Download Quanta and Waves Student booklet II ROR

Advanced Higher
Physics
Quanta and Waves
Student Booklet II
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Page 0
Dick Orr
Wave Phenomena
What Is Wave Motion?
When a wave is moving through a medium, energy is transferred from
one position in the medium to another position in the medium, with no
net transport of mass.
What Does This Mean?
Consider the example of water waves where each water particle moves at
right angles to the direction of travel of the wave. This is illustrated
below:
Direction of the water particles
Direction of the wave
As the wave travels, each water particle is displaced by the same distance
perpendicular to the direction of the wave.
The result is that the water particles themselves do not travel with the
wave.
What is seen is the surface of the water bobbing up and down (in the
absence of any wind or tide).
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Dick Orr
Wave definitions – You already know these, allegedly, make sure you do
now!!!
Wavelength (symbol λ units metres, m).
It is the distance between two successive points on a wave in phase.
Amplitude (symbol A, units metres, m)
It is measured from the centre line to the crest or trough. It is a measure
of how much energy a wave carries.
Frequency (symbol f, unit hertz, Hz)
This is how many waves are produced each second. This is the same as
the number of waves that pass a point in one second.
Period (symbol T unit second s)
This is the time to produce one wave.
Speed (symbol v, unit metres per second, m/s)
This is the distance a wave travels in one second.
Wave equation
v=f×λ
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Dick Orr
Intensity of A Wave
By definition the intensity of a wave is directly proportional to its
amplitude squared.
i.e.
I  A2
This means that
(therefore I = constant x A2)
I1
I2

 constant
A12 A22
where
I is the intensity measured Watts per metre squared (Wm-2)
A is the amplitude measured in metres (m)
Example
If the intensity of a wave is 45 Wm-2 when the amplitude is 2cm, what is
the new intensity of the wave when the amplitude increases to 4cm.
I α A2
I1
I2

A12 A22
45 I2

22 42
45  42
I2 
22
I2  180 Wm2
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Dick Orr
Travelling Wave
A travelling or progressive wave consists of a disturbance moving from a
source to surrounding places as a result of which energy is transferred
from one point to another with no net transfer of mass.
There are two types of travelling wave:
1.
The diagram shows a wave on
a slinky, any transverse wave
has the plane of vibration at
right angles to the direction of
energy transfer.
Transverse Wave
The diagram shows a wave on
a slinky, any longitudinal wave
has the plane of vibration in
the direction of energy
transfer.
Waves can be either mechanical or electromagnetic
Mechanical waves are disturbances that occur in materials that have
mass and elasticity. e.g. water waves
Electromagnetic waves are disturbances that consist of varying electric or
magnetic fields. e.g. light and microwaves
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Dick Orr
Wave Superposition
The principle of superposition states:
If a number of waves are travelling through a medium, then the resulting
disturbance at a point in the medium is the algebraic sum of the
individual disturbances produced by each wave at that point.
As a consequence of this, waves can pass through one another without
being affected in any way. For example if two stones are dropped into a
pool of calm water, two sets of circular waves are produced. These two
waves pass through each other, but at any particular point in time the
disturbance at that point is the algebraic sum.
e.g. if a crest of one wave meets a trough of another wave with equal
amplitude then the water will remain calm at the point that they meet.
And when they meet……..
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Dick Orr
Phase difference
A phase difference exists between two points on the same wave.
Consider the snapshots below of a wave travelling to the right in the
positive x-direction.
Points O and D have a phase difference of 2π radians.
They are separated by one wavelength (λ).
Points O and B have a phase difference of π radians.
They are separated by λ/2.
Notice that points A and B have a phase difference of π/2.
Essentially any phase difference can be defined as either a separation
[wavelength] or an angle [radians]
Since a separation of λ is equal to a phase difference of 2π radians, it is
possible to derive a relationship for any phase difference (φ) in terms of
separation.
phase difference 2

separation

 2

x 
2x
 

where:
 is the phase difference (angle) measured in radians
x is the separation between two points on a wave measured in metres
 is the wavelength measured in metres
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Dick Orr
Wave equation for a travelling wave
It is possible to represent the displacement produced by a travelling wave
by using the relationship below
x
y  Asin2(ft  )

where the symbols have their usual meanings.
This represents a wave travelling in the positive direction. Were the wave
travelling in the negative direction it would be represented by the
expression
x
y  Asin2(ft  )

Applying your knowledge from N5, Higher and unit 1 it is possible to
obtain alternative versions of these relationships.
Divide through bracket by f
x
y  Asin2(t  )
v
Since fλ = v
Also 2πf = ω, which gives
y  Asin(t 
2x
)

You are expected to be able to obtain values for speed, amplitude,
wavelength, frequency, angular frequency and period from the wave
equation.
You may be asked to write an expression for a travelling wave for
particular parameters.
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Dick Orr
Stationary waves
If two waves of equal frequency and amplitude but travelling in opposite
directions combine then a stationary wave will form. This is an
interference effect. The pattern produced will look like that below. The
positions of the nodes and antinodes will not change, hence the term
stationary wave.
The equation of such a wave is
y  Asin2ftcos
2x

Time dependant part describes how the displacement y varies at any
position x along the wave.
Position dependent part describes the maximum amplitude at that
position.
Nodes occur at a separation of

along the wave.
2
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Dick Orr
Interference
Hopefully you can remember studying the phenomena of interference at
Higher. Interference is the test for wave properties.
Interference can only occur if coherent waves combine. Waves are
coherent if they have a constant phase difference. This can only occur if
they have the same frequency, since both positional and angular phase
difference must remain constant.
Path difference
This is the difference in distance travelled from the source of the waves to
the point of combination X.
X
S1
S2
Path difference = S2X –S1X
Constructive interference will occur if path difference is an integer
number of wavelengths
path diff = mλ : m = 0, 1, 2…..
Destructive interference when
path diff = (m + ½) λ : m = 0, 1, 2…..
However if one path travels through a different medium from the other
then we have to take this into account. The reason for this is that the
wavelength will change in the new medium, so the phase difference is no
longer simply related to the distance the wave has travelled. The path
length through the medium is known as the optical path length.
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Dick Orr
Optical path difference
Consider the path of two rays of light as shown below. The left hand
shaded box, P, represents a medium of refractive index n. The right hand
box, Q, represents a path through air refractive index = 1.
A
P
Q
X
B
Consider the time taken for the ray to travel distance X.
X
c
v
c
c
n  air 
so vmed 
vmed vmed
n
X nX
tP  
c c
n
Effective distance travelled:
tQ 
X
Q  c  X
c
nX
P  c   nX
c
optical path length is n  geometric path length
We must always consider the optical path length when deciding if
constructive or destructive interference takes place. This allows for one
path to be in a medium and the other in air.
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Dick Orr
Interference by division of wavelength
This is when a single source produces a wave which is split and follows
different paths to a point. Since a single wave front is divided the light
passing through the slits must be coherent. Most commonly, a double slit
[Young’s slits] or a diffraction grating is used.
When light is shone through the double slit an interference pattern is
formed on the screen.
screen
light source
Δx
d
D
The slit separation is d metres
The slit to screen distance is D metres
The fringe separation is Δx metres
These quantities are related to the wavelength of light used, λ, by the
relationship
D
d
This set up can be used to determine the wavelength of laser light.
x 
Increasing the distance D will reduce the uncertainty in the measured
value of D. The separation of maxima will also increase reducing the
uncertainty in the measured value of Δx.
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Dick Orr
Interference by division of amplitude
This is when a ray is sent by more than one path through a medium. The
fringes exist when they are observed using a lens, camera or eye. They
cannot be projected onto a screen.
Wedge fringes:
Two glass slides formed into a wedge. Length l and height d
d
t
l
Consider light incident at near normal incidence on an air wedge as
shown above. A ray splits at the lower boundary of the first glass slide,
one ray reflects from that surface, the other passes through and reflects
from the top of the second glass slide. The path difference for the light
rays as shown above will be 2×t. The angles are exaggerated in the
diagram.
When a wave reflects at a boundary between a less optically dense
material to a more optically dense material [air to glass] there is a phase
change of π.
This means that instead of a maximum when path difference = mλ,
this set up will give a minimum when path difference = mλ.
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Dick Orr
If we consider the horizontal distance between two successive observed
minima then we can construct the diagram below.
min(n + 1)
minn
θ
t
Δx
The horizontal distance Δx between minima will result in a vertical height

difference of . One wavelength increase in path difference will result in
2
the next minimum.

tan  2
x
The triangle formed by the glass slides (p 12) will be a ‘similar triangle' to
the one above.
tan 
d
l

d
equating gives 2 
x l
so x 
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l
2d
Dick Orr
Thin film interference
If light is incident on a thin film, thickness t and refractive index n, then
interference can take place.
Consider the diagram below.
A phase change of π will take place at the upper surface. No phase
change will take place at the lower surface.
Air
t
film
The condition for a minimum will again be, path difference = mλ.
However since the path difference is due to the path through the film we
need to take the optical path difference which will be 2nt in this case.
Equating gives the condition for destructive interference
2nt = mλ
If the film is able to change thickness, such as a soap film then the varying
thickness will result in varying interference for different wavelength. This
explains the rainbow effect seen on soap bubbles.
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Dick Orr
Non- Reflecting coating.
Some lenses are coated with a film that reduces reflection. The coating is
chosen to have a refractive index between that of air and the glass it
covers. This means that there will be a phase change of π at both
reflections.
Air
t
Coating
Glass
The minimum thickness of coating will give rise to the first minimum, no
reflection if destructive interference takes place.
optical path difference = (m + ½ )λ
2nt = (m + ½ )λ for first min, m = 0

2nt =
2

t=
4n
The non – reflection will only occur 100% at one wavelength. This
wavelength is normally chosen to be in the middle of the visible spectrum
(green). This means that red and blue are reflected giving the lenses a
purplish hue.
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Dick Orr
Polarisation
Light from the sun and other sources is unpolarised and consists of
‘vibrations’ in every plane perpendicular to the direction of travel. This
can be illustrated using the diagram below.
The arrows represent the plane of vibration of the electric field vector E.
In all electromagnetic waves the electric field E and the magnetic field B
vary in size and are at right angles to each other and to the direction of
travel, hence ElectroMagnetic waves.
https://www.ndeed.org/EducationResources/CommunityCollege/RadiationSafety/Graphics/elec_mag_field.gif
If the plane of the electric field does not change, that is it always vibrates
in one plane, it is said to be linearly polarised. Illustrated by the diagram
below.
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Dick Orr
Polarisation By Reflection
Plane polarised light can be produced naturally when light is reflected
from any electrical insulator (e.g. glass, water).
Polarising angle
At one particular angle of incidence, known as Brewster’s angle, the
reflected light is completely Linearly Polarised.
Hence the statement can be made that…
At an angle of incidence equal to Brewster’s Angle the reflected light is
linearly polarised.
When light is incident at a boundary between air and an electrical
insulator, the polarising angle ip is the incident angle in air which causes
the reflected light to be linearly polarised.
air
iP
iP
insulator
r
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Dick Orr
The refractive index of the insulator is taken as being n.
siniP
sinr
but iP + r + 90 = 180
n
so r  (90  iP )
but sin(90  iP )  cosi
siniP
so n 
 taniP
cosiP
iP is known as Brewster’s angle.
How do we know if light is in fact polarised?
We need to use a polarising filter. The filter is rotated and if the light
passing through it is linearly polarised there will be no transmission at a
particular angle of alignment.
If the incident light is unpolarised, then rotating the filter will result in no
change in the transmitted light level.
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Dick Orr