##### Document Text Contents

Page 1

Instructor’s Manual for

FUNDAMENTALS OF PHYSICS

Seventh Edition

by David Halliday, Robert Resnick, and Jearl Walker

Prepared by

J. Richard Christman

Professor Emeritus

United States Coast Guard Academy

with the assistance of

Stanley A. Williams

Iowa State University

Walter Eppenstein

Rensselaer Polytechnic Institute

Page 137

use the data to calculate the wavelength. A microcomputer can be used to take data and

plot the intensity of a visible light pattern.

c. Meiners Experiment 13—6: The Michelson Interferometer. An interferometer is used to

measure the wavelengths of light frommercury and a laser and to find the index of refraction

of a glass pane and air. Good practical applications.

Chapter 36 DIFFRACTION

BASIC TOPICS

I. Qualitative discussion of single-slit diffraction.

A. Shine coherent monochromatic light on a single slit and project the pattern on the wall.

Point out the broad central bright region and the narrower, less bright regions on either

side, with dark regions between. Also point out that light is diffracted into the geometric

shadow.

B. Remark that diffraction can be discussed in terms of Huygens wavelets emanating from

points in the slit. Explain that they not only spread into the shadow region but that

they arrive at any selected point with a distribution of phases and interfere to produce the

pattern. Explain that for quantitative work, this chapter deals with Fraunhofer diffraction,

with the screen far from the slit.

C. Draw a single slit with a plane wave incident normal to it. Also draw parallel rays from

equally spaced points within the slit, all making the same angle θ with the forward direction.

Point out that all wavelets are in phase at the slit. The first minimum can be located by

selecting θ so that, at the observation point, the ray from the top of the slit is 180◦ out

of phase with the ray from the middle of the slit. All wavelets then cancel in pairs. Show

that this leads to a sin θ = λ, where a is the slit width. Point out that this value of θ

determines the width of the central bright region and that this region gets wider as the

slit width narrows. Use sin θ ≈ tan θ ≈ θ (in radians) to show that the linear width of the

central region on a screen a distance D away is 2Dλ/a. Use a variable width slit or a series

of slits to demonstrate the effect.

D. By dividing the slit into fourths, eighths, etc. and showing that in each case the wavelets

cancel in pairs if θ is properly selected, find the locations of other minima. Show that

a sin θ = mλ for a minimum.

E. Explain that for a < λ, the central maximum covers the whole forward direction. No point

of zero intensity can be observed. Also remark that the intensity becomes more uniform as

a decreases from λ. This was the assumption made in the last chapter when the interference

of only one wavelet from each slit was considered.

F. Qualitatively discuss the intensity. Draw a phasor diagram showing ten or so phasors

representing wavelets from equally spaced points in the slit. Show that each wavelet at

the observation point is out of phase with its neighbor by the same amount. First, show

the phasors with zero phase difference (θ = 0), then show them for a larger value of θ.

Show that they approximate a circle at the first minimum and then, as θ increases, they

wrap around to form another maximum, with less intensity than the central maximum.

Point out that as θ increases, the pattern has successive maxima and minima and that the

maxima become successively less intense.

II. The intensity.

A. Draw a diagram showing ten or so phasors along the arc of a circle and let φ be the phase

difference between the first and last. See Fig. 36—8. Explain that you will take the limit

as the number of wavelets increases without bound and draw the phasor addition diagram

Lecture Notes: Chapter 36 131

Page 138

as an arc. Use geometry to show that Eθ = Em(sinα)/α, where α = φ/2. Point out that

the intensity can be written Iθ = Im(sin

2 α)/α2, where Im is the intensity for θ = 0. By

examining the path difference for the rays from the top and bottom of the slit, show that

α = (πa/λ) sin θ. Explain that these expressions give the intensity as a function of the

angle θ.

B. Sketch the intensity as a function of θ (see Fig. 36—7) and show mathematically that the

expression just derived predicts the positions of the minima as found earlier.

C. (Optional) Set the derivative of (sinα)/α equal to 0 and show that tanα = α at an intensity

maximum. State that the first two solutions are α = 4.493 rad and 7.725 rad. Use these

results to show that the intensity at the first two secondary maxima are 4.72 × 10−2 and

1.65 × 10−2, relative to the intensity for θ = 0. You might also want to pick a wavelength

and slit width, then find the angular positions of the first two secondary maxima. Remark

that they are close to but not precisely at midpoints between zeros of intensity.

III. Double-slit diffraction.

A. Consider the double-slit arrangement discussed in the previous chapter. Point out that

the electric field for the light from each of the slits obeys the equation developed for

single-slit diffraction and these two fields are superposed. They have the same amplitude,

Em(sinα)/α, and differ in phase by (2πd/λ) sin θ, where d is the center-to-center slit separa-

tion. The result for the intensity is Iθ = Im(cos

2 β)(sin2 α)/α2, the product of the single-slit

diffraction equation and the double-slit interference equation. Here β = (πd/λ) sin θ.

B. Sketch Iθ versus θ for a double slit and point out that the single-slit pattern forms an

envelope for the double-slit interference pattern. Remark that this is so because d must be

greater than a. See Fig. 36—14.

C. Show how to calculate the number of interference fringes within the central diffraction

maximum and remark that the result depends on the ratio d/a but not on the wavelength.

D. Discuss missing maxima. Point out that the first diffraction minimum on either side of

the central single-slit diffraction maximum might coincide with a double-slit interference

maximum, in which case the maximum would not be seen. Show that the maximum of

order m is missing if d/a = m.

IV. Diffraction gratings.

A. Make or purchase a set of multiple-slit barriers with 3, 4, and 5 slits, all with the same slit

width and spacing. Multiple slits can be made using razor blades and a lamp blackened

microscope slide. Use a laser to show the patterns in order of increasing number of slits.

Finish with a commercial grating.

B. Qualitatively describe the pattern produced as the number of slits is increased. Point out

the principle maxima and, if possible, the secondary maxima. Remark that the principle

maxima narrow and that the number of secondary maxima increases as the number of slits

increases. Remark that for gratings with a large number of rulings, the principal maxima

are called lines. For each barrier, sketch a graph of the intensity as a function of angle.

Explain that the single-slit diffraction pattern forms an envelope for the pattern.

C. Remark that you will assume the slits are so narrow that the patterns you will consider

lie well within the central maximum of the single-slit diffraction pattern and you need to

consider only one wave from each slit. Explain that lines occur whenever the path difference

for rays from two adjacent slits is an integer multiple of the wavelength: d sin θ = mλ.

Remark that m is called the order of the line. Also remark that the angular positions of

the lines depend only on the ratio d/λ and not on the number of slits or their width.

D. Consider N phasors of equal magnitude that form a regular polygon and remark this is the

configuration for an interference minimum adjacent to a principal maximum. Show that for

one of these minima the phase difference for waves from adjacent slits is 2π(m+1/N) and

132 Lecture Notes: Chapter 36

Page 274

Chapter 27

Solution Manual: 5, 9, 13, 19, 21, 31, 33, 35, 45, 53, 55, 57, 61, 67, 73, 77, 81, 85, 91, 93, 101

Study Guide: 3, 7, 11, 17, 23, 25, 29, 37, 41, 43, 47, 51, 59, 69, 71, 75, 83, 99, 103, 111

Chapter 28

Solution Manual: 1, 9, 17, 23, 29, 33, 37, 39, 41, 45, 47, 51, 55, 59, 61, 63, 67, 71, 75, 79, 81, 85,

89, 89

Study Guide: 5, 7, 11, 13, 19, 21, 25, 27, 31, 35, 43, 53, 57, 65, 69, 77, 83, 87

Chapter 29

Solution Manual: 3, 5, 7, 13, 15, 17, 19, 23, 27, 35, 43, 45, 47, 49, 53, 59, 61, 75, 77, 83, 85, 89, 91,

93

Study Guide: 1, 9, 11, 21, 25, 29, 31, 33, 37, 39, 41, 51, 57, 63, 67, 73, 79, 81, 87

Chapter 30

Solution Manual: 5, 9, 19, 25, 27, 33, 35, 41, 49, 53, 57, 61, 63, 67, 69, 71, 77, 85, 87, 89, 99

Study Guide: 3, 11, 13, 15, 23, 31, 37, 45, 47, 51, 59, 65, 73, 75, 79, 83, 93, 95, 97

Chapter 31

Solution Manual: 7, 15, 19, 27, 33, 39, 45, 47, 53, 55, 57, 61, 65, 71, 75, 77, 89

Study Guide: 1, 11, 13, 17, 25, 29, 31, 35, 37, 41, 43, 49, 59, 63, 73, 77, 83, 91, 97

Chapter 32

Solution Manual: 3, 5, 13, 19, 21, 27, 29, 31, 35, 37, 43, 49, 53, 57, 63, 69, 71

Study Guide: 1, 9, 11, 17, 23, 25, 33, 39, 41, 45, 47, 55, 61, 65, 75

Chapter 33

Solution Manual: 5, 19, 23, 25, 27, 33, 43, 49, 53, 61, 65, 75, 85, 89, 93, 95, 105, 109

Study Guide: 3, 13, 17, 21, 29, 35, 39, 47, 51, 57, 63, 67, 71, 73, 79, 83, 87, 91, 101, 107

Chapter 34

Solution Manual: 5, 9, 11, 13, 25, 31, 35, 39, 43, 49, 53, 57, 65, 73, 77, 83, 89, 91, 95, 103, 105, 111

Study Guide: 3, 15, 17, 21, 27, 39, 47, 55, 61, 71, 75, 85, 93, 97, 101, 109, 115, 121, 125, 131,

135

Chapter 35

Solution Manual: 7, 15, 17, 19, 21, 27, 29, 39, 43, 45, 49, 55, 61, 65, 69, 71, 75, 79, 81, 91, 95, 107,

111, 115, 121, 123

Study Guide: 5, 11, 23, 25, 33, 37, 41, 51, 59, 67, 73, 77, 83, 87, 97, 99, 105, 109, 113, 121

268 Problems in the Student Solution Manual, in the Student’s Companion, and on the Wiley Website

Page 275

Chapter 36

Solution Manual: 3, 5, 7, 13, 15, 17, 21, 25, 33, 39, 41, 43, 47, 51, 63, 69, 73, 79, 83, 93, 103, 105,

107, 109

Study Guide: 1, 9, 11, 19, 27, 29, 35, 37, 49, 55, 57, 59, 65, 77, 81, 89, 95

Chapter 37

Solution Manual: 1, 9, 11, 13, 17, 19, 21, 27, 29, 31, 35, 39, 41, 49, 51, 53, 55, 63, 73, 75, 81, 83,

87, 93, 99

Study Guide: 3, 5, 15, 23, 33, 37, 43, 45, 47, 59, 65, 71, 77, 89, 91

Chapter 38

Solution Manual: 9, 13, 19, 25, 29, 33, 39, 41, 45, 49, 51, 55, 57, 61, 65, 73, 83

Study Guide: 7, 15, 17, 21, 23, 31, 37, 43, 51, 59, 63, 69, 75, 79

Chapter 39

Solution Manual: 3, 9, 13, 17, 21, 25, 29, 33, 37, 41, 43, 45, 47, 53, 57

Study Guide: 5, 11, 15, 19, 23, 27, 31, 35, 39, 51, 55

Chapter 40

Solution Manual: 9, 11, 15, 17, 23, 27, 33, 35, 37, 47, 59, 63

Study Guide: 3, 7, 13, 19, 21, 25, 31, 39, 43, 45, 53, 57, 67

Chapter 41

Solution Manual: 5, 9, 15, 19, 21, 27, 31, 35, 37, 45

Study Guide: 1, 11, 13, 17, 23, 25, 29, 33, 41, 47

Chapter 42

Solution Manual: 11, 15, 19, 25, 31, 37, 41, 45, 51, 57, 65, 69, 75, 77

Study Guide: 3, 13, 23, 33, 39, 43, 49, 53, 61, 67, 79, 85, 87

Chapter 43

Solution Manual: 3, 9, 11, 15, 19, 21, 23, 27, 29, 39, 45

Study Guide: 5, 7, 13, 17, 23, 31, 35, 37, 43, 47, 51, 53

Chapter 44

Solution Manual: 7, 11, 15, 23, 33, 37, 39, 43

Study Guide: 1, 5, 13, 17, 27, 31, 35, 41

Problems in the Student Solution Manual, in the Student’s Companion, and on the Wiley Website 269

Instructor’s Manual for

FUNDAMENTALS OF PHYSICS

Seventh Edition

by David Halliday, Robert Resnick, and Jearl Walker

Prepared by

J. Richard Christman

Professor Emeritus

United States Coast Guard Academy

with the assistance of

Stanley A. Williams

Iowa State University

Walter Eppenstein

Rensselaer Polytechnic Institute

Page 137

use the data to calculate the wavelength. A microcomputer can be used to take data and

plot the intensity of a visible light pattern.

c. Meiners Experiment 13—6: The Michelson Interferometer. An interferometer is used to

measure the wavelengths of light frommercury and a laser and to find the index of refraction

of a glass pane and air. Good practical applications.

Chapter 36 DIFFRACTION

BASIC TOPICS

I. Qualitative discussion of single-slit diffraction.

A. Shine coherent monochromatic light on a single slit and project the pattern on the wall.

Point out the broad central bright region and the narrower, less bright regions on either

side, with dark regions between. Also point out that light is diffracted into the geometric

shadow.

B. Remark that diffraction can be discussed in terms of Huygens wavelets emanating from

points in the slit. Explain that they not only spread into the shadow region but that

they arrive at any selected point with a distribution of phases and interfere to produce the

pattern. Explain that for quantitative work, this chapter deals with Fraunhofer diffraction,

with the screen far from the slit.

C. Draw a single slit with a plane wave incident normal to it. Also draw parallel rays from

equally spaced points within the slit, all making the same angle θ with the forward direction.

Point out that all wavelets are in phase at the slit. The first minimum can be located by

selecting θ so that, at the observation point, the ray from the top of the slit is 180◦ out

of phase with the ray from the middle of the slit. All wavelets then cancel in pairs. Show

that this leads to a sin θ = λ, where a is the slit width. Point out that this value of θ

determines the width of the central bright region and that this region gets wider as the

slit width narrows. Use sin θ ≈ tan θ ≈ θ (in radians) to show that the linear width of the

central region on a screen a distance D away is 2Dλ/a. Use a variable width slit or a series

of slits to demonstrate the effect.

D. By dividing the slit into fourths, eighths, etc. and showing that in each case the wavelets

cancel in pairs if θ is properly selected, find the locations of other minima. Show that

a sin θ = mλ for a minimum.

E. Explain that for a < λ, the central maximum covers the whole forward direction. No point

of zero intensity can be observed. Also remark that the intensity becomes more uniform as

a decreases from λ. This was the assumption made in the last chapter when the interference

of only one wavelet from each slit was considered.

F. Qualitatively discuss the intensity. Draw a phasor diagram showing ten or so phasors

representing wavelets from equally spaced points in the slit. Show that each wavelet at

the observation point is out of phase with its neighbor by the same amount. First, show

the phasors with zero phase difference (θ = 0), then show them for a larger value of θ.

Show that they approximate a circle at the first minimum and then, as θ increases, they

wrap around to form another maximum, with less intensity than the central maximum.

Point out that as θ increases, the pattern has successive maxima and minima and that the

maxima become successively less intense.

II. The intensity.

A. Draw a diagram showing ten or so phasors along the arc of a circle and let φ be the phase

difference between the first and last. See Fig. 36—8. Explain that you will take the limit

as the number of wavelets increases without bound and draw the phasor addition diagram

Lecture Notes: Chapter 36 131

Page 138

as an arc. Use geometry to show that Eθ = Em(sinα)/α, where α = φ/2. Point out that

the intensity can be written Iθ = Im(sin

2 α)/α2, where Im is the intensity for θ = 0. By

examining the path difference for the rays from the top and bottom of the slit, show that

α = (πa/λ) sin θ. Explain that these expressions give the intensity as a function of the

angle θ.

B. Sketch the intensity as a function of θ (see Fig. 36—7) and show mathematically that the

expression just derived predicts the positions of the minima as found earlier.

C. (Optional) Set the derivative of (sinα)/α equal to 0 and show that tanα = α at an intensity

maximum. State that the first two solutions are α = 4.493 rad and 7.725 rad. Use these

results to show that the intensity at the first two secondary maxima are 4.72 × 10−2 and

1.65 × 10−2, relative to the intensity for θ = 0. You might also want to pick a wavelength

and slit width, then find the angular positions of the first two secondary maxima. Remark

that they are close to but not precisely at midpoints between zeros of intensity.

III. Double-slit diffraction.

A. Consider the double-slit arrangement discussed in the previous chapter. Point out that

the electric field for the light from each of the slits obeys the equation developed for

single-slit diffraction and these two fields are superposed. They have the same amplitude,

Em(sinα)/α, and differ in phase by (2πd/λ) sin θ, where d is the center-to-center slit separa-

tion. The result for the intensity is Iθ = Im(cos

2 β)(sin2 α)/α2, the product of the single-slit

diffraction equation and the double-slit interference equation. Here β = (πd/λ) sin θ.

B. Sketch Iθ versus θ for a double slit and point out that the single-slit pattern forms an

envelope for the double-slit interference pattern. Remark that this is so because d must be

greater than a. See Fig. 36—14.

C. Show how to calculate the number of interference fringes within the central diffraction

maximum and remark that the result depends on the ratio d/a but not on the wavelength.

D. Discuss missing maxima. Point out that the first diffraction minimum on either side of

the central single-slit diffraction maximum might coincide with a double-slit interference

maximum, in which case the maximum would not be seen. Show that the maximum of

order m is missing if d/a = m.

IV. Diffraction gratings.

A. Make or purchase a set of multiple-slit barriers with 3, 4, and 5 slits, all with the same slit

width and spacing. Multiple slits can be made using razor blades and a lamp blackened

microscope slide. Use a laser to show the patterns in order of increasing number of slits.

Finish with a commercial grating.

B. Qualitatively describe the pattern produced as the number of slits is increased. Point out

the principle maxima and, if possible, the secondary maxima. Remark that the principle

maxima narrow and that the number of secondary maxima increases as the number of slits

increases. Remark that for gratings with a large number of rulings, the principal maxima

are called lines. For each barrier, sketch a graph of the intensity as a function of angle.

Explain that the single-slit diffraction pattern forms an envelope for the pattern.

C. Remark that you will assume the slits are so narrow that the patterns you will consider

lie well within the central maximum of the single-slit diffraction pattern and you need to

consider only one wave from each slit. Explain that lines occur whenever the path difference

for rays from two adjacent slits is an integer multiple of the wavelength: d sin θ = mλ.

Remark that m is called the order of the line. Also remark that the angular positions of

the lines depend only on the ratio d/λ and not on the number of slits or their width.

D. Consider N phasors of equal magnitude that form a regular polygon and remark this is the

configuration for an interference minimum adjacent to a principal maximum. Show that for

one of these minima the phase difference for waves from adjacent slits is 2π(m+1/N) and

132 Lecture Notes: Chapter 36

Page 274

Chapter 27

Solution Manual: 5, 9, 13, 19, 21, 31, 33, 35, 45, 53, 55, 57, 61, 67, 73, 77, 81, 85, 91, 93, 101

Study Guide: 3, 7, 11, 17, 23, 25, 29, 37, 41, 43, 47, 51, 59, 69, 71, 75, 83, 99, 103, 111

Chapter 28

Solution Manual: 1, 9, 17, 23, 29, 33, 37, 39, 41, 45, 47, 51, 55, 59, 61, 63, 67, 71, 75, 79, 81, 85,

89, 89

Study Guide: 5, 7, 11, 13, 19, 21, 25, 27, 31, 35, 43, 53, 57, 65, 69, 77, 83, 87

Chapter 29

Solution Manual: 3, 5, 7, 13, 15, 17, 19, 23, 27, 35, 43, 45, 47, 49, 53, 59, 61, 75, 77, 83, 85, 89, 91,

93

Study Guide: 1, 9, 11, 21, 25, 29, 31, 33, 37, 39, 41, 51, 57, 63, 67, 73, 79, 81, 87

Chapter 30

Solution Manual: 5, 9, 19, 25, 27, 33, 35, 41, 49, 53, 57, 61, 63, 67, 69, 71, 77, 85, 87, 89, 99

Study Guide: 3, 11, 13, 15, 23, 31, 37, 45, 47, 51, 59, 65, 73, 75, 79, 83, 93, 95, 97

Chapter 31

Solution Manual: 7, 15, 19, 27, 33, 39, 45, 47, 53, 55, 57, 61, 65, 71, 75, 77, 89

Study Guide: 1, 11, 13, 17, 25, 29, 31, 35, 37, 41, 43, 49, 59, 63, 73, 77, 83, 91, 97

Chapter 32

Solution Manual: 3, 5, 13, 19, 21, 27, 29, 31, 35, 37, 43, 49, 53, 57, 63, 69, 71

Study Guide: 1, 9, 11, 17, 23, 25, 33, 39, 41, 45, 47, 55, 61, 65, 75

Chapter 33

Solution Manual: 5, 19, 23, 25, 27, 33, 43, 49, 53, 61, 65, 75, 85, 89, 93, 95, 105, 109

Study Guide: 3, 13, 17, 21, 29, 35, 39, 47, 51, 57, 63, 67, 71, 73, 79, 83, 87, 91, 101, 107

Chapter 34

Solution Manual: 5, 9, 11, 13, 25, 31, 35, 39, 43, 49, 53, 57, 65, 73, 77, 83, 89, 91, 95, 103, 105, 111

Study Guide: 3, 15, 17, 21, 27, 39, 47, 55, 61, 71, 75, 85, 93, 97, 101, 109, 115, 121, 125, 131,

135

Chapter 35

Solution Manual: 7, 15, 17, 19, 21, 27, 29, 39, 43, 45, 49, 55, 61, 65, 69, 71, 75, 79, 81, 91, 95, 107,

111, 115, 121, 123

Study Guide: 5, 11, 23, 25, 33, 37, 41, 51, 59, 67, 73, 77, 83, 87, 97, 99, 105, 109, 113, 121

268 Problems in the Student Solution Manual, in the Student’s Companion, and on the Wiley Website

Page 275

Chapter 36

Solution Manual: 3, 5, 7, 13, 15, 17, 21, 25, 33, 39, 41, 43, 47, 51, 63, 69, 73, 79, 83, 93, 103, 105,

107, 109

Study Guide: 1, 9, 11, 19, 27, 29, 35, 37, 49, 55, 57, 59, 65, 77, 81, 89, 95

Chapter 37

Solution Manual: 1, 9, 11, 13, 17, 19, 21, 27, 29, 31, 35, 39, 41, 49, 51, 53, 55, 63, 73, 75, 81, 83,

87, 93, 99

Study Guide: 3, 5, 15, 23, 33, 37, 43, 45, 47, 59, 65, 71, 77, 89, 91

Chapter 38

Solution Manual: 9, 13, 19, 25, 29, 33, 39, 41, 45, 49, 51, 55, 57, 61, 65, 73, 83

Study Guide: 7, 15, 17, 21, 23, 31, 37, 43, 51, 59, 63, 69, 75, 79

Chapter 39

Solution Manual: 3, 9, 13, 17, 21, 25, 29, 33, 37, 41, 43, 45, 47, 53, 57

Study Guide: 5, 11, 15, 19, 23, 27, 31, 35, 39, 51, 55

Chapter 40

Solution Manual: 9, 11, 15, 17, 23, 27, 33, 35, 37, 47, 59, 63

Study Guide: 3, 7, 13, 19, 21, 25, 31, 39, 43, 45, 53, 57, 67

Chapter 41

Solution Manual: 5, 9, 15, 19, 21, 27, 31, 35, 37, 45

Study Guide: 1, 11, 13, 17, 23, 25, 29, 33, 41, 47

Chapter 42

Solution Manual: 11, 15, 19, 25, 31, 37, 41, 45, 51, 57, 65, 69, 75, 77

Study Guide: 3, 13, 23, 33, 39, 43, 49, 53, 61, 67, 79, 85, 87

Chapter 43

Solution Manual: 3, 9, 11, 15, 19, 21, 23, 27, 29, 39, 45

Study Guide: 5, 7, 13, 17, 23, 31, 35, 37, 43, 47, 51, 53

Chapter 44

Solution Manual: 7, 11, 15, 23, 33, 37, 39, 43

Study Guide: 1, 5, 13, 17, 27, 31, 35, 41

Problems in the Student Solution Manual, in the Student’s Companion, and on the Wiley Website 269