S PB WavesOptics09 T 3

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.

Copyright © 2009 Laying the Foundation®, Inc., Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org

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. 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. Free Response Questions

2008 Question 6 2008 Form B Question 5 2007 Question 6 2007 Form B Question 6 2006 Question 4 2006 Form B Question 4 2005 Question 4 2005 Form B Question 4 2004 Question 4 2004 Form B Question 3 2003 Question 4 2003 Form B Question 3 2002 Question 4 2002 Form B Question 4 2001 Question 4 2000 Question 4 1999 Question 6

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


http://apcentral.collegeboard.com/

Waves & Optics

What I Absolutely Have to Know to Survive the AP* Exam

Properties of Waves Critical Angle Single Slit Ray Diagrams Total Internal Reflection Double Slit Law of Reflection Converging lenses & mirrors Diffraction Grating Snell’s Law Diffraction & Interference Thin Film Interference

Key Formulas and Relationships

Waves and Sound

1 where is the frequency and is the period

where is the velocity and is wavelength

m331 0.6 so at 20 C, the speed of sound is 343 s

f f TT

v f vT

v T

λ λ λ

=

= =

= + °

any positive integer (mode of oscillation) (vibrating string and open tube)2

any odd positive integer (closed tube) and is the length of the vibrating column of air4

n

n

vf n nLvf n n LL

⎛ ⎞= =⎜ ⎟⎝ ⎠⎛ ⎞= =⎜ ⎟⎝ ⎠

Optics

where the angle of relection as measured from the normal equals the angle of incidence1 where f is the focal length and is the radius of curvature2

where is the image hei

r i r i

i ii

o o

f R f R

h d hh d

θ θ θ=

=

=−

θ

ght and object height and is the image distance and object distance

1 1 1

where is the magnfication

o i o

o i

i

o

h d d

d d fdm md

+ =

= −


Waves and Optics

8

1 1 2 2 1 2

2

1

m where =3.00 10 for electromagnetic waves (light)s

where is the index of refraction

sin sin where is the angle of incidence and is the angle of refraction

sin where c C

c f c

cn nv

n nnn

λ

θ θ θ θ

θ θ

= ×

=

=

= is the critical angle

Double slit

sin , 0,1,2,3,... where m is the integer representing the order of a fringe

angle of spread of the light passing through a double slit

tan where distance from the center of the central

m md

y yL

λθ

θ

θ

= =

=

= = bright line produced

on a screen by the interference of light and the center of another bright line ( ) distance from the double slit or diffraction grating to the viewing screen

antinodeL =

Single Slitmsin , 1, 2,3,.. width of the single slit but the single slit formulaW

describes minima (dark spots) rather than maxima

Thin Film Interference

where thickness of a thin vacuumfilm

m W

tn

λθ

λλ

= = =

= = film

12 net change , 0,1,2,3,..(destructive interference)2

2 net change =m from m=1,2,3...(constructive interference)

film

film

t phase m m

t phase

λ

λ

⎛ ⎞+ = + =⎜ ⎟⎝ ⎠

+


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

½ of a wavelength, 1 wavelength, 1½ wavelengths, 2 wavelengths, or 2 3 4 5, , , ,2 2 2 2 2λ λ λ λ λ in a string or

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

increments and the frequencies of the harmonics will be the odd Integers, 3 5 7, , ,4 4 4 4λ λ λ λ .

• 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.


Waves and Optics

(a)

(b)

(c)

(d)

Each segment is equal to half a wavelength. In general for a given harmonic, the wavelength λ is

2Ln

λ =

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.

masslength

μ =

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 where f is the frequency of the wave. For a stretched string: f= λ

2Lfvn

=

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:

Tv =μ


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.

• 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.

Incidentlight ray

Reflectedlight ray

Refractedlight 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: 1 2sin sini rn nθ θ=

Mirror

Θi Θr

Normal line

Incident light ray Reflected

light ray


Waves and Optics

• The quantity i is the index of refraction for the medium in which the light was incident. The quantity r 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 .

nn

v

vcn =

When light travels

from a less dense to more dense medium from a more dense to less dense medium • Light travels slower • Light travels faster • The frequency is unchanged • The frequency is unchanged • Thus wavelength is shorter • Thus wavelength is longer • So light bends toward the normal • 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.

2

1

sin Cnn

θ =

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 , the image distance , and the focal length of a lens or mirror is given by the fundamental lens/mirror equation.

od id f


Waves and Optics

iddf111

0

+=

Convex Lens & Concave Mirror (converging optical devices)

Object placed at:

Real or virtual

Upright or inverted

Enlarged, reduced, or same size

Real Inverted Reduced fd 20⟩

Real Inverted Same Size d 20 = f

Real Inverted Enlarged fdf 20⟨⟨

No image No image No image fd =0

Virtual Upright Enlarged fd ⟨0

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.


Waves and Optics

Ray Diagrams for Convex Lens

Object

2F´ F´

F 2F

Image

Object

2F´ F´

F 2F

Image

Object

2F´ F´

F 2F

Image

Object

2F´ F´

F 2F

Object

2F´ F´

F 2F

Image


Waves and Optics

Ray Diagrams for Concave Mirror

Object

Image

FC

Object

Image

FC

Object

Image FC

Object

F

C

Object

FC

Image


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.

•

sin , 0,1,2,3,... where m is the integer representing the order of a fringe

angle of spread of the light passing through a single or double slit

tan where distance from the center of th

m md

y yL

λθ

θ

θ

= =

=

= = e central bright line produced

on a screen by the interference of light and the center of another bright line ( ) distance from the double slit or diffraction grating and the screen

antinodeL =

Dark

Bright

Dark

Bright

y

m = 0

m = 1 m =2

θ

L

d


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

light reflects off a surface of higher index of refraction, a phase shift of 2λ occurs in the reflected

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

the light undergoes phase shift of 2λ but it also undergoes a phase shift of

2λ when reflected from

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.

•

where thickness of a thin film

12 net change , 0,1,2,3,..(destructive interference)2

2 net change =m from m=1,2,3...(constructive interference

vacuumfilm

film

film

tn

t phase m m

t phase

λλ

λ

λ

= =

⎛ ⎞+ = + =⎜ ⎟⎝ ⎠

+


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.

1 point for noting that the spacing between two successive resonance

frequencies is 2λ

(2 points max) Closed pipes have only the odd harmonics,

the first occurs at 4λ and the third occurs at

34λ so the spacing between two successive

resonance frequencies is 2λ

1 point for the correct answer including correct units

0.322

0.64 m

mλ

λ

=

=


Waves and Optics

B. Determine the frequency of the unknown tuning fork

1 point for finding the speed of the sound wave in air

(3 points max)

m m331 0.6 343s s

v T= + = 1 point for the correct frequency equation

m343s 536 Hz

0.64mvfλ

= = = 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)

( )( )( )( )

where n is any positive odd Integer4

For n = 11 343 /

0.16 m4 4 536

nvfL

m snvLf 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

( )( )( )( )

where n is any positive odd Integer4

For n = 33 343 /

0.48 m4 4 536

nvfL

m snvLf Hz

=

= = =

Or Alternate solution: Since the spacing between the successive resonances is 0.32 m 0.16 m + 0.32 m = 0.48 m

1 point for the correct answer for the third harmonic including correct units and reasonable number of significant digits


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).

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

(2 points max)

( )( )( )( )

where n is any positive Integer2

For n = 11 343 /

0.32 m2 2 536

nvfL

m snvLf Hz

=

= = =

1 point for the correct answer including correct units and reasonable number of significant digits


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


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

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

4cycles 40 Hz0.10s

f = =

The wavelength is 1.0 m. 1 point For the correct application of equation finding frequency ( )( ) m40 Hz 1.0 m 40

sv f λ= = =

1 point For an indication that the

wavelength is 1.0 m

4.0 m m 40 0.10 s s

m40 s

1.0 m

40 Hz

d vtdvt

v

v fvf

f

f

λ

λ

=

=

= =

=

=

=

=

1 point For an answer consistent with part B(i), including appropriate units 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 1 point For an indication that the wavelength is 1.0 m 1 point For an answer consistent with part B(ii) , including appropriate units


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.

5 N m10 0.05 kg/m s

TFvμ

= = = 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.

10 m/s 0.25 m40 Hz

vf

λ = = =

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.

1 point For an indication that the wavelength will increase and no incorrect statements The wavelength must increase with

increasing tension, since wavelength is proportional to wave speed.

1 point For a correct justification


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


Waves and Optics

Question 3 (10 pts)

Yellow

Violet

Glass30˚

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.

88

88

3.00 10 m/s 1.95 10 m/s1.54

3.00 10 m/s 1.85 10 m/s1.62

YY

VV

c xvn

c xv xn

= = =

= = =

x

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


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.

1 point For a correct equation for the wavelength of the light in the glass

77

77

5.80 10 m 3.76 10 m1.54

4.20 10 m 2.59 10 m1.62

airY

Y

airV

V

x xn

x xn

λλ

λλ

−−

−−

= = =

= = =

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.

8

147

814

7

3.00 10 m/s 5.17 10 Hz5.80 10 m

3.00 10 m/s 7.14 10 Hz4.20 10 m

Yair

Vair

c xfx

c xfx

λ

λ

−

−

= = =

= = =

x

x

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


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

Violet

Glass30˚

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


Waves and Optics

Question 4 (10 pts)

180 150 120 90 60 30 30 60 90 120 150 180cm

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.

1 point For the correct answer: left side of the mirror

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 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.


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

i of d d= +

1 point For recognition that the candle must be placed between f (30 cm) and 2f (60 cm).

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 point For the correct answer with units corresponding to the image distance

1 1 1

1 130cm 40cm

120cm

i o

i

i

f d d

dd

= +

= +

=

1


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.

180 150 120 90 60 30 30 60 90 120 150 180cm

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


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.

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.

1 point for stating the wave property of light, diffraction, or constructive and destructive interference

Color Wavelength (nm) red 664 orange 622 yellow 580 green 520


Waves and Optics

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


Waves and Optics

D. Determine the color of the light which was passed through the slits?

The angle θ can be found by

21 1 2.07 10 mθ tan tan 0.47

2.50 myL

−− − ⎛ ⎞×⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

°

1 point for finding the correct angle using the tangent or the small angle

approximation for Ly

Then the wavelength of the light is

( ) ( )4

7

1.40 10 m sin 0.47sinθ2

5.80 10 m

dm

λ

λ

−

−

×= =

= ×

°

1 point for using the correct equation to find the wavelength of the light

The color of the light is yellow.

1 point for the correct answer, yellow

E. Determine the frequency of the light which was passed through the slits?

8

14-7

3.0 10 / =5.2 10 Hz5.8 10

c m sfmλ

×= = ×

×

1 point for using the correct equation to find the frequency of the light 1 point for the correct answer, including units


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

L = 12 cm

2cm

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


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

B) 2A

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

The frequency remains the same. When a wave undergoes refraction, its speed changes along with its wavelength. C


https://webcms.rpclearning.com/ArtInfo.aspx?SourceArtID=62624

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) Nodes are formed at both ends of the pipe. B) Antinodes are formed at both ends of the pipe. C) An antinode is formed at the closed end of the pipe and a node is formed at the open end. D) An antinode is formed at the open end of the pipe and a node is formed at the closed end. E) 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 . The tension is then relaxed to half its initial tension. The speed of the wave is now

A) ov2

1

B) ov2 C) D) ov2 E) ov4

TFv ∝ , so halving the tension force FT gives a speed of 2ov

. A


Waves and Optics

6. Which phenomenon proves that light has wave properties? A) Images formed by lenses B) Interference patterns formed when light passes through a diffraction grating C) Light reflecting off a mirror D) The photoelectric effect E) 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) Reflection B) Refraction C) Interference D) Dispersion E) Diffraction

A wave diffracts when it must go around an obstacle or through an opening. E



Waves and Optics

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) A only B) B only C) C only D) A and B only E) B and C only

Constructive interference occurs whenever waves intersect in phase with one another. Rarefaction meets rarefaction or compression meets compression. D Also note that both points A and B are along the central line of symmetry where there is always constructive interference.


Waves and Optics

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) Medium 1 is more dense than medium 2. B) Medium 1 is less dense than medium 2 C) The speed of the light in medium 2 is greater than the speed of the light in medium 1. D) The frequency of the light is greater in medium 1. E) The wavelength is the same in medium 1 and medium 2.

The light bends toward the normal when it enters medium 2 which indicates it is more optically dense than medium 1. B

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) I only B) II only C) I and III only D) II and III only E) I and IV only

A convex lens and a concave mirror both produce magnified upright images when the object is placed inside the focal point. E


Waves and Optics

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) real, upright, and enlarged B) real, inverted, and enlarged C) real, inverted, and smaller D) virtual, upright, and enlarged E) virtual, upright, and smaller

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. C

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) is roughly zero since light behaves like particles (photons) B) is smaller for red light than green light C) is larger for a larger d D) is larger for a smaller d E) depends on λ only

D Hence, a smaller d will result in a larger angular separation .


Waves and Optics

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) 10° B) 30° C) 40° D) 60°

E) 90°

D


Waves and Optics

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) The thickness of the film must be greater than the wavelength. B) The wavelength must be equal to half the thickness of the film C) The wavelength must be equal to 4 times the thickness of the film. D) The wavelength must be a multiple of twice the thickness of the film. E) The thickness of the film must be less than the wavelength.

The equation that governs the bright reflection (constructive interference) is 2t = mλ. D

15. A plane mirror will produce a virtual image A) when the object distance is greater than the image distance. B) when the object distance is less than the image distance. C) when the object is on the principal axis of the mirror.

D) when the rays converge at the focal point of the mirror E) at all distances from the mirror.

A plane (flat) mirror will reflect light and produce an image at any distance from the object. E


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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.

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. C

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) 30 cm B) 15 cm C) 6 cm D) 6 cm

E) 15 cm

B


Waves and Optics

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) 0.40 B) 0.67 C) 1.0 D) 1.5 E) 2.4

The angle of incidence = 30º and the angle of refraction = 12º. The index of refraction can be found by Snell’s law:

( )1 1 2 2

2

2

sin sin1 sin 30 sin12

2.4

n nn

n

θ θ=

=

=

E

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) decreases, then increases B) decreases, then decreases again C) increases, then decreases D) increases, then increases again E) the velocity of the ray of light remains the same

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.

A

20. If light is passed through a narrow, single-slit opening onto a screen, the pattern of light produced on the screen is A) alternating bright and dark lines of equal width B) a bright central band of light with much smaller, dimmer bands toward the edges C) concentric circles of light D) one circle of light E) 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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21. When white light shines on a glass prism it disperses into a rainbow of colors as shown above. This occurs because A) Each frequency of light has a different index of refraction B) There is constructive and destructive interference between the different colors C) White light is made up of all the colors of the rainbow D) Violet light travels faster through the prism than red light does E) Impurities in the glass absorb the white light and produce the colors

Dispersion is caused by slightly different indices of refraction for each frequency of light. A



Waves and Optics

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) I only B) II only C) I and II only D) I and III only E) II and III only

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. D



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