Optics

Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in Fig. E35.5. (a) At the observer’s location, what is the path difference for waves from the two speakers? (b) Will the sound waves interfere constructively or destructively at the observer’s location—or something in between constructive and destructive? (c) Suppose the observer now increases her distance from the closest speaker to 17.0 m, staying directly in front of the same speaker as initially. Answer the questions of parts (a) and (b) for this new situation.

The two sources ${\mathbf{S}}_{\mathbf{1}}\mathbf{ }\mathbf{and}\mathbf{ }{\mathbf{S}}_{\mathbf{2}}$ shown in Fig. 35.3 emit waves of the same wavelength $\mathbf{\lambda }$ and are in phase with each other. Suppose ${\mathbf{S}}_{\mathbf{1}}$ is a weaker source, so that the waves emitted by ${\mathbf{S}}_{\mathbf{1}}$ have half the amplitude of the waves emitted by ${\mathbf{S}}_{\mathbf{2}}$. How would this affect the positions of the antipodal lines and nodal lines? Would there be total reinforcement at points on the antipodal curves? Would there be total cancellation at points on the nodal curves? Explain your answers. 

Two light sources can be adjusted to emit monochromatic light of any visible wavelength. The two sources are coherent, $\mathbf{2}\mathbf{.}\mathbf{04}\mathbf{\hspace{0.33em}}\mathbf{\mu m}$ apart, and in line with an observer, so that one source is $\mathbf{2}\mathbf{.}\mathbf{04}\mathbf{\hspace{0.33em}}\mathbf{\mu m}$ farther from the observer than the other. (a) For what visible wavelengths (380 to 750 nm) will the observer see the brightest light, owing to constructive interference? (b) How would your answers to part (a) be affected if the two sources were not in line with the observer, but were still arranged so that one source is $\mathbf{2}\mathbf{.}\mathbf{04}\mathbf{\hspace{0.33em}}\mathbf{\mu m}$ farther away from the observer than the other? (c) For what visible wavelengths will there be destructive interference at the location of the observer?

 Could the Young two-slit interference experiment be performed with gamma rays? If not, why not? If so, discuss differences in the experimental design compared to the experiment with visible light. 

 Coherent light with wavelength $\mathit{\lambda }$ falls on two narrow slits separated by a distance $\mathit{d}$. If $\mathit{d}$ is less than some minimum value, no dark fringes are observed. Explain. In terms of $\mathit{\lambda }$, what is this minimum value of $\mathit{d}$?

 A fellow student, who values memorizing equations above understanding them, combines Eq. (35.4) and (35.13) to “prove” that $\mathit{\varphi }$ can only equal $2\pi m$. How would you explain to this student that $\varphi$ can have values other than $\mathbf{2}\mathit{\pi }\mathit{m}$?

 If the monochromatic light shown in Fig. 35.5a were replaced by white light, would a two-slit interference pattern be seen on the screen? Explain.

 In using the superposition principle to calculate intensities in interference patterns, could you add the intensities of the waves instead of their amplitudes? Explain.

 A glass windowpane with a thin film of water on it reflects less than when it is perfectly dry. Why

 A very thin soap film (n = 1.33), whose thickness is much less than a wavelength of visible light, looks black; it appears to reflect no light at all. Why? By contrast, an equally thin layer of soapy water (n = 1.33) on glass (n = 1.50) appears quite shiny. Why is there a difference?

Interference can occur in thin films. Why is it important that the films be thin? Why don’t you get these effects with a relatively thick film? Where should you put the dividing line between “thin” and “thick”? Explain your reasoning.

 If we shine white light on an air wedge like that shown in Fig. 35.12, the colours that are weak in the light reflected from any point along the wedge are strong in the light transmitted through the wedge. Explain why this should be so.

 Monochromatic light is directed at normal incidence on a thin film. There is destructive interference for the reflected light, so the intensity of the reflected light is very low. What happened to the energy of the incident light?

When a thin oil film spreads out on a puddle of water, the thinnest part of the film looks dark in the resulting interference pattern. What does this tell you about the relative magnitudes of the refractive indexes of oil and water?

Young’s experiment is performed with light from excited helium atoms ( $\mathbf{\lambda }$= 502 nm). Fringes are measured carefully on a screen 1.20 m away from the double slit, and the center of the 20th fringe (not counting the central bright fringe) is found to be 10.6 mm from the center of the central bright fringe. What is the separation of the two slits?

Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away, the distance between dark fringes is 3.90 mm. What is the slit separation?

Two slits spaced 0.450 mm apart are placed 75.0 cm from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 500 nm?

 If the entire apparatus of Exercise 35.9 (slits, screen, and space in between) is immersed in water, what then is the distance between the second and third dark lines?

Two thin parallel slits that are $\mathbf{0}\mathbf{.}\mathbf{0116}$ mm apart are illuminated by a laser beam of wavelength $\mathbf{585}$ mm. (a) On a very large distant screen, what is the total number of bright fringes (those indicating complete constructive interferences), including the central fringe and those on both sides of it? Solve this problem without calculating all the angles! (Hint: What is the largest that $\mathbf{sin\theta }$ can be? What does this tell you is the largest value of $\mathit{m}$?) (b) At what angle, relative to the original direction of the beam, will the fringe that is most distant from the central bright fringe occur?

Coherent light with wavelength $\mathbf{400}$ nm passes through two very narrow slits that are separated by  $\mathbf{0}\mathbf{.}\mathbf{200}$nm, and the interference pattern is observed on a screen $\mathbf{4}\mathbf{.}\mathbf{00}$nm from the slits. (a) What is the width (in nm) of the central interference maximum? (b) What is the width of the first-order bright fringe?

Two very narrow slits are spaced $\mathbf{1}\mathbf{.}\mathbf{80}\mathbf{ }\mathit{\mu }\mathit{m}$ apart and are placed $\mathbf{35}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$ from a screen. What is the distance between the first and second dark lines of the interference pattern when the slits are illuminated with coherent light with $\mathit{\lambda }\mathbf{=}\mathbf{550}\mathit{n}\mathit{m}$? (Hint: The angle $\mathit{\theta }$ in Eq. ( 35.5 ) is not small.)

Coherent light that contains two wavelengths, 660nm (red) and 470nm (blue), passes through two narrow slits that are separated by 0.300mm. Their interference pattern is observed on a screen 4.00m from the slits. What is the distance on the screen between the first-order bright fringes for the two wavelengths?

Coherent light that contains two wavelengths, 600nm (red) and 470nm (blue), passes through two narrow slits that are separated by 0.300mm. Their interference pattern is observed on a screen 3.00m from the slits. The first-order bright fringe is at 4.84mm from the center of the central bright fringe. For what wavelength of light will the first-order dark fringe be observed at this same point on the screen?

Coherent light of frequency $\mathbf{6}\mathbf{.}\mathbf{32}\mathbf{*}{\mathbf{10}}^{\mathbf{14}}\mathbf{ }\mathit{H}\mathit{z}$ passes through two thin slits and falls on a screen 85.0cm away. You observe that the third bright fringe occurs at $\mathbf{\pm }\mathbf{3}\mathbf{.}\mathbf{11}\mathit{c}\mathit{m}$ on either side of the central bright fringe. (a) How far apart are the two slits? (b) At what distance from the central bright fringe will the third dark fringe occur?

In a two-slit interference pattern, the intensity at the peak of the central maximum is ${\mathit{I}}_{\mathbf{0}}$. (a) At a point in the pattern where the phase difference between the waves from the two slits is $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{°}$, what is the intensity? (b) What is the path difference for $\mathbf{480}\mathbf{-}\mathit{n}\mathit{m}$ light from the two slits at a point where the phase difference is $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{°}$?

Coherent sources A and B emit electromagnetic waves with wavelength 2.00cm . Point P is 4.86m from A and 5.24m from B. What is the phase difference at P between these two waves?

Coherent light with wavelength 500 nm passes through narrow slits separated by 0.340mm. At a distance from the slits large compared to their separation, what is the phase difference (in radians) in the light from the two slits at an angle of $\mathbf{23}\mathbf{.}\mathbf{0}\mathbf{°}$ from the centerline?

Two slits spaced 0.260mm apart are 0.990m from a screen and illuminated by coherent light of wavelength . The intensity at the center of the central maximum $\left(\theta =0°\right)$is ${\mathit{I}}_{\mathbf{0}}$. What is the distance on the screen from the center of the central maximum (a) to the first minimum; (b) to the point where the intensity has fallen to $\frac{{\mathbf{I}}_{\mathbf{0}}}{\mathbf{2}}$?

Consider two antennas separated by 9.00m that radiate in phase at 120MHz, as described in Exercise . A receiver placed 150m from both antennas measures an intensity ${\mathit{I}}_{\mathbf{0}}$.The receiver is moved so that it is closer to one antenna than to the other. (a) What is the phase difference $\mathit{\varphi }$ between the two radio waves produced by this path difference? (b) In terms of ${\mathit{I}}_{\mathbf{0}}$, what is the intensity measured by the receiver at its new position?

Two slits spaced 0.0720mm apart are 0.800m from a screen. Coherent light of wavelength $\mathit{\lambda }$passes through the two slits. In their interference pattern on the screen, the distance from the center of the central maximum to the first minimum is 3.00mm . If the intensity at the peak of the central maximum is $\mathbf{0}\mathbf{.}\mathbf{600}\mathit{W}\mathbf{/}{\mathit{m}}^{\mathbf{2}}$, what is the intensity at points on the screen that are (a) 2.00mm and (b) 1.50mm from the center of the central maximum?

What is the thinnest film of a coating with \(n = 1.42\) on glass \((n = 1.52)\) for which destructive interference of the red component \((650nm)\) of an incident white light beam in air can take place by reflection?

When viewing a paper of art that is behind glass, one often is affected by the light that is reflected off the front of the glass (called glare), which can make it difficult to see the art clearly. One solution is to coat the outer surface of the glass with a film to cancel part of the glare. (a) If the glass has a refractive index 1.62 and you use $\mathit{T}\mathit{i}{\mathit{O}}_{\mathbf{2}}$ , which has an index of refraction of 2.62 , as the coating, what is the minimum film thickness that will cancel light of wavelength 505nm? (b) If this coating is too thin to stand up to wear, what other thickness would also work? Find only the three thinnest ones.

Two rectangular pieces of plane glass are laid one upon the other on a table. A thin strip of paper is placed between them at one edge so that a very thin wedge of air is formed. The plates are illuminated at normal incidence by 546 -nm  light from a mercury-vapor lamp. Interference fringes are formed, with 15.0 fringes per centimeter. Find the angle of the wedge.

A plate of glass 9.00 cm long is placed in contact with a second plate and is held at a small angle with it by a metal strip 0.0800 mm thick placed under one end. The space between the plates is filled with air. The glass is illuminated from above with light having a wavelength in air of 656 nm. How many interference fringes are observed per centimeter in the reflected light?

A researcher measures the thickness of a layer of benzene 1n = 1.502 floating on water by shining monochromatic light onto the film and varying the wavelength of the light. She finds that light of wavelength 575 nm is reflected most strongly from the film. What does she calculate for the minimum thickness of the film?

A uniform film of TiO2 , 1036 nm thick and having index of refraction 2.62, is spread uniformly over the surface of crown glass of refractive index 1.52. Light of wavelength 520.0 nm falls at normal incidence onto the film from air. You want to increase the thickness of this film so that the reflected light cancels. 

(a) What is the minimum thickness of TiO2 that you must add so the reflected light cancels as desired?

(b) After you make the adjustment in part (a), what is the path difference between the light reflected off the top of the film and the light that cancels it after traveling through the film? Express your answer in (i) nanometers and (ii) wavelengths of the light in the TiO2 film.

The walls of a soap bubble have about the same index of refraction as that of plain water, n = 1.33. There is air both inside and outside the bubble. 

(a) What wavelength (in the air) of visible light is most strongly reflected from a point on a soap bubble where its wall is 290 nm thick? To what color does this correspond (see Fig. 32.4 and Table 32.1)? 

(b) Repeat part (a) for a wall thickness of 340 nm.

A compact disc (CD) is read from the bottom by a semiconductor laser with wavelength 790 nm passing through a plastic substrate of refractive index 1.8. When the beam encounters a pit, part of the beam is reflected from the pit and part from the flat region between the pits, so these two beams interfere with each other (Fig. E35.31). What must the minimum pit depth be so that the part of the beam reflected from a pit cancels the part of the beam reflected from the flat region? (It is this cancellation that allows the player to recognize the beginning and end of a pit.

What is the thinnest soap film (excluding the case of zero thickness) that appears black when illuminated with light with wavelength 480 nm? The index of refraction of the film is 1.33, and there is air on both sides of the film.

How far must the mirror M₂ (see Fig. 35.19) of the Michelson interferometer be moved so that 1800 fringes of He-Ne laser light $\mathit{\lambda }\mathbf{=}\mathbf{633}$ move across a line in the field of view?

Jan first uses a Michelson interferometer with the 606-nm light from a krypton-86 lamp. He displaces the movable mirror away from him, counting 818 fringes moving across a line in his field of view. Then Linda replaces the krypton lamp with filtered 502-nm light from a helium lamp and displaces the movable mirror toward her. She also counts 818 fringes, but they move across the line in her field of view opposite to the direction they moved for Jan. Assume that both Jan and Linda counted to 818 correctly. (a) What distance did each person move the mirror? (b) What is the resultant displacement of the mirror?

Newton’s rings are visible when a plano-convex lens is placed on a flat glass surface. For a particular lens with an index of refraction of n = 1.50 and a glass plate with an index of n = 1.80, the diameter of the third bright ring is 0.640 mm. If water 1n = 1.332 now fills the space between the lens and the glass plate, what is the new diameter of this ring? Assume the radius of curvature of the lens is much greater than the wavelength of the light.

Eyeglass lenses can be coated on the inner surfaces to reduce the reflection of stray light to the eye. If the lenses are medium flint glass of refractive index 1.62 and the coating is fluorite of refractive index 1.432, 

(a) what minimum thickness of film is needed on the lenses to cancel light of wavelength 550 nm reflected toward the eye at normal incidence? 

(b) Will any other wavelengths of visible light be cancelled or enhanced in the reflected light?

After an eye examination, you put some eyedrops on your sensitive eyes. The cornea (the front part of the eye) has an index of refraction of 1.38, while the eyedrops have a refractive index of 1.45. After you put in the drops, your friends notice that your eyes look red, because red light of wavelength 600 nm has been reinforced in the reflected light. (a) What is the minimum thickness of the film of eyedrops on your cornea? (b) Will any other wavelengths of visible light be reinforced in the reflected light? Will any be cancelled? (c) Suppose you had contact lenses, so that the eyedrops went on them instead of on your corneas. If the refractive index of the lens material is 1.50 and the layer of eyedrops has the same thickness as in part (a), what wavelengths of visible light will be reinforced? What wavelengths will be cancelled?

Two flat plates of glass with parallel faces are on a table, one plate on the other. Each plate is 11.0 cm long and has a refractive index of 1.55. A very thin sheet of metal foil is inserted under the end of the upper plate to raise it slightly at that end, in a manner similar to that discussed in Example 35.4. When you view  the glass plates from above with reflected white light, you observe that, at 1.15 mm from the line where the sheets are in contact, the violet light of wavelength 400.0 nm is enhanced in this reflected light, but no visible light is enhanced closer to the line of contact. 

(a) How far from the line of contact will green light (of wavelength 550.0 nm) and orange light (of wavelength 600.0 nm) first be enhanced? 

(b) How far from the line of contact will the violet, green, and orange light again be enhanced in the reflected light? 

(c) How thick is the metal foil holding the ends of the plates apart?

In a setup similar to that of Problem 35.39, the glass has an index of refraction of 1.53, the plates are each 8.00 cm long, and the metal foil is 0.015 mm thick. The space between the plates is filled with a jelly whose refractive index is not known precisely but is known to be greater than that of the glass. When you illuminate these plates from above with light of wavelength 525 nm, you observe a series of equally spaced dark fringes in the reflected light. You measure the spacing of these fringes and find that there are 10 of them every 6.33 mm. What is the index of refraction of the jelly?

A thin uniform film of refractive index 1.750 is placed on a sheet of glass of refractive index 1.50. At room temperature 120.0°C2, this film is just thick enough for light with wavelength 582.4 nm reflected off the top of the film to be canceled by light reflected from the top of the glass. After the glass is placed in an oven and slowly heated to 170°C, you find that the film cancels reflected light with a wavelength of 588.5 nm. What is the coefficient of linear expansion of the film? (Ignore any changes in the refractive index of the film due to the temperature change.)

White light reflects at normal incidence from the top and bottom surfaces of a glass plate 1n = 1.522. There is air above and below the plate. Constructive interference is observed for light whose wavelength in air is 477.0 nm. What is the thickness of the plate if the next longer wavelength for which there is constructive interference is 540.6 nm?

Laser light of wavelength 510 nm is traveling in air and shines at normal incidence onto the flat end of a transparent plastic rod that has n = 1.30. The end of the rod has a thin coating of a transparent material that has refractive index 1.65. What is the minimum (nonzero) thickness of the coating 

(a) for which there is maximum transmission of the light into the rod; 

(b) for which transmission into the rod is minimized?

Red light with a wavelength of 700 nm is passed through a two-slit apparatus. At the same time, monochromatic visible light with another wavelength passes through the same apparatus. As a result, most of the pattern that appears on the screen is a mixture of two colors; however, in the center of the third bright fringe 1m = 32 of the red light appears pure red, with none of the other colors. What are the possible wavelengths of the second type of visible light? Do you need to know the slit spacing to answer this question? Why or why not?