energy level diagram. (b) Add and label the three transitions causing the emitted light. (c) Calculate the values of E1 and E2, which will give these three spectral lines. (d) What colour would you expect a gas discharge lamp of this atom to appear? Examiner tip In spectra, it is common to get pairs of wavelengths very close together. When doing calculations using such numerical values, it is essential to work with four or more significant figures or you end up with the same answer for both! The speed of light to 4 sig. figs. is 2.998 × 108 m s–1. Worked example Examiner tip Questions 181 vi 934 physics.prelims.indd 6 15/11/07 8:06:51 am ons Introduction • Learning objectives – these are taken from the Physics AS specification to highlight what you need to know and to understand. • Key definitions – these are the terms that appear in the specification. You must know the definitions and how to use them. • Terms in bold – these draw attention to terms that you are expected to know. These are important terms used by physicists. You will find each term in bold listed in the glossary at the end of the book • Examiner tips – these will help you avoid making common errors in the examinations. • Worked examples – these show you how calculations should be set out. • How Science Works – this book has been written in a style that reflects the way that scientists work. Certain sections have been highlighted as good examples of How Science Works. • Questions – at the end of each topic are a few questions that you should be able to answer after studying that topic. Reinforce your learning and keep up to date with recent developments in science by taking advantage of Heinemann’s unique partnership with New Scientist. Visit www.heinemann. co.uk/newscientistmagazine for guidance and discounts on subscriptions. In addition, you’ll find an Exam Café CD-ROM in the back of the book, with more questions, revision flashcards, study tips, answers to the exam questions in the book and more. The examination It is useful to know some of the language used by examiners. Often this means little more that just looking closely at the wording used in each question on the paper. • Look at the number of marks allocated to a part question – ensure you write enough points down to gain these marks. The number of marks allocated for any part of a question is a guide to the depth of treatment required for the answer. • Look for words in bold. These are meant to draw your attention. • Look for words in italics. These are often used to emphasise a definition. • Diagrams, tables and equations often communicate your answer better than trying to explain everything in sentences. Look for the action word. Make sure you know what each word means and answer what it asks. The meanings of some action words are listed below. • Define: only a formal statement of a definition is required. • Explain: a supporting argument is required using your knowledge of physics. The depth of treatment should be judged from the mark for the question. • State: a concise answer is expected, with little or no supporting argument. • List: give a number of points with no elaboration. If you are asked for two points then only give two! • Describe: state in words, using diagrams where appropriate, the main points of the topic. • Discuss: give a detailed account of the points involved in the topic. • Deduce/Predict: make a logical connection between pieces of information given. • Outline: restrict the answer to essential detail only. • Suggest: you are expected to apply your knowledge and understanding to a ‘novel’ situation, which may include topics that you have not covered in the specification. • Calculate: a numerical answer is required. In general, working should be shown. • Determine: the quantity cannot be measured directly but is obtained by calculation, substituting values of other quantities into a standard formula. • Sketch: in graphs, the shape of the curve need only be qualitatively correct. When you first read the question, do so very carefully. Once you have read something incorrectly, it is very difficult to get the incorrect wording out of your head! Check your work as you go. If you wait until the end of the examination you may not have time and you will have forgotten the detail of the question. Check that an answer is reasonable; check that its units are possible; check that it answers the question; and if you have time at the end of the examination, read through your descriptive answers to ensure that what you wrote is what you intended to write. 934 physics.prelims.indd 1 15/11/07 8:06:57 am Unit 2 Module 4 Waves Introduction Earthquake seismic waves are not the type of waves that immediately spring to mind when ‘waves’ are mentioned, but they are probably the most dangerous waves on Earth. It was seismic waves that were responsible for the awful tsunami in the Indian Ocean on Boxing Day 2004. The image shows seismic waves (red) radiating out from the focus (or hypocentre, red dot) of an earthquake. The epicentre is the point on the surface above this. It was by analysing seismic waves from earthquakes that scientists were able to determine the structure of the Earth with its mainly molten core, its mantle surrounding the core and consisting of dense, iron-bearing rock, and its rocky crust. The fact that some of the core is liquid is known because seismic waves that must have passed through the core, and would have started out as both transverse and longitudinal waves, arrived at a detector as longitudinal waves only. Transverse seismic waves cannot travel through liquids. Water waves are visible waves. Sound waves are audible waves. Electromagnetic waves encompass a great variety of waves, from radio and microwaves, infrared, light and ultraviolet to X-rays and gamma rays. As a letter in the press recently expressed it: ‘We live in an electrosmog of electromagnetic waves.’ We would all have to get used to a radical change in lifestyle if the only electromagnetic waves available were those of infrared, light and ultraviolet from the Sun and from the Earth itself. The special point to note about man-made radio waves and microwaves, apart from the fact that they travel so quickly, is the way in which they can carry information. The information can be in digital form, in which case the wave is interrupted, often billions of time per second, to give a series of ones and zeroes. With analogue signals, the amplitude or the frequency of the wave is modified rapidly. In this long module you will learn about: • the general principles that apply to all waves • the different types of waves • the meaning of terms such as refraction, interference, diffraction and polarisation • the principle of superposition • stationary waves. Test yourself 1 2 3 4 Namethreetypesofwavethatyourbodycandetect. Whatequationlinkswavelength,frequencyandspeedofawave? Nameadevicethatusesstationarywaves. Stateoneobjectormethodthatcreatescolourwithoutdyesorpaints. 934 physics.U2 M4.indd 134 15/11/07 8:41:28 am Module contents 1 Wave motion 2 Wave terminology 3 Wave speed 4 Wave properties 5 Electromagnetic waves 6 Polarisation 7 Interference 8 The Young double-slit experiment 9 The diffraction grating 10 Stationary waves 11 Stationary wave experiments 12 Stationary longitudinal waves 934 physics.U2 M4.indd 135 15/11/07 8:41:40 am 2.4 1 Wavemotion By the end of this spread, you should be able to . . . ✱ Describeanddistinguishbetweenprogressivelongitudinalandtransversewaves. Introduction When people think of waves they usually picture water waves, such as those rocking a boat or crashing against rocks, or a tsunami. Such images are shown in Figures 1, 2 and 3. Far less dramatic, but still water waves, are the ripples on a puddle (see Figure 4). Waves of this type involve the movement of matter and can be formed on any liquid, not just water. Figure 1 A boat being buffeted by a wave in a rough sea Figure 2 Water waves crashing against a sea wall Figure 3 A tsunami Figure 4 Ripples spreading across the surface of a puddle, picked out by artificial colour There are also other sorts of waves, such as radio waves and microwaves. These waves form part of the electromagnetic spectrum. This is a range of electromagnetic waves that travel in free space at 300 000 km s–1. The spectrum also includes visible light, infrared, ultraviolet, X-rays and gamma rays. Most of our knowledge regarding the parts of the Universe beyond our Solar System comes from analysing the electromagnetic spectrum. Wave movement B A Thepropagationofawave Waves that move away from a source are called progressive waves. Figure 5 shows a wave moving forward, with two ‘snapshots’ taken at a short interval from one another. You can see that particle A is moving upwards while particle B is moving downwards to create the wave movement. All the particles oscillate vertically but they do not move forwards or backwards, even though the wave moves forward. Figure 5 Oscillation of particles producing a wave 136 934 physics.U2 M4.indd 136 15/11/07 8:41:49 am Module 4 Waves Wave motion Transverse waves Time Displacement Displacement Both water waves and waves in the electromagnetic spectrum have the direction of propagation at right angles to the direction of the oscillations, i.e. the direction of the wave travel is at right angles to the direction of the oscillations. Such waves are called transverse waves. In practice the oscillations are anything but regular and smooth. Waves which do have smooth and regular oscillations are called pure waves. The pattern of oscillation is a sine wave. The difference between pure, impure and irregular waves is shown in Figure 6. Time (b) A regular impure wave Displacement (a) A pure wave Time (c) An irregular impure wave Figure 6 Types of transverse waves Longitudinal waves In sound waves, the oscillations of the particles take place in the direction of propagation, rather than at right angles to it – i.e. the movement of the particles is in the same direction as the wave’s movement. Waves of this type are called longitudinal waves. The molecules in air are less firmly bound to one another than in solids, and because of this, pressure waves, such as sound waves, cannot be propagated as transverse waves. Figure 7 shows how the movement of air particles results in the propagation of a sound wave. Regions of high pressure are called compressions and regions of low pressure are called rarefactions. Compression Rarefaction tT t 78 T Time t 68 T t 58 T t 48 T t 38 T t 28 T t 18 T tT Movement of particles Figure 7 Particle movement in a longitudinal wave. The green lines show the oscillation of an individual particle. Notice how the compressions and rarefactions move to the right as time proceeds. Questions 1 How do an oscillation and a wave differ? Hint: Think about motion at one point in space over all time and motion of all points in space at one time. 2 A longitudinal wave can be represented by a graph of displacement against time or pressure difference against time. Sketch both of these graphs on the same time axis for a sinusoidal sound wave, that is, a pure note. How are the two waves the same? How are they different? 137 934 physics.U2 M4.indd 137 15/11/07 8:41:55 am 2.4 2 Waveterminology By the end of this spread, you should be able to . . . ✱ Describeandusethetermsdisplacement,amplitude,wavelength,period,phasedifference andfrequencyofawave. Height Introduction Mean water level 2 4 6 8 10 Distance/m Height Figure 1 Height of water wave plotted against distance Mean water level 5 10 15 One of the major difficulties with describing waves in a book or on an examination paper is that they move. So, representing waves can be awkward. The problem is partially overcome by drawing diagrams or sketch graphs of the waves at a specific instant in time and then stating what happens at a later time. For a sketch graph to make sense, we have to note what is being plotted against what. The x-axis may be labelled as time or distance. Compare Figure 1 with Figure 2. These two graphs are for the same water wave. The graph in Figure 1 has distance plotted on the x-axis. It is effectively a snapshot at a particular moment and tells you that the distance occupied by one wave is 4.0 m. The other graph has time plotted on its x-axis. This shows you how the height of the wave at a particular point changes with time. You can also see from this graph that it takes 10 s for one complete wave to pass that point. These two sketch graphs look similar and indeed do give the same information with regard to the height of the wave. However, the differences between them can be crucial and you must always check exactly which property is plotted on each axis. Termsandsymbolsusedtodescribewaves Time/s Figure 2 Height of water wave plotted against time Examinertip Correct axis labelling is particularly important with longitudinal waves like sound. Students often draw sound waves as transverse waves because of their appearance on an oscilloscope. This is acceptable provided the axes are labelled correctly. On an oscilloscope it is time on the x-axis not displacement and the y-axis measures the output from a microphone, which will usually be the pressure in the sound wave. Wavelengthl, unit:metre(m) The wavelength of a wave is the smallest distance between two points that have the same pattern of oscillation. It is also the distance the wave travels before the pattern repeats itself. In Figure 1 you can see that here the wavelength is 4.0 m, i.e. the wave has travelled 4.0 m before repeating its oscillation pattern. PeriodT,unit:second(s) The period of a wave is the time for one complete pattern of oscillation to take place at any point. Figure 2 shows this period to be 10.0 s, i.e. it has taken 10.0 s for the wave to complete one oscillation pattern. Frequencyf,unit:hertz(Hz) The frequency of a wave is the number of oscillations per unit time at any point. Using Figure 2, the period is 10 s and since frequency = 1/period, the frequency = 1/10 = 0.10 Hz. It is 0.10 oscillations per second, i.e. the wave has completed 0.1 of an oscillation in one second. Displacementx,unit:metre(m) Displacement is the distance any part of the wave has moved from its mean (or rest) position. It can be positive or negative. Amplitudex0,unit:metre(m) Displacement Amplitude is the maximum displacement, i.e. the distance from a peak to the mean (rest) position. It is always positive. Figure 3 Two waves in antiphase Phasedifference,unit:rad Time Phase difference concerns the relationship between the pattern of vibration at two points. Two points that have exactly the same pattern of oscillation are said to be in phase – there is zero phase difference between them. Wavelength can also be defined as the shortest distance between two points that are in phase. If the pattern of movement at the two points is exactly opposite to one another, then the waves are said to be in antiphase as shown in Figure 3. They are half a cycle different from one another. 138 934 physics.U2 M4.indd 138 15/11/07 10:04:51 am Module 4 Waves Height of arrow tip The unit used for phase difference is the radian or rad. There is a strong relationship between circular motion and wave motion. Imagine the vector arrow shown in Figure 4 rotating at a constant rate. As it rotates, the height of the tip of the arrow as it moves above the base line plots out the graph of a wave. This is shown to the right of the circle in Figure 4. The angle for one complete rotation of the vector arrow is 2π radians, so one complete cycle of a wave is given as 2π rad. Waves which are in antiphase are π rad out of phase. This is illustrated in Figure 5 where two waves are π/2 out of phase. Time Wave terminology The radian You need to be familiar with angles measured in radians rather than in degrees. An angle u, measured in radians, is given by dividing the curved distance along the arc of a circle c, by the radius r of the circle: u = c/r. (see Figure 6). This gives: circumference of circle one revolution = __________________ radius of circle 2πr = = ___ 2π radians (2π rad) r Figure 4 Pure wave pattern for rotating vector arrow Displacement From this we can deduce: 360° = 2π rad 180° = π rad 90° = π/2 rad c r Time r Figure 5 Two waves π/2 out of phase Figure 6 Angle u, measured in radians, = c/r Questions Displacement Displacement/10–10 m 1 Figure 7 shows a displacement– position graph at one instant of 5 time of part of a sound wave A B 0 Position/m of frequency 1700 Hz. 0.1 0.2 0.3 –5 (a) Describe in as much detail as you can the motion of the air at position A as the wave passes. Figure 7 (b)How does the motion of the air at position B differ from that at position A, and in what ways is it the same? 2 Figure 8 shows the shape of a wave at a particular instant. Its wavelength is 8.0 cm. The vector arrows, or phasors, corresponding to point 0 and point A on the wave are shown in the circles below the wave profile. (a) (i) What is the phase difference between points 0 and A? Movement of wave (ii)What fraction of a wavelength, the path difference, are points 0 and A A apart? O (iii) Show that the answers to (i) and (ii) satisfy the general relationship: phase difference = 2π/λ (path difference). (b) Draw the vector arrow corresponding to point B on the wave. (c)How far has the wave moved from A when the vector arrow corresponding to point A has rotated once? (d) It takes 0.1 s for the vector arrow to rotate. What is: Phasor at A Phasor at O (i) the frequency of the wave? Figure 8 (ii) the speed of the wave? (e)Copy the diagram and add to it another wave of the same amplitude and wavelength, but having a phase difference of π with the original wave. Position B 139 934 physics.U2 M4.indd 139 15/11/07 8:42:11 am 2.4 3 Wavespeed By the end of this spread, you should be able to . . . ✱ Usethetermspeedofawavetoderiveandusethewaveequationv = fl. ✱ Explainhowaprogressivewavecantransferenergy. ✱ Relatetheintensityofawavetoitsamplitude. Mach numbers When an aircraft achieves mach 1 (i.e. the speed of sound), large forces act on it. Military aircraft are built to withstand these forces, but commercial aircraft are designed to fly at subsonic speeds – Concorde was an exception to this. An aircraft wing is shaped so that the air travelling over the top surface of the wing moves faster than the air beneath the wing, but in a commercial aircraft even this air flow is kept below the speed of sound. Additionally, the temperature of the air through which commercial aircraft fly is around –50 °C, so commercial aircraft travel with an air speed of around 250 m s–1or mach 0.8, which is about 560 mph. Headwinds decrease the aircraft’s speed over the ground, while tail winds increase it. Some military jets can travel at over mach 3. However, space satellites need to reach speeds of 8000 m s–1 in order to keep in a low circular orbit around the Earth. This is over mach 20, but why do you think mach numbers are not used when describing a satellite’s speed? Sun Earth 1.5 1011 m Figure 1 Electromagnetic waves spreading out from the Sun Thewaveequation distance The equation speed = ________ can also be applied to wave movement. time In a time equal to one period T, a wave travels one wavelength L, therefore the speed v for any wave is: L v = __ T 1 One further step uses the relationship: frequency f = __ T to get v=f×L velocity = frequency × wavelength The SI units for this equation are: velocity in m s–1; frequency in Hz; and wavelength in m. Numericalvalues The speed of light is 300 000 000 m s–1 or 3.00 × 108 m s–1. Green light – in the centre of the visible spectrum – has a wavelength of 0.000 000 500 m or 5.00 × 10–7 m. The frequency of this green light is thus the speed divided by the wavelength which gives: 3.00 × 108 m s–1 _______________ = 6.00 × 1014 Hz 5.00 × 10–7 m Sound waves travel approximately a million times slower than light waves. Their speed depends on what they are travelling through and is also temperature-dependent: in air at 0 °C it is 331 m s–1. The frequency for the middle C note on a piano is 264 Hz. This gives a wavelength for this note of 331/264 = 1.254 m. Energytransferencebyaprogressivewave A progressive wave transfers energy from one place to another. Sound travels from a speaker to your ear drum, light from a candle to your eye. These are both transfers of energy. The Earth receives its energy from the Sun, but none in the form of sound. The vacuum between the Earth and the Sun prevents any sound energy travelling between them – even the sound of the Sun’s violent explosions. Some of the Sun’s energy reaches the Earth as highly energetic particles. These cause auroras – diffused coloured light in the upper atmosphere over the polar regions (northern and southern lights) – and can affect radio communications. However, the majority of the Earth’s energy from the Sun is in the form of electromagnetic waves – mostly as infrared, ultraviolet and visible light. The Sun radiates electromagnetic energy at a rate of 3.7 × 1026 J s–1, i.e. it has a power of 3.7 × 1026 W. This radiation spreads out from the Sun in all directions. The distance from the Sun to the Earth is 1.5 × 1011 m. Figure 1 shows how the power of the Sun’s electromagnetic waves is spread over the surface area of a sphere of radius 1.5 × 1011 m. Only a small fraction of this power reaches the Earth. So, each square metre of the upper atmosphere of the Earth receives energy at a rate: power output of Sun 3.7 × 1026 _______________________________________ = ____________ = 1.3 kW m–2 11 surface area of sphere of radius 1.5 × 10 m 4π(1.5 × 1011)2 140 934 physics.U2 M4.indd 140 15/11/07 8:42:16 am Module 4 Waves Wave speed This figure is called the solar constant and is a measure of the intensity of the Sun’s radiation. Note that the unit of intensity will always be a unit of power divided by a unit of area. The planet Neptune is 30 times further away from the Sun than the Earth. Hence, its 1 solar constant will be ____ of the Earth’s solar constant. This is because the surface area 900 of a sphere which has a 30 times larger radius will be 900 times larger. As waves spread out to cover a larger area, the intensity of the energy falls. This can be summarised by the expression: 1 intensity ∝ _________ distance2 This is a statement of a law known as the inverse square law. Many effects reduce in this way. It is a feature of the geometry of three-dimensional space. Figure 2 shows how an area nine times larger is covered by a wave from a point source that travels three times further. If the power is spread over an area nine times larger then the intensity will be reduced to 1/9 of the original value. Point source x 3x Figure 2 Waves spreading out from a single point only have 1/9 of the intensity after travelling three times further Relationship between wave amplitude and intensity A displacement–time graph for two particles in a sound wave is shown in Figure 3. Both particles have the same frequency, but particle A has twice the amplitude of particle B. A tangent has been drawn on each graph at the point when the particles are at their mean positions. The gradients of these tangents give the speed for each particle. intensity ∝ amplitude2 Questions Gradient of B at zero displacement Displacement From the graphs, you can see that the gradient of the tangent for particle A is twice that for particle B. Further analysis confirms that the speed of particle A is twice that of particle B, when at the mean position. A particle with twice the speed, will have four times the kinetic energy. For sound wave particles, four times the energy implies a wave of four times the intensity. This is true for all waves and gives us the following principle: Gradient of A at zero displacement A B Time Figure 3 Displacement–time graph for two sound wave particles: 2 × amplitude: 2 × speed therefore 4 × kinetic energy 1 All electromagnetic waves travel at 3.0 × 108 m s–1 in space and more slowly in other substances. When visible light of wavelength 600 nm passes through glass its speed is about 2 × 108 m s–1. Calculate: (a) the frequency, and (b) the wavelength of this light in glass. 2 (a)The receiving aerial for a UHF television is about 25 cm long. This is one half of the wavelength of the transmission. Calculate the frequency of the transmission. (b)Sound waves in air travel at 340 m s–1 on a warm day. The range of human hearing is 20 Hz to 20 000 Hz for a young person. Calculate the corresponding range of wavelengths. (c) Calculate the speed of the sound wave in question 1 on spread 2.4.2. 3 Two fishing floats, a distance of 4.5 m apart, bob up and down at 20 times per minute. The floats always move in antiphase. There is always at least one wave crest between the floats, but never more than two. (a)Show that the wavelength of the ripples on the river is 3.0 m. Hence, find the speed of the ripples on the surface. (b)Near the bank, the depth of the river halves. The speed v of water waves in shallow water of depth d is given by v = gd, where g is 9.8 m s–2. (i) What is the new frequency and wavelength of the waves near the bank? (ii) What is the depth of the river near the bank? 141 934 physics.U2 M4.indd 141 15/11/07 8:42:22 am 2.4 4 Waveproperties By the end of this spread, you should be able to . . . ✱ Explainwhatismeantbyreflection,refractionanddiffractionofwaves. Reflection While we usually associate the process of reflection with light, most wave types can be reflected. Figure 1 illustrates the reflection of a television signal using a dish aerial, while an echo – involving the reflection of sound waves – is shown in Figure 2. Note how in Figure 2 sound waves spread in all directions from the source, and how those which hit the wall continue to spread as they travel back towards the source. An echo is heard when returning waves reach the original source of the sound. Receiver Parabolic dish aerial Figure 1 Microwaves (TV signal) being reflected at a receiving aerial A single line – known as a ray – has been drawn on the diagram in Figure 2 to illustrate the direction in which the waves travel. Rays are often used to simplify diagrams in which the waves themselves are not shown. Rays are always drawn at right angles to the waves. In illustrations such as this, wavefronts are used to indicate the progress of a wave. These are drawn at the start of each successive wave, with the distance between wavefronts representing wavelength. Notice how the wavelength does not change after the wave has been reflected. Refraction Figure 2 Sound waves being reflected by a wall Air Water Refraction occurs when a wave is not reflected by a surface but actually enters the material it meets. The term refraction describes the change in direction of a wave at the boundary between two materials. It is best illustrated by light, which travels at different speeds in materials of differing optical density (the more optically dense the material, the slower the speed). This causes waves to change direction. Refraction is apparent when we observe reflections on the surface of a lake. Here, the sky often appears darker in the reflections than in reality. This is because some of the light from the sky is refracted into the water so less is reflected. Refraction is also responsible for the properties of lenses (see Figure 4) and, in combination with reflection, for the creation of rainbows (Figure 5). Since a diagram showing both partial reflection and partial refraction looks rather confusing, Figure 3 shows only the effect of refraction. The actual wavelength of light is extremely short, so a wavefront has been drawn about once every 20 000 waves. Several rays are illustrated. Figure 3 Light spreading out from a source – because it travels more slowly when in water its direction of travel changes at the boundary between air and water, i.e. it is refracted Figure 4 A lens refracts light Figure 5 A rainbow is caused by reflection and refraction in spherical raindrops 142 934 physics.U2 M4.indd 142 15/11/07 8:42:29 am Module 4 Waves Wave properties Each is at right angles to a wavefront, with the change in direction of the rays after entry into the water clearly shown. The only ray that does not change direction on entry into the water is the one at a right angle to the surface. Diffraction Changes in the direction of waves may also occur when they encounter an obstacle, or as they pass through an aperture. Known as diffraction, this process is best illustrated using waves in water. Figure 6 shows a wave in a ripple tank approaching a gap in a barrier. Since the gap is wide, most of the wave travels straight through it. In Figure 7, where the gap is much narrower, it acts as a point source of waves, causing a much greater degree of diffraction. The energy of the waves is distributed in all directions, resulting in the formation of an almost semicircular pattern. However, the wavelength does not change. It is diffraction effects that are responsible for the vivid colours to be seen when handling a CD. We will deal with diffraction in more detail in spreads 2.4.8 and 2.4.9. Mirror Figure 8 Reflection of plane waves at a mirror Air Dense material Figure 9 Refraction of plane waves as they enter a denser material from air. Note the slower speed of the wave gives a shorter wavelength Figure 6 Little diffraction of water waves at a wide gap Figure 7 Considerable diffraction of water waves at a narrow gap Questions 1 Copy Figure 1, drawing the waves that correspond to the rays shown on the diagram. 2 The direction of travel of some sea waves 100 m from the coast is almost parallel to the shore. However, as the waves move into shallower water, they slow and turn towards the shore. (a)Explain why this happens and why the effect is most pronounced on flat sandy beaches (steep beaches usually require breakwaters to reduce erosion of the shore). (b)Imagine a theoretical beach where the depth of water decreases steadily towards the shore. A plane wavefront to represent an approaching wave is drawn in Figure 10. Copy the diagram and add further lines to represent the position and shape of the wavefront at equal intervals of time as it moves towards the shore. Beach Shallow water A B Deeper water Figure 10 143 934 physics.U2 M4.indd 143 15/11/07 8:42:40 am 2.4 FurtherquestionsD Wavemotion Wave motion 1 Figure 1 shows, at one instant of time, the shape of a stretched string along which a transverse wave is travelling with wavelength 50 mm and amplitude 10 mm. Five points on the string are labelled V, W, X, Y and Z. W V X 0.25 m X 0.5 m Z Figure 3 Figure 1 (a) Copy the diagram. (i) Label on it the wavelength and amplitude. (ii) Add a sketch of the shape of the string after it has moved 12.5 mm. (iii) Draw arrows to show the direction in which the particles of the string at W and Y have moved. (b) Which of the labelled points on the string are moving (i) π out of phase; (ii) π/2 out of phase; and (iii) π/4 out of phase with each other? (c) The wave is travelling along the string at 0.5 m s–1. (i) How many wavelengths pass point X in one second? (ii) What is the frequency of the wave? 2 Figure 2 shows part of a transverse progressive wave moving from left to right at a particular time. Axes have been included. Displacement along a string at a speed of 2.0 m s–1 towards X. Direction in which wave is travelling Y 0 3 A wave pulse of the shape shown in Figure 3 is travelling Position The leading edge is at a distance of 1.0 m from X, where the string is fixed to a rigid wall. The diagram shows the pulse at time t = 0. Draw a graph of the displacement of a point on the string at a distance of 0.50 m from the wall over a period from t = 0 to t = 0.75 s. Mark the scales on both axes clearly. Waveintensity 4 The intensity of the quietest sound that can be heard is 10–12 W m–2. Estimate the greatest distance at which, theoretically, you might just hear the broadcast from a 10 W speaker. Assume that the speaker is acting as a point source emitting energy equally in all directions. Remember that the surface area of a sphere of radius r is 4πr2. Suggest reasons why this distance is unlikely in practice. 5 The minimum energy of visible light of wavelength 500 to 600 nm that can just be detected by the eye is 10–17 W. Assume that only 1% of the power input to a 60 W light bulb is emitted between these wavelengths. Estimate the maximum distance that it might be possible to see the lit bulb. The pupil of the eye will be at its maximum diameter, say 8 mm. At its maximum sensitivity the eye only detects 8% of the light incident on it. Figure 2 Sketch the figure three times, labelling the new sketches (i), (ii) and (iii). Add a wave of the same amplitude and same speed of propagation but twice the frequency to (i); same frequency but twice the speed of propagation to (ii); same frequency and speed but with a phase difference of π/2 behind the original wave to (iii). 144 934 physics.U2 M4.indd 144 15/11/07 8:42:45 am Module 4 Waves Further questions D Wave properties 6 Refer back to the wave pulse in Figure 3 of question 3 above. Sketch the pulse shape you would expect to see at t = 1.0 s after the pulse is reflected at X. Label the point X clearly on your diagram. 7 Diffraction is a property of all waves, but is only a significant effect when the wavelength of the diffracted waves is about the same size as the aperture. Explain why the diffraction of sound is easily observed in everyday life but the diffraction of light is not. 8 Here is one method of measuring the focal length of a lens. A small light source is mounted at the centre of a white screen. This is placed in front of a thin lens behind which is a plane mirror, as shown in Figure 4. When the source is at a distance of the focal length, 150 mm in this case, from the lens, a focused image of the light source is formed on the screen close to the source. 150 mm Small light source Principal axis Screen 150 mm Lens Mirror Figure 4 The distance from the focal point to the lens is called the focal length of the lens. Light from the focal point passing through the lens is refracted into a parallel beam. Copy the diagram. On it, sketch two wavefronts of the light from the small source (i) between the source and the lens at distances of 50 mm and 100 mm from the source and (ii) between the lens and mirror at 50 mm and 100 mm from the mirror. Add three rays of light which pass from the source to the mirror. Your diagram should be such that the reflected paths of all rays are the same as the outgoing ones. 145 934 physics.U2 M4.indd 145 15/11/07 8:42:50 am 2.4 5 Electromagneticwaves By the end of this spread, you should be able to . . . ✱ Statetypicalvaluesforthewavelengthsofthedifferentregionsoftheelectromagneticspectrum. ✱ Statethatallelectromagneticwavestravelatthesamespeedinavacuum. ✱ Describedifferencesandsimilaritiesbetweendifferentregionsoftheelectromagneticspectrum. ✱ Describesomeofthepracticalusesofelectromagneticwaves. ✱ DescribethecharacteristicsanddangersofUV-A,UV-BandUV-Cradiationsandexplaintherole ofsunscreen. Commonpropertiesofelectromagneticwaves Some properties of electromagnetic waves were described in the introduction to waves in spread 2.4.1. In this spread the electromagnetic spectrum will be looked at in more detail. It is called a spectrum because it has a range of values in the same way that a rainbow has a range of colours. Indeed the colours in a rainbow are one tiny part of the whole electromagnetic spectrum. Wavelengths for visible light range from approximately 370 nm for violet to 740 nm for deep red. This means red light has twice the wavelength of violet. The difference between the two is termed an octave. Visible light represents a tiny part of the electromagnetic spectrum, which spans a range of wavelengths from 10–16 m for gamma rays at one extreme to 104 m for radio waves at the other. All electromagnetic waves share the following common properties, however: • They can all travel through a vacuum. • All possess both a magnetic wave and an electrical wave interlocked and at right angles to each other. • In free space, they all travel at a speed of exactly 299 792 458 m s–1. • They are all transverse waves. Region Wavelength/m –16 → 10 –9 Frequency/Hz Methodofproduction Methodofdetection Uses 3 × 10 → 3 × 10 nuclear decay or in a nuclear accelerator photographic film, Geiger tube Diagnosis and cancer treatment (radiotherapy) 24 17 gamma (γ)-rays 10 X-rays 10–12 → 10–7 3 × 1020 → 3 × 1015 bombarding metals with high-energy electrons photographic film, fluorescence CT scans, X-ray photography, crystal structure analysis ultraviolet 10–9 → 3.7 × 10–7 3 × 1017 → 8.0 × 1014 from high-temperature solids and gases photographic film, phosphors, sunburn disco lights, tanning studios, counterfeit detection, by detergents visible light 3.7 × 10–7 → 7.4 × 10–7 8.0 × 1014 → 4.0 × 1014 from high-temperature solids and gases, lasers photographic film, retina of eye Sight, communication infrared 7.4 × 10–7 → 10–3 4.0 × 1014 → 3 × 1011 oscillation of molecules, from all objects at any temperature above absolute zero photographic film, thermopile, heating of skin heaters, night vision equipment, remote controls microwaves 10–4 → 10–1 3 × 1012 → 3 × 109 magnetron, klystron oscillators, using electrons to set up oscillations in a cavity heating effect, electronic circuits radar, mobile phones, microwave ovens, satellite navigation radio waves 10–1 → 104 3 × 109 → 3 × 104 electrons oscillated by electric fields in aerials resonance in electronic circuits television, radio, telecommunications Table 1 Electromagnetic spectrum 146 934 physics.U2 M4.indd 146 15/11/07 8:42:53 am Module 4 Waves Categories of electromagnetic waves Electromagnetic waves Division of the electromagnetic spectrum into different categories is arbitrary. There is a gradual change across the whole spectrum rather than there being exact wavelengths where one type of wave becomes another. Categorisations are often based on method of production of the wave rather than wavelength. This is why there is considerable overlap between the wavelengths given in Table 1 and, except for visible light, why only the order of magnitude is given. Even with light there is no sharp division between wavelengths that are visible and those that are not. At the deep red end of the visible spectrum (Figure 1), for example, some people are able to see the radiation, while others are not. Visibility is also dependent on the intensity of the radiation. Uses of electromagnetic radiation Figure 1 The spectrum of visible light Many of the uses of electromagnetic radiation mentioned in Table 1 are obvious. For example, the use of radio waves in radio and television, the use of X-ray photographs to examine a broken bone, the use of visible light for sight. Electromagnetic radiation is so much part of everyday life that it is difficult to imagine life without it. Yet until the middle of the nineteenth century, with the exception of light and heat radiation (infrared), little was known about the electromagnetic spectrum. It was James Clerk Maxwell who formulated a set of equations, now called the Maxwell equations, that related electric and magnetic fields and theoretically showed that electromagnetic waves were possible. Heinrich Hertz confirmed Maxwell’s equations experimentally by becoming the first person to produce radio waves. This early work paved the way for the discovery of the remainder of the electromagnetic spectrum: X-rays in 1895, γ-rays in 1896, microwaves, leading to radar, with the magnetron in the 1930s. X-rays X-ray photography has been used for over a century now, but many technical advances in the field are relatively recent. Where once X-rays were essentially shadows, useful only for confirming bone fractures, the use of computers has enabled vast improvements in image contrast. By linking X-ray machines to computers, we are now able to construct three-dimensional images from a series of cross-sectional planes. Known as computed tomography (CT) scans, these allow much more accurate and varied diagnoses. For example, now X-ray photographs of the alimentary canal can be taken and every twist and turn of the large intestine can be seen with great clarity (see Figure 2). Figure 2 A false-colour X-ray (CT) scan of the large intestine Ultraviolet Substances known as phosphors glow when subjected to ultraviolet radiation (see Figure 3), making UV radiation visible to the naked eye. Manufacturers of washing powders incorporate phosphors in their products, which is why white clothing glows a bright blue-white under the UV lights used in night clubs. Ultraviolet radiation emitted by the Sun is often Figure 3 Coral in a fish tank divided into three regions: photographed under UV light. Coral is • UV-A: Wavelength 315–400 nm; causes tanning fluorescent when skin is exposed to the sun (accounts for 99% of UV light). • UV-B: Wavelength 280–315 nm; causes damage such as sunburn and skin cancer. • UV-C: Wavelength 100–280 nm; is filtered out by the atmosphere and does not reach the surface of the Earth. Sunscreens contain chemicals designed to filter out UV-B, preventing sunburn and skin damage. Glass is an efficient absorber of ultraviolet – which is why you do not get sunburnt indoors, even if you sit for long periods in the sun. Questions 1 In December 1901 Marconi succeeded in sending the first radio signals through the atmosphere 3200 km across the Atlantic. Many scientists at the time predicted that this experiment was impossible. Their prediction would have been true for television signals. Why did Marconi’s experiment work? Why was it not possible to send a television signal the same distance through the atmosphere until the 1960s? 147 934 physics.U2 M4.indd 147 15/11/07 8:43:01 am 2.4 6 Polarisation By the end of this spread, you should be able to . . . ✱ Explainwhatismeantbyplanepolarisedwavesandthatpolarisationisaphenomenon associatedwithtransversewavesonly. ✱ Describewaysofproducingpolarisedlight. ✱ RecallandapplyMalus’lawfortransmittedintensityoflightfromapolarisingfilter. Plane-polarisedwaves Direction of oscillation Direction of travel Figure 1 Light wave shining directly onto an eye For a transverse wave, its displacement is at right angles to its direction of travel. However, this does not determine the direction of the displacement. Look at Figure 1. Here a light wave is shown shining directly on an eye. The oscillations of the particles in the wave can be in any direction – up/down, left/right, 45° – as long as this movement is at right angles to the wave’s direction of travel. Some crystalline materials can cause the oscillation to occur in one plane only. These are known as polarising filters, for example ‘Polaroid’ filters. A wave that oscillates only in one plane is called a plane-polarised wave. Note that only transverse waves can be polarised, as longitudinal waves do not have oscillations at right angles to their direction of travel. The oscillations of these waves are in line with the wave’s direction of travel. For a wave to be polarised, it must be a transverse wave. Rotatingtheplaneofpolarisation Polariser Unpolarised light A normal light wave from a light bulb is shown passing through a sheet of Polaroid in Figure 2. The first Polaroid sheet is labelled a polariser. It produces plane-polarised light, i.e. polarised in the vertical direction. The light wave then carries on to a second sheet of Polaroid, called the analyser, which is rotated through an angle u relative to the polariser sheet. The analyser sheet polarises the light Light polarised in in a direction parallel to its long edge, i.e. the plane of polarisation will direction of plane of polarisation of have been rotated by an angle u. analyser Light vertically polarised Analyser Figure 2 Normal light passing through a polariser and then through an analyser that has its plane of polarisation at an angle u to that of the polariser The above experiment is shown as a front view in Figure 3. When the amplitude of the light wave approaching the analyser sheet is x, the amplitude, after it has had its plane of polarisation rotated by an angle u, will be x cos u. Since the intensity of a wave is proportional to amplitude squared, the intensity after the analyser is proportional to cos2 u. The intensity of the wave is reduced as it passes through the second filter. This is known as Malus’s law after Etienne-Louis Malus. It states that when a perfect polariser is placed in a polarised beam of light the intensity, I, of light that passes through it is given by: I = Imax cos2 u Polarised wave approaching the analyser Analyser where I is the intensity transmitted at angle θ and Imax is the maximum intensity transmitted (at u = 0). This law shows that if the analyser is at right angles to the polariser, then u = 90° and no light will pass through. This situation is known as crossed Polaroids. Polariser Polarised wave leaving the analyser Figure 3 Front view of Figure 2 148 934 physics.U2 M4.indd 148 15/11/07 8:43:05 am Module 4 Waves Uses of polarisation Polarisation Reflected and scattered light Although sunlight and light from a lamp bulb are not polarised, some naturally occurring light can be partially polarised. This means that there is more light with the direction of oscillation in one direction than there is in any other direction. This mainly occurs in light that has been reflected. The light reflected from the surface of a lake, for example, is partially polarised, and so is the blue light from the sky. Blue light is more easily scattered by the dust particles and water vapour in the atmosphere than red light. It is the scattered blue light that is partially polarised. (It is because of blue light being scattered and causing blue skies that sunsets are red.) An angler usually cannot see the fish under the water’s surface because of the reflected light. However, if he wears Polaroid glasses they cut out the partially horizontally polarised light reflected from the water surface, but allow the partially vertically polarised light reflected from objects below the surface to pass. So any fish under the water become visible. Similarly, photographers often use Polaroid filters to enhance the colour of the sky. The filters remove some of the polarised light from a blue sky, so that the sky seems more intense as shown in Figure 4. Figure 4 This photograph was taken using a polarising filter over the lens, to emphasise the colour of the sky Strain analysis Another technique that also makes use of polarisation is strain analysis. Certain plastics, such as those used for making rulers, protractors and even Sellotape, are able to rotate the plane of polarisation. When these plastics are placed between crossed Polaroids, coloured images are produced which change as the plastics are stretched or squashed. This effect is shown in Figure 5 for a plastic model of a hip bone. Apart from being attractive, these models can be used to analyse the stresses in the bone. Furthermore, detailed analysis of crystal shapes can also be obtained by using this technique (see Figure 6). Figure 5 Stress pattern in a hip joint becomes apparent when a plastic model hip joint is placed between crossed Polaroids Television transmission So far, we have only discussed the applications of polarised light. However, radio waves can also be polarised. Television main transmitters send out horizontally polarised signals. In order to cover the whole country, low-power infill transmitters are used to give signals in valleys. So many of these infill transmitters are needed that their signals could interfere with those from the main transmitters. To overcome this, many of the infill transmitters are vertically polarised. The vertically polarised signals cannot interfere with the horizontally polarised ones from the main transmitters. We will discuss interference in more detail in spread 2.4.7. Questions 1 Suppose the intensity of a beam of unpolarised light incident on a linear polariser is I. Explain why the maximum possible intensity of the transmitted light, which is plane-polarised, is ½ I. We call this a perfect polariser. (In reality, when an unpolarised beam of light is shone onto a typical sheet of Polaroid, the transmitted beam is only about 30 to 35% of the incident intensity.) Figure 6 The shape of a cholesterol crystal is shown in polarised light 149 934 physics.U2 M4.indd 149 15/11/07 8:43:15 am 2.4 7 Interference By the end of this spread, you should be able to . . . ✱ Stateandusetheprincipleofsuperpositionandillustratethisgraphically. ✱ Describeconstructiveanddestructiveinterference. ✱ Describeanexperimenttodemonstratetwo-sourceinterferenceusingsound. ✱ Explainthetermsinterference,coherence,pathdifferenceandphasedifference. Theprincipleofsuperposition So far we have considered the behaviour of single waves, but what happens when two or more waves of the same type are present at the same time in the same place? The principle of superposition, illustrated in Figure 1, can be applied to calculate the resultant wave (i.e. the net displacement) at any time. • The principle of superposition states that when two or more waves of the same type exist at the same place, the resultant wave will be found by adding the displacements of each individual wave. Remember that displacement is a vector, and take care when adding negative and positive displacements. For example, if a displacement of –3 is added to a displacement of +5, the resultant is +2. Displacement Figure 1 Two waves are superposed – the resultant wave is the sum of the two individual waves Time Wave A and B added together Wave B Wave A Signal generator Loudspeakers Interference If two waves, A and B, exist at the same point and are travelling in phase, the amplitude of the resultant wave will be twice that of the individual waves (see Figure 2a). This is known as constructive interference. If, on the other hand, the two waves are in antiphase, they will cancel each other out, and the resultant wave will have an amplitude of zero (see Figure 2b). This is known as destructive interference. Figure 2a Constructive interference Time Displacement Displacement Path of person walking Figure 3 Interference with sound waves can be demonstrated using two loudspeakers connected to the same signal generator. As you walk along in front of the loudspeakers you will hear a loud sound where the sound waves reinforce one another and a quiet sound where the waves partially cancel one another. This variation is clearer if you cover one ear. (Why is this?) The distance between the loud and quiet regions is larger for low frequencies than for high frequencies Time Figure 2b Destructive interference 150 934 physics.U2 M4.indd 150 15/11/07 8:43:19 am Module 4 Waves Coherence Interference Displacement Displacement Of course, in practice things are rather more complicated than the two situations just described. The two waves may have different amplitudes and shapes, or may change their phase relationship with one another. In order to calculate a meaningful resultant using the principle of superposition, the two waves must display a constant phase difference – that is, Time they must be coherent. This is true of the waves in Figures 4a and 4b, but notice that the waves in Figure 4c do not follow a regular pattern. One has a ‘blip’ in the middle, so while their Same amplitude, same frequency, constant phase difference, coherent phase difference is constant initially, it changes as they travel, Figure 4a Coherent waves and these waves are not coherent. Small discontinuities such as this are particularly common in light waves, for which achieving coherence is impossible unless the waves originate Time from the same source. In radar systems, microwaves of wavelength 5.0 cm travel in tubes known as waveguides. Figure 5 shows a system in which the microwave energy is split along two different paths, A and B, before joining again. By adjusting the length of the two paths, it is possible to create either constructive or destructive interference when the waves rejoin. For example, if tube A is 35 cm long and tube B is 45 cm long, waves in A will travel 7.0 wavelengths and waves in B will travel 9.0 wavelengths before the two sets rejoin. They have a path difference of 10 cm or 2.0 wavelengths, and will therefore rejoin in phase, resulting in constructive interference. Different amplitude, same frequency, constant phase difference, coherent Figure 4b Coherent waves Displacement Interference with microwaves Time Same amplitude, same basic frequency, varying phase difference, not coherent Figure 4c Non-coherent waves By altering the dimensions of the waveguides, it is possible to create destructive interference. Imagine, for example, that tube A is 37.5 cm long and tube B is 45 cm long. In this case, waves following path A will travel 7.5 wavelengths before they rejoin waves following path B, which have travelled 9.0 wavelengths. They have a path difference of 7.5 cm, or 1.5 wavelengths, and so will be in antiphase, cancelling each other out. While the actual procedure employed in radar systems is more complex than this, the principle is the same. This method is used to prevent outgoing waves from swamping weak returning echoes. Path, or phase, difference can be used to determine whether interference is constructive or destructive, as shown in Table 1. Path difference Phase difference in degrees in radians constructive interference a whole number of wavelengths e.g. 0, λ, 2, 3 . . . 0, 360, 720 . . . 0, 2π, 4π, . . . destructive interference an odd number of half wavelengths e.g. ½, 1½, 2½ ... 180, 540, 900 . . . π, 3π, 5π . . . 5 cm waves Path A Path B Figure 5 Waveguides allow microwaves two routes before rejoining Table 1 Question 1 Explain: (a) what is meant by coherence; (b) why it is much easier to produce two radio waves that are coherent than two light waves that are coherent. 151 934 physics.U2 M4.indd 151 15/11/07 8:43:25 am 2.4 8 TheYoungdouble-slitexperiment By the end of this spread, you should be able to . . . ✱ DescribetheYoungdouble-slitexperimentandexplainhowitisaclassicalconfirmationofthe wavenatureoflight. ✱ Usetheequationl = ax/D. ✱ Describeanexperimenttodeterminethewavelengthofmonochromaticlightusingalaser. Interferenceusinglight The coloured patterns you see in a soap bubble or in an oil spill on water are a result of light interference. In order to make measurements of the wavelength of light, however, two conditions need to be satisfied: • The light needs to be monochromatic light – all the light has the same wavelength. • There needs to be an accurate method of obtaining a very small path difference, and of measuring this path difference. When we discussed microwaves in spread 2.4.7, we could use waves of wavelength 5.0 cm; light waves have a much smaller wavelength. TheYoungdouble-slitexperiment Figure 1 Light interference in a soap bubble Thomas Young was the first person to successfully measure the wavelength of light under these conditions, in 1801, thus establishing the wave theory of light. Until then, many people, including Newton, thought of light as a stream of tiny particles called corpuscles. The apparatus that Young used is illustrated in Figure 2. He used a monochromatic red light source which he placed behind a single slit in a black obstacle, X. Light passing through the slit spreads out by diffraction, until it reaches another obstacle, Y, in which there are two parallel narrow slits. The light from these two slits is coherent. This is because it starts from the same source and is in phase at the double slit. From here it spreads out again by diffraction, until it reaches a screen. You can see how the two sets of waves overlap near the centre of the pattern. B X Source of waves Y a = 17 mm x = 39 mm A O A' D = 160 mm B' Figure 2 Thomas Young’s double-slit experiment but shown with a wavelength of 4 mm 152 934 physics.U2 M4.indd 152 15/11/07 10:58:09 am Module 4 Waves In Figure 2, the waves are drawn with a wavelength 4.0 mm. The spacing between the double slits is 17 mm. The centre of the pattern has been marked at point O. At this point, waves from both slits have travelled the same distance, so the two waves arriving there are in phase and will constructively interfere. The Young double-slit experiment If you count the number of wavelengths from the top slit to point A, you will find there are 40½ wavelengths. However, the number of wavelengths from the bottom slit to point A is 40. The path difference is half a wavelength and therefore the waves destructively interfere. For point B, the number of wavelengths from the bottom slit is 42, and from the top slit is 41. So, as the path difference is one – a whole number – constructive interference takes place. This continues as you move further away from point O. The overall pattern, using light, is shown in the photographs in Figures 3a and 3b. You can see that, if you use red light (Figure 3a), the spacing of the bright regions – the fringe width – is greater than when you use green light (see Figure 3b), showing that red light has a longer wavelength than green light. A simplified theory using data from this experiment is shown in Figure 4. To create the first bright fringe, light from slit Q must travel one wavelength further than that from slit P. For small values of the fringe width x, the two shaded triangles are a similar shape, so: D a __ = __ where D is the distance of the screen from the double slits and a is the slit spacing. x l ax This gives the equation for the wavelength as: l = ___ . D 17 mm × 39 mm Using the data from Figure 2: l = _______________ = 4.1 mm 160 mm The wavelength of red light is around 650 nm and of green light, 550 nm. Nowadays this experiment can be done using a laser instead of a lamp as the source of monochromatic light. Figure 3a Double-slit interference in red light from horizontal slits positioned one above the other 1st bright fringe Figure 4 Theory using double-slit experiment x Slit P O a Slit Q D Screen The double-slit experiment with microwaves The same experiment can be carried out with microwaves instead of light. A microwave generator is placed 20 cm or so in front of a metal sheet with two slits about 5 cm apart. A microwave detector can be moved across the wave pattern to find the positions of maximum and minimum. Questions 1 Figures 5a–c show how the displacement varies with position for two waveforms, at the same instant of time, travelling along a string. Trace each set of curves and add the resultant shape of the string at that instant. Figure 5a Figure 5b Figure 3b Double-slit interference in green light, again from horizontal slits positioned one above the other Figure 5c 2 Which of the pairs of waves in Figures 5a–c are coherent? State the phase difference between the coherent pairs. 153 934 physics.U2 M4.indd 153 15/11/07 8:43:40 am 2.4 Wavessummary Radio 360° = 2� rad Microwave IR Visible UV X-rays Gamma (Increasing frequency) Radian : phase difference (rad) l: wavelength (m) T : period (s) f : frequency (Hz) xO: amplitude (m) D a = L x Vacuum Uses c = 3 108 ms–1 1 f= T em waves Constructive or destructive Young’s double-slit experiment v = fL Interference Properties Energy transfer Waves Intensity � amplitude2 Intensity � Diffraction grating nL = d sin U 1 Distance2 Longitudinal Compressions and rarefactions Transverse Reflection Pipes Polarising Diffraction Strings Nodes and antinodes Refraction Strain analysis, TV, reflection Coherence L sin U = d Stationary waves e.g. water and all em waves Types e.g. sound Path difference Harmonics Fundamental I = Imax cos2 U 164 934 physics.U2 M4.indd 164 15/11/07 8:44:43 am Module 4 Waves P r a c t ic e q u e s t io n s Practice questions 1 A pulse is travelling to the right along a stretched string in Figure 1a. A Figure 1a B P x P D (b)Explain why a wave of maximum amplitude travels in towards the shore every 30 seconds. (c)Calculate the ratio x P 4 (a)Figure 3a shows a string stretched between two points A and B. E x energy carried by a wave of maximum amplitude __________________________________________ . energy carried by a wave of minimum amplitude C x (a)Wave A has an amplitude of 1.2 m while wave B has an amplitude of 0.8 m. State the maximum and minimum amplitudes in m of the resultant wave. P x The pulse is reflected at point P. Five alternative versions are shown of the displacement of the string at some instant after the pulse is reflected. P x P Figure 1b Which of A, B, C, D or E in Figure 1b is the most likely when (i) the point P is fixed and (ii) the point P is free to move up and down along the black vertical line? (b)Draw diagrams to illustrate how plane water waves are diffracted when they pass through a gap about (i) 2 wavelengths wide and (ii) 10 wavelengths wide. (c)Suggest why the diffraction of light waves cannot usually be observed except under laboratory conditions. B Figure 3a (i)State how you would set up a standing wave on the string. (ii)Copy the diagram and draw on it the lowest frequency standing wave that can be formed. 2 (a)State what is meant by the diffraction of waves. A (b)Figure 3b shows the appearance of another standing wave formed on the same string. A B Figure 3b The distance between A and B is 2.4 m. Use Figure 3b to calculate (i) the distance in m between neighbouring nodes and (ii) the wavelength in m of the standing wave. 3 Surfers try to ride big waves as they come in towards the 5 The diffraction grating is illuminated with a parallel beam of shore. light. Diffracted beams are produced on the other side of the grating as shown in Figure 4. P Bright angled beams Parallel beam of light 16° Grating Figure 2a A surfer waits at point P some distance off shore in Figure 2a. Two wave trains A and B travelling towards the beach superimpose to produce a resultant wave. (a)The angle between the first order diffracted beam and a line perpendicular to the grating is 16o. Show that the spacing of the slits in the grating is 2.1 × 10–6 m. The wavelength of the light is 5.9 × 10–7 m. (b)The corresponding angle for the second order diffracted beam is 33.5o. Show that the value for the spacing of the slits is confirmed by this result. (c)Gratings are normally labelled with the number of lines per mm. Calculate the number of lines per mm for this grating. Rotating phasors for the two waves, A and B, are shown in Figure 2b. Phasor for wave A One rotation every 5 seconds 1.2 m Phasor for wave B 0.8 m One rotation every 6 seconds Figure 4 Figure 2b 165 934 physics.U2 M4.indd 165 15/11/07 8:44:54 am 2.4 1 Examinationquestions1 (b) Waves with a peak-to-trough height of 1.8 m approach a beach as shown in Figure 3. Figure 1 shows two sine waves, A and B, plotted against time. The rotating vectors, called phasors, used to generate the waves are shown at t = 0. 1.8 m B A Figure 3 Time 0 t1 A A B B (i) State the amplitude x in m of these waves. (ii) Calculate the energy � in J m–2 carried by these waves per m2 of sea surface, using the equation given in (a). Take R = 1030 kg m–3 and g = 9.8 N kg–1. [2] (iii) The wave energy is being carried onto the beach with the group of waves shown in Figure 2 at a velocity of 1.2 m s–1. Show that the energy arriving per second on a 1.0 m length of the beach is about 49 kW. [2] (iv) Calculate the power delivered by these waves to a 0.5 km length of beach. Express your answer in megawatt. [2] (v) Suggest one possible consequence, or use, of this wave power being delivered to the shore. [1] (OCR 2861 Jan05) Figure 1 (a) State the phase difference between A and B. [1] (b) Draw the positions of the rotating phasors for A and B at time t1. [2] (OUDLE 7731 Jun00) 2 This question is about wave energy. Figure 2 shows a group of waves travelling across the sea towards a beach. Velocity of group of waves 12 m s1 [1] 3 This question is about water waves travelling in a large water tank. Wave machine A B C D 0.4 m 1.8 m Shore Each 1 m2 of the sea surface carries energy towards the shore at 12 m s1 Figure 2 (a) The energy � carried by every 1 m2 of surface of the sea is given by � = ½gRx2 where g is the gravitational field strength R is the density of the sea water and x is the amplitude of the waves in the group. Show that ½gRx2 has the units J m–2. Take the units of g as N kg–1. [2] 50 m 25 m 25 m Not to scale Figure 4 Figure 4 shows a wave tank in a research laboratory. A wave machine, situated at one end of the tank, generates waves that travel from one end of the tank to the other. (a) The wave machine produces waves that travel from A to B in 12 s. (i) Calculate the velocity in m s–1 of the waves between A and B. [1] (ii) The wave machine produces 32 waves per minute. Calculate the frequency in Hz of the waves. [1] 166 934 physics.U2 M4.indd 166 15/11/07 8:45:00 am Module 4 Waves Examination questions 1 (b) The velocity of the water waves in this tank is given by the equation v2 = gd where g is the acceleration due to gravity and d is the depth of water. (i) Use the equation to calculate the velocity of the waves at the points C and D shown in Figure 4. Copy and complete the table. g = 9.8 m s–2. [2] position in tank water depth d/m velocity of waves/m s–1 B 1.8 4.2 5 T his question is about electromagnetic waves in free space. Figure 6 shows a short section of the electromagnetic wave being transmitted from a vertical aerial a large distance away. The wave is travelling in the x-direction. Its wavelength is 0.6 m. Transmitter aerial z y L x C D P 0.4 (ii) Using information from the table, describe and explain what happens to the frequency and wavelength as waves travel from B to D. [3] (iii)In Figure 5, the same tank is shown from above. A B C D Figure 5 barrier is placed across the tank at B. The gap in A the barrier is about four wavelengths wide. Waves are shown between A and B, where the depth of the tank is constant. Copy the diagram and complete it by drawing the waves between B and D. [4] (OCR 2861 Jun06) 4 (a) Identify the types of electromagnetic radiation having the frequencies given: (i) 1010 Hz, (ii) 1013 Hz and (iii) 1018 Hz. [3] (b) State briefly how each kind of radiation may be produced. Figure 6 The electric field accelerates electrons up and down the aerial, causing a current in the wire. There is a magnetic field around the wire associated with the current. (a) Suggest a suitable length, L, for the transmitting aerial. See Figure 6. [1] (b) (i) At the instant shown, in which direction is the electric field at point P? Use the coordinate axes shown in Figure 6. (ii) In which direction is the magnetic field at P? (iii)How many wavelengths long is the section of electromagnetic wave shown in Figure 6? [3] (c) Calculate the frequency of the wave. [2] (d) State how the magnitude and direction of the electric and magnetic fields at point P compare with those in Figure 6 at a time 1.0 × 10–9 s later. Justify your answer. [2] (e) The transmitter aerial is now rotated at a steady rate about the x-axis. (i) Describe what will happen to the wave. (ii) State how the signal amplitude detected by a vertical receiving aerial will vary with time. Justify your answer. [4] (O&C 5637 Jun99) [6] (c) I n the case of radiation of type (iii) it is necessary to shield workers from irradiation. This is not necessary in the other two cases. Explain why this is so. [2] (d) Choose one of these types of radiation and discuss its technological importance. [5] (O&C 137 Jun96) 167 934 physics.U2 M4.indd 167 15/11/07 8:45:06 am 2.4 1 Examinationquestions2 (a) State what is meant by the principle of superposition of waves. [2] (b) Figure 1 shows an arrangement which can be used to determine the speed of sound in air. 2 (a) Standing waves have nodes and antinodes. State what is meant by (i) a node and (ii) an antinode. [3] (b) Using a labelled sketch to illustrate your answer, describe an experiment to demonstrate how a standing wave can be produced in an air column. In your answer • state whether the wave is transverse or longitudinal • mark on your diagram the position of a node, label this N, and an antinode, label this A. [4] 3 This question is about two-source interference experiments. (a) What conditions are necessary for interference fringes (regions of maximum to minimum signal) to be observed in the resultant signal received from two sources? [3] (b) Explain why interference fringes cannot be observed from the two headlights of a motor car but can be heard from two loudspeakers connected to the same signal generator. [3] (c) Describe and explain the results of the experiment where the terminals of two small loudspeakers are connected in the same sense in parallel to a signal generator, operating at a frequency of 2.0 kHz. The speakers are placed 0.50 m apart. A microphone connected through an amplifier to an oscilloscope moves along a line parallel to the line joining the speakers at a distance of 2.5 m. See Figure 3. Take the [6] speed of sound in air to be 340 m s–1. Ruler P moving microphone Loudspeaker Microphones Q fixed microphone Signal adder c.r.o. Figure 1 The loudspeaker emits a sinusoidal sound wave. The electrical signals from the two microphones P and Q are added together in the electronic ‘signal adder’ and the resultant signal is displayed on the cathode-ray oscilloscope (c.r.o.) screen. This process may be regarded as equivalent to the superposition of the waves. Microphone Q is fixed and microphone P is slowly moved back along the edge of the ruler. (i) Figure 2 shows the appearance of the trace on the c.r.o. when both microphones are at the left hand end of the ruler i.e. the same distance from the loudspeaker. Pre-amplifier c.r.o. O Signal generator Microphone Loudspeakers Figure 3 1 cm Figure 2 The time-base setting of the c.r.o. is 0.2 ms/cm. Determine for the sound wave 1. the period in s, and 2. the frequency in Hz. [4] (ii) As P is moved slowly along the edge of the ruler, the amplitude of the trace is seen to decrease, then increase, then decrease and so on. Explain 1. why the amplitude is a maximum when P and Q are at the left hand of the ruler 2. why the amplitude of the trace varies. [4] (iii) The first minimum of the amplitude occurs when P is at a distance of 6.8 cm from the left hand end of the ruler. Determine 1. the wavelength in m of the sound and 2. the speed in m s–1 of the sound in air. [4] (OCR 2823 Jun03) (d) The microphone is placed at the central point O, in Figure 3. Describe what happens to the oscilloscope screen pattern, giving your reasons, when (i) one loudspeaker is disconnected, (ii) the loudspeaker is then reconnected with the leads reversed. [4] (O&C 137 Jun97) 168 934 physics.U2 M4.indd 168 15/11/07 8:45:12 am Module 4 Waves Examination questions 2 4 This question is about a method of finding the wavelength of light from a laser. A thin parallel beam of light of a single wavelength falls on a diffraction grating, as shown in Figure 4a. Screen Thin beam Grating 5 This question is about a radar speed trap. A microwave transmitter T, emitting radiation of wavelength 0.030 m, is placed adjacent to a receiver R. See Figure 5. Some of the output of T is fed directly to R and some is reflected from a metal sheet M. Figure 4a Light passes through the grating and a regular pattern of light and dark regions is observed on the screen. M Intensity A B Central maximum (c) S uggest one change that could be made to the experimental arrangement to improve the accuracy of the wavelength measurement. Explain your reasoning. [2] (OCR 2861 Jan05) Position on screen Figure 4b (a) F igure 4b shows how the intensity pattern varies across the central region of the screen. (i) Describe two important features of the intensity pattern shown in Figure 4b. [2] (ii) Explain the difference in intensity between points A and B in the pattern using the ideas of superposition. [2] (b) F igure 4c shows the experimental arrangement in more detail. Not to scale Grating Figure 5 (a) T he position of M is adjusted until the signal detected by R is a maximum. M is then slowly moved towards T and R. It is observed that the signal drops, reaching a minimum when M has been moved 0.0075 m. Explain why (i) the signal has decreased from a maximum, (ii) a minimum occurs but the signal is not zero. [4] (b) T he sheet M is removed and the device pointed towards a car which is moving straight towards it at constant speed. The signal detected by R is observed to fluctuate in amplitude at a frequency of 1.2 kHz. Calculate the speed of the car. [3] (O&C 137 Jun98) Screen First-order maximum Central maximum 80 lines per mm 1.20 m First-order maximum Figure 4c (i) The grating has 80 lines per mm. Show that the spacing between the lines on the grating is 1.25 × 10–5 m. [1] (ii) The distance between the central maximum and the first order maximum is measured on the screen and found to be 60 mm. The screen is 1.20 m away from the grating. Show that the first order maximum is observed at an angle θ of about 3o to the straight through direction. [2] (iii)Calculate the wavelength of the light, using the information above. [2] 169 934 physics.U2 M4.indd 169 15/11/07 8:45:21 am Glossary absorptionspectrum Aspectrumofdarklinesacrossthe patternofspectralcoloursproducedwhenlightpasses throughagasandthegasabsorbscertainfrequencies dependingontheelementsinthegas. cannotbecreatedordestroyed,justtransformedfromone formintoanotherortransferredfromoneplacetoanother. Thisisthesituationinanyclosedsystem. acceleration(a) Therateofchangeofvelocity,measuredin metrespersecondsquared(ms–2);avectorquantity. conventionalcurrent Amodelusedtodescribethe movementofchargeinacircuit.Conventionalcurrent travelsfrom+to–. accelerationoffreefall(g) Theaccelerationofabody fallingundergravity.OnEarthithasthevalueof9.81ms–2. coulomb Unitofelectriccharge(C),e.g.1.6×10–19C.1C= 1A×1s. ammeter Adeviceusedtomeasureelectriccurrent, connectedinserieswiththecomponents. couple Twoforcesthatareequalandoppositetoeach otherbutnotinthesamestraightline. amountofsubstance SIquantity,measuredinmoles(mol). ampere SIunitforelectriccurrent,e.g.4A. amplitude(xo) Themaximumdisplacementofawavefrom itsmean(orrest)position,measuredinmetres(m). antinode Apointofmaximumamplitudealongastationary wavecausedbyconstructiveinterference. area(A) Aphysicalquantityrepresentingthesizeofpartofa surface,measuredinmetressquared(m2). crumplezone Anareaofavehicledesignedtoincreasethe distanceoverwhichthevehicledeceleratesandsoreduce theaverageforceacting. current seeelectriccurrent. deBroglieequation Anequationexpressingthewavelength ofaparticleasaratioofPlanck’sconstantandthe particle’smomentum,mv. averagespeed Ameasureofthetotaldistancetravelledina degreeCelsius Unitfortemperature,e.g.100°C(nottheSI unittime. unit;seekelvin). brakingdistance Thedistanceavehicletravelswhile deceleratingtoastop. brittle Amaterialthatdistortsverylittleevenwhensubjectto alargestressanddoesnotexhibitanyplasticdeformation; forexample,concrete. density(R) Themassperunitvolume,measuredin kilogramspercubicmetre(kgm–3);ascalarquantity. diffraction Whenawavespreadsoutafterpassingaround anobstacleorthroughagap. CelsiusseedegreeCelsius. displacement(s or x) Thedistancetravelledinaparticular direction,measuredinmetres(m),e.g.3m;avector quantity. centreofgravity Thepointatwhichtheentireweightofan objectcanbeconsideredtoact. displacement–timegraph Amotiongraphshowing displacementagainsttimeforagivenbody. centreofmass seecentreofgravity(N.Balthoughthereisa distance(d). Howfaronepositionisfromanother, measuredinmetres(m),e.g.12m;ascalarquantity. technicaldifferenceitisnotrequiredatthislevel). drag Theresistiveforcethatactsonabodywhenitmoves charge seeelectriccharge. throughafluid. coherence Twowaveswithaconstantphaserelationship. components Partsofelectriccircuits,includingbulbs, LDRs,thermistors,etc. dragcoefficient Acharacteristicthatdeterminesthe amountofdragthatactsonanobject. componentsofavector Theresultsfromresolvingasingle vectorintohorizontalandverticalparts. driftvelocity Theaveragevelocityofanelectronasittravels throughawireduetoap.d. compressiveforce Twoormoreforcesthathavetheeffect ofreducingthevolumeoftheobjectonwhichtheyare acting,orreducingthelengthofaspring. ductile Materialsthathavealargeplasticregion(therefore theycanbedrawnintoawire);forexample,copper. conductor Amaterialwithahighnumberdensityof conductionelectronsandthereforealowresistance. conservationofcharge Physicallawstatingchargeis conservedinallinteractions;itcannotbecreatedor destroyed. conservationofenergy Physicallawstatingthatenergy dynamo Adevicethatconvertskineticenergyintoelectrical energy. efficiency Theratioofusefuloutputenergytototalinput energy. elasticdeformation Theobjectwillreturntoitsoriginal shapewhenthedeformingforceisremoved. 204 934 physics.endmatter.indd 204 15/11/07 8:51:21 am Glossary elastic limit The point at which elastic deformation becomes plastic deformation. frequency (f) The number of oscillations per unit time, measured in hertz (Hz), e.g. 50 Hz. elastic potential energy The energy stored in a stretched or compressed object (for example a spring), measured in joules (J); a scalar quantity. fundamental frequency The lowest frequency in a harmonic series where a stationary wave forms. electric charge (Q or q) = current × time, measured in coulombs (C); a scalar quantity. electric current (I) A flow of charge. An SI quantity, measure in amperes (A); a vector quantity. electrolyte A fluid that contains ions that are free to move and hence conduct electricity. electromagnetic wave A self-propagating transverse wave that does not require a medium to travel through. electromotive force, e.m.f. The electrical energy transferred per unit charge when one form of energy is converted into electrical energy, measured in volts (V). electron Negatively charged sub-atomic particle. Conduction electrons travel around circuits creating an electric current. electron diffraction The process of diffracting an electron through a gap (usually between atoms in a crystal structure, for example graphite). An example of wave–particle duality. electron flow The movement of electrons (usually around a circuit), from – to +. electronvolt One electronvolt is the energy change of an electron when it moves through a potential difference of one volt. Its value is 1.60 × 10–19 J. emission spectrum A pattern of colours of light, each colour having a specific wavelength. energy (E) The stored ability to do work, measured in joules (J); a scalar quantity. energy levels One of the specific energies an electron can have when in an atom. equations of motion The equations used to describe displacement, acceleration, initial velocity, final velocity and time when a body undergoes a constant acceleration. fuse An electrical component designed to heat up, melt and break the circuit (hence stop the current) when a specified amount of electric current passes through it. Used as a safety device. g, acceleration of free fall The acceleration of a body under gravity, 9.81 m s–2. gamma rays A form of electromagnetic wave with wavelengths between 10–16 m and 10–9 m. Used in cancer treatment. global positioning system A network of satellites used to determine an object’s position on the Earth’s surface. Used in navigation. gradient of a graph The change in y-axis over the change in the x-axis (rise over step). gravitational force The force due a gravitational field acting on an object’s mass. gravitational potential energy The energy stored in an object (the work an object can do) by virtue of the object being in a gravitational field, measured in joules (J); a scalar quantity. harmonics Whole number multiples of the fundamental frequency of a stationary wave. Hooke’s law The extension of an elastic body is proportional to the force that causes it. infrared A form of electromagnetic wave with wavelengths between 7.4 × 10–7 and 10–3 m. Used in remote controls. insulator A material with a small number density of conduction electrons and therefore a very high resistance. instantaneous speed The speed of an object at a given moment in time. equilibrium When there is zero resultant force acting on an object. intensity The energy incident per square metre of a surface per second, measured in watts per metre squared (W m–2). extension (x) The change in length of an object when subjected to a tension, measured in metres (m). interference The addition of two or more waves (superposition) that results in a new wave pattern. fluid A material that can flow from one place to another (i.e. liquids and gases). force (F) A push or a pull on an object, measured in newtons (N); a vector quantity. force constant (k) The constant of proportionality in Hooke’s law, measured in newtons per metre (N m–1). free fall When an object is accelerating under gravity (i.e. at 9.81 m s–2). internal resistance (r) The resistance of a battery or cell, measured in ohms (Ω). I–V characteristic A graph to show how the electric current through a component varies with the potential difference across it. joule Unit of energy (J), e.g. 1200 J. 1 J is the work done when a force of 1 N moves its point of application 1 m in the direction of the force. 205 934 physics.endmatter.indd 205 15/11/07 8:51:23 am kelvin SI unit of temperature (K), e.g. 373 K. kilowatt Unit of power (kW), e.g. 3.5 kW. 1k W = 1000 W. kilowatt-hour Unit of energy (kWh), e.g. 3 kWh. Used by electricity companies when charging for electricity. 1 kWh = 1000 W for 3600 s = 3.6 MJ. kinetic energy The work an object can do by virtue of its speed, measured in joules (J); A scalar quantity. Kirchhoff’s first law The sum of the currents entering any junction is always equal to the sum of the currents leaving the junction (a form of conservation of charge). Kirchhoff’s second law The sum of the e.m.f.s is equal to the sum of the p.d.s in a closed loop (a form of conservation of energy). parallel circuit A type of circuit where the components are connected in two or more branches and therefore provide more than one path for the electric current. π perpendicular At right angles (90° or __ rad) to. 2 period (T) The time taken for one complete pattern of oscillation, measured in seconds (s). phase difference (F) The difference by which one wave leads or lags behind another. For example, in-phase waves are in step with each other. In waves that are completely out phase one wave is half a wavelength in front of the other. Measured in radians (rad). photocell A component that reduces its resistance when light shines on it due to photoelectric emission of electrons. light dependent resistor, LDR A component that changes its resistance with changes in the light intensity (dark = high resistance, light = low resistance). photoelectric effect The emission of electrons from the surface of material when electromagnetic radiation is incident on the surface. light emitting diode, LED A component that only allows electric current to pass through it in one direction and that emits light when a p.d. is applied across it. photon A quantum of light, often described as a particle of light. line spectrum A spectrum produced by a material that contains only certain frequencies due to electron transitions between energy levels. longitudinal wave A wave where the oscillations are parallel to the direction of wave propagation, e.g. sound. Malus’ law A physical law describing the change in intensity of a transverse wave passing through a Polaroid analyser. mass (m) SI quanity, measured in kilograms (kg), e.g. 70 kg; a scalar quantity. microwaves A form of electromagnetic wave with wavelengths between 10–4 and 10–1 m. Used in mobile phones. moment of a force The turning effect due to a single force, calculated from the force multiplied by the perpendicular distance from a given point, measured in newton metres (N m), e.g. 4 N m; a vector quantity. monochromatic light Light waves with a single frequency (or wavelength). newton Unit of force (N), e.g. 4000 N. 1 N is the force which gives a mass of 1 kg an acceleration of 1 m s–2. node A point that always has zero amplitude along a stationary wave caused by destructive interference. ohm Unit of resistance (Ω), e.g. 24 Ω. 1 Ω = 1 V A–1. Ohm’s law The electric current through a conductor is proportional to the potential difference across it, provided physical conditions, such as temperature, remain constant. Planck constant (h) Constant used in quantum physics; 6.63 × 10–34 J s. plane polarised wave A transverse wave oscillating in only one plane. plastic deformation The object will not return to its original shape when the deforming force is removed, it becomes permanently distorted. polarisation The process of turning an unpolarised wave into a plane polarised wave (for example, light passing through a Polaroid filter). polymeric material A material made of many smaller molecules bonded together, often making tangled long chains. These materials often exhibit very large strains (e.g. 300%), for example rubber. potential difference, p.d. The electrical energy transferred per unit charge when electrical energy is converted into some other form of energy. potential divider A type of circuit containing two components designed to divide up the p.d. in proportion to the resistances of the components. potential energy A form of stored energy (see gravitational potential energy, elastic potential energy and spread 1.3.3). power (P) The rate of doing work, measured in watts (W); a scalar quantity. pressure (p) Force per unit area, measured in pascals (Pa), e.g. 100 000 Pa. 1 Pa = 1 N m–1; a scalar quantity. principle of moments For a body in rotational equilibrium the sum of the clockwise moments equals the sum of the anticlockwise moments. progressive wave A wave that travels from one place to another. 206 934 physics.endmatter.indd 206 15/11/07 8:51:24 am Glossary quantum A discrete, indivisible quantity. radian (rad) Unit of angle or phase difference, e.g. 3π rad. One radian is the angle subtended at the centre of a circle by an arc of circumference that is equal in length to the radius of the circle. � 2π = 360°. radio waves A form of electromagnetic wave with wavelengths between 10–1 and 104 m. Used in telecommunications. reflection When waves rebound from a barrier, changing direction but remaining in the same medium. refraction When waves change direction when they travel from one medium to another due to a difference in the wave speed in each medium. resistance (R) A property of a component that regulates the electric current through it. Measured in ohms (Ω), e.g. 24 Ω. resistivity (R) The ratio of the product of resistance and cross-sectional area of a component and its length (best RA defined by using the equation R = ___ ). resolution of vectors Splitting a vector into horizontal and vertical components (use to aid vector arithmetic). resultant force The overall force when combining two or more forces. resultant velocity The overall velocity when combing two or more velocities. scalar A physical property with magnitude (size) but not direction; for example, speed, distance, pressure, potential difference, etc. semiconductor A material with a lower number density of conduction electrons than a conductor and therefore a higher resistance. series circuit A type of circuit where the components are connected end to end and therefore provide only one path for the electric current. spectral line A line relating to a specific frequency either missing from an absorption spectrum or present in an emission spectrum. spectrum A collection of waves with a range of frequencies, for example, visible spectrum and electromagnetic spectrum. speed (s) The distance travelled per unit time, measured in metres per second (m s–1), e.g. 12 m s–1; a scalar quantity. spring constant force per unit extension. a vehicle from seeing the need to stop to vehicle becoming stationary). strain The extension per unit length. stress The force per unit cross-sectional area, measured in pascals (Pa). superposition The principle that states that when two or more waves of the same type exist at the same place the resultant wave will be found by adding the displacements of each individual wave. temperature (T or U) SI quantity, measured in kelvin (K), e. g. 273 K. Also measured in degrees Celsius (°C). tensile force Usually two equal and opposite forces acting on a wire in order to stretch it. When both forces have the value T, the tensile force is also T, not 2T. tensile stress The tensile force per unit cross-sectional area. terminal velocity The velocity at which an object’s drag equals its accelerating force. Therefore there is no resultant force and zero acceleration. thermistor A component that changes its resistance depending on its temperature. An NTC thermistor’s resistance reduces as the temperature increases. thinking distance The distance travelled from seeing the need to stop to applying the brakes. threshold frequency The lowest frequency of electromagnetic radiation that will result in the emission of photoelectrons from a specified metal surface. thrust A type of force due to an engine. time interval (t) SI quantity, measured in seconds (s), e.g. 60 s; a scalar quantity. torque The turning effect due to a couple, measured in newton metres (N m). transverse wave A wave where the oscillations are perpendicular to the direction of wave propagation, e.g. water waves, electromagnetic waves, etc. triangle of forces If three forces acting at a point can be represented by the sides of a triangle, the forces are in equilibrium�. turning forces One or more forces that if unbalanced will cause a rotation. standing wave An alternative name for a stationary wave. ultimate tensile strength The maximum tensile force that can be applied to an object before it breaks. stationary wave A wave formed by the interference of two waves travelling in opposite directions. ultimate tensile stress The maximum stress that can be applied to an object before it breaks. stopping distance The sum of the thinking distance and the braking distance (i.e. the total distance required to stop ultraviolet A form of electromagnetic wave with wavelengths between 10–9 and 3.7 × 10–7 m. Causes sun tanning. 207 934 physics.endmatter.indd 207 15/11/07 8:51:26 am upthrust A force on an object due to a difference in pressure when immersed in a fluid. vector A physical quantity that has both magnitude (size) and direction. For example, velocity, force, acceleration, electric current, etc. velocity (v) The displacement per unit time, measured in metres per second (m s–1), e.g. 330 m s–1; a vector quantity. a wave and the identical point on the next wave (e.g. the distance from one peak to the next peak), measured in metres (m). wave–particle duality The theory that states all objects can exhibit both wave and particle properties. weight (w) The gravitational force on a body, measured in newtons. velocity–time graph A motion graph showing velocity against time for a given body. work (W) The product of force and the distance moved in the direction of the force, it can also be considered as the energy converted from one form into another, measured in joules (J); a scalar quantity. volt Unit of potential difference and e.m.f (V), e.g. 230 V. 1 V = 1 J C–1. work function energy (F) The minimum energy required to release an electron from a material, measured in joules (J). voltmeter Device used to measure the p.d. across a component. It is connected in parallel across a component. volume (V) A physical quantity representing how much 3D space an object occupies, measured in metres cubed (m3). X-rays A form of electromagnetic wave with wavelengths between 10–12 and 10–7 m. Used in X-ray photography. watt Unit of power (W), e.g. 60 W. 1 W = 1 J s–1. Young’s double slit An experiment to demonstrate the wave nature of light via superposition and interference. wave A series of vibrations that transfer energy from one place to another. Young modulus (Y) The ratio between stress and strain, measured in pascals (Pa). wavelength (l) The smallest distance between one point on 208 934 physics.endmatter.indd 208 15/11/07 8:51:27 am Index abbreviations 97 absorptionlinespectra 184–5 acceleration constantacceleration 18–19,20–3 displacement–timegraphs 16 force 30–1 freefall 20–3,32 kinematicdefinitions 15 non-constantacceleration 32–3 turningforces 38 velocity–timegraphs 17 accuracy 190–1 aircolumns 162–3 ammeters 84 amperes(A) 82–3,99 amplitude 138,141 angletodirectionofmovementforces 56–7 antinodes 159,162 antiphase 138–9 appliancecosts,electrical 64–5 atoms energylevels 180–1 structure 179,185 batteries 118–19,124 BigBangtheory 185 Bohr,Niels 181,185 brittlematerials 75 carbatteries 119,124 categoriesofmaterials 74–5 centreofgravity 36–7 charge 82–3,94,97 chemicalenergy 58 circuits analysis 122–5 components 100–1,122 directcurrentcircuits 118–33 symbols 94 closedpipes/tubes 162,163 coherencewaves 151 components 100–1,122 compression 68,137 Compton,Arthur 178 computedtomography(CT)scans 147 conductionelectrons 83 conductors 83,87 conservationofenergy 58–61,118,120–1,176 constantacceleration 18–19,20–3 constanttemperature 98–9 constructiveinterference 150–1 costsofelectricalappliances 64–5 costsforelectricalenergy 110–11 coulomb 82–3 couple 38–9 crossedPolaroids 148 currentseeelectriccurrent DeBrogliediffraction 178 deformationofmaterials 68–9,70–1 density 32 destructiveinterference 150–1 diffraction 143,152–5,178–9 digitalammeters 84 directcurrent(DC)circuits 118–33 circuitanalysis 122–5 parallelcircuits 120–1,122–5 seriescircuits 118–19,122–5,126–7 displacement 14–15,16,138,159 distance 7 domesticelectricalsupply 108–11 Dopplerprinciple 154 double-slitexperiment 152–3 drag 30,32–3 drawingcircuitdiagrams 94 driftvelocity 86–7 ductilematerials 74–5 efficiency 58,66–7,119,122–3 Einstein’sphotoelectricequation 176–7 elasticdeformation 68–9,70 elasticlimit 69,71 elasticpotentialenergy 70–1,75 electriccharge 82–3,94,97 electriccurrent abbreviations 97 charge 82–3 circuitanalysis 122–4 directcurrent 118–33 electromotiveforce 94–5,96–7,122–5 electrondriftvelocity 86–7 electronflow 84–5 I–Vcharacteristics 98–9,100,101 Kirchhoff’sfirstlaw 84–5,118,124 resistivity 102 electricdischargetubes 185 electricalabbreviations 97 electricalappliancecosts 64–5 electricalcircuitdiagrams 94 electricalenergy,charging 110–11 electricalforces 30 electricalpotentialenergy 58 210 934 physics.endmatter.indd 210 15/11/07 8:51:29 am Index electrical power 106–7 electrical supply 108–11 electrolytes/electrolysis 86 electromagnetic radiation energy 58 measurement method 22–3 particulate nature 172–3 photon energy 172–3 Sun 140–1 waves 140–1, 146–7 see also light electromagnetic spectrum 146–7 electromotive force (e.m.f.) 94–5, 96–7, 122–5 electron diffraction 178–9 electrons conduction electrons 83 drift velocity 86–7 energy levels 105 flow 84–5, 86–7 polycrystalline graphite 178, 179 wavelengths 178–9 electronvolt 172–3, 174 electroscopes 174 elementary charge (e) 84 energy abbreviations 97 atoms 180–1 conservation 58–61, 118, 120–1, 176 efficiency 58, 66–7 electrical energy charges 110–11 electromotive force 96–7 falling objects 61 forces 30 levels 180–1, 184–5 photoelectrons 174–5 photon 172–3 potential difference 96–7 progressive waves 140–1 spectral patterns 184–5 equations of motion 18–19 equilibrium 34–5, 38, 39 estimated physical quantities 6–7 F = ma 30–1 falling objects 20–3, 32, 61 filament lamps 87, 100, 104, 184 force constant 70–1 forces angle to direction of movement 56–7 centre of gravity 36–7 compressive forces 68 deformation of materials 68–9 electrical 30 F = ma 30–1 mass and acceleration 30–1 newton 31, 32 tensile forces 68 triangle of forces 34–5 turning forces 38–9 types 30 units 31 formation of stationary waves 158 Fraunhofer lines 180–1, 184 free fall 20–3, 32, 61 frequency f 138, 146, 175–6 fringe widths 153 fundamental modes 160–2 fuses 108–9 gamma rays 146 global positioning systems (GPS) 49 gradients 16 graphite 178, 179 graphs of motion 16–17 gratings 154–5 gravity centre of gravity 36–7 gravitational forces 30–1, 32 gravitational potential energy 58, 60 harmonics 160–2 Hertz, Heinrich 147, 174, 175 Hooke’s law 70–1 household circuits 108–9 Hubble, Edwin 154, 185 hydrogen atoms 181 I–V characteristics 98–9, 100, 101 illumination control 127 infrared radiation 146, 147 instantaneous velocity 16 insulators 87 intensity 141 interference 150–3 internal energy 58 internal resistance 118–19, 120–1, 122–3 joules 56–7, 64, 108 kelvins 104 kilowatt-hours (kWh) 64–5, 108, 110–11 kilowatts (kW) 64–5 kinematics 14–15 kinetic energy 58, 60–1, 174 Kirchhoff’s first law 84–5, 118, 124 Kirchhoff’s second law 118, 120–1 211 934 physics.endmatter.indd 211 15/11/07 8:51:32 am LDRs (light-dependent resistors) 127 LEDs (light-emitting diodes) 101, 105, 173 light bulbs 87, 100, 104, 184 intensity monitoring 127 speed 140 Young’s double-slit experiment 152–3 see also electromagnetic radiation light-dependent resistors (LDRs) 127 light-emitting diodes (LEDs) 101, 105, 173 line spectra 180–1, 184–5 longitudinal waves 137, 162–3 Mach numbers 140 magnetic forces 30 Malus’s law 148 mass 7, 20, 30–2, 36 mass–acceleration relationship 30–1 mass–weight relationship 32 material categories 74–5 material deformation 68–9, 70–1 Maxwell equations 147 mean drift velocity 86–7 measurement of free fall acceleration g 20–3, 32 metals 83, 104–5 microwaves 146–7 interference 151 properties 142 stationary waves 160 Young’s double-slit experiment 153 moments 38, 39 motion constant acceleration 18–19, 20–3 free fall 20–3, 32, 61 graphs 16–17 kinematic definitions 14–15 non-constant acceleration 32–3 physical quantities 4–7 scalars 8 units 4–5 vectors 9–13 moving coil ammeters 84 multiple slits 152–5 multiple-flash photography 21 negative temperature coefficient (NTC) thermistors 105 net moment 38 newtons 30, 31, 32 nodes 159, 162 non-constant (non-linear) acceleration 32–3 NTC (negative temperature coefficient) thermistors 105 nuclear energy 58 nucleus diameters 179 objects on ramps 56–7 Ohm’s law 98–9 open pipes/tubes 162–3 order of magnitude 6 oscillations see waves parallel circuits 120–1, 122–5 particulate nature of electromagnetic radiation 172–3, 178–9 path difference 151 p.d. see potential difference phase difference 138–9, 151, 153 photo currents 174, 176–7 photoelectric effect 174–8 photoelectron energy 174–5 photon energy 172–3 physical quantities 4–7 pipes/tubes 162–3 Planck constant 173 plane polarised waves 148 plastic deformation 68–9, 71 polarisation 148–9 Polaroids 148 polycrystalline graphite 178, 179 polymeric materials 76 potential difference (p.d.) 96–7 circuit analysis 124–5 potential divider circuits 126–7 resistance of filament lamps 100 series circuits 118 potential divider circuits 126–7 potential energy 58, 60–1, 70–1, 75 power 7 abbreviations 97 circuit analysis 122 electrical power 106–7 forces 30 series circuits 118 watts 64–5 prefixes, SI units 5 pressure 30 principle of moments 38, 39 principle of superposition 150–1 progressive waves 140–1, 158 propagation of waves 136–7 Pythagoras’ theorem 12 quantum physics 172–89 radians 139 radio waves 146, 147, 149, 175 ramps 56–7 212 934 physics.endmatter.indd 212 15/11/07 8:51:32 am Index rarefaction 137 red shift 185 reflection 142, 149, 162–3 refraction 142–3 representing vectors 10 resistance circuit analysis 122–5 circuit components 100–1 domestic electrical supply 108–11 electrical power 106–7 electromotive force 94–5 fuses 108–9 Ohm’s law 98–9 parallel circuits 120–1 potential difference 96–7 resistivity 102–5 resistors at constant temperature 98–9 resistors in parallel 120–1 series circuits 118–19 resolving vectors 12–13 resultant forces 10–13, 34 right angles and vector resolution 12–13 rotating the plane of polarisation 148 Sankey diagrams 59, 67, 110–11 scalars 8 scattering 149 second, defining 184 semiconductors 87, 105 series circuits 118–19, 122–5 SI units 4–5, 56 significant figures 190–1 sky diving 33 slits, diffraction 152–5 solar constant 141 sound 58, 140, 160–3 spectra 87, 100, 104, 146–7, 180–1, 184–5 spectrometers 185 speed kinematic definitions 14–15 light 140 sound 140, 163 waves 140–1 stars, spectra 184–5 stationary waves 158–63 stopping potentials 174, 176–7 strain 72–5, 149 stress 72–5, 149 stretching wires 68–75 strings 160–1 Sun 140–1, 184 supply, electrical 108–11 Systèm International (SI) units 4–5, 56 television transmission 149 temperature 98–9, 104–5, 127 tensile forces 68 terminal velocity 33 thermistors 105, 127 Thompson, Sir Joseph 185 threshold frequency 175, 176 thrust 30 time 7, 16, 97 torque 38–9 transverse waves 137, 148, 160 trap door experiments 22–3 triangle of forces 34–5 triangles, vectors 11–13 tubes 162–3 tungsten filament lamps 87, 100, 104, 184 turning forces 38–9 ultraviolet radiation 146, 147 units 4–5, 31 Universe expansion 154, 185 variable resistors 126–7 vectors arithmetic 10–11 calculations 10–13 quantities 9 radians 139 representation 10 resolution 12–13 triangle of forces 34–5 triangles 11–13 velocity electron drift velocity 86–7 kinematic definitions 14–15 terminal velocity 33 velocity–time graphs 16–17, 18 vibration 160–2 voltage 94–5, 98–101, 122–3 voltmeters 96–7 volts 94–7, 99 watts 64–5 wave equation 140–1 wave–particle duality 172–3, 178–9 wavelengths 138 determinations 153–5 electromagnetic waves 146 electrons 178–9 equation 153 stationary waves 160 213 934 physics.endmatter.indd 213 15/11/07 8:51:34 am waves motion 136–7 properties 142–3 speed 140–1 terminology 138–9 see also electromagnetic radiation weight 30, 32, 36, 38 wire stretching experiments 68–75 work 56–7, 58, 64, 70–1X-rays 146, 147, 178 Young modulus 72–3 Young’s double-slit experiment 152–3 214 934 physics.endmatter.indd 214 15/11/07 8:51:34 am