Content check — College Physics, Chapter 27: Wave Optics

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Chapter 27: Wave Optics

51 problems · 5 figures

Show that when light passes from air to water, its wavelength decreases to 0.750 times its original value.

Answer

1 / 1.333 =

0.750

27-3 §27.1 source ↗

What is the index of refraction of a material for which the wavelength of light is 0.671 times its value in a vacuum? Identify the likely substance.

Answer

1.49, Polystyrene

27-5 §27.1 source ↗

What is the ratio of thicknesses of crown glass and water that would contain the same number of wavelengths of light?

Answer

0.877 glass to water

27-6 §27.3 source ↗

At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?

Answer

0 . 516º

27-8 §27.3 source ↗

What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of

30 . 0º

Answer

1 . 22 \times 10^{- 6} m

27-10 §27.3 source ↗

Calculate the wavelength of light that has its third minimum at an angle of

30 . 0º

when falling on double slits separated by

3.00 μm

. Explicitly, show how you follow the steps in Problem-Solving Strategies for Wave Optics.

Answer

600 nm

27-12 §27.3 source ↗

At what angle is the fourth-order maximum for the situation in Exercise 27.6?

Answer

2 . 06º

27-14 §27.3 source ↗

Find the largest wavelength of light falling on double slits separated by

1.20 μm

for which there is a first-order maximum. Is this in the visible part of the spectrum?

Answer

1200 nm (not visible)

27-16 §27.3 source ↗

(a) What is the smallest separation between two slits that will produce a second-order maximum for any visible light? (b) For all visible light?

Answer

(a) 760 nm

(b) 1520 nm

27-18 §27.3 source ↗

Figure 27.56 shows a double slit located a distance

x

from a screen, with the distance from the center of the screen given by

y

. When the distance

d

between the slits is relatively large, there will be numerous bright spots, called fringes. Show that, for small angles (where

sin θ \approx θ

, with

θ

in radians), the distance between fringes is given by

Δ y = xλ / d

The figure shows a schematic of a double slit experiment. A double slit is at the left and a screen is at the right. The slits are separated by a distance d. From the midpoint between the slits, a horizontal line labeled x extends to the screen. From the same point, a line angled upward at an angle theta above the horizontal also extends to the screen. The distance between where the horizontal line hits the screen and where the angled line hits the screen is marked y, and the distance between adjacent fringes is given by delta y, which equals x times lambda over d. — Figure 27.56 in the book ↗

Answer

For small angles $sin θ -$ $tan θ \approx$ $θ (in radians)$ .

For two adjacent fringes we have,

d sin θ_{m} = mλ

and

d sin θ_{m + 1} = m + 1 λ

Subtracting these equations gives

\begin{matrix} d sin θ_{m + 1} - sin θ_{m} = m + 1 - m λ \\ d θ_{m + 1} - θ_{m} = λ \\ tan θ_{m} = \frac{y_{m}}{x} \approx θ_{m} \Rightarrow d \frac{y_{m + 1}}{x} - \frac{y_{m}}{x} = λ \\ d \frac{Δ y}{x} = λ \Rightarrow Δ y = \frac{xλ}{d} \end{matrix}

27-20 §27.3 source ↗

Using the result of the problem two problems prior, find the wavelength of light that produces fringes 7.50 mm apart on a screen 2.00 m from double slits separated by 0.120 mm (see Figure 27.56).

Answer

450 nm

27-21 §27.4 source ↗

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?

Answer

5 . 97º

27-23 §27.4 source ↗

How many lines per centimeter are there on a diffraction grating that gives a first-order maximum for 470-nm blue light at an angle of

25 . 0º

Answer

8 . 99 \times 10^{3}

27-25 §27.4 source ↗

Calculate the wavelength of light that has its second-order maximum at

45 . 0º

when falling on a diffraction grating that has 5000 lines per centimeter.

Answer

707 nm

27-27 §27.4 source ↗

(a) What do the four angles in the above problem become if a 5000-line-per-centimeter diffraction grating is used? (b) Using this grating, what would the angles be for the second-order maxima? (c) Discuss the relationship between integral reductions in lines per centimeter and the new angles of various order maxima.

Answer

(a) $\begin{matrix} 11.8º, 12.5º, 14.1º, 19.2º \end{matrix}$

(b) $\begin{matrix} 24.2º, 25.7º, 29.1º, 41.0º \end{matrix}$

(c) Decreasing the number of lines per centimeter by a factor of x means that the angle for the x‐order maximum is the same as the original angle for the first-order maximum.

27-29 §27.4 source ↗

The yellow light from a sodium vapor lamp seems to be of pure wavelength, but it produces two first-order maxima at

36.093 º

and

36.129 º

when projected on a 10,000 line per centimeter diffraction grating. What are the two wavelengths to an accuracy of 0.1 nm?

Answer

589.1 nm and 589.6 nm

27-31 §27.4 source ↗

Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600-nm light?

Answer

28.7º

27-33 §27.4 source ↗

At what angle does a diffraction grating produces a second-order maximum for light having a first-order maximum at

20 . 0º

Answer

43 . 2º

27-35 §27.4 source ↗

If a diffraction grating produces a first-order maximum for the shortest wavelength of visible light at

30 . 0º

, at what angle will the first-order maximum be for the longest wavelength of visible light?

Answer

90 . 0º

27-37 §27.4 source ↗

(a) Show that a 30,000-line-per-centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a first-order maximum? (c) What is the greatest number of lines per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?

Answer

(a) The longest wavelength is 333.3 nm, which is not visible.

(b) 333 nm (UV)

27-39 §27.4 source ↗

The analysis shown in the figure below also applies to diffraction gratings with lines separated by a distance

d

. What is the distance between fringes produced by a diffraction grating having 125 lines per centimeter for 600-nm light, if the screen is 1.50 m away?

Answer

1 . 13 \times 10^{- 2} m

27-41 §27.4 source ↗

Unreasonable Results

(a) What visible wavelength has its fourth-order maximum at an angle of $25.0º$ when projected on a 25,000-line-per-centimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Answer

(a) 42.3 nm

(b) Not a visible wavelength

The number of slits in this diffraction grating is too large. Etching in integrated circuits can be done to a resolution of 50 nm, so slit separations of 400 nm are at the limit of what we can do today. This line spacing is too small to produce diffraction of light.

27-43 §27.5 source ↗

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width

1.00 μm

? (b) Will there be a second minimum?

Answer

(a) $33 . 4º$

(b) No

27-45 §27.5 source ↗

(a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of

28 . 0º

? (b) At what angle will the second minimum be?

Answer

(a) $1 . 35 \times 10^{- 6} m$

(b) $69 . 9º$

27-47 §27.5 source ↗

Find the wavelength of light that has its third minimum at an angle of

48 . 6º

when it falls on a single slit of width

3.00 μm

Answer

750 nm

27-49 §27.5 source ↗

(a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width

7.50 μm

. At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

Answer

(a) $9 . 04º$

(b) 12

27-51 §27.5 source ↗

(a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width

2.00 μm

. (b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit? (c) Discuss the ease or difficulty of measuring such a distance.

Answer

(a) $0 . 0150º$

(b) 0.262 mm

(c) This distance is not easily measured by human eye, but under a microscope or magnifying glass it is quite easily measurable.

27-53 §27.5 source ↗

(a) If a single slit produces a first minimum at

14 . 5º

, at what angle is the second-order minimum? (b) What is the angle of the third-order minimum? (c) Is there a fourth-order minimum? (d) Use your answers to illustrate how the angular width of the central maximum is about twice the angular width of the next maximum (which is the angle between the first and second minima).

Answer

(a) $30 . 1º$

(b) $48 . 7º$

(d) ${2θ}_{1} =$ $(2) (14.5º) =$ $29º, θ_{2} -$ $θ_{1} =$ $30 . 05º -$ $14 . 5º = 15 . 56º$ . Thus, $29º \approx$ $(2) (15 . 56º) =$ $31 . 1º$ .

27-55 §27.5 source ↗

Integrated Concepts

A water break at the entrance to a harbor consists of a rock barrier with a 50.0-m-wide opening. Ocean waves of 20.0-m wavelength approach the opening straight on. At what angle to the incident direction are the boats inside the harbor most protected against wave action?

Answer

23 . 6º

and

53 . 1º

27-57 §27.6 source ↗

The 300-m-diameter Arecibo radio telescope pictured in Figure 27.28 detects radio waves with a 4.00 cm average wavelength.

(a) What is the angle between two just-resolvable point sources for this telescope?

(b) How close together could these point sources be at the 2 million light year distance of the Andromeda galaxy?

The figure shows a photograph from above looking into the Arecibo Telescope in Puerto Rico. It is a huge bowl-shaped structure lined with reflecting material. The diameter of the bowl is three times as long as a football field. Trees can be seen around the bowl, but they do not shade the bowl significantly. — Figure 27.28 in the book ↗

Answer

(a) $1 . 63 \times 10^{- 4} rad$

(b) 326 ly

27-59 §27.6 source ↗

Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally 5.00 cm in diameter with an average wavelength of 600 nm.

Answer

1 . 46 \times 10^{- 5} rad

27-61 §27.6 source ↗

A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the Moon.

(a) If this is done with the Mount Wilson telescope, producing a 2.54-m-diameter beam of 633-nm light, what is the minimum angular spread of the beam (the angle between the beam's maximum and its first zero)?

(b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the Moon, assuming a lunar distance of $3.84 \times 10^{8} m$ ?

Answer

(a) $3 . 04 \times 10^{- 7} rad$

(b) Diameter of $235 m$

27-63 §27.6 source ↗

What is the minimum diameter mirror on a telescope that would allow you to see details as small as 5.00 km on the Moon some 384,000 km away? Assume an average wavelength of 550 nm for the light received.

Answer

5.15 cm

27-65 §27.6 source ↗

(a) The planet Pluto and its Moon Charon are separated by 19,600 km. Neglecting atmospheric effects, should the 5.08-m-diameter Mount Palomar telescope be able to resolve these bodies when they are $4 . 50 \times 10^{9} km$ from Earth? Assume an average wavelength of 550 nm.

(b) In actuality, it is just barely possible to discern that Pluto and Charon are separate bodies using an Earth-based telescope. What are the reasons for this?

Answer

(a) Yes. Should easily be able to discern.

(b) The fact that it is just barely possible to discern that these are separate bodies indicates the severity of atmospheric aberrations.

27-70 §27.7 source ↗

A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?

Answer

532 nm (green)

27-72 §27.7 source ↗

Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.

Answer

83.9 nm

27-74 §27.7 source ↗

A film of soapy water (

n = 1 . 33

) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?

Answer

620 nm (orange)

27-76 §27.7 source ↗

Suppose you have a lens system that is to be used primarily for 700-nm red light. What is the second thinnest coating of magnesium fluoride that would be non-reflective for this wavelength? Refer to Table 25.1 for the index of refraction.

Table 25.1 Index of Refraction in Various Media in the book ↗

Medium	n
*Gases at $0ºC$ , 1 atm*
Air	1.000293
Carbon dioxide	1.00045
Hydrogen	1.000139
Oxygen	1.000271
Liquids at $20ºC$
Benzene	1.501
Carbon disulfide	1.628
Carbon tetrachloride	1.461
Ethanol	1.361
Glycerine	1.473
Water, fresh	1.333
Solids at $0ºC$
Diamond	2.419
Fluorite	1.434
Glass, crown	1.52
Glass, flint	1.66
Ice at $0ºC$	1.309
Magnesium Fluoride	1.38
Polystyrene	1.49
Plexiglas	1.51
Quartz, crystalline	1.544
Quartz, fused	1.458
Sodium chloride	1.544
Zircon	1.923

Answer

380 nm

27-78 §27.7 source ↗

A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.

Answer

33.9 nm

27-80 §27.7 source ↗

Figure 27.34 shows two 7.50-cm-long glass slides illuminated by pure 589-nm wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are 1.00 mm apart?

Figure A shows two microscope slides that have been pressed together. Multicolor swirling rainbow bands are visible coming from the slides. Figure B shows a cross section of two glass slides stacked one on top of the other. The lower slide is horizontal and the upper slide is tilted up at an angle that is larger than the actual angle between slides would be. Two rays come from above and impinge upon the slides. Their refraction and partial reflection is shown at each glass air interface. — Figure 27.34 in the book ↗

Answer

4 . 42 \times 10^{- 5} m

27-82 §27.7 source ↗

Repeat Exercise 27.71, but take the light to be incident at a

45º

angle.

Answer

The oil film will appear black, since the reflected light is not in the visible part of the spectrum.

27-84 §27.8 source ↗

What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?

Answer

45 . 0º

27-86 §27.8 source ↗

If you have completely polarized light of intensity

150 W / m^{2}

, what will its intensity be after passing through a polarizing filter with its axis at an

89 . 0º

angle to the light’s polarization direction?

Answer

45.7 {mW/m}^{2}

27-88 §27.8 source ↗

At the end of Example 27.8, it was stated that the intensity of polarized light is reduced to

90 . 0%

of its original value by passing through a polarizing filter with its axis at an angle of

18 . 4º

to the direction of polarization. Verify this statement.

Example 27.8 Calculating Intensity Reduction by a Polarizing Filter in the book ↗

Answer

90 . 0%

27-90 §27.8 source ↗

Prove that, if

I

is the intensity of light transmitted by two polarizing filters with axes at an angle

θ

and

I'

is the intensity when the axes are at an angle

90.0º - θ,

then

I + I' = I_{0,}

the original intensity. (Hint: Use the trigonometric identities

cos (90.0º - θ) =

sin θ

and

{cos}^{2} θ +

{sin}^{2} θ =

1.

)

Answer

I_{0}

27-92 §27.8 source ↗

What is Brewster’s angle for light traveling in water that is reflected from crown glass?

Answer

48 . 8º

27-94 §27.8 source ↗

At what angle is light inside crown glass completely polarized when reflected from water, as in a fish tank?

Answer

41 . 2º

27-96 §27.8 source ↗

(a) Light reflected at

62 . 5º

from a gemstone in a ring is completely polarized. Can the gem be a diamond? (b) At what angle would the light be completely polarized if the gem was in water?

Answer

(a) 1.92, not diamond (Zircon)

(b) $55 . 2º$

27-98 §27.8 source ↗

Integrated Concepts

If a polarizing filter reduces the intensity of polarized light to $50 . 0%$ of its original value, by how much are the electric and magnetic fields reduced?

Answer

B_{2} = 0 . 707 B_{1}

27-100 §27.8 source ↗

Integrated Concepts

(a) On a day when the intensity of sunlight is $1.00 kW / m^{2}$ , a circular lens 0.200 m in diameter focuses light onto water in a black aluminum beaker. Two polarizing sheets of plastic are placed in front of the lens with their axes at an angle of $20 . 0º.$ Assuming the sunlight is unpolarized and the polarizers are $100%$ efficient, what is the initial rate of heating of the water in $ºC / s$ , assuming it is $80 . 0%$ absorbed? The aluminum beaker has a mass of 30.0 grams and contains 250 grams of water. (b) Do the polarizing filters get hot? Explain.

Answer

(a) $2.07 \times 10^{-2}$ °C/s

(b) Yes, the polarizing filters get hot because they absorb some of the lost energy from the sunlight.

27-101 §27.8 source ↗

Critical Thinking A semiconductor laser emits light at 400.0 nm. The light is passed through a grating where 1,000 lines are etched covering a space of 10.00 mm. The light is directed directly at the grating. (a) What is the angle the light beam diffracts for first order diffraction? (b) What is the angle the light beam diffracts for second order diffraction? (c) The laser is changed out. The first order angle of diffraction is now

23.9 °

. What wavelength of light does it produce? (d) If a new grating with the spacing between the lines doubled were to replace the one above, what would the new angle be for part a? (e) Would the magnitude of the angle of diffraction be different going left from going right?

Answer

(a) $\sin θ =$ $\frac{4.00 \times 10^{- 7}}{10^{- 6}} =$ $0.400 θ =$ $0.400 =$ $23.6 °$

(b) $\sin θ =$ $\frac{(2) (4.00 \times 10^{- 7})}{10^{- 6}} 0.800 θ =$ $0.800 =$ $53.1 °$

(d) $\sin θ =$ $\frac{4.00 \times 10^{- 7}}{2.00 \times 10^{- 6}} =$ $0.200 θ =$ $0.200 =$ $11.5 °$

(e) No.

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Problems and answers are adapted from College Physics 2e by Paul Peter Urone, Roger Hinrichs, Kim Dirks and Manjula Sharma, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.