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AP PHYSICS B
Waves & Optics
Teacher Packet
AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not
involved in the production of this material.
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Waves & Optics
Objective
To review the student on the concepts, processes and problem solving strategies necessary to
successfully answer questions on waves and optics.
Standards
Waves and Optics are addressed in the topic outline of the College Board AP* Physics Course
Description Guide as described below.
IV. Waves and Optics
A. Wave Motion (including sound)
1. Traveling Waves
2. Wave Propagation
3. Standing Waves
4. Superposition
B. Physical Optics
1. Interference and Diffraction
2. Dispersion of Light and the Electromagnetic Spectrum
C. Geometric Optics
1. Reflection and Refraction
2. Mirrors and Lenses
AP Physics Exam Connections
Topics relating to waves and optics are tested every year on the multiple choice and in most years
on the free response portion of the exam. The list below identifies free response questions that
have been previously asked over waves and optics. These questions are available from the
College Board and can be downloaded free of charge from AP Central.
http://apcentral.collegeboard.com.
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
Free Response Questions
Question 6
2008 Form B Question 5
Question 6
2007 Form B Question 6
Question 4
2006 Form B Question 4
Question 4
2005 Form B Question 4
Question 4
2004 Form B Question 3
Question 4
2003 Form B Question 3
Question 4
2002 Form B Question 4
Question 4
Question 4
Question 6
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Waves & Optics
What I Absolutely Have to Know to Survive the AP* Exam
Properties of Waves
Ray Diagrams
Law of Reflection
Snell’s Law
Critical Angle
Total Internal Reflection
Converging lenses & mirrors
Diffraction & Interference
Single Slit
Double Slit
Diffraction Grating
Thin Film Interference
Key Formulas and Relationships
Waves and Sound
f =
v=
1
T
λ
T
where f is the frequency and T is the period
= f λ where v is the velocity and λ is wavelength
v = 331 + 0.6T so at 20° C, the speed of sound is 343
m
s
⎛ v ⎞
fn = n ⎜
⎟ n = any positive integer (mode of oscillation) (vibrating string and open tube)
⎝ 2L ⎠
⎛ v ⎞
fn = n ⎜
⎟ n = any odd positive integer (closed tube) and L is the length of the vibrating column of air
⎝ 4L ⎠
Optics
θ r = θi where the angle of relection θ r as measured from the normal equals the angle of incidenceθ i
1
R where f is the focal length f and R is the radius of curvature
2
hi
d
= i where hi is the image height and ho object height and di is the image distance and d o object distance
− ho d o
f =
1 1 1
+ =
d o di f
m=−
di
where m is the magnfication
do
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Waves and Optics
c = f λ where c =3.00 × 108
m
for electromagnetic waves (light)
s
c
where n is the index of refraction
v
n1 sin θ1 = n2 sin θ 2 where θ1 is the angle of incidence and θ 2 is the angle of refraction
n=
sin θ c =
n2
where θC is the critical angle
n1
Double slit
mλ
sin θ =
, m = 0,1, 2,3,... where m is the integer representing the order of a fringe
d
θ = angle of spread of the light passing through a double slit
y
tan θ = where y = distance from the center of the central bright line produced
L
on a screen by the interference of light and the center of another bright line (antinode)
L = distance from the double slit or diffraction grating to the viewing screen
Single Slit
mλ
sinθ =
, m = 1, 2,3,.. W = width of the single slit but the single slit formula
W
describes minima (dark spots) rather than maxima
Thin Film Interference
λ film =
λvacuum
n
where t = thickness of a thin film
1⎞
⎛
2t + net phase change = ⎜ m + ⎟ λ film , m = 0,1, 2,3,..(destructive interference)
2⎠
⎝
2t + net phase change =mλ film from m=1,2,3...(constructive interference)
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Waves and Optics
Important Concepts
Basic Wave Vocabulary:
• Wavelength: The distance from the crest of one wave to the crest of the adjacent wave. Wavelength is
symbolized with the Greek letter, lambda, λ. It is measured in units of length.
• Frequency: The number of waves per unit time. The symbol for frequency is “f” unless it is
electromagnetic frequency. In that case, it is the Greek letter, nu, which looks like ν or c. Frequency is
measured in hertz (Hz) or 1/second. It is sometimes seen as s-1 which is equivalent to 1/second.
• Amplitude: The displacement of the wave medium from the equilibrium position.
• Period: The time required for one complete wave cycle. Period is symbolized by “T”, and is measured
in seconds.
• Harmonics: Strings and open pipes may resonate with more than one frequency. The possible
wavelengths for a system will be in ½ wavelength increments. In other words, there will be waves with
λ 2λ 3λ 4λ 5λ
in a string or
½ of a wavelength, 1 wavelength, 1½ wavelengths, 2 wavelengths, or , , , ,
2 2 2 2 2
open pipe. The lowest possible frequency in a vibrating system is called the fundamental. The others
are harmonics of this fundamental and the frequencies of the harmonics will be whole number multiples
or integers of the fundamental. Closed pipes (pipes that are closed on one end and open on the other end)
may resonate with more than one frequency. The possible wavelengths will be in ¼ wavelength
λ 3λ 5λ 7λ
.
increments and the frequencies of the harmonics will be the odd Integers, , , ,
4 4 4 4
• Waves transport energy.
•
The speed of a wave is controlled by the medium that the wave is passing through. It is the
product of the wavelength and the frequency. This is the fundamental wave equation v = f λ .
•
Refraction is the behavior of a wave as it passes from one medium into a second medium at an angle.
•
Diffraction is the behavior of a wave as it passes through a small opening or around a small obstacle.
•
Interference is the behavior of two or more waves occupying the same space at the same time. When
two or more waves occupy the same location in a medium at the same time the medium responds to the
waves by moving in a manner that is the algebraic sum of each individual disturbance. The point of no
disturbance is called a node and the point where the interference is maximum constructive is called an
antinode.
•
When a stretched string is plucked it will vibrate in its fundamental mode in a single segment with
nodes on each end and an antinode in the center. If the string is driven at this fundamental frequency, a
standing wave is formed. Standing waves also form if the string is driven at any integer multiple of the
fundamental frequency. These higher frequencies are called the harmonics.
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Waves and Optics
(a)
(c)
(b)
(d)
Each segment is equal to half a wavelength. In general for a given harmonic, the wavelength λ is
λ=
2L
n
where L is the length of the string and n is the number of segments in the string.
The linear mass density of the string can be directly measured by weighing a known length of the string. The
density is the mass of the string per unit length.
μ=
mass
length
The linear mass density of the string can also be found by studying the relationship between the tension,
frequency, length of the string, and the number of segments in the standing wave.
The velocity of any wave is given by v = λf where f is the frequency of the wave. For a stretched string:
v=
2 Lf
n
The velocity of a wave traveling in a string is also dependent on the tension, T, in the string and the linear mass
density, µ, of the string:
v=
®
T
μ
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Waves and Optics
•
When a ray of light is incident on a mirror, the angle of incidence of the ray is equal to the angle at
which it is reflected from the surface of the mirror. This phenomenon is called the law of reflection.
Both angles are measured from a line drawn perpendicular from the surface of the mirror called the
normal line.
Normal line
Incident
light ray
Reflected
light ray
Θi
Θr
Mirror
•
When you see your face in a plane mirror, the image is upright and left-right reversed, but no larger or
smaller than your actual face. We say that the image is formed behind the mirror and is virtual. A
virtual image is one that cannot be projected onto a screen, and can be located by drawing rays
representing the light and extending them to a point where they intersect.
When light encounters a boundary (a change in optical medium) some of the light reflects back obeying the law
of reflection and some of the light is transmitted into the new medium. The transmitted light does not travel in
the same direction as the original light. Instead it is bent (refracted) at the boundary and travels in a different
direction. This phenomenon is called refraction.
Incident
light ray
Reflected
light ray
Refracted
light ray
The refraction of light at the interface between two materials is described mathematically by Snell’s Law. In
the diagram above, the long dashed line represents the normal, a line perpendicular to the surface. The angle θ1
measures the angle of incidence relative to the normal. The angle θ2 measures the angle of refraction relative to
the normal. Snell’s Law states: ni sin θ1 = nr sin θ 2
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Waves and Optics
•
The quantity ni is the index of refraction for the medium in which the light was incident. The quantity
nr is the index of refraction for the medium in which the light was refracted. The index of refraction n
of a material is a measure of the speed of light in that medium. It is defined as the ratio of the speed of
light in vacuum c to the speed of light in the medium v .
n=
c
v
When light travels
from a less dense to more dense medium
• Light travels slower
• The frequency is unchanged
• Thus wavelength is shorter
• So light bends toward the normal
from a more dense to less dense medium
• Light travels faster
• The frequency is unchanged
• Thus wavelength is longer
• So light bends away from the
normal
Total Internal Reflection
• If the incident angle is of a certain size it will result in a 90o angle of refraction. This incident angle is
called the critical angle. At incident angles larger than the critical angle the light reflects back into the
substance. Thus the light at the critical angle or greater is totally internally reflected.
sin θ C =
n2
n1
Lenses and Mirrors
•
Concave mirrors and convex lenses cause light to converge. A convex lens converges parallel rays that
pass through it such that they intersect at the focal point. Likewise, a concave mirror causes incident
light rays parallel to its principal axis to converge at the focal point. Hence converging lenses and
mirrors produce real images, virtual images, or no images. Light rays actually pass through real images,
which can be projected onto a viewing screen. Virtual images cannot be projected onto a screen and in
the case of a mirror appear behind the plane of the mirror, whereas in the case of a lens a virtual image
appears on the same side of the lens as does the object.
•
Both converging lenses (convex) and mirrors (concave) create images that can be upright or inverted,
larger or smaller in size, or the same size as the object. The position and characteristics of the image
depend upon the location of the object relative to the focal length of the lens.
•
The relationship between the object distance d o , the image distance di , and the focal length f of a
lens or mirror is given by the fundamental lens/mirror equation.
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Waves and Optics
1 1 1
=
+
f d0 di
Convex Lens & Concave Mirror (converging optical devices)
Object
placed at:
Real or
virtual
Upright or
inverted
Enlarged, reduced, or
same size
d0 〉 2 f
Real
Inverted
Reduced
d0 = 2 f
Real
Inverted
Same Size
f 〈d0 〈2 f
Real
Inverted
Enlarged
d0 = f
No image
No image
No image
d0 〈 f
Virtual
Upright
Enlarged
Rules for Lenses: Remember light goes through the lens and refracts. (convex lens cause light to converge
while a concave lens causes light to diverge)
• Rays travelling parallel to the principal axis, either converge on the far focal point or diverge from
the near focal point.
• Rays that go through the center of the lens do not bend, but travel in straight lines.
Rules for mirrors: Remember light reflects off the mirror. (concave mirrors cause light to converge while
a convex mirror causes light to diverge)
• Rays travelling parallel to the principal axis, either converge on the near focal point or diverge from
the far focal point.
• Rays drawn through the object and the focal point, reflect parallel to the principal axis.
• Rays that go through the center of curvature reflect (C = 2F) straight back.
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Waves and Optics
Ray Diagrams for Convex Lens
Object
F
2F´
2F
Image
F´
Object
F
2F´
2F
F´
Image
Object
F
2F´
2F
Image
F´
Object
2F´
Image
F
2F
F
2F
F´
Object
2F´
F´
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Waves and Optics
Ray Diagrams for Concave Mirror
Object
F
C
Image
Object
F
C
Image
Object
Image
F
C
Object
C
F
Image
Object
C
F
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Waves and Optics
•
Light diffracting through a diffraction grating or double-slit opening will interfere constructively
and destructively, producing bright fringes, or antinodes, and dark fringes, or nodes, respectively.
m =2
m=1
θ
d
m=0
y
Bright
Dark
L
Bright
Dark
mλ
, m = 0,1, 2,3,... where m is the integer representing the order of a fringe
d
θ = angle of spread of the light passing through a single or double slit
y
tan θ = where y = distance from the center of the central bright line produced
L
on a screen by the interference of light and the center of another bright line (antinode)
L = distance from the double slit or diffraction grating and the screen
sin θ =
•
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Waves and Optics
•
When light enters a thin transparent film, reflects off of a surface beneath the film, and emerges
from the film once again, the light waves may be in phase or out of phase, depending on the extra
distance traveled by the light or twice the thickness of the film and whether or not a phase change
has occurred. The pattern produced is called thin-film interference. An important consideration in
determining whether these waves interfere constructively or destructively is the fact that whenever
λ
occurs in the reflected
light reflects off a surface of higher index of refraction, a phase shift of
2
wave. There is no phase shift whenever light reflects off a surface of lower index of refraction. In
the example below the net phase change is zero, since at the interface between the air and the film
λ
λ
but it also undergoes a phase shift of
when reflected from
the light undergoes phase shift of
2
2
the interface of the film and the mirror so that the net phase change is 0. Hence only the extra
distance (2t) traveled by the second ray of light determines the conditions for constructive or
destructive interference.
λ film =
•
λvacuum
n
where t = thickness of a thin film
1⎞
⎛
2t + net phase change = ⎜ m + ⎟ λ film , m = 0,1, 2,3,..(destructive interference)
2⎠
⎝
2t + net phase change =mλ film from m=1,2,3...(constructive interference
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Waves and Optics
Free Response
Question 1 (10 pts)
A student strikes a tuning fork and holds it over the open end of a pipe that is closed at
the opposite end by a container of water. When the length of the pipe is just right, the
student hears the first loud resonance frequency, the fundamental. Slowly the student
adjusts the length of the pipe and hears the next successive resonance frequency when the
spacing between the two resonances is 0.32 m. The air inside the pipe is at normal room
temperature 20˚C.
L=
L=
A. Determine the wavelength.
(2 points max)
Closed pipes have only the odd harmonics,
λ
the first occurs at
and the third occurs at
4
3λ
so the spacing between two successive
4
λ
resonance frequencies is
2
λ
= 0.32m
2
λ = 0.64 m
®
1 point for noting that the spacing
between two successive resonance
frequencies is
λ
2
1 point for the correct answer including
correct units
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Waves and Optics
B. Determine the frequency of the unknown tuning fork
(3 points max)
m
m
v = 331 + 0.6T = 343
s
s
m
343
v
s = 536 Hz
f = =
λ 0.64m
1 point for finding the speed of the
sound wave in air
1 point for the correct frequency
equation
1 point for the correct answer
including correct units and reasonable
number of significant digits consistent
with answer in part A
C. What were the lengths of the vibrating columns of air when the student heard the
fundamental or first resonance frequency and the next successive resonance frequency or
harmonic?
(3 points max)
nv
where n is any positive odd Integer
4L
For n = 1
f =
nv (1)( 343m / s )
L=
=
= 0.16 m
4f
( 4 )( 536 Hz )
nv
where n is any positive odd Integer
4L
For n = 3
f =
L=
nv ( 3)( 343m / s )
=
= 0.48 m
4f
( 4 )( 536 Hz )
1 point for a statement relating the
length of the vibrating column of air to
the frequency, speed, and harmonic
number
1 point for the correct answer for the
fundamental or first harmonic
including correct units and reasonable
number of significant digits
1 point for the correct answer for the
third harmonic including correct units
and reasonable number of significant
digits
Or Alternate solution:
Since the spacing between the successive
resonances is 0.32 m
0.16 m + 0.32 m = 0.48 m
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Waves and Optics
D. The closed pipe is replaced with a pipe which is open at both ends. Determine the
length of the open pipe in order for the original tuning fork to resonate at its fundamental
frequency (first harmonic).
(2 points max)
nv
where n is any positive Integer
f =
2L
For n = 1
L=
nv (1)( 343m / s )
=
= 0.32 m
2f
( 2 )( 536 Hz )
®
1 point For recognition that the first
resonance condition in an open pipe
requires the length of the pipe to be
one-half the wavelength
1 point for the correct answer
including correct units and reasonable
number of significant digits
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Waves and Optics
Question 2 (15 pts)
One end of a 2-meter string is attached to a fixed wall, and the other end is attached to a
vibrating machine, sending waves through the string toward the wall, as shown below.
A. On the diagram above, sketch the shape of the standing wave in the string after
the wave reaches the fixed point on the wall and indicate the direction of the
reflected wave using arrows.
1 point For indicating that
the wave reflects on the
opposite side of the
baseline from the incident
wave
1 point For a reflection
pattern matching the
symmetry of the incident
pattern
1 point For indicating that
the direction of the
reflected wave is away
from the wall
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Waves and Optics
B. The time it takes for the wave to travel down the string and back to its starting
point is 0.10 s.
(i) Calculate the frequency of the wave
(ii) Calculate the speed of the wave as it travels through the string
The wave vibrates 4 complete cycles during
its round trip. The frequency is
4 cycles
f =
= 40 Hz
0.10s
The wavelength is 1.0 m.
v = f λ = ( 40 Hz )(1.0 m ) = 40
m
s
1 point For an indication that the
number of vibrations in a round trip is
4 or the period is the total time
divided by 4 vibrations or equivalent
1 point For the correct application of
equation finding frequency
1 point For an indication that the
wavelength is 1.0 m
1 point For an answer consistent with
part B(i), including appropriate units
d = vt
d
v=
t
4.0 m
m
v=
= 40
0.10 s
s
v= fλ
v
f =
Alternate solution:
1 point For using a distance
consistent with the time of travel
1 point For the correct calculation of
speed consistent with distance and
time used previously
λ
m
s
f =
1.0 m
40
1 point For an indication that the
wavelength is 1.0 m
f = 40 Hz
1 point For an answer consistent with
part B(ii) , including appropriate units
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Waves and Optics
The string remains attached to the wall and vibrating machine, but the tension in the
string is now decreased to 5 N. The linear density of the string is 0.05 kg/m.
C. Determine the speed of the wave in the string.
v=
FT
μ
=
5 N
m
= 10
0.05 kg/m
s
1 point For an indication that the
speed of a wave depends upon the
physical properties of the medium and
no incorrect statements
1 point For a correct answer including
units
D. Determine the wavelength of the wave in the string.
λ=
v 10 m/s
=
= 0.25 m
f 40 Hz
1 point For any indication that the
frequency of the machine does not
change
1 point For a correct answer
including units
The string remains attached to the wall and vibrating machine, but the tension in the
string is now increased so that it is greater than at the beginning of the experiment.
E. Describe in words how the wavelength of the wave will be different than in part
A. Justify your answer.
The speed of the wave will increase with
increasing tension in the string.
Frequency does not change.
The wavelength must increase with
increasing tension, since wavelength is
proportional to wave speed.
®
1 point For an indication that the
wavelength will increase and no
incorrect statements
1 point For a correct justification
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Waves and Optics
F. On the diagram below, make a sketch illustrating a possible standing wave pattern
in the string of greater tension.
1 point For any standing wave
pattern
1 point For showing a longer
wavelength, i.e., one, two, or
three antinodes or must be
consistent with answer in part A
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Waves and Optics
Question 3 (10 pts)
Yellow
Glass
30˚
Violet
30˚
The glass plate shown above has an index of refraction that depends on the wavelength of
the light that enters it. The index of refraction is 1.54 for yellow light of wavelength 5.80
x 10-9 m in the air and 1.62 for violet light of wavelength 4.20 x 10-9 m in the air. Both
the yellow and violet beams of light enter the glass from the left at the same angle of 30º
above the normal, are refracted inside the glass, and exit the glass on the right.
A. Determine
i. the speed of the yellow beam of light in the glass.
ii. the speed of the violet beam of light in the glass.
vY =
c 3.00 x108 m/s
=
= 1.95 x108 m/s
nY
1.54
vV =
c 3.00 x108 m/s
=
= 1.85 x108 m/s
nV
1.62
®
1 point For the correct equation using
the index of refraction to determine
the speed of the light beams in the
glass
1 point For the correct answers with
units
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Waves and Optics
B. Determine
i. the wavelength of the yellow beam of light in the glass.
ii. the wavelength of the violet beam of light in the glass.
λY =
λV =
λair
nY
λair
nV
=
5.80 x10−7 m
= 3.76 x10−7 m
1.54
1 point For a correct equation for the
wavelength of the light in the glass
=
4.20 x10−7 m
= 2.59 x10−7 m
1.62
1 point For the correct answers with
units
C. Determine
i. the frequency of the yellow beam of light in the glass.
ii. the frequency of the violet beam of light in the glass.
fY =
fV =
c
λair
c
λair
3.00 x108 m/s
=
= 5.17 x1014 Hz
−7
5.80 x10 m
=
3.00 x108 m/s
= 7.14 x1014 Hz
−7
4.20 x10 m
1 point For a correct equation for the
frequency of the light
1 point For recognition that the
frequency of each color is the same
in the glass and in the air
1 point For the correct answer with
units
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Waves and Optics
D. On the figure below, sketch the approximate paths of both the yellow and the violet
rays as they pass through the glass and then exit into the air.
Yellow
30˚
Glass
Violet
30˚
1 point For both beams bending
toward the normal inside the glass
1 point For both beams bending away
from the normal when they exit the
glass into the air
1 point For the violet beam bending
more toward the normal inside the
glass than the yellow beam
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Waves and Optics
Question 4 (10 pts)
cm
180 150 120 90 60 30
30
60
90
120 150 180
The concave mirror shown above has a focal length of 30.0 centimeters. You are given a
candle 5.0 centimeters high. You wish to produce an image on a screen that has a
magnification of -3.
A. State whether you would place the candle on the left side or the right side of the
mirror shown above. Explain your choice.
The candle must be placed on the left side
of the mirror, since the left side is the
concave (converging) side of the mirror. A
convex mirror produces virtual images
only. A concave mirror may produce a real
image if the candle is located outside the
focal point.
®
1 point For the correct answer: left
side of the mirror
1 point For a correct explanation that a
concave mirror may produce a real
image if the candle is located outside
the focal point or causes light to
converge.
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Waves and Optics
B. Choose an appropriate object distance for the candle and calculate the
corresponding distance from the mirror at which the screen should be placed in
order to see an image 15.0 cm tall on the screen.
1 point For a correct equation to find
the image distance
1 1 1
= +
f di do
In order to produce an enlarged real image,
the candle would need to be placed at an
object distance between f and 2f. Choosing
40 cm as the object distance, we can find
the corresponding image distance.
1 1 1
= +
f di do
1 point For recognition that the candle
must be placed between f (30 cm) and
2f (60 cm).
1 point For the correct answer with
units corresponding to the image
distance
1
1
1
=
+
30cm 40cm d i
d i = 120cm
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Waves and Optics
C. Draw a ray diagram on the figure below which verifies your calculation. Be sure
to label the image formed in your ray diagram.
cm
180 150 120 90 60 30
30
60
90
120 150 180
1 point For the candle located
between 30 cm and 60 cm on the left
hand side of the mirror
1 point For drawing one principal
ray parallel to the principal axis and
reflecting through the focal point
1 point For drawing one principal
ray passing through the center of
curvature (other principal rays could
have been used)
1 point For locating the image at the
intersection of two principal rays
1 point For drawing the image such
that it is inverted relative to the
object and magnified
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Waves and Optics
Question 5 (10 pts)
A particular color of light is passed through the double-slit apparatus shown above. The
distance between the slits d = 1.40 × 10−4 m, and the length from the slits to the screen is
L = 2.50 m. The second-order bright fringe is measured to be y = 2.07 × 10−2 m from the
bright central antinode. The wavelengths of several colors of light are listed below as
well.
Color
red
orange
yellow
green
Wavelength (nm)
664
622
580
520
A. Which property of light is best illustrated by a double slit apparatus?
The interference pattern illustrates the wave
property of light, or constructive and
destructive interference, or diffraction.
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1 point for stating the wave property of
light, diffraction, or constructive and
destructive interference
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B. Sketch the representation of the light intensity pattern which appears on the screen
opposite the double slit on the diagram above.
1 point for sketching the
classic double slit pattern
with a central maxima and
other maxima at different
heights
C. At which point in the diagram, P or Q, is there a maximum in the interference
pattern? Determine the path difference between the light arriving at this point
from the two slits.
Point P is located at the second-order
bright line, or m = 2. A line drawn from
the lower slit to point P differs in path
length from a line drawn from the other
slit to point P by a path difference of mλ,
or in this case 2λ which is 1160 nm.
1 point for the correct answer point P
1 point for indicating that the distance
of point P from the center of the
interference pattern is twice the
spacing of the pattern
1 point for the correct answer of 2λ or
1160 nm or an answer consistent with
the point chosen
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D. Determine the color of the light which was passed through the slits?
The angle θ can be found by
⎛ 2.07 × 10−2 m ⎞
⎛ y⎞
θ = tan −1 ⎜ ⎟ = tan −1 ⎜
⎟ = 0.47°
⎝ L⎠
⎝ 2.50 m ⎠
Then the wavelength of the light is
−4
d sin θ (1.40 × 10 m ) sin ( 0.47° )
=
λ=
2
m
−7
λ = 5.80 × 10 m
The color of the light is yellow.
1 point for finding the correct angle
using the tangent or the small angle
y
approximation for
L
1 point for using the correct
equation to find the wavelength of
the light
1 point for the correct answer,
yellow
E. Determine the frequency of the light which was passed through the slits?
f =
c
λ
=
3.0 × 108 m / s
= 5.2 × 1014 Hz
5.8 × 10-7 m
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1 point for using the correct equation
to find the frequency of the light
1 point for the correct answer,
including units
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Waves and Optics
Multiple Choice
1. The diagram below represents a wave propagating along a string at a speed of 320 cm/s. The
frequency of the wave is
2cm
L = 12 cm
A)
40 Hz
B)
80 Hz
C) 320 Hz
D) 640 Hz
E) 1280 Hz
From the diagram, the wavelength is 4 cm since there are three complete
wavelengths in a distance of 12 cm.
B
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Waves and Optics
2. Two waves approach each other in the same spring at the same time, as shown below. When
the two waves are exactly at point P , the vertical displacement of the spring will be
A) 0
A
2
B)
C) A
D) 2A
E) 3A
The two waves are in phase (crest on crest), and so they will constructively
interfere and produce a larger wave of twice the amplitude at P.
D
3. Consider the following properties of waves.
I. Speed
II. Wavelength
III. Frequency
Which of the above properties change when a wave is refracted?
A) I only
B) II only
C) I and II only
D) II and III only
E) I, II, and III
C
The frequency remains the same. When a wave undergoes refraction, its
speed changes along with its wavelength.
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Waves and Optics
4. Which of the following is true of a sound that is resonating in a pipe which is closed at one
end (in terms of pressure nodes and antinodes)?
A)
B)
C)
D)
E)
Nodes are formed at both ends of the pipe.
Antinodes are formed at both ends of the pipe.
An antinode is formed at the closed end of the pipe and a node is formed at the open end.
An antinode is formed at the open end of the pipe and a node is formed at the closed end.
A sound wave cannot resonate in a pipe which is closed at only one end.
The wave is reflected off the closed end, creating a pressure antinode at the
closed end and a pressure node at the open end.
C
5. A wave source of constant frequency sends a wave through a tight string of uniform density
with a speed vo and wavelength
speed of the wave is now
A)
B)
1
2
. The tension is then relaxed to half its initial tension. The
vo
2v o
C)
D) 2vo
E) 4vo
A
v ∝ FT , so halving the tension force FT gives a speed of
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vo
2
.
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Waves and Optics
6. Which phenomenon proves that light has wave properties?
A)
B)
C)
D)
E)
Images formed by lenses
Interference patterns formed when light passes through a diffraction grating
Light reflecting off a mirror
The photoelectric effect
The refraction of light as it passes from air into water
Only waves form interference patterns.
B
7. A barrier wall protects a seaport from most ocean waves. The waves that pass through the
opening in the barrier wall change direction as shown in the diagram below. This is an
example of
A)
B)
C)
D)
E)
Reflection
Refraction
Interference
Dispersion
Diffraction
A wave diffracts when it must go around an obstacle or through an opening.
E
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8. The diagram above shows sound waves emanating from two loud speakers. Compressions in
the sound waves are indicated by the semicircular solid lines. Rarefactions are indicated by
dashed lines. At which locations will the sound waves produce constructive interference?
A)
B)
C)
D)
E)
A only
B only
C only
A and B only
B and C only
D
Constructive interference occurs whenever waves intersect in phase with one
another. Rarefaction meets rarefaction or compression meets compression.
Also note that both points A and B are along the central line of symmetry
where there is always constructive interference.
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9. Light passes from medium 1 to medium 2 and bends along the path shown below. Which of
the following statements is correct?
1
2
A)
B)
C)
D)
E)
Medium 1 is more dense than medium 2.
Medium 1 is less dense than medium 2
The speed of the light in medium 2 is greater than the speed of the light in medium 1.
The frequency of the light is greater in medium 1.
The wavelength is the same in medium 1 and medium 2.
B
The light bends toward the normal when it enters medium 2 which indicates it
is more optically dense than medium 1.
10. An object is placed 5 cm away from an optical device. The image is magnified and has the
same orientation as the object. The optical device could be a
I. Convex lens
II. Plane mirror
III. Concave lens
IV. Concave mirror
A)
B)
C)
D)
E)
I only
II only
I and III only
II and III only
I and IV only
E
A convex lens and a concave mirror both produce magnified upright images
when the object is placed inside the focal point.
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11. A candle is placed on the principal axis of a convex lens at a distance of 30 cm from the lens.
The focal length of the lens is 10 cm. The image formed will be
A)
B)
C)
D)
E)
real, upright, and enlarged
real, inverted, and enlarged
real, inverted, and smaller
virtual, upright, and enlarged
virtual, upright, and smaller
C
The candle is placed at a distance greater than twice the focal length, and so
the image formed will be real, inverted, and smaller than the candle.
12. In Young’s double slit experiment, the distance between the two slits is d, the wavelength of
the light is , and the distance from the viewing screen to the double slit apparatus is L. The
angular separation , between the interference fringes
A)
B)
C)
D)
E)
is roughly zero since light behaves like particles (photons)
is smaller for red light than green light
is larger for a larger d
is larger for a smaller d
depends on λ only
D
Hence, a smaller d will result in a larger angular separation .
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13. The speed of light in a particular piece of glass is 2.0 x 108 m/s, and the speed of light in
water is 2.3 x 108 m/s. Find the critical angle for the light passing from the glass to the water.
A)
B)
C)
D)
E)
10°
30°
40°
60°
90°
D
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14. Light is incident on a thin film which covers a mirrored surface. In order to see the brightest
reflection of light after passing through the film, which of the following must be true?
A)
B)
C)
D)
E)
The thickness of the film must be greater than the wavelength.
The wavelength must be equal to half the thickness of the film
The wavelength must be equal to 4 times the thickness of the film.
The wavelength must be a multiple of twice the thickness of the film.
The thickness of the film must be less than the wavelength.
D
The equation that governs the bright reflection (constructive
interference) is 2t = mλ.
15. A plane mirror will produce a virtual image
A)
B)
C)
D)
E)
when the object distance is greater than the image distance.
when the object distance is less than the image distance.
when the object is on the principal axis of the mirror.
when the rays converge at the focal point of the mirror
at all distances from the mirror.
E
A plane (flat) mirror will reflect light and produce an image at any distance
from the object.
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16. Two beams of light, red and blue, enter a prism as shown below. Which of the following
statements is true concerning the light as it passes through the prism?
Red
Blue
A) The blue light will bend more than the red light, since the blue light has a longer
wavelength.
B) The red light will bend more than the blue light, since the red light has a longer
wavelength.
C) The blue light will bend more than the red light, since the blue light has a shorter
wavelength.
D) The red light will bend more than the blue light, since the red light has a shorter
wavelength.
E) The red and blue light will bend by the same amount, since all colors of light
refract equally.
C
Blue light has a higher frequency than red light and thus a shorter wavelength
and will bend more than the red light as it passes through the prism.
17. A converging lens has a focal length of 30 cm. A 5 cm tall candle is placed at a distance of
10 cm in front of the lens. Determine the image distance.
A)
B)
C)
D)
E)
30 cm
15 cm
6 cm
6 cm
15 cm
B
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18. A beam of light enters the flat surface of a diamond at an angle of 30º from the normal. The
angle of refraction in the diamond is measured to be 12º from the normal. (sin 30° = 0.50,
sin 12° = 0.21) Determine the index of refraction for the diamond.
A)
B)
C)
D)
E)
0.40
0.67
1.0
1.5
2.4
The angle of incidence
= 30º and the angle of refraction
index of refraction can be found by Snell’s law:
E
= 12º. The
n1 sin θ1 = n2 sin θ 2
(1) sin 30 = n2 sin12
n2 = 2.4
19. A ray of light passes through air (n = 1.00), glass (n = 1.50), and then water (n = 1.33). Select
the answer that best describes what happens to the velocity of the ray of light as it passes from
air into glass and then out of the glass into water.
A)
B)
C)
D)
E)
decreases, then increases
decreases, then decreases again
increases, then decreases
increases, then increases again
the velocity of the ray of light remains the same
A
As the light passes from a less dense medium to a more dense medium it
slows down (air to glass), and speeds up as it goes from more dense to less
dense (glass to water). Since the refractive index is the ratio of the speed of
light in a medium to the speed of light in a vacuum, the higher the value, the
more dense the medium.
20. If light is passed through a narrow, single-slit opening onto a screen, the pattern of light
produced on the screen is
A)
B)
C)
D)
E)
alternating bright and dark lines of equal width
a bright central band of light with much smaller, dimmer bands toward the edges
concentric circles of light
one circle of light
one band of light
B
The single-slit diffraction pattern creates a central antinode which is much
larger than the fringes.
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Waves and Optics
21. When white light shines on a glass prism it disperses into a rainbow of colors as shown
above. This occurs because
A)
B)
C)
D)
E)
Each frequency of light has a different index of refraction
There is constructive and destructive interference between the different colors
White light is made up of all the colors of the rainbow
Violet light travels faster through the prism than red light does
Impurities in the glass absorb the white light and produce the colors
A
Dispersion is caused by slightly different indices of refraction for each
frequency of light.
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22. Monochromatic light passing through two closely spaced slits forms a light and dark
pattern on a screen as shown above. This demonstrates which property of light?
I. Interference of light
II. Particle properties of light
III. Diffraction of light
A)
B)
C)
D)
E)
I only
II only
I and II only
I and III only
II and III only
D
The double-slit produces a light and dark pattern which is due to constructive
and destructive interference of light and demonstrates the wave property of
light.
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