Chapter 24

To study wave interference, a student uses two speakers driven by the same sound wave of wavelength \(0.50 \mathrm{~m}\). If the distances from a point to the speakers differ by \(0.75 \mathrm{~m},\) will the waves interfere constructively or destructively at that point? What if the distances differ by \(1.0 \mathrm{~m} ?\)

In the development of Young's double-slit experiment, a small-angle approximation \((\tan \theta \approx \sin \theta)\) was used to find the lateral displacements of the maxima (bright) and minima (dark) positions. How good is this approximation? For example, what is the percentage error for \(\theta=10^{\circ} ?\)

Two parallel slits \(0.075 \mathrm{~mm}\) apart are illuminated with monochromatic light of wavelength \(480 \mathrm{nm}\). Find the angle between the center of the central maximum and the center of the first side maximum.

When two parallel slits are illuminated with monochromatic light of wavelength \(632.8 \mathrm{nm}\), the angle between the center of the central maximum and the center of the second side maximum is \(0.45^{\circ} .\) What is the distance between the parallel slits?

In a double-slit experiment that uses monochromatic light, the angular separation between the central maximum and the second-order maximum is \(0.160^{\circ} .\) What is the wavelength of the light if the distance between the slits is \(0.350 \mathrm{~mm} ?\)

Monochromatic light passes through two narrow slits and forms an interference pattern on a screen. (a) If the wavelength of light used increases, will the distance between the maxima (1) increase, (2) remain the same, or (3) decrease? Explain. (b) If the slit separation is \(0.25 \mathrm{~mm}\) the screen is \(1.5 \mathrm{~m}\) away from the slits, and light of wavelength \(550 \mathrm{nm}\) is used, what is the distance from the center of the central maximum to the center of the third-order maximum? (c) What if the wavelength is \(680 \mathrm{nm}\) ?

(a) If the wavelength used in a double-slit experiment is decreased, the distance between adjacent maxima will (1) increase, (2) decrease, (3) remain the same. Explain. (b) If the separation between the two slits is \(0.20 \mathrm{~mm}\) and the adjacent maxima of the interference pattern on a screen \(1.5 \mathrm{~m}\) away from the slits are \(0.45 \mathrm{~cm}\) apart, what is the wavelength and color of the light? (c) If the wavelength is \(550 \mathrm{nm}\), what is the distance between adjacent maxima?

In a double-slit experiment using monochromatic light, a screen is placed \(1.25 \mathrm{~m}\) away from the slits, which have a separation distance of \(0.0250 \mathrm{~mm}\). The position of the third-order maximum is \(6.60 \mathrm{~cm}\) from the center of the central maximum. Find (a) the wavelength of the light and (b) the position of the second-order maximum.

In a double-slit experiment with monochromatic light and a screen at a distance of \(1.50 \mathrm{~m}\) from the slits, the angle between the second- order maximum and the central maximum is 0.0230 rad. If the separation distance of the slits is \(0.0350 \mathrm{~mm}\), what are (a) the wavelength and color of the light and (b) the lateral displacement of this maximum?

Two parallel slits are illuminated with monochromatic light, and an interference pattern is observed on a screen. (a) If the distance between the slits were decreased, would the distance between the maxima (1) increase, (2) remain the same, or (3) decrease? Explain. (b) If the slit separation is \(1.0 \mathrm{~mm}\), the wavelength is \(640 \mathrm{nm}\), and the distance from the slits to the screen is \(3.00 \mathrm{~m}\), what is the separation between adjacent interference maxima? (c) What if the slit separation is \(0.80 \mathrm{~mm} ?\)

(a) In a double-slit experiment, if the distance from the double slits to the screen is increased, the separation between the adiacent maxima will (1) increase, (2) decrease, (3) remain the same. Explain. (b) Yellow-green light \((\lambda=550 \mathrm{nm})\) illuminates a double-slit separated by \(1.75 \times 10^{-4} \mathrm{~m} .\) If the screen is located \(2.00 \mathrm{~m}\) from the slits, determine the separation between the adjacent maxima. (c) What if the screen is located \(3.00 \mathrm{~m}\) from the slits?

(a) Derive a relationship that gives the locations of the minima in a Young's double-slit experiment. What is the distance between adjacent minima? (b) For a thirdorder minimum (the third side dark position from the central maximum), what is the path length difference between that location and the two slits?

When a double-slit setup is illuminated with light of wavelength \(632.8 \mathrm{nm}\), the distance between the center of the central bright position and the second side dark position is \(4.5 \mathrm{~cm}\) on a screen that is \(2.0 \mathrm{~m}\) from the slits. What is the distance between the slits?

Light of two different wavelengths is used in a double-slit experiment. The location of the third-order maximum for the first light, yellow-orange light \((\lambda=600 \mathrm{nm})\) coincides with the location of the fourth-order maximum for the other color's light. What is the wavelength of the other light?

Light of wavelength \(550 \mathrm{nm}\) in air is normally incident on a glass plate \((n=1.5)\) whose thickness is \(1.1 \times 10^{-5} \mathrm{~m} .\) (a) What is the thickness of the glass expressed in terms of the wavelength of light in glass? (b) How many reflected waves will experience the \(180^{\circ}\) phase shift? (c) Will the reflected waves interfere constructively or destructively?

A film of index of refraction of 1.4 and thickness of \(1.2 \times 10^{-5} \mathrm{~m}\) is on a lens with an index of refraction of 1.6. Light of wavelength \(600 \mathrm{nm}\) is incident normally from air to the film. Consider only reflections from the top and bottom surfaces of the film. (a) How many reflected waves will experience the \(180^{\circ}\) phase shift? (b) What is the path length difference between the two reflected waves? (c) Will the reflected waves interfere constructively or destructively?

A lens with an index of refraction of 1.60 is to be coated with a material \((n=1.40)\) that will make the lens nonreflecting for yellow-orange light \((\lambda=515 \mathrm{nm})\) normally incident on the lens. What is the minimum required thickness of the coating?

Magnesium fluoride \((n=1.38)\) is frequently used as a lens coating to make nonreflecting lenses. What is the difference in the minimum film thickness required for maximum transmission of blue light \((\lambda=400 \mathrm{nm})\) and of red light \((\lambda=700 \mathrm{nm}) ?\)

A solar cell is designed to have a nonreflective film of a transparent material for a wavelength of \(550 \mathrm{nm}\). (a) Will the thickness of the film depend on the index of refraction of the underlying material in the solar cell? Discuss the possible scenarios. (b) If \(n_{\text {solar }}>n_{\text {film }}\) and \(n_{\mathrm{film}}=1.22,\) what is the minimum thickness of the film? (c) Repeat the calculation in (b) if \(n_{\text {solar }}

A thin layer of oil \((n=1.50)\) floats on water. Destructive interference is observed for reflected light of wavelengths \(480 \mathrm{nm}\) and \(600 \mathrm{nm}\), each at a different location. (a) If the order number is the same for both wavelengths, which wavelength is at a greater thickness: (1) \(480 \mathrm{nm},\) or (2) \(600 \mathrm{nm} ?\) Explain. (b) Write the general condition of destructive intereference for reflected light. (c) Find the two minimum thicknesses of the oil film, assuming normal incidence.

A slit of width \(0.15 \mathrm{~mm}\) is illuminated with monochromatic light of wavelength \(632.8 \mathrm{nm}\). At what angle will the first minimum occur?

In a single-slit diffraction pattern using light of wavelength \(550 \mathrm{nm}\), the second-order minimum is measured to be at \(0.32^{\circ} .\) What is the slit width?

A slit \(0.025 \mathrm{~mm}\) wide is illuminated with red light \((\lambda=680 \mathrm{nm})\). How wide are (a) the central maximum and (b) the side maxima of the diffraction pattern formed on a screen \(1.0 \mathrm{~m}\) from the slit?

At what angle will the second-order maximum be seen from a diffraction grating of spacing \(1.25 \mu \mathrm{m}\) when illuminated by light of wavelength \(550 \mathrm{nm} ?\)

A venetian blind is essentially a diffraction grating-not for visible light, but for waves with much longer wavelengths. If the spacing between the slats of a blind is 2.5 \(\mathrm{cm},\) (a) for what wavelength would there be a first-order maximum at an angle of \(10^{\circ},\) and (b) what type of radiation is this?

A single slit is illuminated with monochromatic light, and a screen is placed behind the slit to observe the diffraction pattern. (a) If the width of the slit is increased, will the width of the central maximum (1) increase, (2) remain the same, or (3) decrease? Why? (b) If the width of the slit is \(0.50 \mathrm{~mm}\), the wavelength is \(680 \mathrm{nm},\) and the screen is \(1.80 \mathrm{~m}\) from the slit, what would be the width of the central maximum? (c) What if the width of the slit is \(0.60 \mathrm{~mm}\) ?

(a) If the wavelength used in a single-slit diffraction experiment increases, will the width of the central maximum (1) increase, (2) remain the same, or (3) decrease? Why? (b) If the width of the slit is \(0.45 \mathrm{~mm}\) the wavelength is \(400 \mathrm{nm},\) and the screen is \(2.0 \mathrm{~m}\) from the slit, what would be the width of the central maximum? (c) What if the wavelength is \(700 \mathrm{nm} ?\)

A teacher standing in a doorway \(1.0 \mathrm{~m}\) wide blows a whistle with a frequency of \(1000 \mathrm{~Hz}\) to summon children from the playground (v Fig. 24.31 ). Two boys are playing on the swings \(20 \mathrm{~m}\) away from the school building. One boy is at an angle of \(0^{\circ}\) and another one at \(19.6^{\circ}\) from a line normal to the doorway. Taking the speed of sound in air to be \(335 \mathrm{~m} / \mathrm{s}\), which boy may not hear the whisle? Prove your answer.

A diffraction grating is designed to have the secondorder maxima at \(10^{\circ}\) from the central maximum for red light \((\lambda=700 \mathrm{nm})\). How many lines per centimeter does the grating have?

Find the angles of the blue \((\lambda=420 \mathrm{nm})\) and red \((\lambda=680 \mathrm{nm})\) components of the first- and second-order maxima in a pattern produced by a diffraction grating with 7500 lines \(/ \mathrm{cm}\).

A certain crystal gives a deflection angle of \(25^{\circ}\) for the first-order maximum of monochromatic X-rays with a frequency of \(5.0 \times 10^{17} \mathrm{~Hz}\). What is the lattice spacing of the crystal?

(a) Only a limited number of maxima can be observed with a diffraction grating. The factor(s) that limit(s) the number of maxima seen is (are) (a) (1) the wavelength, (2) the grating spacing, (3) both. Explain. (b) How many maxima appear when monochromatic light of wavelength 560 nm illuminates a diffraction grating that has 10000 lines \(/ \mathrm{cm},\) and what are their order numbers?

A diffraction grating with 6000 lines \(/ \mathrm{cm}\) is illuminated with a red light from a He-Ne laser \((\lambda=632.8 \mathrm{nm})\). How many side maxima are formed in the diffraction pattern, and at what angles are they observed?

In a particular diffraction grating pattern, the red component \((700 \mathrm{nm})\) in the second-order maximum is deviated at an angle of \(20^{\circ} .\) (a) How many lines per centimeter does the grating have? (b) If the grating is illuminated with white light, how many maxima of the complete visible spectrum would be produced?

The commonly used CD (Compact Disc) consists of many closely spaced tracks that can be used as reflecting gratings. The industry standard for the track- to-track distance is \(1.6 \mu \mathrm{m} .\) If a He-Ne laser with a wavelength of \(632.8 \mathrm{nm}\) is incident normally onto a \(\mathrm{CD}\), calculate the angles for all the visible maxima.

White light of wavelength ranging from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) is used for a diffraction grating with 6500 lines per centimeter. (a) In a particular order of maximum, red color will have (1) a larger, (2) the same, or (3) a smaller angle than blue color. Explain. (b) Calculate the angles for \(400 \mathrm{nm}\) and \(700 \mathrm{nm}\) in the second-order maximum. (c) What is the angular width of the whole spectrum in the second order?

White light whose components have wavelengths from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) illuminates a diffraction grating with 4000 lines \(/ \mathrm{cm} .\) Do the first- and second-order spectra overlap? Justify your answer.

Show that for a diffraction grating, the violet \((\lambda=400 \mathrm{nm})\) portion of the third-order maximum overlaps the yellow-orange \((\lambda=600 \mathrm{nm})\) portion of the second-order maximum, regardless of the grating's spacing.

Unpolarized light is incident on a polarizeranalyzer pair that can have their transmission axes at an angle of either \(30^{\circ}\) or \(45^{\circ} .\) (a) The \(30^{\circ}\) angle will allow (1) more, (2) the same, or (3) less light to go through. (b) Calculate the percentage of light that goes through the polarizer-analyzer pair in terms of the incident light intensity.

When unpolarized light is incident on a polarizer-analyzer pair, \(30 \%\) of the original light intensity passes the analyzer. What is the angle between the transmission axes of the polarizer and analyzer?

Some types of glass have a range of indices of refraction of about 1.4 to 1.7 . What is the range of the polarizing (Brewster) angle for these glasses when light is incident on them from air?

Light is incident on a certain material in air. (a) If the index of refraction of the material increases, the polarizing (Brewster) angle will (1) also increase, (2) decrease, (3) remain the same. Explain. (b) What are the polarizing angles if the index of refraction is 1.6 and \(1.8 ?\)

Unpolarized light of intensity \(I_{\mathrm{o}}\) is incident on a polarizer- analyzer pair. (a) If the angle between the polarizer and analyzer increases in the range of \(0^{\circ}\) to \(90^{\circ}\) the transmitted light intensity will (1) also increase, (2) decrease, (3) remain the same. Explain. (b) If the angle between the polarizer and analyzer is \(30^{\circ},\) what light intensity would be transmitted through the polarizer and the analyzer, respectively? (c) What if the angle is \(60^{\circ}\) ?

A beam of light is incident on a glass plate \((n=1.62)\) in air and the reflected ray is completely polarized. What is the angle of refraction for the beam?

The critical angle for total internal reflection in a certain media boundary is \(45^{\circ} .\) What is the polarizing (Brewster) angle for light externally incident on the same boundary?

The polarizing (Brewster) angle for a certain media boundary is \(33^{\circ} .\) What is the critical angle for total internal reflection for the same boundary?

The angle of incidence is adjusted so there is maximum linear polarization for the reflected light from a transparent piece of plastic in air. (a) There is (1) no, (2) maximum, or (3) some light transmitted through the plastic. Explain. (b) If the index of refraction of the plastic is 1.40 , what would be the angle of refraction in the plastic?

(a) The polarizing (Brewster) angle of a piece of flint glass \((n=1.66)\) in water is (1) greater than, (2) less than, (3) the same as that of the glass in air. Explain. (b) What are the polarizing angles when it is in air and submerged in water, respectively?

Sunlight is reflected off a vertical plate-glass window \((n=1.55)\). What would the Sun's altitude (angle above the horizon) have to be for the reflected light to be completely polarized?

A plate of crown glass \((n=1.52)\) is covered with a layer of water. A beam of light traveling in air is incident on the water and partially transmitted. Is there any angle of incidence for which the light reflected from the water- glass interface will have maximum linear polarization? Justify your answer mathematically.