(a) Show that the values of a at which intensity maxima

The distance between the first and fifth minima of a singleslit diffraction pattern is 0.35 mm with the screen 40 cm away from the slit, when light of wavelength 550 nm is used. (a) Find the slit width. (b) Calculate the angle u of the first diffraction minimum.

What must be the ratio of the slit width to the wavelength for a single slit to have the first diffraction minimum at u ! 45.0?

A plane wave of wavelength 590 nm is incident on a slit with a width of a ! 0.40 mm.A thin converging lens of focal length '70 cm is placed between the slit and a viewing screen and focuses the light on the screen. (a) How far is the screen from the lens? (b) What is the distance on the screen from the center of the diffraction pattern to the first minimum?

In conventional television, signals are broadcast from towers to home receivers. Even when a receiver is not in direct view of a tower because of a hill or building, it can still intercept a signal if the signal diffracts enough around the obstacle, into the obstacles shadow region. Previously, television signals had a wavelength of about 50 cm, but digital television signals that are transmitted from towers have a wavelength of about 10 mm. (a) Did this change in wavelength increase or decrease the diffraction of the signals into the shadow regions of obstacles? Assume that a signal passes through an opening of 5.0 m width between two adjacent buildings. What is the angular spread of the central diffraction maximum (out to the first minima) for wavelengths of (b) 50 cm and (c) 10 mm?

A single slit is illuminated by light of wavelengths la and lb, chosen so that the first diffraction minimum of the la component coincides with the second minimum of the lb component. (a) If lb ! 350 nm, what is la? For what order number mb (if any) does a minimum of the lb component coincide with the minimum of the la component in the order number (b) ma ! 2 and (c) ma ! 3?

Monochromatic light of wavelength 441 nm is incident on a narrow slit. On a screen 2.00 m away, the distance between the second diffraction minimum and the central maximum is 1.50 cm. (a) Calculate the angle of diffraction u of the second minimum. (b) Find the width of the slit.

Light of wavelength 633 nm is incident on a narrow slit. The angle between the first diffraction minimum on one side of the central maximum and the first minimum on the other side is 1.20. What is the width of the slit?

Sound waves with frequency 3000 Hz and speed 343 m/s diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length d ! 100 m.The opening, which has a horizontal width of 30.0 cm, faces a wall 100 m away (Fig. 36-36). Along that wall, how far from the central axis will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)

A slit 1.00 mm wide is illuminated by light of wavelength 589 nm. We see a diffraction pattern on a screen 3.00 m away.What is the distance between the first two diffraction minima on the same side of the central diffraction maximum?

Manufacturers of wire (and other objects of small dimension) sometimes use a laser to continually monitor the thickness of the product. The wire intercepts the laser beam, producing a diffraction pattern like that of a single slit of the same width as the wire diameter (Fig. 36-37). Suppose a helium neon laser, of wavelength 632.8 nm, illuminates a wire, and the diffraction pattern appears on a screen at distance L ! 2.60 m. If the desired wire diameter is 1.37 mm, what is the observed distance between the two tenth-order minima (one on each side of the central maximum)?

A 0.10-mm-wide slit is illuminated by light of wavelength 589 nm. Consider a point P on a viewing screen on which the diffraction pattern of the slit is viewed; the point is at 30 from the central axis of the slit. What is the phase difference between the Huygens wavelets arriving at point P from the top and midpoint of the slit? (Hint: See Eq. 36-4.)

Figure 36-38 gives a versus the sine of the angle u in a single-slit diffraction experiment using light of wavelength 610 nm. The vertical axis scale is set by as ! 12 rad. What are (a) the slit width, (b) the total number of diffraction minima in the pattern (count them on both sides of the center of the diffraction pattern), (c) the least angle for a minimum, and (d) the greatest angle for a minimum?

Monochromatic light with wavelength 538 nm is incident on a slit with width 0.025 mm. The distance from the slit to a screen is 3.5 m. Consider a point on the screen 1.1 cm from the central maximum. Calculate (a) u for that point, (b) a, and (c) the ratio of the intensity at that point to the intensity at the central maximum.

In the single-slit diffraction experiment of Fig.36-4,let the wavelength of the light be 500 nm, the slit width be 6.00 mm, and the viewing screen be at distance D ! 3.00 m. Let a y axis extend upward along the viewing screen, with its origin at the center of the diffraction pattern. Also let IP represent the intensity of the diffracted light at point P at y ! 15.0 cm. (a) What is the ratio of IP to the intensity Im at the center of the pattern? (b) Determine where point P is in the diffraction pattern by giving the maximum and minimum between which it lies,or the two minima between which it lies.

The full width at half-maximum (FWHM) of a central diffraction maximum is defined as the angle between the two points in the pattern where the intensity is one-half that at the center of the pattern. (See Fig. 36-8b.) (a) Show that the intensity drops to one-half the maximum value when sin2 a ! a2 /2. (b) Verify that a ! 1.39 rad (about 80) is a solution to the transcendental equation of (a). (c) Show that the FWHM is 2u ! 2 sin&1 (0.443l/a), where a is the slit width. Calculate the FWHM of the central maximum for slit width (d) 1.00l,(e) 5.00l,and (f) 10.0l

Babinets principle. A monochromatic beam of parallel light is incident on a collimating hole of diameter . Point P lies in the geometrical shadow region on a distant screen (Fig. 36-39a). Two diffracting objects, shown in Fig. 36-39b, are placed in turn over the collimating hole. Object A is an opaque circle with a hole in it, and B is the photographic negative of A. Using superposition concepts, show that the intensity at P is identical for the two diffracting objects A and B

(a) Show that the values of a at which intensity maxima for single-slit diffraction occur can be found exactly by differentiating Eq. 36-5 with respect to a and equating the result to zero, obtaining the condition tan a ! a. To find values of a satisfying this relation, plot the curve y ! tan a and the straight line y ! a and then find their intersections, or use a calculator to find an appropriate value of a by trial and error. Next, from , determine the values of m associated with the maxima in the singleslit pattern. (These m values are not integers because secondary maxima do not lie exactly halfway between minima.) What are the (b) smallest a and (c) associated m, the (d) second smallest a and (e) associated m, and the (f) third smallest a and (g) associated m?

The wall of a large room is covered with acoustic tile in which small holes are drilled 5.0 mm from center to center. How far can a person be from such a tile and still distinguish the individual holes, assuming ideal conditions, the pupil diameter of the observers eye to be 4.0 mm, and the wavelength of the room light to be 550 nm?

(a) How far from grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is 1.5 mm, the grains are spherical with radius 50 mm, and the light from the grains has wavelength 650 nm? (b) If the grains were blue and the light from them had wavelength 400 nm, would the answer to (a) be larger or smaller?

The radar system of a navy cruiser transmits at a wavelength of 1.6 cm, from a circular antenna with a diameter of 2.3 m. At a range of 6.2 km, what is the smallest distance that two speedboats can be from each other and still be resolved as two separate objects by the radar system?

Estimate the linear separation of two objects on Mars that can just be resolved under ideal conditions by an observer on Earth (a) using the naked eye and (b) using the 200 in. (! 5.1 m) Mount Palomar telescope. Use the following data: distance to Mars ! 8.0 ( 107 km, diameter of pupil ! 5.0 mm, wavelength of light ! 550 nm.

Assume that Rayleighs criterion gives the limit of resolution of an astronauts eye looking down on Earths surface from a typical space shuttle altitude of 400 km. (a) Under that idealized assumption, estimate the smallest linear width on Earths surface that the astronaut can resolve. Take the astronauts pupil diameter to be 5 mm and the wavelength of visible light to be 550 nm. (b) Can the astronaut resolve the Great Wall of China (Fig. 36-40), which is more than 3000 km long, 5 to 10 m thick at its base, 4 m thick at its top, and 8 m in height? (c) Would the astronaut be able to resolve any unmistakable sign of intelligent life on Earths surface?

The two headlights of an approaching automobile are 1.4 m apart. At what (a) angular separation and (b) maximum distance will the eye resolve them? Assume that the pupil diameter is 5.0 mm, and use a wavelength of 550 nm for the light. Also assume that diffraction effects alone limit the resolution so that Rayleighs criterion can be applied.

Entoptic halos. If someone looks at a bright outdoor lamp in otherwise dark surroundings, the lamp appears to be surrounded by bright and dark rings (hence halos) that are actually a circular diffraction pattern as in Fig. 36-10, with the central maximum overlapping the direct light from the lamp. The diffraction is produced by structures within the cornea or lens of the eye (hence entoptic). If the lamp is monochromatic at wavelength 550 nm and the first dark ring subtends angular diameter 2.5 in the observers view, what is the (linear) diameter of the structure producing the diffraction?

Find the separation of two points on the Moons surface that can just be resolved by the 200 in. ( 5.1 m) telescope at Mount Palomar, assuming that this separation is determined by diffraction effects. The distance from Earth to the Moon is 3.8 ( 105 km.Assume a wavelength of 550 nm for the light.

The telescopes on some commercial surveillance satellites can resolve objects on the ground as small as 85 cm across (see Google Earth), and the telescopes on military surveillance satellites reportedly can resolve objects as small as 10 cm across. Assume first that object resolution is determined entirely by Rayleighs criterion and is not degraded by turbulence in the atmosphere.Also assume that the satellites are at a typical altitude of 400 km and that the wavelength of visible light is 550 nm. What would be the required diameter of the telescope aperture for (a) 85 cm resolution and (b) 10 cm resolution? (c) Now, considering that turbulence is certain to degrade resolution and that the aperture diameter of the Hubble Space Telescope is 2.4 m, what can you say about the answer to (b) and about how the military surveillance resolutions are accomplished?

If Superman really had x-ray vision at 0.10 nm wavelength and a 4.0 mm pupil diameter, at what maximum altitude could he distinguish villains from heroes, assuming that he needs to resolve points separated by 5.0 cm to do this?

The wings of tiger beetles (Fig. 36-41) are colored by interference due to thin cuticle-like layers. In addition, these layers are arranged in patches that are 60 mm across and produce different colors. The color you see is a pointillistic mixture of thin-film interference colors that varies with perspective. Approximately what viewing distance from a wing puts you at the limit of resolving the different colored patches according to Rayleighs criterion? Use 550 nm as the wavelength of light and 3.00 mm as the diameter of your pupil.

(a) What is the angular separation of two stars if their images are barely resolved by the Thaw refracting telescope at the Allegheny Observatory in Pittsburgh? The lens diameter is 76 cm and its focal length is 14 m. Assume l ! 550 nm. (b) Find the distance between these barely resolved stars if each of them is 10 light-years distant from Earth. (c) For the image of a single star in this telescope, find the diameter of the first dark ring in the diffraction pattern, as measured on a photographic plate placed at the focal plane of the telescope lens. Assume that the structure of the image is associated entirely with diffraction at the lens aperture and not with lens errors.

Floaters. The floaters you see when viewing a bright, featureless background are diffraction patterns of defects in the vitreous humor that fills most of your eye. Sighting through a pinhole sharpens the diffraction pattern. If you also view a small circular dot, you can approximate the defects size. Assume that the defect diffracts light as a circular aperture does. Adjust the dots distance L from your eye (or eye lens) until the dot and the circle of the first minimum in the diffraction pattern appear to have the same size in your view. That is, until they have the same diameter D# on the retina at distance L# ! 2.0 cm from the front of the eye, as suggested in Fig. 36-42a, where the angles on the two sides of the eye lens are equal. Assume that the wavelength of visible light is l ! 550 nm. If the dot has diameter D ! 2.0 mm and is distance L 45.0 cm from the eye and the defect is x 6.0 mm in front of the retina (Fig. 36-42b), what is the diameter of the defect? ! ! (a) (b) D' D' D Eye Circular lens dot Retina Retina L L' x Defect __1 2 D' __1 2 Figure 36-42 Problem 30

Millimeter-wave radar generates a narrower beam than conventional microwave radar, making it less vulnerable to antiradar missiles than conventional radar. (a) Calculate the angular width 2u of the central maximum, from first minimum to first minimum, produced by a 220 GHz radar beam emitted by a 55.0-cmdiameter circular antenna. (The frequency is chosen to coincide with a low-absorption atmospheric window.) (b) What is 2u for a more conventional circular antenna that has a diameter of 2.3 m and emits at wavelength 1.6 cm?

(a) A circular diaphragm 60 cm in diameter oscillates at a frequency of 25 kHz as an underwater source of sound used for submarine detection. Far from the source, the sound intensity is distributed as the diffraction pattern of a circular hole whose diameter equals that of the diaphragm.Take the speed of sound in water to be 1450 m/s and find the angle between the normal to the diaphragm and a line from the diaphragm to the first minimum. (b) Is there such a minimum for a source having an (audible) frequency of 1.0 kHz?

Nuclear-pumped x-ray lasers are seen as a possible weapon to destroy ICBM booster rockets at ranges up to 2000 km. One limitation on such a device is the spreading of the beam due to diffraction, with resulting dilution of beam intensity. Consider such a laser operating at a wavelength of 1.40 nm. The element that emits light is the end of a wire with diameter 0.200 mm. (a) Calculate the diameter of the central beam at a target 2000 km away from the beam source. (b) What is the ratio of the beam intensity at the target to that at the end of the wire? (The laser is fired from space, so neglect any atmospheric absorption.)

A circular obstacle produces the same diffraction pattern as a circular hole of the same diameter (except very near u ! 0).Airborne water drops are examples of such obstacles.When you see the Moon through suspended water drops, such as in a fog, you intercept the diffraction pattern from many drops. The composite of the central diffraction maxima of those drops forms a white region that surrounds the Moon and may obscure it. Figure 36-43 is a photograph in which the Moon is obscured. There are two faint, colored rings around the Moon (the larger one may be too faint to be seen in your copy of the photograph). The smaller ring is on the outer edge of the central maxima from the drops; the somewhat larger ring is on the outer edge of the smallest of the secondary maxima from the drops (see Fig. 36-10).The color is visible because the rings are adjacent to the diffraction minima (dark rings) in the patterns. (Colors in other parts of the pattern overlap too much to be visible.) (a) What is the color of these rings on the outer edges of the diffraction maxima? (b) The colored ring around the central maxima in Fig. 36-43 has an angular diameter that is 1.35 times the angular diameter of the Moon, which is 0.50. Assume that the drops all have about the same diameter.Approximately what is that diameter?

Suppose that the central diffraction envelope of a double-slit diffraction pattern contains 11 bright fringes and the first diffraction minima eliminate (are coincident with) bright fringes. How many bright fringes lie between the first and second minima of the diffraction envelope?

A beam of light of a single wavelength is incident perpendicularly on a double-slit arrangement, as in Fig. 35-10. The slit widths are each 46 mm and the slit separation is 0.30 mm. How many complete bright fringes appear between the two first-order minima of the diffraction pattern?

In a double-slit experiment, the slit separation d is 2.00 times the slit width w. How many bright interference fringes are in the central diffraction envelope?

In a certain two-slit interference pattern, 10 bright fringes lie within the second side peak of the diffraction envelope and diffraction minima coincide with two-slit interference maxima. What is the ratio of the slit separation to the slit width?

Light of wavelength 440 nm passes through a double slit, yielding a diffraction pattern whose graph of intensity I versus angular position u is shown in Fig. 36-44. Calculate (a) the slit width and (b) the slit separation. (c) Verify the displayed intensities of the m ! 1 and m ! 2 interference fringes.

Figure 36-45 gives the parameter b of Eq. 36-20 versus the sine of the angle u in a two-slit interference experiment using light of wavelength 435 nm.The vertical axis scale is set by bs ! 80.0 rad.What are (a) the slit separation, (b) the total number of interference maxima (count them on both sides of the patterns center), (c) the smallest angle for a maxima, and (d) the greatest angle for a minimum? Assume that none of the interference maxima are completely eliminated by a diffraction minimum.

In the two-slit interference experiment of Fig. 35-10, the slit widths are each 12.0 mm,their separation is 24.0 mm,the wavelength is 600 nm, and the viewing screen is at a distance of 4.00 m. Let IP represent the intensity at point P on the screen, at height y ! 70.0 cm. (a) What is the ratio of IP to the intensity Im at the center of the pattern? (b) Determine where P is in the two-slit interference pattern by giving the maximum or minimum on which it lies or the maximum and minimum between which it lies. (c) In the same way, for the diffraction that occurs,determine where point P is in the diffraction pattern.

(a) In a double-slit experiment, what largest ratio of d to a causes diffraction to eliminate the fourth bright side fringe? (b) What other bright fringes are also eliminated? (c) How many other ratios of d to a cause the diffraction to (exactly) eliminate that bright fringe?

(a) How many bright fringes appear between the first diffraction-envelope minima to either side of the central maximum in a double-slit pattern if l ! 550 nm, d ! 0.150 mm, and a ! 30.0 mm? (b) What is the ratio of the intensity of the third bright fringe to the intensity of the central fringe?

Perhaps to confuse a predator, some tropical gyrinid beetles (whirligig beetles) are colored by optical interference that is due to scales whose alignment forms a diffraction grating (which scatters light instead of transmitting it). When the incident light rays are perpendicular to the grating, the angle between the firstorder maxima (on opposite sides of the zeroth-order maximum) is about 26 in light with a wavelength of 550 nm. What is the grating spacing of the beetle?

A diffraction grating 20.0 mm wide has 6000 rulings. Light of wavelength 589 nm is incident perpendicularly on the grating. What are the (a) largest, (b) second largest, and (c) third largest values of u at which maxima appear on a distant viewing screen?

Visible light is incident perpendicularly on a grating with 315 rulings/mm. What is the longest wavelength that can be seen in the fifth-order diffraction?

A grating has 400 lines/mm. How many orders of the entire visible spectrum (400700 nm) can it produce in a diffraction experiment, in addition to the m ! 0 order?

A diffraction grating is made up of slits of width 300 nm with separation 900 nm. The grating is illuminated by monochromatic plane waves of wavelength l ! 600 nm at normal incidence. (a) How many maxima are there in the full diffraction pattern? (b) What is the angular width of a spectral line observed in the first order if the grating has 1000 slits?

Light of wavelength 600 nm is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by sin u ! 0.2 and sin u ! 0.3. The fourth-order maxima are missing. (a) What is the separation between adjacent slits? (b) What is the smallest slit width this grating can have? For that slit width, what are the (c) largest, (d) second largest, and (e) third largest values of the order number m of the maxima produced by the grating?

With light from a gaseous discharge tube incident normally on a grating with slit separation 1.73 mm, sharp maxima of green light are experimentally found at angles u ! ,17.6, 37.3, &37.1, 65.2, and &65.0. Compute the wavelength of the green light that best fits these data.

A diffraction grating having 180 lines/mm is illuminated with a light signal containing only two wavelengths, l1 400 nm and l2 ! 500 nm. The signal is incident perpendicularly on the grating. (a) What is the angular separation between the secondorder maxima of these two wavelengths? (b) What is the smallest angle at which two of the resulting maxima are superimposed? (c) What is the highest order for which maxima for both wavelengths are present in the diffraction pattern?

A beam of light consisting of wavelengths from 460.0 nm to 640.0 nm is directed perpendicularly onto a diffraction grating with 160 lines/mm. (a) What is the lowest order that is overlapped by another order? (b) What is the highest order for which the complete wavelength range of the beam is present? In that highest order, at what angle does the light at wavelength (c) 460.0 nm and (d) 640.0 nm appear? (e) What is the greatest angle at which the light at wavelength 460.0 nm appears?

A grating has 350 rulings/mm and is illuminated at normal incidence by white light. A spectrum is formed on a screen 30.0 cm from the grating. If a hole 10.0 mm square is cut in the screen, its inner edge being 50.0 mm from the central maximum and parallel to it, what are the (a) shortest and (b) longest wavelengths of the light that passes through the hole?

Derive this expression for the intensity pattern for a three-slit grating: where and a 5 l

A source containing a mixture of hydrogen and deuterium atoms emits red light at two wavelengths whose mean is 656.3 nm and whose separation is 0.180 nm. Find the minimum number of lines needed in a diffraction grating that can resolve these lines in the first order.

(a) How many rulings must a 4.00-cm-wide diffraction grating have to resolve the wavelengths 415.496 and 415.487 nm in the second order? (b) At what angle are the second-order maxima found?

Light at wavelength 589 nm from a sodium lamp is incident perpendicularly on a grating with 40 000 rulings over width 76 nm. What are the first-order (a) dispersion D and (b) resolving power R, the second-order (c) D and (d) R,and the third-order (e) D and (f) R?

A grating has 600 rulings/mm and is 5.0 mm wide. (a) What is the smallest wavelength interval it can resolve in the third order at l ! 500 nm? (b) How many higher orders of maxima can be seen?

A diffraction grating with a width of 2.0 cm contains 1000 lines/cm across that width. For an incident wavelength of 600 nm, what is the smallest wavelength difference this grating can resolve in the second order?

The D line in the spectrum of sodium is a doublet with wavelengths 589.0 and 589.6 nm. Calculate the minimum number of lines needed in a grating that will resolve this doublet in the second-order spectrum.

With a particular grating the sodium doublet (589.00 nm and 589.59 nm) is viewed in the third order at 10 to the normal and is barely resolved. Find (a) the grating spacing and (b) the total width of the rulings.

A diffraction grating illuminated by monochromatic light normal to the grating produces a certain line at angle u. (a) What is the product of that lines half-width and the gratings resolving power? (b) Evaluate that product for the first order of a grating of slit separation 900 nm in light of wavelength 600 nm.

Assume that the limits of the visible spectrum are arbitrarily chosen as 430 and 680 nm. Calculate the number of rulings per millimeter of a grating that will spread the first-order spectrum through an angle of 20.0.

What is the smallest Bragg angle for x rays of wavelength 30 pm to reflect from reflecting planes spaced 0.30 nm apart in a calcite crystal?

An x-ray beam of wavelength A undergoes first-order reflection (Bragg law diffraction) from a crystal when its angle of incidence to a crystal face is 23, and an x-ray beam of wavelength 97 pm undergoes third-order reflection when its angle of incidence to that face is 60. Assuming that the two beams reflect from the same family of reflecting planes,find (a) the interplanar spacing and (b) the wavelength A.

An x-ray beam of a certain wavelength is incident on an NaCl crystal, at 30.0 to a certain family of reflecting planes of spacing 39.8 pm. If the reflection from those planes is of the first order, what is the wavelength of the x rays?

Figure 36-46 is a graph of intensity versus angular position u for the diffraction of an x-ray beam by a crystal. The horizontal scale is set by us ! 2.00.The beam consists of two wavelengths, and the spacing between the reflecting planes is 0.94 nm. What are the (a) shorter and (b) longer wavelengths in the beam?

If first-order reflection occurs in a crystal at Bragg angle 3.4, at what Bragg angle does second-order reflection occur from the same family of reflecting planes?

X rays of wavelength 0.12 nm are found to undergo secondorder reflection at a Bragg angle of 28 from a lithium fluoride crystal. What is the interplanar spacing of the reflecting planes in the crystal?

In Fig. 36-47, first-order reflection from the reflection planes shown occurs when an x-ray beam of wavelength 0.260 nm makes an angle u ! 63.8 with the top face of the crystal. What is the unit cell size a0?

In Fig. 36-48, let a beam of x rays of wavelength 0.125 nm be incident on an NaCl crystal at angle u 45.0 to the top face of the crystal and a family of reflecting planes. Let the reflecting planes have separation d ! 0.252 nm. The crystal is turned through angle f around an axis perpendicular to the plane of the page until these reflecting planes give diffraction maxima. What are the (a) smaller and (b) larger value of f if the crystal is turned clockwise and the (c) smaller and (d) larger value of f if it is turned counterclockwise?

In Fig. 36-48, an x-ray beam of wavelengths from 95.0 to 140 pm is incident at u ! 45.0 to a family of reflecting planes with spacing d ! 275 pm.What are the (a) longest wavelength l and (b) associated order number m and the (c) shortest l and (d) associated m of the intensity maxima in the diffraction of the beam?

Consider a two-dimensional square crystal structure, such as one side of the structure shown in Fig. 36-28a.The largest interplanar spacing of reflecting planes is the unit cell size a0. Calculate and sketch the (a) second largest, (b) third largest, (c) fourth largest, (d) fifth largest, and (e) sixth largest interplanar spacing. (f) Show that your results in (a) through (e) are consistent with the general formula where h and k are relatively prime integers (they have no common factor other than unity).

An astronaut in a space shuttle claims she can just barely resolve two point sources on Earths surface, 160 km below. Calculate their (a) angular and (b) linear separation, assuming ideal conditions.Take l ! 540 nm and the pupil diameter of the astronauts eye to be 5.0 mm.

Visible light is incident perpendicularly on a diffraction grating of 200 rulings/mm. What are the (a) longest, (b) second longest, and (c) third longest wavelengths that can be associated with an intensity maximum at u ! 30.0?

A beam of light consists of two wavelengths, 590.159 nm and 590.220 nm, that are to be resolved with a diffraction grating. If the grating has lines across a width of 3.80 cm, what is the minimum number of lines required for the two wavelengths to be resolved in the second order?

In a single-slit diffraction experiment, there is a minimum of intensity for orange light (l 600 nm) and a minimum of intensity for blue-green light (l ! 500 nm) at the same angle of 1.00 mrad. For what minimum slit width is this possible?

A double-slit system with individual slit widths of 0.030 mm and a slit separation of 0.18 mm is illuminated with 500 nm light directed perpendicular to the plane of the slits. What is the total number of complete bright fringes appearing between the two first-order minima of the diffraction pattern? (Do not count the fringes that coincide with the minima of the diffraction pattern.)

A diffraction grating has resolving power R ! lavg/2l ! Nm. (a) Show that the corresponding frequency range f that can just be resolved is given by 2f ! c/Nml. (b) From Fig. 36-22, show that the times required for light to travel along the ray at the bottom of the figure and the ray at the top differ by 2t ! (Nd/c) sinu. (c) Show that (2f)(2t) ! 1, this relation being independent of the various grating parameters.Assume N 4 1.

The pupil of a persons eye has a diameter of 5.00 mm. According to Rayleighs criterion, what distance apart must two small objects be if their images are just barely resolved when they are 250 mm from the eye? Assume they are illuminated with light of wavelength 500 nm.

Light is incident on a grating at an angle c as shown in Fig. 36-49. Show that bright fringes occur at angles u that satisfy the equation d(sin c ' sin u) ! ml, for m ! 0, 1, 2, . . . . (Compare this equation with Eq. 36-25.) Only the special case c ! 0 has been treated in this chapter.

A grating with d ! 1.50 mm is illuminated at various angles of incidence by light of wavelength 600 nm. Plot, as a function of the angle of incidence (0 to 90), the angular deviation of the firstorder maximum from the incident direction. (See Problem 81.)

In two-slit interference, if the slit separation is 14 mm and the slit widths are each 2.0 mm, (a) how many two-slit maxima are in the central peak of the diffraction envelope and (b) how many are in either of the first side peak of the diffraction envelope?

In a two-slit interference pattern, what is the ratio of slit separation to slit width if there are 17 bright fringes within the central diffraction envelope and the diffraction minima coincide with two-slit interference maxima?

A beam of light with a narrow wavelength range centered on 450 nm is incident perpendicularly on a diffraction grating with a width of 1.80 cm and a line density of 1400 lines/cm across that width. For this light, what is the smallest wavelength difference this grating can resolve in the third order?

A beam of light with a narrow wavelength range centered on 450 nm is incident perpendicularly on a diffraction grating with a width of 1.80 cm and a line density of 1400 lines/cm across that width. For this light, what is the smallest wavelength difference this grating can resolve in the third order?

Two yellow flowers are separated by 60 cm along a line perpendicular to your line of sight to the flowers. How far are you from the flowers when they are at the limit of resolution according to the Rayleigh criterion? Assume the light from the flowers has a single wavelength of 550 nm and that your pupil has a diameter of 5.5 mm.

In a single-slit diffraction experiment, what must be the ratio of the slit width to the wavelength if the second diffraction minima are to occur at an angle of 37.0 from the center of the diffraction pattern on a viewing screen?

A diffraction grating 3.00 cm wide produces the second order at 33.0 with light of wavelength 600 nm. What is the total number of lines on the grating?

A single-slit diffraction experiment is set up with light of wavelength 420 nm, incident perpendicularly on a slit of width 5.10 mm. The viewing screen is 3.20 m distant. On the screen, what is the distance between the center of the diffraction pattern and the second diffraction minimum?

A diffraction grating has 8900 slits across 1.20 cm. If light with a wavelength of 500 nm is sent through it, how many orders (maxima) lie to one side of the central maximum?

In an experiment to monitor the Moons surface with a light beam, pulsed radiation from a ruby laser (l ! 0.69 mm) was directed to the Moon through a reflecting telescope with a mirror radius of 1.3 m. A reflector on the Moon behaved like a circular flat mirror with radius 10 cm, reflecting the light directly back toward the telescope on Earth. The reflected light was then detected after being brought to a focus by this telescope. Approximately what fraction of the original light energy was picked up by the detector? Assume that for each direction of travel all the energy is in the central diffraction peak.

In June 1985, a laser beam was sent out from the Air Force Optical Station on Maui, Hawaii, and reflected back from the shuttle Discovery as it sped by 354 km overhead.The diameter of the central maximum of the beam at the shuttle position was said to be 9.1 m, and the beam wavelength was 500 nm.What is the effective diameter of the laser aperture at the Maui ground station? (Hint: A laser beam spreads only because of diffraction; assume a circular exit aperture.)

A diffraction grating 1.00 cm wide has 10 000 parallel slits. Monochromatic light that is incident normally is diffracted through 30 in the first order. What is the wavelength of the light?

If you double the width of a single slit, the intensity of the central maximum of the diffraction pattern increases by a factor of 4, even though the energy passing through the slit only doubles. Explain this quantitatively.

When monochromatic light is incident on a slit 22.0 mm wide, the first diffraction minimum lies at 1.80 from the direction of the incident light.What is the wavelength?

A spy satellite orbiting at 160 km above Earths surface has a lens with a focal length of 3.6 m and can resolve objects on the ground as small as 30 cm. For example, it can easily measure the size of an aircrafts air intake port.What is the effective diameter of the lens as determined by diffraction consideration alone? Assume l ! 550 nm.

Suppose that two points are separated by 2.0 cm. If they are viewed by an eye with a pupil opening of 5.0 mm, what distance from the viewer puts them at the Rayleigh limit of resolution? Assume a light wavelength of 500 nm.

A diffraction grating has 200 lines/mm. Light consisting of a continuous range of wavelengths between 550 nm and 700 nm is incident perpendicularly on the grating. (a) What is the lowest order that is overlapped by another order? (b) What is the highest order for which the complete spectrum is present?

A diffraction grating has 200 rulings/mm, and it produces an intensity maximum at u ! 30.0". (a) What are the possible wavelengths of the incident visible light? (b) To what colors do they correspond?

Show that the dispersion of a grating is D ! (tan u)/l

Monochromatic light (wavelength ! 450 nm) is incident perpendicularly on a single slit (width ! 0.40 mm). A screen is placed parallel to the slit plane, and on it the distance between the two minima on either side of the central maximum is 1.8 mm. (a) What is the distance from the slit to the screen? (Hint: The angle to either minimum is small enough that sin u tan u.) (b) What is the distance on the screen between the first minimum and the third minimum on the same side of the central maximum?

Light containing a mixture of two wavelengths, 500 and 600 nm, is incident normally on a diffraction grating. It is desired (1) that the first and second maxima for each wavelength appear at u 30", (2) that the dispersion be as high as possible, and (3) that the third order for the 600 nm light be a missing order. (a) What should be the slit separation? (b) What is the smallest individual slit width that can be used? (c) For the values calculated in (a) and (b) and the light of wavelength 600 nm, what is the largest order of maxima produced by the grating?

A beam of x rays with wavelengths ranging from 0.120 nm to 0.0700 nm scatters from a family of reflecting planes in a crystal. The plane separation is 0.250 nm. It is observed that scattered beams are produced for 0.100 nm and 0.0750 nm. What is the angle between the incident and scattered beams?

Show that a grating made up of alternately transparent and opaque strips of equal width eliminates all the even orders of maxima (except m ! 0)

Light of wavelength 500 nm diffracts through a slit of width 2.00 mm and onto a screen that is 2.00 m away. On the screen, what is the distance between the center of the diffraction pattern and the third diffraction minimum?

If, in a two-slit interference pattern, there are 8 bright fringes within the first side peak of the diffraction envelope and diffraction minima coincide with two-slit interference maxima, then what is the ratio of slit separation to slit width?

White light (consisting of wavelengths from 400 nm to 700 nm) is normally incident on a grating. Show that, no matter what the value of the grating spacing d, the second order and third order overlap.

If we make d ! a in Fig. 36-50, the two slits coalesce into a single slit of width 2a. Show that Eq. 36-19 reduces to give the diffraction pattern for such a slit

Derive Eq. 36-28, the expression for the half-width of the lines in a gratings diffraction pattern.

Prove that it is not possible to determine both wavelength of incident radiation and spacing of reflecting planes in a crystal by measuring the Bragg angles for several orders.

How many orders of the entire visible spectrum (400700 nm) can be produced by a grating of 500 lines/mm?

An acoustic double-slit system (of slit separation d and slit width a) is driven by two loudspeakers as shown in Fig. 36-51. By use of a variable delay line, the phase of one of the speakers may be varied relative to the other speaker. Describe in detail what changes occur in the double-slit diffraction pattern at large distances as the phase difference between the speakers is varied from zero to 2p. Take both interference and diffraction effects into account.

Two emission lines have wavelengths l and l ' 2l, respectively, where 2l l. Show that their angular separation 2u in a grating spectrometer is given approximately by where d is the slit separation and m is the order at which the lines are observed. Note that the angular separation is greater in the higher orders than the lower orders