A mica sheet that is thick is suspended in air. In

A phase difference due to a path-length difference is observed for monochromatic visible light. Which phase difference requires the least (minimum) path-length difference? (a) (b) (c) (d) The answer depends on the wavelength of the light.

Which of the following pairs of light sources are coherent: (a) two candles, (b) one point source and its image in a plane mirror, (c) two pinholes uniformly illuminated by the same point source, (d) two headlights of a car, (e) two images of a point source due to reflection from the front and back surfaces of a soap film?

The spacing between Newton’s rings decreases rapidly as the diameter of the rings increases. Explain why this result occurs.

If the angle of a wedge-shaped air film such as the angle in Example 33-2 is too large, fringes are not observed. Why?

Why must a film that is used to observe interference colors be thin?

A loop of wire is dipped in soapy water and held up so that the soap film is vertical. (a) Viewed by reflection using white SSM 270, 90, 180, light, the top of the film appears black. Explain why. (b) Below the black region are colored bands. Is the first band red or violet?

A two-slit interference pattern is formed using monochromatic laser light that has a wavelength of At the second maximum from the central maximum, what is the path-length difference between the light coming from each of the slits? (a) (b) (c) (d)

A two-slit interference pattern is formed using monochromatic laser light that has a wavelength of At the first minimum from the central maximum, what is the path-length difference between the light coming from each of the slits? (a) (b) (c) (d)

Atwo-slit interference pattern is formed using monochromatic laser light that has a wavelength of What happens to the distance between the first maximum and the central maximum as the two slits are moved closer together? (a) The distance increases. (b) The distance decreases. (c) The distance remains the same.

A two-slit interference pattern is formed using two different monochromatic lasers, one green and one red. Which color light has its first maximum closer to the central maximum? (a) green, (b) red, (c) Both maxima are at the same location.

A single-slit diffraction pattern is formed using monochromatic laser light that has a wavelength of What happens to the distance between the first maximum and the central maximum as the slit is made narrower? (a) The distance increases. (b) The distance decreases. (c) The distance remains the same.

Equation 33-2, which is and Equation 33-11, which is are sometimes confused. For each equation, define the symbols and explain the equations application.

When a diffraction grating is illuminated by white light, the first-order maximum of green light (a) is closer to the central maximum than the first-order maximum of red light, (b) is closer to the central maximum than the first-order maximum of blue light, (c) overlaps the second-order maximum of red light, (d) overlaps the second-order maximum of blue light.

Adouble-slit interference experiment is set up in a chamber that can be evacuated. Using light from a heliumneon laser, an interference pattern is observed when the chamber is open to air. As the chamber is evacuated, one will note that (a) the interference fringes remain fixed, (b) the interference fringes move closer together, (c) the interference fringes move farther apart, (d) the interference fringes disappear completely.

True or false: (a) When waves interfere destructively, the energy is converted into heat. (b) Interference patterns are observed only if the relative phases of the waves that superpose remain constant. (c) In the Fraunhofer diffraction pattern for a single slit, the narrower the slit, the wider the central maximum of the diffraction pattern. (d) A circular aperture can produce both a Fraunhofer diffraction pattern and a Fresnel diffraction pattern. (e) The ability to resolve two point sources depends on the wavelength of the light.

You observe two very closely spaced sources of white light through a circular opening using various filters. Which color filter is most likely to prevent your resolving the images on your retinas as coming from two distinct sources? (a) red, (b) yellow,(c) green, (d) blue, (e) The filter choice is irrelevant.

Explain why the ability to distinguish the two headlights of an oncoming car at a given distance is easier for a human eye at night than during the daytime. Assume the headlights of the oncoming car are on both during both daytime and nighttime hours.

It is claimed that the Great Wall of China is the only human-made object that can be seen from space using no equipment. Check to see if this claim is true, based on the resolving power of the human eye. Assume the observers are in a low-Earth orbit that has an altitude of about

(a) Estimate how close an approaching car at night on a flat, straight part of a highway must be before its headlights can be distinguished from the single headlight of a motorcycle. (b) Estimate how far ahead of you a car is if its two red taillights merge to look as if they were one.

A small loudspeaker is located at a large distance to the east from you. The loudspeaker is driven by a sinusoidal current whose frequency can be varied. Estimate the lowest frequency for which one of your ears would receive the sound waves exactly out of phase with that received by your other ear when you are facing north.

Estimate the maximum distance at which a binary star system could be resolvable by the human eye. Assume the two stars are about fifty times farther apart than Earth and the Sun are. Neglect any atmospheric effects. (A test similar to this eye test was used in ancient Rome to test for eyesight acuity before entering the army. A person who had normal eyesight could just barely resolve two well-known stars that appear close in the sky. Anyone who could not tell there were two stars failed the test.)

Light that has a wavelength of is incident normally on a film of water thick. (a) What is the wavelength of the light in the water? (b) How many wavelengths are contained in the distance where is the thickness of the film? (c) The film has air on both sides. What is the phase difference between the wave reflected from the front surface and the wave reflected from the back surface in the region where the two reflected waves superpose?

Two coherent microwave sources both produce waves that each have a wavelength equal to The sources are located in the plane, one at and the other at If the sources are in phase, find the difference in phase between the two waves for a receiver located at the origin.

Awedge-shaped film of air is made by placing a small slip of paper between the edges of two flat plates of glass. Light that has a wavelength of is incident normally on the glass plates, and interference fringes are observed by reflection. (a) Is the first fringe near the point of contact of the plates dark or bright? Why? (b) If there are five dark fringes per centimeter, what is the angle of the wedge?

The diameters of fine fibers can be accurately measured using interference patterns. Two optically flat pieces of glass that each have a length are arranged with the fiber between them, as shown in Figure 33-40. The setup is illuminated by monochromatic light, and the resulting interference fringes are observed. Suppose that is and that yellow sodium light is used for illumination. If 19 bright fringes are seen along this distance, what are the limits on the diameter of the fiber? Hint: The nineteenth fringe might not be right at the end, but you do not see a twentieth fringe at all.

Light that has a wavelength equal to is used to illuminate two glass plates at normal incidence. The plates are in length, touch at one end, and are separated at the other end by a wire that has a radius equal to How many bright fringes appear along the total length of the plates? 0.025 mm.

A thin film having an index of refraction of 1.50 is surrounded by air. It is illuminated normally by white light. Analysis of the reflected light shows that the wavelengths are the only missing wavelengths in or near the visible portion of the spectrum. That is, for those wavelengths, there is destructive interference. (a) What is the thickness of the film? (b) What visible wavelengths are brightest in the reflected interference pattern? (c) If this film were resting on glass that has an index of refraction of 1.60, what wavelengths in the visible spectrum would be missing from the reflected light?

A drop of oil (refractive index of 1.22) floats on water (refractive index of 1.33). When reflected light is observed from above, as shown in Figure 33-41, what is the thickness of the drop at the point where the second red fringe, counting from the edge of the drop, is observed? Assume red light has a wavelength of

A film of oil that has an index of refraction of 1.45 rests on an optically flat piece of glass that has an index of refraction of 1.60. When illuminated by white light at normal incidence, light of wavelengths and is predominant in the reflected light. Determine the thickness of the oil film.

A film of oil that has an index of refraction equal to 1.45 floats on water. When illuminated by white light at normal incidence, light of wavelengths and is predominant in the reflected light. Determine the thickness of the oil film.

ANewtons ring apparatus consists of a plano-convex glass lens that has a radius of curvature and rests on a flat glass plate, as shown in Figure 33-42. The thin film is air of variable thickness. The apparatus is illuminated from above by light from a sodium lamp that has a wavelength equal to The pattern is viewed by reflected light. (a) Show that for a thickness the condition for a bright (constructive) interference ring is where (b) Show that for the radius of a fringe is related to by (c) For a radius of curvature of and a lens diameter of how many bright fringes would you see in the reflected light? (d) What would be the diameter of the sixth bright fringe? (e) If the glass used in the apparatus has an index of refraction and water replaces the air between the two pieces of glass, explain qualitatively the changes that will take place in the bright-fringe pattern.

A plano-convex glass lens of radius of curvature rests on an optically flat glass plate. The arrangement is illuminated from above using monochromatic light that has a wavelength. The indices of refraction of the lens and plate are 1.60. Determine the radii of the first and second bright fringe from the center in the reflected light.

Suppose that before the lens of Problem 32 is placed on the plate, a film of oil that has a refractive index equal to 1.82 is deposited on the plate. What will then be the radii of the first and second bright fringes?

Two narrow slits separated by are illuminated by light that has a wavelength, and the interference pattern is viewed on a screen away. Calculate the number of bright fringes per centimeter on the screen in the region near the center fringe.

Using a conventional two-slit apparatus and light that has a wavelength, 28 bright fringes per centimeter are observed near the center of a screen away. What is the slit separation?

Light that has a wavelength and is from a heliumneon laser is shone normal to a plane having two slits. The first interference maximum is from the central maximum on a screen away. (a) Find the separation of the slits. (b) How many interference maxima is it, in principle, possible to observe?

Two narrow slits are separated by a distance Their interference pattern is to be observed on a screen a large distance away. (a) Calculate the spacing between successive maxima near the center fringe for light that has a wavelength, when is and is (b) Would you expect to be able to observe the interference of light on the screen for this situation? (c) How close together should the slits be placed for the maxima to be separated by for this wavelength and screen distance?

Light is incident at an angle with the normal to a vertical plane that has two slits of separation (Figure 33-43). Show that the interference maxima are located at angles given by

White light falls at an angle of to the normal of a plane that has a pair of slits separated by What visible wavelengths give a bright interference maximum in the transmitted light in the direction normal to the plane? (See Problem 38.)

Two small loudspeakers are separated by a distance of as shown in Figure 33-44. The speakers are driven in phase with a sine wave signal of frequency A small microphone is placed a distance away from the speakers on the axis running through the middle of the two speakers, and the microphone is then moved perpendicular to the axis. Where does the microphone record the first minimum and the first maximum of the interference pattern from the speakers? The speed of sound in air is

Light that has a wavelength is incident on a long narrow slit. Find the angle of the first diffraction minimum if the width of the slit is (a) (b) and (c)

Plane microwaves are incident on a thin metal sheet that has a long, narrow slit of width in it. The microwave radiation strikes the sheet at normal incidence. The first diffraction minimum is observed at What is the wavelength of the microwaves?

Measuring the distance to the moon (lunar ranging) is routinely done by firing short-pulse lasers and measuring the time it takes for the pulses to reflect back from the moon. Apulse is fired from Earth. To send the pulse out, the pulse is expanded so that it fills the aperture of a telescope. Assuming the only thing spreading the beam out is diffraction and that the light wavelength is how large will the beam be when it reaches the moon, 3.82 _ 105 km away? SSM

How many interference maxima will be contained in the central diffraction maximum in the interferencediffraction pattern of two slits if the separation of the slits is exactly 5 times their width? How many will there be if the slit separation is an integral multiple of the slit width (that is, for any value of

A two-slit Fraunhofer interferencediffraction pattern is observed using light that has a wavelength equal to The slits have a separation of and an unknown width. (a) Find the width if the fifth interference maximum is at the same angle as the first diffraction minimum. (b) For that case, how many bright interference fringes will be seen in the central diffraction maximum?

A two-slit Fraunhofer interferencediffraction pattern is observed using light that has a wavelength equal to The slits have widths of and are separated by How many bright fringes will be seen in the central diffraction maximum?

Suppose that the central diffraction maximum for two slits has 17 interference fringes for some wavelength of light. How many interference fringes would you expect in the diffraction maximum adjacent to one side of the central diffraction maximum?

Light that has a wavelength equal to illuminates two slits that both have widths equal to and separations equal to (a) How many interference maxima fall within the full width of the central diffraction maximum? (b) What is the ratio of the intensity of the third interference maximum to one side of the center interference maximum to the intensity of the center interference maximum?

Find the resultant of the two waves whose electric fields at a given location vary with time as follows: and

Find the resultant of the two waves whose electric fields at a given location vary with time as follows: and

Monochromatic light is incident on a sheet with a long narrow slit (Figure 33-45). Let be the intensity at the central maximum of the diffraction pattern on a distant screen, and let I be the intensity at the second intensity maximum from the central intensity maximum. The distance from this second intensity maximum to the far edge of the slit is longer than the distance from the second intensity maximum to the near edge of the slit by approximately 2.5 wavelengths. What is the ratio of I to I0?

Monochromatic light is incident on a sheet that has three long narrow parallel equally spaced slits a distance apart. (a) Show that the positions of the interference minima on a screen a large distance away from the sheet that has the three equally spaced slits (spacing with are given approximately by where that is, is not a multiple of 3. (b) For a screen distance of a light wavelength of and a source spacing of calculate the width of the principal interference maxima (the distance between successive minima) for three sources.

Monochromatic light is incident on a sheet that has four long narrow parallel equally spaced slits a distance apart. (a) Show that the positions of the interference minima on a screen a large distance away from the sheet that has the four equally spaced slits (spacing with are given approximately by where that is, is not a multiple of 4. (b) For a screen distance of a light wavelength of and a source spacing of calculate the width of the principal interference maxima (the distance between successive minima) for four sources. Compare the width with that for two sources with the same spacing.

Light that has a wavelength equal to falls normally on four slits. Each slit is wide and the center-to-center separation between it and the next slit is (a) Find the angular width of the central intensity maximum of the single-slit diffraction pattern on a distant screen. (b) Find the angular position of all interference intensity maxima that lie inside the central diffraction maximum. (c) Find the angular width of the central interference maximum. That is, find the angle between the first interference intensity minima on either side of the central interference maximum. (d) Sketch the relative intensity as a function of the sine of the angle.

Three slits, each separated from its neighbor by are illuminated at the central intensity maximum by a coherent light source that has a wavelength equal to The slits are extremely narrow. A screen is located from the slits. The intensity is Consider a location from the central maximum. (a) Draw a phasor diagram suitable for the addition of the three harmonic waves at that location. (b) From the phasor diagram, calculate the intensity of light at that location.

In single-slit Fraunhofer diffraction, the intensity pattern (Figure 33-11) consists of a broad central maximum with a sequence of secondary maxima to either side of the central maximum. The intensity is given by where is the phase difference between the wavelets arriving from the opposite edges of the slits. Calculate the values of for the first three seconday maxima to one side of the central maximum by finding the values of for which is equal to zero. Check your results by comparing your answers with approximate values for of and . (That these values for are approximatly correct at the secondary intensity maxima is discussed in the discussion surrounding Figure 33-27.)

Light that has a wavelength equal to is incident on a pinhole of diameter (a) What is the angle between the central maximum and the first diffraction minimum for a Fraunhofer diffraction pattern? (b) What is the distance between the central maximum and the first diffraction minimum on a screen away?

Two sources of light that both have wavelengths equal to are away from the pinhole of Problem 57. How far apart must the sources be for their diffraction patterns to be resolved by Rayleighs criterion?

Two sources of light that both have wavelengths equal to are separated by a horizontal distance They are from a vertical slit of width What is the smallest value of for which the diffraction pattern of the sources can be resolved by Rayleighs criterion?

The ceiling of your lecture hall is probably covered with acoustic tile, which has small holes separated by about (a) Using light that has a wavelength of how far could you be from this tile and still resolve the holes? Assume the diameter of the pupil of each of your eyes is about (b) Could you resolve the holes better using red light or violet light? Explain your answer.

The telescope on Mount Palomar has a diameter of Suppose a double star were 4.00 light-years away. Under ideal conditions, what must be the minimum separation of the two stars for their images to be resolved using light that has a wavelength equal to

The star Mizar in Ursa Major is a binary system of stars that have nearly equal magnitudes. The angular separation between the two stars is 14 seconds of arc. What is the minimum diameter of the pupil that allows resolution of the two stars using light that has a wavelength equal to

Adiffraction grating that has 2000 slits per centimeter is used to measure the wavelengths emitted by hydrogen gas. (a) At what angles in the first-order spectrum would you expect to find the two violet lines that have wavelengths of and (b) What are the angles if the grating has per centimeter?

Using a diffraction grating that has 2000 slits per centimeter, two lines in the first-order hydrogen spectrum are found at angles of and What are the wavelengths of the lines?

The colors of many butterfly wings and beetle carapaces are due to the effects of diffraction. The Morpho butterfly has structural elements on its wings that effectively act as a diffraction grating that has an spacing. At what angle will the first diffraction maximum occur for normally incident light diffracted by the butterflys wings? Assume the light is blue and has a wavelength of

A diffraction grating that has 2000 slits per centimeter is used to analyze the spectrum of mercury. (a) Find the angular separation in the first-order spectrum of the two lines that have wavelengths equal to and (b) How wide must the beam on the grating be for the lines to be resolved?

A diffraction grating that has 4800 lines per centimeter is illuminated at normal incidence using white light (wavelength range of to How many orders of complete spectra can one observe in the transmitted light? Do any of these orders overlap? If so, describe the overlapping regions.

A square diffraction grating that has an area of has a resolution of in the fourth order. At what angle should you look to see a wavelength of in the fourth order?

Sodium light that has a wavelength equal to falls normally on a diffraction grating ruled with 4000 lines per centimeter. The Fraunhofer diffraction pattern is projected onto a screen a distance of from the grating by a focal-length lens that is placed immediately in front of the grating. Find (a) the distances of the first and second order intensity maxima from the central intensity maximum, (b) the width of the central maximum, and (c) the resolution in the first order. (Assume the entire grating is illuminated.)

The spectrum of neon is exceptionally rich in the visible region. Among the many lines are two lines at wavelengths of 519.313 nm and 519.322 nm. If light from a neon discharge tube is normally incident on a transmission grating that has 8400 lines per centimeter and the spectrum is observed in second order, what must be the width of the grating that is illuminated, so that the two lines can be resolved?

Mercury has several stable isotopes, among them and The strong spectral line of mercury, at about is a composite of spectral lines from the various mercury isotopes. The wavelengths of the line for and are and respectively. What must be the resolving power of a grating capable of resolving the two isotopic lines in the third-order spectrum? If the grating is illuminated over a region, what must be the number of lines per centimeter of the grating?

A diffraction grating has lines per unit length. Show that the angular separation of two lines of wavelengths and is approximately where is the order number.

For a diffraction grating in which all the surfaces are normal to the incident radiation, most of the energy goes into the zeroth order, which is useless from a spectroscopic point of view, because in zeroth order all the wavelengths are at Therefore, modern reflection gratings have shaped, or blazed, grooves, as shown in Figure 33-46. This shifts the specular reflection, which contains most of the energy, from the zeroth order to some higher order. (a) Calculate the blaze angle in terms of the groove separation the wavelength and the order number in which specular reflection is to occur for (b) Calculate the proper blaze angle for the specular reflection to occur in the second order for light of wavelength incident on a grating with lines per centimeter. SSM

In this problem, you will derive the relation (Equation 33-27) for the resolving power of a diffraction grating having slits separated by a distance To do this, you will calculate the angular separation between the intensity maximum and intensity minimum for some wavelength and set it equal to the angular separation of the maximum for two nearby wavelengths. (a) First show that the phase difference waves between the waves from two adjacent slits is given by (b) Next, differentiate that expression to show that a small change in angle results in a change in phase of given by (c) Then for slits, the angular separation between an interference maximum and an interference minimum corresponds to a phase change of Use that to show that the angular separation between the intensity maximum and intensity minimum for some wavelength is given by (d) Next, use the fact that the angle of the interference maximum for wavelength is specified by (Equation 33-26). Compute the differential of each side of the equation to show that angular separation of the mth-order maximum for two nearly equal wavelengths differing by is given by (e) According to Rayleighs criterion, two wavelengths will be resolved in the order if the angular separation of the wavelengths, given by the Part (d) result, equals the angular separation of the interference maximum and the interference minimum given by the Part (c) result. Use this to arrive at (Equation 33-27) for the resolving power of a grating.

Naturally occuring coronas (brightly colored rings) are sometimes seen around the moon or the Sun when viewed through a thin cloud. (Warning: When viewing a Sun corona, be sure that the entire Sun is blocked by the edge of a building, a tree, or a traffic pole to safeguard your eyes.) These coronas are due to diffraction of light by small water droplets in the cloud. A typical angular diameter for a coronal ring is about From this, estimate the size of the water droplets in the cloud. Assume that the water droplets can be modeled as opaque disks that have the same radius as the droplet, and that the Fraunhofer diffraction pattern from an opaque disk is the same as the pattern from an aperture of the same diameter. (This last statement is known as Babinets principle.)

An artificial corona (see Problem 75) can be made by placing a suspension of polystyrene microspheres in water. Polystyrene microspheres are small, uniform spheres that are made of plastic and have indices of refraction equal to 1.59. Assuming that the water has an index of refraction equal to 1.33, what is the angular diameter of such an artificial corona if particles are illuminated by light from a heliumneon laser that has a wavelength in air of

Coronas (see Problem 75) can be caused by pollen grains, typically of birch or pine. Such grains are irregular in shape, but they can be treated as if they had an average diameter of about What is the angular diameter (in degrees) of the corona for blue light? What is the angular diameter (in degrees) of the corona for red light?

Light from a HeNe laser is directed upon a human hair, in an attempt to measure its diameter by examining the diffraction pattern. The hair is mounted in a frame from a wall, and the central diffraction maximum is measured to be wide. What is the diameter of the hair? (The diffraction pattern of a hair that has a diameter is the same as the diffraction pattern of a single slit that has a width See Babinets principle discussed in Problem 75.)

A long, narrow horizontal slit lies above a plane mirror, which is in the horizontal plane. The interference pattern produced by the slit and its image is viewed on a screen from the slit. The wavelength of the light is (a) Find the distance from the mirror to the first maximum. (b) How many dark bands per centimeter are seen on the screen?

A radio telescope is situated at the edge of a lake. The telescope is looking at light from a radio galaxy that is just rising over the horizon. If the height of the antenna is above the surface of the lake, at what angle above the horizon will the radio galaxy be when the telescope is centered in the first intensity interference maximum of the radio waves? Assume the wavelength of the radio waves is Hint: The interference is caused by the light reflecting off the lake and remember that this reflection will result in a phase shift.

The diameter of the radio telescope at Arecibo, Puerto Rico, is What is the smallest angular separation of two objects that this telescope can detect when it is tuned to detect microwaves of 3.2-cm wavelength

A thin layer of a transparent material that has an index of refraction of 1.30 is used as a nonreflective coating on the surface of glass that has an index of refraction of 1.50. What should the minimum thickness of the material be for the material to be nonreflecting for light that has a wavelength of

A FabryPerot interferometer (Figure 33-47) consists of two parallel, half-silvered mirrors that face each other and are separated by a small distance A half-silvered mirror is one that transmits of the incident intensity and reflects of the incident intensity. Show that when light is incident on the interferometer at an angle of incidence the transmitted light will have maximum intensity when 2a _ ml cos u. SSM

A mica sheet that is thick is suspended in air. In reflected light, there are gaps in the visible spectrum at 421, 474, 542, and 633 nm. Find the index of refraction of the mica sheet.

A camera lens is made of glass that has an index of refraction of 1.60. This lens is coated with a magnesium fluoride film (index of refraction equal to 1.38) to enhance its light transmission. The purpose of this film is to produce zero reflection for light that has a wavelength of Treat the lens surface as a flat plane and the film as a uniformly thick flat film. (a) What minimum thickness of this film will accomplish its objective? (b) Would there be destructive interference for any other visible wavelengths? (c) By what factor would the reflection for light of wavelength be reduced by the presence of this film? Neglect the variation in the reflected light amplitudes from the two surfaces.

In a pinhole camera, the image is fuzzy because of geometry (rays arrive at the film after passing through different parts of the pinhole) and because of diffraction. As the pinhole is made smaller, the fuzziness due to geometry is reduced, but the fuzziness due to diffraction is increased. The optimum size of the pinhole for the sharpest possible image occurs when the spread due to diffraction equals the spread due to the geometric effects of the pinhole. Estimate the optimum size of the pinhole if the distance from the pinhole to the film is and the wavelength of the light is

The Impressionist painter Georges Seurat used a technique called pointillism, in which his paintings are composed of small, closely spaced dots of pure color, each about in diameter. The illusion of the colors blending together smoothly is produced in the eyes of the viewer by diffraction effects. Calculate the minimum viewing distance for this effect to work properly. Use the wavelength of visible light that requires the greatest distance between dots, so that you are sure the effect will work for all visible wavelengths. Assume the pupil of the eye has a diameter of 3.0 mm. SSM