Chapter 27

In a Young's double-slit experiment, the angle that locates the second-order bright fringe is \(2.0^{\circ} .\) The slit separation is \(3.8 \times 10^{-5} \mathrm{~m}\). What is the wavelength of the light?

In a Young's double-slit experiment, the wavelength of the light used is \(520 \mathrm{nm}\) (in vacuum), and the separation between the slits is \(1.4 \times 10^{-6} \mathrm{~m}\). Determine the angle that locates (a) the dark fringe for which \(m=0,\) (b) the bright fringe for which \(m=1,\) (c) the dark fringe for which \(m=1,\) and \((\mathrm{d})\) the bright fringe for which \(m=2\)

Two in-phase sources of waves are separated by a distance of \(4.00 \mathrm{~m}\). These sources produce identical waves that have a wave length of \(5.00 \mathrm{~m}\). On the line between them, there are two places at which the same type of interference occurs. (a) Is it constructive or destructive interference, and (b) where are the places located?

The dark fringe for \(m=0\) in a Young's double-slit experiment is located at an angle of \(\theta=15^{\circ} .\) What is the angle that locates the dark fringe for \(m=1 ?\)

A flat observation screen is placed at a distance of \(4.5 \mathrm{~m}\) from a pair of slits. The separation on the screen between the central bright fringe and the first-order bright fringe is \(0.037 \mathrm{~m}\). The light illuminating the slits has a wavelength of \(490 \mathrm{~nm}\). Determine the slit separation.

At most, how many bright fringes can be formed on either side of the central bright fringe when light of wavelength \(625 \mathrm{~nm}\) falls on a double slit whose slit separation is \(3.76 \times 10^{-6} \mathrm{~m} ?\)

In a Young's double-slit experiment the separation \(y\) between the second- order bright fringe and the central bright fringe on a flat screen is \(0.0180 \mathrm{~m}\) when the light has a wavelength of 425 \(\mathrm{nm}\). Assume that the angles that locate the fringes on the screen are small enough so that \(\sin \theta \approx \tan \theta\). Find the separation \(y\) when the light has a wavelength of \(585 \mathrm{~nm}\).

Refer to Interactive Solution 27.7 at for help in solving this problem. In a Young's double-slit experiment the separation \(y\) between the second-order bright fringe and the central bright fringe on a flat screen is \(0.0180 \mathrm{~m}\) when the light has a wavelength of 425 nm. Assume that the angles that locate the fringes on the screen are small enough so that \(\sin \theta \approx \tan \theta .\) Find the separation \(y\) when the light has a wavelength of \(585 \mathrm{nm}\)

In Young's experiment a mixture of orange light \((611 \mathrm{nm})\) and blue light \((471 \mathrm{nm})\) shines on the double slit. The centers of the first- order bright blue fringes lie at the outer edges of a screen that is located \(0.500 \mathrm{~m}\) away from the slits. However, the first-order bright orange fringes fall off the screen. By how much and in which direction (toward or away from the slits) should the screen be moved, so that the centers of the first-order bright orange fringes just appear on the screen? It may be assumed that \(\theta\) is small, so that \(\sin \theta \approx \tan \theta\).

A sheet that is made of plastic \((n=1.60)\) covers one slit of a double slit (see the drawing). When the double slit is illuminated by monochromatic light \(\left(\lambda_{\text {vacuum }}=586 \mathrm{nm}\right),\) the center of the screen appears dark rather than bright. What is the minimum thickness of the plastic?

A mix of red light \(\left(\lambda_{\text {vacuum }}=661 \mathrm{nm}\right)\) and green light \(\left(\lambda_{\text {vacuum }}=551 \mathrm{nm}\right)\) is directed perpendicularly onto a soap film \((n=1.33)\) that has air on either side. What is the minimum nonzero thickness of the film, so that destructive interference causes it to look red in reflected light?

Light of wavelength \(691 \mathrm{nm}\) (in vacuum) is incident perpendicularly on a soap film \((n=1.33)\) suspended in air. What are the two smallest nonzero film thicknesses (in \(\mathrm{nm})\) for which the reflected light undergoes constructive interference?

A tank of gasoline \((n=1.40)\) is open to the air \((n=1.00)\). A thin film of liquid floats on the gasoline and has a refractive index that is between \(1.00\) and \(1.40\). Light that has a wavelength of \(625 \mathrm{~nm}\) (in vacuum) shines perpendicularly down through the air onto this film, and in this light the film looks bright due to constructive interference. The thickness of the film is \(242 \mathrm{~nm}\) and is the minimum nonzero thickness for which constructive interference can occur. What is the refractive index of the film?

A transparent film \((n=1.43)\) is deposited on a glass plate \((n=1.52)\) to form a nonreflecting coating. The film has a thickness that is \(1.07 \times 10^{-7} \mathrm{~m}\). What is the longest possible wavelength (in vacuum) of light for which this film has been designed?

Interactive LearningWare 27.2 at provides some pertinent background for this problem. A transparent film \((n=1.43)\) is deposited on a glass plate \((n=1.52)\) to form a nonreflecting coating. The film has a thickness that is \(1.07 \times 10^{-7} \mathrm{~m}\). What is the longest possible wavelength (in vacuum) of light for which this film has been designed?

Orange light \(\left(\lambda_{\text {vacuum }}=611 \mathrm{nm}\right.\) ) shines on a soap film \((n=1.33)\) that has air on either side of it. The light strikes the film perpendicularly. What is the minimum thickness of the film for which constructive interference causes it to look bright in reflected light?

A film of oil lies on wet pavement. The refractive index of the oil exceeds that of the water. The film has the minimum nonzero thickness such that it appears dark due to destructive interference when viewed in red light (wavelength \(=640.0 \mathrm{~nm}\) in vacuum). Assuming that the visible spectrum extends from 380 to \(750 \mathrm{~nm}\), for which visible wavelength(s) (in vacuum) will the film appear bright due to constructive interference?

Consult Interactive Solution \(\underline{27} .17\) at to review a model for solving this problem. A film of oil lies on wet pavement. The refractive index of the oil exceeds that of the water. The film has the minimum nonzero thickness such that it appears dark due to destructive interference when viewed in red light (wavelength \(=640.0 \mathrm{nm}\) in vacuum). Assuming that the visible spectrum extends from 380 to \(750 \mathrm{nm}\), for which visible wavelength(s) (in vacuum) will the film appear bright due to constructive interference?

A uniform layer of water \((n=1.33)\) lies on a glass plate \((n=1.52)\). Light shines perpendicularly on the layer. Because of constructive interference, the layer looks maximally bright when the wavelength of the light is \(432 \mathrm{~nm}\) in vacuum and also when it is \(648 \mathrm{~nm}\) in vacuum. (a) Obtain the minimum thickness of the film. (b) Assuming that the film has the minimum thickness and that the visible spectrum extends from 380 to \(750 \mathrm{~nm}\), determine the visible wavelength(s) (in vacuum) for which the film appears completely dark.

A diffraction pattern forms when light passes through a single slit. The wavelength of the light is \(675 \mathrm{~nm}\). Determine the angle that locates the first dark fringe when the width of the slit is (a) \(1.8 \times 10^{-4} \mathrm{~m}\) and (b) \(1.8 \times 10^{-6} \mathrm{~m}\).

(a) As Section \(17.3\) discusses, high-frequency sound waves exhibit less diffraction than low-frequency sound waves do. However, even high-frequency sound waves exhibit much more diffraction under normal circumstances than do light waves that pass through the same opening. The highest frequency that a healthy ear can typically hear is \(2.0 \times 10^{4} \mathrm{~Hz}\). Assume that a sound wave with this frequency travels at \(343 \mathrm{~m} / \mathrm{s}\) and passes through a doorway that has a width of \(0.91 \mathrm{~m}\). Determine the angle that locates the first minimum to either side of the central maximum in the diffraction pattern for the sound. This minimum is equivalent to the first dark fringe in a single-slit diffraction pattern for light. (b) Suppose that yellow light (wave length \(=580 \mathrm{~nm}\), in vacuum) passes through a doorway and that the first dark fringe in its diffraction pattern is located at the angle determined in part (a). How wide would this hypothetical doorway have to be?

Light that has a wavelength of \(668 \mathrm{~nm}\) passes through a slit \(6.73 \times 10^{-6} \mathrm{~m}\) wide and falls on a screen that is \(1.85 \mathrm{~m}\) away. What is the distance on the screen from the center of the central bright fringe to the third dark fringe on either side?

A flat screen is located \(0.60 \mathrm{~m}\) away from a single slit. Light with a wavelength of 510 \(\mathrm{nm}\) (in vacuum) shines through the slit and produces a diffraction pattern. The width of the central bright fringe on the screen is \(0.050 \mathrm{~m}\). What is the width of the slit?

Light shines through a single slit whose width is \(5.6 \times 10^{-4} \mathrm{~m}\). A diffraction pattern is formed on a flat screen located \(4.0 \mathrm{~m}\) away. The distance between the middle of the central bright fringe and the first dark fringe is \(3.5 \mathrm{~mm}\). What is the wave length of the light?

How many dark fringes will be produced on either side of the central maximum if light \((\lambda=651 \mathrm{~nm})\) is incident on a single slit that is \(5.47 \times 10^{-6} \mathrm{~m}\) wide?

ssm The central bright fringe in a single-slit diffraction pattern has a width that equals the distance between the screen and the slit. Find the ratio \(\lambda / W\) of the wavelength of the light to the width of the slit.

The central bright fringe in a single-slit diffraction pattern has a width that equals the distance between the screen and the slit. Find the ratio \(\lambda / W\) of the wavelength of the light to the width of the slit.

In a single-slit diffraction pattern on a flat screen, the central bright fringe is \(1.2 \mathrm{~cm}\) wide when the slit width is \(3.2 \times 10^{-5} \mathrm{~m}\). When the slit is replaced by a second slit, the wavelength of the light and the distance to the screen remaining unchanged, the central bright fringe broadens to a width of \(1.9 \mathrm{~cm}\). What is the width of the second slit? It may be assumed that \(\theta\) is so small that \(\sin \theta \approx \tan \theta\).

In a single-slit diffraction pattern, the central fringe is 450 times as wide as the slit. The screen is 18000 times farther from the slit than the slit is wide. What is the ratio \(\lambda / W\) where \(\lambda\) is the wavelength of the light shining through the slit and \(W\) is the width of the slit? Assume that the angle that locates a dark fringe on the screen is small, so that \(\sin \theta \approx \tan \theta\)

A hunter who is a bit of a braggart claims that, from a distance of \(1.6 \mathrm{~km}\), he can selectively shoot either of two squirrels who are sitting ten centimeters apart on the same branch of a tree. What's more, he claims that he can do this without the aid of a telescopic sight on his rifle. (a) Determine the diameter of the pupils of his eyes that would be required for him to be able to resolve the squirrels as separate objects. In this calculation use a wavelength of \(498 \mathrm{~nm}\) (in vacuum) for the light. (b) State whether his claim is reasonable, and provide a reason for your answer. In evaluating his claim, consider that the human eye automatically adjusts the diameter of its pupil over a typical range of 2 to \(8 \mathrm{~mm}\), the larger values coming into play as the lighting becomes darker. Note also that under dark conditions, the eye is most sensitive to a wavelength of \(498 \mathrm{~nm}\).

Two stars are \(3.7 \times 10^{11} \mathrm{~m}\) apart and are equally distant from the earth. A telescope has an objective lens with a diameter of \(1.02 \mathrm{~m}\) and just detects these stars as separate objects. Assume that light of wavelength \(550 \mathrm{~nm}\) is being observed. Also assume that diffraction effects, rather than atmospheric turbulence, limit the resolving power of the telescope. Find the maximum distance that these stars could be from the earth.

Multiple-Concept Example 7 illustrates how the concepts needed in this problem are applied. The largest refracting telescope in the world is at the Yerkes Observatory in Williams Bay, Wiscon sin. The objective of the telescope has a diameter of \(1.02 \mathrm{~m}\). Two objects are \(3.75 \times 10^{4} \mathrm{~m}\) from the telescope. With light of wavelength \(565 \mathrm{~nm}\), how close can the objects be to each other so that they are just resolved by the telescope?

Astronomers have discovered a planetary system orbiting the star Upsilon Andromedae, which is at a distance of \(4.2 \times 10^{17} \mathrm{~m}\) from the earth. One planet is believed to be located at a distance of \(1.2 \times 10^{11} \mathrm{~m}\) from the star. Using visible light with a vacuum wavelength of \(550 \mathrm{~nm}\), what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

At the surface of the moon, which is \(3.77 \times 10^{8} \mathrm{~m}\) away, the light strikes a reflector left there by astronauts. The reflected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about \(0.20 \mathrm{~m}\), and diffraction causes the beam to spread as the light travels to the moon. In effect, the first circular dark fringe in the diffraction pattern defines the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

A spotlight sends red light (wavelength \(=694.3 \mathrm{nm}\) ) to the moon. At the surface of the moon, which is \(3.77 \times 10^{8} \mathrm{~m}\) away, the light strikes a reflector left there by astronauts. The reflected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about \(0.20 \mathrm{~m}\), and diffraction causes the beam to spread as the light travels to the moon. In effect, the first circular dark fringe in the diffraction pattern defines the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

Two concentric circles of light emit light whose wavelength is \(555 \mathrm{~nm}\). The larger circle has a radius of \(4.0 \mathrm{~cm}\), and the smaller circle has a radius of \(1.0 \mathrm{~cm}\). When taking a picture of these lighted circles, a camera admits light through an aperture whose diameter is \(12.5 \mathrm{~mm}\). What is the maximum distance at which the camera can (a) distinguish one circle from the other and (b) reveal that the inner circle is a circle of light rather than a solid disk of light?

The wavelength of the laser beam used in a compact disc player is \(780 \mathrm{~nm}\). Suppose that a diffraction grating produces first-order tracking beams that are \(1.2 \mathrm{~mm}\) apart at a distance of \(3.0 \mathrm{~mm}\) from the grating. Estimate the spacing between the slits of the grating.

The light shining on a diffraction grating has a wavelength of \(495 \mathrm{nm}\) (in vacuum). The grating produces a second-order bright fringe whose position is defined by an angle of \(9.34^{\circ} .\) How many lines per centimeter does the grating have?

For a wavelength of \(420 \mathrm{~nm}\), a diffraction grating produces a bright fringe at an angle of \(26^{\circ}\). For an unknown wavelength, the same grating produces a bright fringe at an angle of \(41^{\circ} .\) In both cases the bright fringes are of the same order \(m\). What is the unknown wavelength?

A diffraction grating is \(1.50 \mathrm{~cm}\) wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of \(18.0^{\circ} .\) What is the wavelength (in \(\mathrm{nm}\) )?

A diffraction grating has 2604 lines per centimeter, and it produces a principal maximum at \(\theta=30.0^{\circ} .\) The grating is used with light that contains all wavelengths between 410 and \(660 \mathrm{~nm}\). What is (are) the wavelength(s) of the incident light that could have produced this maximum?

Violet light (wavelength \(=410 \mathrm{~nm}\) ) and red light (wavelength \(=660 \mathrm{~nm}\) ) lie at opposite ends of the visible spectrum. (a) For each wavelength, find the angle \(\theta\) that locates the first-order maximum produced by a grating with 3300 lines \(/ \mathrm{cm}\). This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for (b) the second-order maximum and (c) the third-order maximum, (d) From your results, decide whether there is an overlap between any of the "rainbows" and, if so, specify which orders overlap.

Three, and only three, bright fringes can be seen on either side of the central maximum when a grating is illuminated with light \((\lambda=510 \mathrm{~nm}) .\) What is the maximum number of lines \(/ \mathrm{cm}\) for the grating?

Two gratings \(A\) and \(B\) have slit separations \(d_{A}\) and \(d_{B}\), respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating \(\mathrm{B}\), it is observed that the first-order maximum of \(\mathrm{A}\) is exactly replaced by the secondorder maximum of B. (a) Determine the ratio \(d_{\mathrm{B}} / d_{\mathrm{A}}\) of the spacings between the slits of the gratings. (b) Find the next two principal maxima of grating \(\mathrm{A}\) and the principal maxima of B that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.

Two gratings \(A\) and \(B\) have slit separations \(d_{\mathrm{A}}\) and \(d_{\mathrm{B}},\) respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating \(\mathrm{B},\) it is observed that the first-order maximum of \(\mathrm{A}\) is exactly replaced by the secondorder maximum of B. (a) Determine the ratio \(d_{\mathrm{B}} / d_{\mathrm{A}}\) of the spacings between the slits of the gratings. (b) Find the next two principal maxima of grating A and the principal maxima of \(\mathrm{B}\) that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.

In a Young's double-slit experiment, the seventh dark fringe is located \(0.025 \mathrm{~m}\) to the side of the central bright fringe on a flat screen, which is \(1.1 \mathrm{~m}\) away from the slits. The separation between the slits is \(1.4 \times 10^{-4} \mathrm{~m}\). What is the wavelength of the light being used?

The transmitting antenna for a radio station is \(7.00 \mathrm{~km}\) from your house. The frequency of the electromagnetic wave broad cast by this station is \(536 \mathrm{kHz}\). The station builds a second transmitting antenna that broadcasts an identical electromagnetic wave in phase with the original one. The new antenna is \(8.12 \mathrm{~km}\) from your house. Does constructive or destructive interference occur at the receiving antenna of your radio? Show your calculations.

A nonreflective coating of magnesium fluoride \((n=1.38)\) covers the glass \((n=1.52)\) of a camera lens. Assuming that the coating prevents reflection of yellow-green light (wavelength in vacuum \(=565 \mathrm{~nm}\) ), determine the minimum nonzero thickness that the coating can have.

A single slit has a width of \(2.1 \times 10^{-6} \mathrm{~m}\) and is used to form a diffraction pattern. Find the angle that locates the second dark fringe when the wavelength of the light is (a) 430 \(\mathrm{nm}\) and (b) \(660 \mathrm{~nm}\).

Two parallel slits are illuminated by light composed of two wave lengths, one of which is \(645 \mathrm{~nm}\). On a viewing screen, the light whose wavelength is known produces its third dark fringe at the same place where the light whose wavelength is unknown produces its fourth-order bright fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wave length?