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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
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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
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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 = KˆAv2
where K is a constant depending on the shape of the
raindrop
ˆ 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 =
KˆA
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
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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.
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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
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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
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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
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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
tT
t  78 T
Time
t  68 T
t  58 T
t  48 T
t  38 T
t  28 T
t  18 T
tT
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?
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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.
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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
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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
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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?
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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
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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
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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).
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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.
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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
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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?
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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
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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
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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
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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.
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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
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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.
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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
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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
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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 s1
[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 s1
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]
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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)
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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)
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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]
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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