Red laser light from a He-Ne laser (A = 632.8 nm) isused

If 680-nm light falls on a slit 0.0365 mm wide, what is the angular width of the central diffraction peak?

Monochromatic light falls on a slit that is 2.60 X 10-3 mm wide. If the angle between the first dark fringes on either side of the central maximum is 32.0 (dark fringe to dark fringe), what is the wavelength of the light used?

Light of wavelength 580 nm falls on a slit that is 3.80 X 10_3mm wide. Estimate how far the first brightest diffraction fringe is from the strong central maximum if the screen is 10.0 m away.

Consider microwaves which are incident perpendicular to a metal plate which has a 1.6-cm slit in it. Discuss the angles at which there are diffraction minima for wavelengths of (a) 0.50 cm, (b) 1.0 cm, and (c) 3.0 cm.

If parallel light falls on a single slit of width D at a 23.0 angle to the normal, describe the diffraction pattern.

Monochromatic light of wavelength 633 nm falls on a slit. If the angle between the first bright fringes on either side of the central maximum is 35, estimate the slit width.

If a slit diffracts 580-nm light so that the diffraction maximum is 6.0 cm wide on a screen 2.20 m away, what will be the width of the diffraction maximum for light with a wavelength of 460 nm?

(a) For a given wavelength A, what is the minimum slit width for which there will be no diffraction minima? (b) What is the minimum slit width so that no visible light exhibits a diffraction minimum?

When blue light of wavelength 440 nm falls on a single slit, the first dark bands on either side of center are separated by 55.0. Determine the width of the slit.

A single slit 1.0 mm wide is illuminated by 450-nm light. What is the width of the central maximum (in cm) in the diffraction pattern on a screen 5.0 m away?

Coherent light from a laser diode is emitted through a rectangular area 3.0 /mi X 1.5 fjum (horizontal-by-vertical). If the laser light has a wavelength of 780 nm, determine the angle between the first diffraction minima (a) above and below the central maximum, (b) to the left and right of the central maximum.

If you double the width of a single slit, the intensity of the light passing through the slit is doubled, (a) Show, however, that the intensity at the center of the screen increases by a factor of 4. (b) Explain why this does not violate conservation of energy.

Light of wavelength 750 nm passes through a slit 1.0 jam wide and a single-slit diffraction pattern is formed vertically on a screen 25 cm away. Determine the light intensity I 15 cm above the central maximum, expressed as a fraction of the central maximums intensity 70.

(a) Explain why the secondary maxima in the single-slit diffraction pattern do not occur precisely at j6/2 = (ra + 5)77 where ra = 1,2, 3,---. (b) By differentiating Eq. 35-7 with respect to j8 show that the secondary maxima occur when j8/2 satisfies the relation tan(/3/2) = /3/2. (c) Carefully and precisely plot the curves y = j6/2 and y = tan/3/2. From their intersections, determine the values of /3 for the first and second secondary maxima. What is the percent difference from /3/2 = (ra + \)tt1

If a double-slit pattern contains exactly nine fringes in the central diffraction peak, what can you say about the slit width and separation? Assume the first diffraction minimum occurs at an interference minimum.

Design a double-slit apparatus so that the central diffraction peak contains precisely seventeen fringes. Assume the first diffraction minimum occurs at (a) an interference miminum, (b) an interference maximum.

605-nm light passes through a pair of slits and creates an interference pattern on a screen 2.0 m behind the slits. The slits are separated by 0.120 mm and each slit is 0.040 mm wide. How many constructive interference fringes are formed on the screen? [Many of these fringes will be of very low intensity.]

In a double-slit experiment, if the central diffraction peak contains 13 interference fringes, how many fringes are contained within each secondary diffraction peak (between ra = +1 and +2 in Eq. 35-2). Assume the first diffraction minimum occurs at an interference minimum.

Two 0.010-mm-wide slits are 0.030 mm apart (center to center). Determine (a) the spacing between interference fringes for 580 nm light on a screen 1.0 m away and (b) the distance between the two diffraction minima on either side of the central maximum of the envelope.

Suppose d = D in a double-slit apparatus, so that the two slits merge into one slit of width 2D. Show that Eq. 35-9 reduces to the correct equation for single-slit diffraction.

In a double-slit experiment, let d = 5.00 D = 40.0 A. Compare (as a ratio) the intensity of the third-order interference maximum with that of the zero-order maximum.

How many fringes are contained in the central diffraction peak for a double-slit pattern if (a) d = 2.00Z), (b) d = 12.0D, (c) d = 4.50D, (d) d = 7.20D.

(a) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of 8 = 27rdsin0/A where d is the distance between adjacent slits and assume the slit width D A. (b) Show that there is only one secondary maximum between principal peaks.

What is the angular resolution limit (degrees) set by diffraction for the 100-inch (254-cm mirror diameter) Mt. Wilson telescope (A = 560 nm)?

Two stars 16 light-years away are barely resolved by a 66-cm (mirror diameter) telescope. How far apart are the stars? Assume A = 550 nm and that the resolution is limited by diffraction.

The nearest neighboring star to the Sun is about 4 light-years away. If a planet happened to be orbiting this star at an orbital radius equal to that of the Earth-Sun distance, what minimum diameter would an Earth-based telescopes aperture have to be in order to obtain an image that resolved this star-planet system? Assume the light emitted by the star and planet has a wavelength of 550 nm.

If you shine a flashlight beam toward the Moon, estimate the diameter of the beam when it reaches the Moon. Assume that the beam leaves the flashlight through a 5.0-cm aperture, that its white light has an average wavelength of 550 nm, and that the beam spreads due to diffraction only.

Suppose that you wish to construct a telescope that can resolve features 7.5 km across on the moon, 384,000 km away. You have a 2.0-m-focal-length objective lens whose diameter is 11.0 cm. What focal-length eyepiece is needed if your eye can resolve objects 0.10 mm apart at a distance of 25 cm? What is the resolution limit set by the size of the objective lens (that is, by diffraction)? Use A = 560 nm.

The normal lens on a 35-mm camera has a focal length of 50.0 mm. Its aperture diameter varies from a maximum of 25 mm (//2) to a minimum of 3.0 mm (//16). Determine the resolution limit set by diffraction for (//2) and (//16). Specify as the number of lines per millimeter resolved on the detector or film. Take A = 560 nm.

At what angle will 480-nm light produce a second-order maximum when falling on a grating whose slits are 1.35 X 10-3 cm apart?

A source produces first-order lines when incident normally on a 12,000-line/cm diffraction grating at angles 28.8, 36.7, 38.6, and 47.9. What are the wavelengths?

A 3500-line/cm grating produces a third-order fringe at a 26.0 angle. What wavelength of light is being used?

A grating has 6800 lines/cm. How many spectral orders can be seen (400 to 700 nm) when it is illuminated by white light?

How many lines per centimeter does a grating have if the third order occurs at a 15.0 angle for 650-nm light?

Red laser light from a He-Ne laser (A = 632.8 nm) is used to calibrate a diffraction grating. If this light creates a second-order fringe at 53.2 after passing through the grating, and light of an unknown wavelength A creates a first-order fringe at 20.6, find A

White light containing wavelengths from 410 nm to 750 nm falls on a grating with 7800 lines/cm. How wide is the first-order spectrum on a screen 2.80 m away?

A diffraction grating has 6.0 X 105 lines/m. Find the angular spread in the second-order spectrum between red light of wavelength 7.0 X 10_7m and blue light of wavelength 4.5 X 10-7 m.

A tungsten-halogen bulb emits a continuous spectrum of ultraviolet, visible, and infrared light in the wavelength range 360 nm to 2000 nm. Assume that the light from a tungsten-halogen bulb is incident on a diffraction grating with slit spacing d and that the first-order brightness maximum for the wavelength of 1200 nm occurs at angle 0. What other wavelengths within the spectrum of incident light will produce a brightness maximum at this same angle 0? [Optical filters are used to deal with this bothersome effect when a continuous spectrum of light is measured by a spectrometer.]

Show that the second- and third-order spectra of white light produced by a diffraction grating always overlap. What wavelengths overlap?

Two first-order spectrum lines are measured by a 9650 line/cm spectroscope at angles, on each side of center, of +2638', +4102' and -2618', -4027'. Calculate the wavelengths based on these data.

Suppose the angles measured in Problem 40 were produced when the spectrometer (but not the source) was submerged in water. What then would be the wavelengths (in air)?

The first-order line of 589-nm light falling on a diffraction grating is observed at a 16.5 angle. How far apart are the slits? At what angle will the third order be observed?

White light passes through a 610-line/mm diffraction grating. First-order and second-order visible spectra (rainbows) appear on the wall 32 cm away as shown in Fig. 35-47. Determine the widths ^ and l2 of the two rainbows (400 nm to 700 nm). In which order is the rainbow dispersed over a larger distance?FIGURE 35-47 Problem 43.

Missing orders occur for a diffraction grating when a diffraction minimum coincides with an interference maximum. Let D be the width of each slit and d the separation of slits, (a) Show that if d = 2D, all even orders (ra = 2, 4, 6, ) are missing. (b) Show there will be missing orders whenever d, _ m1 D ra2 where m1 and ra2 are integers, (c) Discusss the case d = D, the limit in which the space between slits becomes negligible.

Monochromatic light falls on a transmission diffraction grating at an angle <f> to the normal, (a) Show that Eq. 35-13 for diffraction maxima must be replaced by d(sin</> + sin0) = raA. ra = 0,1, 2, . (b) Explain the sign, (c ) Green light with a wavelength of 550 nm is incident at an angle of 15 to the normal on a diffraction grating with 5000 lines/cm. Find the angles at which the first-order maxima occur.

A 6500-line/cm diffraction grating is 3.18 cm wide. If light with wavelengths near 624 nm falls on the grating, how close can two wavelengths be if they are to be resolved in any order? What order gives the best resolution?

A diffraction grating has 16,000 rulings in its 1.9 cm width. Determine (a) its resolving power in first and second orders, and (b) the minimum wavelength resolution (AA) it can yield for A = 410 nm.

Let 580-nm light be incident normally on a diffraction grating for which d = 3.00D = 1050 nm. (a) How many orders (principal maxima) are present? (b) If the grating is 1.80 cm wide, what is the full angular width of each principal maximum?

X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.285 nm apart. At what angle <f> (relative to the surface, Fig. 35-28) must the X-rays be directed if the first diffraction maximum is to be observed?

First-order Bragg diffraction is observed at 26.8 relative to the crystal surface, with spacing between atoms of 0.24 nm. (a) At what angle will second order be observed? (b) What is the wavelength of the X-rays?

If X-ray diffraction peaks corresponding to the first three orders ( m = 1,2, and 3) are measured, can both the X-ray wavelength A and lattice spacing d be determined? Prove your answer

Two polarizers are oriented at 65 to one another. Unpolarized light falls on them. What fraction of the light intensity is transmitted?

Two Polaroids are aligned so that the light passing through them is a maximum. At what angle should one of them be placed so the intensity is subsequently reduced by half?

What is Brewsters angle for an air-glass (n = 1.58) surface?

What is Brewsters angle for a diamond submerged in water if the light is hitting the diamond (n = 2.42) while traveling in the water?

The critical angle for total internal reflection at a boundary between two materials is 55. What is Brewsters angle at this boundary? Give two answers, one for each material.

At what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light to (a) \ , (b) ^ ?

Two polarizers are oriented at 36.0 to one another. Light polarized at an 18.0 angle to each polarizer passes through both. What is the transmitted intensity (%)?

What would Brewsters angle be for reflections off the surface of water for light coming from beneath the surface? Compare to the angle for total internal reflection, and to Brewsters angle from above the surface.

Unpolarized light passes through six successive Polaroid sheets each of whose axis makes a 45 angle with the previous one. What is the intensity of the transmitted beam?

Two polarizers A and B are aligned so that their transmission axes are vertical and horizontal, respectively. A third polarizer is placed between these two with its axis aligned at angle 6 with respect to the vertical. Assuming vertically polarized light of intensity /0 is incident upon polarizer A, find an expression for the light intensity I transmitted through this three-polarizer sequence. Calculate the derivative dl/d0\ then use it to find the angle 6 that maximizes I.

The percent polarization P of a partially polarized beam of light is defined as Anax + Anin where /max and /min are the maximum and minimum intensities that are obtained when the light passes through a polarizer that is slowly rotated. Such light can be considered as the sum of two unequal plane-polarized beams of intensities /max and /mjn perpendicular to each other. Show that the light transmitted by a polarizer, whose axis makes an angle (f> to the direction in which /max is obtained, has intensity 1 + p cos 2(f> i + P /max where p = P/ 100 is the fractional polarization.

When violet light of wavelength 415 nm falls on a single slit, it creates a central diffraction peak that is 8.20 cm wide on a screen that is 2.85 m away. How wide is the slit?

A series of polarizers are each placed at a 10 interval from the previous polarizer. Unpolarized light is incident on this series of polarizers. How many polarizers does the light have to go through before it is \ of its original intensity?

The wings of a certain beetle have a series of parallel lines across them. When normally incident 480-nm light is reflected from the wing, the wing appears bright when viewed at an angle of 56. How far apart are the lines?

A teacher stands well back from an outside doorway 0.88 m wide, and blows a whistle of frequency 850 Hz. Ignoring reflections, estimate at what angle(s) it is not possible to hear the whistle clearly on the playground outside the doorway. Assume 340 m/s for the speed of sound.

Light is incident on a diffraction grating with 7600 lines/cm and the pattern is viewed on a screen located 2.5 m from the grating. The incident light beam consists of two wavelengths, Ai = 4.4 X 10_7m and A2 = 6.8 X 10_7m. Calculate the linear distance between the first-order bright fringes of these two wavelengths on the screen.

How many lines per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength?

When yellow sodium light, A = 589 nm, falls on a diffraction grating, its first-order peak on a screen 66.0 cm away falls 3.32 cm from the central peak. Another source produces a line 3.71 cm from the central peak. What is its wavelength? How many lines/cm are on the grating?

Two of the lines of the atomic hydrogen spectrum have wavelengths of 656 nm and 410 nm. If these fall at normal incidence on a grating with 8100 lines/cm, what will be the angular separation of the two wavelengths in the first-order spectrum?

(a) How far away can a human eye distinguish two car headlights 2.0 m apart? Consider only diffraction effects and assume an eye diameter of 6.0 mm and a wavelength of 560 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about V of arc. Why is it not equal to your answer in (b)l

A laser beam passes through a slit of width 1.0 cm and is pointed at the Moon, which is approximately 380,000 km from the Earth. Assume the laser emits waves of wavelength 633 nm (the red light of a He-Ne laser). Estimate the width of the beam when it reaches the Moon.

A He-Ne gas laser which produces monochromatic light of wavelength A = 6.328 X 10_7m is used to calibrate a reflection grating in a spectroscope. The first-order diffraction line is found at an angle of 21.5 to the incident beam. How many lines per meter are there on the grating?

The entrance to a boys bedroom consists of two doorways, each 1.0 m wide, which are separated by a distance of 3.0 m. The boys mother yells at him through the two doors as shown in Fig. 35-48, telling him to clean up his room. Her voice has a frequency of 400 Hz. Later, when the mother discovers the room is still a mess, the boy says he never heard her telling him to clean his room. The velocity of sound is 340 m/s. (a) Find all of the angles 0 (Fig. 35-48) at which no sound will be heard within the bedroom when the mother yells. Assume sound is completely absorbed when it strikes a bedroom wall. (b) If the boy was at the position shown when his mother yelled, does he have a good explanation for not having heard her? Explain. FIGURE 35-48 Problem 74.

At what angle above the horizon is the Sun when light reflecting off a smooth lake is polarized most strongly?

Unpolarized light falls on two polarizer sheets whose axes are at right angles, (a) What fraction of the incident light intensity is transmitted? (b) What fraction is transmitted if a third polarizer is placed between the first two so that its axis makes a 66 angle with the axis of the first polarizer? (c) What if the third polarizer is in front of the other two?

At what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light by an additional factor (after the first Polaroid cuts it in half) of (a) 4, (b) 10, (c) 100?

Four polarizers are placed in succession with their axes vertical, at 30.0 to the vertical, at 60.0 to the vertical, and at 90.0 to the vertical, (a) Calculate what fraction of the incident unpolarized light is transmitted by the four polarizers, (b) Can the transmitted light be decreased by removing one of the polarizers? If so, which one? (c) Can the transmitted light intensity be extinguished by removing polarizers? If so, which one(s)?

Spy planes fly at extremely high altitudes (25 km) to avoid interception. Their cameras are reportedly able to discern features as small as 5 cm. What must be the minimum aperture of the camera lens to afford this resolution? (Use A = 580 nm.)

Two polarizers are oriented at 48 to each other and planepolarized light is incident on them. If only 25% of the light gets through both of them, what was the initial polarization direction of the incident light?__________________________

X-rays of wavelength 0.0973 nm are directed at an unknown crystal. The second diffraction maximum is recorded when the X-rays are directed at an angle of 23.4 relative to the crystal surface. What is the spacing between crystal planes?

X-rays of wavelength 0.10 nm fall on a microcrystalline powder sample. The sample is located 12 cm from the photographic film. The crystal structure of the sample has an atomic spacing of 0.22 nm. Calculate the radii of the diffraction rings corresponding to first- and second-order scattering. Note in Fig. 35-28 that the X-ray beam is deflected through an angle 2<f>.

The Hubble Space Telescope with an objective diameter of 2.4 m, is viewing the Moon. Estimate the minimum distance between two objects on the Moon that the Hubble can distinguish. Consider diffraction of light with wavelength 550 nm. Assume the Hubble is near the Earth.

The Earth and Moon are separated by about 400 X 106 m. When Mars is 8 X 1010m from Earth, could a person standing on Mars resolve the Earth and its Moon as two separate objects without a telescope? Assume a pupil diameter of 5 mm and A = 550 nm.

plate. Light with wavelength A = 650 nm passes through this slit and forms a single-slit diffraction pattern on a screen a distance = 2.0 m away. Defining x to be the distance between the first minima (ra = +1 and ra = -1 ) in this diffraction pattern, find the change Ax in this distance when the temperature T of the metal plate is changed by amount AT = 55 C. [Hint: Since A D, the first minima occur at a small angle.]

A student shined a laser light onto a single slit of width 0.04000 mm. He placed a screen at a distance of 1.490 m from the slit to observe the diffraction pattern of the laser light. The accompanying Table shows the distances of the dark fringes from the center of the central bright fringe for different orders. Order number,ra: 1 2 3 4 5 6 7 8 Distance (m) 0.0225 0.0445 0.0655 0.0870 0.1105 0.1320 0.1540 0.1775 Determine the angle of diffraction, 0, and sin 0 for each order. Make a graph of sin 0 vs. order number, ra, and find the wavelength, A, of the laser from the best-fit straight line.

Describe how to rotate the plane of polarization of a plane-polarized beam of light by 90 and produce only a 10% loss in intensity, using polarizers. Let N be the number of polarizers and 0 be the (same) angle between successive polarizers.

The full-width at half-maximum (FWHM) of the central peak for single-slit diffraction is defined as the angle A0 between the two points on either side of center where the intensity is \ I 0. (a) Determine A0 in terms of (A/D). Use graphs or a spreadsheet to solve sin a = a / v 2 . (b) Determine A0 (in degrees) for D = A and for D = 100A.