A grating with 1 500 slits per centimeter is illuminated

A laser beam (   632.8 nm) is incident on two slits 0.200 mm apart. How far apart are the bright interference fringes on a screen 5.00 m away from the double slits?

In a Youngs double-slit experiment, a set of parallel slits with a separation of 0.100 mm is illuminated by light having a wavelength of 589 nm, and the interference pattern is observed on a screen 4.00 m from the slits. (a) What is the difference in path lengths from each of the slits to the location of a third-order bright fringe on the screen? (b) What is the difference in path lengths from the two slits to the location of the third dark fringe on the screen, away from the center of the pattern?

A pair of narrow, parallel slits separated by 0.250 mm is illuminated by the green component from a mercury vapor lamp (   546.1 nm). The interference pattern is observed on a screen 1.20 m from the plane of the parallel slits. Calculate the distance (a) from the central maximum to the rst bright region on either side of the central maximum and (b) between the rst and second dark bands in the interference pattern.

Light of wavelength 460 nm falls on two slits spaced 0.300 mm apart. What is the required distance from the slit to a screen if the spacing between the rst and second dark fringes is to be 4.00 mm?

In a location where the speed of sound is 354 m/s, a 2 000-Hz sound wave impinges on two slits 30.0 cm apart. (a) At what angle is the rst maximum located? (b) If the sound wave is replaced by 3.00-cm microwaves, what slit separation gives the same angle for the rst maximum? (c) If the slit separation is 1.00 m, what frequency of light gives the same rst maximum angle?

White light spans the wavelength range between about 400 nm and 700 nm. If white light passes through two slits 0.30 mm apart and falls on a screen 1.5 m from the slits, nd the distance between the rst-order violet and the rst-order red fringes.

Two radio antennas separated by 300 m, as shown in Figure P24.7, simultaneously transmit identical signals of the same wavelength. A radio in a car traveling due north receives the signals. (a) If the car is at the position of the second maximum, what is the wavelength of the signals? (b) How much farther must the car travel to encounter the next minimum in reception? [Hint: Determine the path difference between the two signals at the two locations of the car.]

If the distance between two slits is 0.050 mm and the distance to a screen is 2.50 m, nd the spacing between the rst- and second-order bright fringes for yellow light of 600-nm wavelength.

Waves from a radio station have a wavelength of 300 m. They travel by two paths to a home receiver 20.0 km from the transmitter. One path is a direct path, and the second is by reection from a mountain directly behind the home receiver. What is the minimum distance from the mountain to the receiver that produces destructive interference at the receiver? (Assume that no phase change occurs on reection from the mountain.)

A pair of slits, separated by 0.150 mm, is illuminated by light having a wavelength of   643 nm. An interference pattern is observed on a screen 140 cm from the slits. Consider a point on the screen located at y  1.80 cm from the central maximum of this pattern. (a) What is the path difference  for the two slits at the location y? (b) Express this path difference in terms of the wavelength. (c) Will the interference correspond to a maximum, a minimum, or an intermediate condition?

A riverside warehouse has two open doors, as in Figure P24.11. Its interior is lined with a sound-absorbing material. A boat on the river sounds its horn. To person A, the sound is loud and clear. To person B, the sound is barely audible. The principal wavelength of the sound waves is 3.00 m. Assuming person B is at the position of the rst minimum, determine the distance between the doors, center to center.

The waves from a radio station can reach a home receiver by two different paths. One is a straight-line path from the transmitter to the home, a distance of 30.0 km. The second path is by reection from a storm cloud. Assume that this reection takes place at a point midway between receiver and transmitter. If the wavelength broadcast by the radio station is 400 m, nd the minimum height of the storm cloud that will produce destructive interference between the direct and reected beams. (Assume no phase changes on reection.)

Radio waves from a star, of wavelength 250 m, reach a radio telescope by two separate paths, as shown in Figure P24.13. One is a direct path to the receiver, which is situated on the edge of a cliff by the ocean. The second is by reection off the water. The rst minimum of destructive interference occurs when the star is 25.0 above the horizon. Find the height of the cliff. (Assume no phase change on reection.)

Determine the minimum thickness of a soap lm (n  1.330) that will result in constructive interference of (a) the red H  line (   656.3 nm); (b) the blue H  line (   434.0 nm).

Suppose the lm shown in Figure 24.7 has an index of refraction of 1.36 and is surrounded by air on both sides. Find the minimum thickness that will produce constructive interference in the reected light when the lm is illuminated by light of wavelength 500 nm.

A thin lm of glass (n  1.50) oats on a liquid of n  1.35 and is illuminated by light of   580 nm incident from air above it. Find the minimum thickness of the glass, other than zero, that will produce destructive interference in the reected light.

A coating is applied to a lens to minimize reections. The index of refraction of the coating is 1.55, and that of the lens is 1.48. If the coating is 177.4nm thick, what wavelength is minimally reected for normal incidence in the lowest order?

A transparent oil with index of refraction 1.29 spills on the surface of water (index of refraction 1.33), producing a maximum of reection with normally incident orange light (wavelength 600 nm in air). Assuming the maximum occurs in the rst order, determine the thickness of the oil slick.

A possible means for making an airplane invisible to radar is to coat the plane with an antireective polymer. If radar waves have a wavelength of 3.00 cm and the index of refraction of the polymer is n  1.50, how thick would you make the coating?

A beam of light of wavelength 580 nm passes through two closely spaced glass plates, as shown in Figure P24.20. For what minimum non-zero value of the plate separation d will the transmitted light be bright? This arrangement is often used to measure the wavelength of light and is called a FabryPerot interferometer.

Astronomers observe the chromosphere of the sun with a lter that passes the red hydrogen spectral line of wavelength 656.3 nm, called the H  line. The lter consists of a transparent dielectric of thickness d held between two partially aluminized glass plates. The lter is kept at a constant temperature. (a) Find the minimum value of d that will produce maximum transmission of perpendicular H  light if the dielectric has an index of refraction of 1.378. (b) If the temperature of the lter increases above the normal value increasing its thickness, what happens to the transmitted wavelength? (c) The dielectric will also pass what near-visible wavelength? One of the glass plates is colored red to absorb this light.

Two rectangular optically at plates (n  1.52) are in contact along one end and are separated along the other end by a 2.00 m-thick spacer (Fig. P24.22). The top plate is illuminated by monochromatic light of wavelength 546.1 nm. Calculate the number of dark parallel bands crossing the top plate (including the dark band at zero thickness along the edge of contact between the plates).

An air wedge is formed between two glass plates separated at one edge by a very ne wire, as in Figure P24.22. When the wedge is illuminated from above by 600-nm light, 30 dark fringes are observed. Calculate the radius of the wire.

A planoconvex lens with radius of curvature R  3.0 m is in contact with a at plate of glass. A light source and the observers eye are both close to the normal, as shown in Figure 24.8a. The radius of the 50th bright Newtons ring is found to be 9.8 mm. What is the wavelength of the light produced by the source?

A planoconvex lens rests with its curved side on a at glass surface and is illuminated from above by light of wavelength 500 nm. (See Fig. 24.8.) A dark spot is observed at the center, surrounded by 19 concentric dark rings (with bright rings in between). How much thicker is the air wedge at the position of the 19th dark ring than at the center?

Nonreective coatings on camera lenses reduce the loss of light at the surfaces of multilens systems and prevent internal reections that might mar the image. Find the minimum thickness of a layer of magnesium uoride (n  1.38) on int glass (n  1.66) that will cause destructive interference of reected light of wavelength 550 nm near the middle of the visible spectrum.

A thin lm of MgF2 (n  1.38) with thickness 1.00
105 cm is used to coat a camera lens. Are any wavelengths in the visible spectrum intensied in the reected light?

A at piece of glass is supported horizontally above the at end of a 10.0-cm-long metal rod that has its lower end rigidly xed. The thin lm of air between the rod and the glass is observed to be bright when illuminated by light of wavelength 500 nm. As the temperature is slowly increased by 25.0C, the lm changes from bright to dark and back to bright 200 times. What is the coefcient of linear expansion of the metal?

Heliumneon laser light (   632.8 nm) is sent through a 0.300-mm-wide single slit. What is the width of the central maximum on a screen 1.00 m from the slit?

Light of wavelength 600 nm falls on a 0.40-mm-wide slit and forms a diffraction pattern on a screen 1.5 m away. (a) Find the position of the rst dark band on each side of the central maximum. (b) Find the width of the central maximum.

Light of wavelength 587.5 nm illuminates a slit of width 0.75 mm. (a) At what distance from the slit should a screen be placed if the rst minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum.

Microwaves of wavelength 5.00 cm enter a long, narrow window in a building that is otherwise essentially opaque to the incoming waves. If the window is 36.0 cm wide, what is the distance from the central maximum to the rstorder minimum along a wall 6.50 m from the window?

A slit of width 0.50 mm is illuminated with light of wavelength 500 nm, and a screen is placed 120 cm in front of the slit. Find the widths of the rst and second maxima on each side of the central maximum.

A screen is placed 50.0 cm from a single slit, which is illuminated with light of wavelength 680 nm. If the distance between the rst and third minima in the diffraction pattern is 3.00 mm, what is the width of the slit?

Three discrete spectral lines occur at angles of 10.1, 13.7, and 14.8, respectively, in the rst-order spectrum of a diffraction-grating spectrometer. (a) If the grating has 3 660 slits/cm, what are the wavelengths of the light? (b) At what angles are these lines found in the secondorder spectra?

Intense white light is incident on a diffraction grating that has 600 lines/mm. (a) What is the highest order in which the complete visible spectrum can be seen with this grating? (b) What is the angular separation between the violet edge (400 nm) and the red edge (700 nm) of the rstorder spectrum produced by the grating?

The hydrogen spectrum has a red line at 656 nm and a violet line at 434 nm. What angular separation between these two spectral lines obtained with a diffraction grating that has 4 500 lines/cm?

A grating with 1 500 slits per centimeter is illuminated with light of wavelength 500 nm. (a) What is the highestorder number that can be observed with this grating? (b) Repeat for a grating of 15 000 slits per centimeter.

A light source emits two major spectral lines: an orange line of wavelength 610 nm and a blue-green line of wavelength 480 nm. If the spectrum is resolved by a diffraction grating having 5 000 lines/cm and viewed on a screen 2.00 m from the grating, what is the distance (in centimeters) between the two spectral lines in the second-order spectrum?

White light is spread out into its spectral components by a diffraction grating. If the grating has 2 000 lines per centimeter, at what angle does red light of wavelength 640 nm appear in the rst-order spectrum?

Sunlight is incident on a diffraction grating that has 2 750 lines/cm. The second-order spectrum over the visible range (400700 nm) is to be limited to 1.75 cm along a screen that is a distance L from the grating. What is the required value of L?

Light containing two different wavelengths passes through a diffraction grating with 1 200 slits/cm. On a screen 15.0 cm from the grating, the third-order maximum of the shorter wavelength falls midway between the central maximum and the rst side maximum for the longer wavelength. If the neighboring maxima of the longer wavelength are 8.44 mm apart on the screen, what are the wavelengths in the light? [Hint: Use the smallangle approximation.]

A beam of 541-nm light is incident on a diffraction grating that has 400 lines/mm. (a) Determine the angle of the second-order ray. (b) If the entire apparatus is immersed in water, determine the new second-order angle of diffraction. (c) Show that the two diffracted rays of parts (a) and (b) are related through the law of refraction.

Light from a heliumneon laser (   632.8 nm) is incident on a single slit. What is the maximum width for which no diffraction minima are observed? [Hint: Values of sin   1 are not possible.]

The angle of incidence of a light beam in air onto a reecting surface is continuously variable. The reected ray is found to be completely polarized when the angle of incidence is 48.0. (a) What is the index of refraction of the reecting material? (b) If some of the incident light (at an angle of 48.0) passes into the material below the surface, what is the angle of refraction?

Unpolarized light passes through two polaroid sheets. The axis of the rst is vertical, and that of the second is at 30.0 to the vertical. What fraction of the initial light is transmitted?

The index of refraction of a glass plate is 1.52. What is the Brewsters angle when the plate is (a) in air? (b) in water? (See Problem 51.)

At what angle above the horizon is the Sun if light from it is completely polarized upon reection from water?

A light beam is incident on heavy int glass (n  1.65) at the polarizing angle. Calculate the angle of refraction for the transmitted ray.

The critical angle for total internal reection for sapphire surrounded by air is 34.4. Calculate the Brewster angle for sapphire if the light is incident from the air.

Equation 24.14 assumes that the incident light is in air. If the light is incident from a medium of index n1 onto a medium of index n2, follow the procedure used to derive Equation 24.14 to show that tan 

Plane-polarized light is incident on a single polarizing disk, with the direction of E0 parallel to the direction of the transmission axis. Through what angle should the disk be rotated so that the intensity in the transmitted beam is reduced by a factor of (a) 3.00, (b) 5.00, (c) 10.0?

Three polarizing plates whose planes are parallel are centered on a common axis. The directions of the transmission axes relative to the common vertical direction are shown in Figure P24.53. A linearly polarized beam of light with plane of polarization parallel to the vertical reference direction is incident from the left onto the rst disk with intensity Ii  10.0 units (arbitrary). Calculate the transmitted intensity If when  1  20.0,  2  40.0, and  3  60.0. [Hint: Make repeated use of Maluss law.]

Light of intensity I0 and polarized parallel to the transmission axis of a polarizer, is incident on an analyzer. (a) If the transmission axis of the analyzer makes an angle of 45 with the axis of the polarizer, what is the intensity of the transmitted light? (b) What should the angle between the transmission axes be to make I/I0  1/3?

Light with a wavelength in vacuum of 546.1 nm falls perpendicularly on a biological specimen that is 1.000 m thick. The light splits into two beams polarized at right angles, for which the indices of refraction are 1.320 and 1.333, respectively. (a) Calculate the wavelength of each component of the light while it is traversing the specimen. (b) Calculate the phase difference between the two beams when they emerge from the specimen.

A beam containing light of wavelengths  1 and  2 is incident on a set of parallel slits. In the interference pattern, the fourth bright line of the  1 light occurs at the same position as the fth bright line of the  2 light. If  1 is known to be 540 nm, what is the value of  2?

Light of wavelength 546 nm (the intense green line from a mercury source) produces a Youngs interference pattern in which the second minimum from the central maximum is along a direction that makes an angle of 18.0 min of arc with the axis through the central maximum. What is the distance between the parallel slits?

The two speakers are placed 35.0 cm apart. A single oscillator makes the speakers vibrate in phase at a frequency of 2.00 kHz. At what angles, measured from the perpendicular bisector of the line joining the speakers, would a distant observer hear maximum sound intensity? Minimum sound intensity? (Take the speed of sound to be 340 m/s.)

Interference effects are produced at point P on a screen as a result of direct rays from a 500-nm source and reected rays off a mirror, as in Figure P24.59. If the source is 100 m to the left of the screen and 1.00 cm above the mirror, nd the distance y (in millimeters) to the rst dark band above the mirror.

Many cells are transparent and colorless. Structures of great interest in biology and medicine can be practically invisible to ordinary microscopy. An interference microscope reveals a difference in refractive index as a shift in interference fringes, to indicate the size and shape of cell structures. The idea is exemplied in the following problem: An air wedge is formed between two glass plates in contact along one edge and slightly separated at the opposite edge. When the plates are illuminated with monochromatic light from above, the reected light has 85 dark fringes. Calculate the number of dark fringes that appear if water (n  1.33) replaces the air between the plates.

A thin layer of oil (n  1.25) is oating on water. How thick is the oil in the region that strongly reects green light (   525 nm)?

Three polarizers, centered on a common axis and with their planes parallel to each other, have transmission axes oriented at angles of  1,  2, and  3 from the vertical, as shown in Figure P24.53. Light of intensity Ii, polarized with its plane of polarization oriented vertically, is incident from the left onto the rst polarizer. What is the ratio If/Ii of the nal transmitted intensity to the incident intensity if (a)  1  45,  2  90, and  3  0? (b)  1  0,  2  45, and  3  90?

Figure P24.63 shows a radio-wave transmitter and a receiver, both h  50.0 m above the ground and d  600 m apart. The receiver can receive signals directly from the transmitter, and indirectly, from signals that bounce off the ground. If the ground is level between the transmitter and receiver and a  /2 phase shift occurs upon reection, determine the longest wavelengths that interfere (a) constructively and (b) destructively.

A planoconvex lens (at on one side, convex on the other) with index of refraction n rests with its curved side (radius of curvature R) on a at glass surface of the same index of refraction with a lm of index nlm between them. The lens is illuminated from above by light of wavelength  . Show that the dark Newton rings which appear have radii of where m is an integer.

The transmitting antenna on a submarine is 5.00 m above the water when the ship surfaces. The captain wishes to transmit a message to a receiver on a 90.0-m-tall cliff at the ocean shore. If the signal is to be completely polarized by reection off the ocean surface, how far must the ship be from the shore?

(a) If light is incident at an angle  from a medium of index n1 on a medium of index n2 so that the angle between the reected ray and refracted ray is  , show that Hint: Use the trigonometric identity: sin(A B)  sin A cos B cos A sin B (b) Show that the foregoing equation for tan  reduces to Brewsters law when   90, n1  1, and n2 =n.

A diffraction pattern is produced on a screen 140 cm from a single slit, using monochromatic light of wavelength 500 nm, The distance from the center of the central maximum to the rst-order maximum is 3.00 mm. Calculate the slit width. [Hint: Assume that the rst-order maximum is halfway between the rst- and second-order minima.]

A glass plate (n  1.61) is covered with a thin, uniform layer of oil (n  1.20). A light beam of variable wavelength is normally incident from air onto the oil surface. Observation of the reected beam shows destructive interference at 500 nm and constructive interference at 750 nm. From this information, calculate the thickness of the oil lm.

The condition for constructive interference by reection from a thin lm in air, as developed in Section 24.4, assumes nearly normal incidence. (a) Show that for large angles of incidence, the condition for constructive interference of light reecting from a thin lm of thickness t, with index ofrefraction n, and surrounded by air may be written as where  2 is the angle of refraction. (b) Calculate the minimum thickness for constructive interference if sodium light (   590 nm) is incident at an angle of 30.0 on a lm with an index of refraction of 1.38.

Figure P24.70 illustrates the formation of an interference pattern by the Lloyds mirror method. Light from source S reaches the screen via two different pathways. One is a direct path, and the second is by reection from a horizontal mirror. The effect is as if light from two different sources S and S had interfered as in the Youngs double-slit arrangement. Assume that the actual source S and the virtual source S are in a plane 25 cm to the left of the mirror and the screen is a distance L  120 cm to the right of that plane. Source S is a distance h  2.5 mm above the top surface of the mirror, and the light is monochromatic with   620 nm. Determine the distance of the rst bright fringe above the surface of the mirror.

A piece of transparent material having an index of refraction n is cut into the shape of a wedge as shown in Figure P24.71. The angle of the wedge is small. Monochromatic light of wavelength  is normally incident from above, and viewed from above. Let h represent the height of the wedge and
its width. Show that bright fringes occur at the positions x  
(m

Place a clear dish or plate on a black surface, such as a sheet of black construction paper. Now add a thin layer of water to the glass, and place a few drops of kerosene or light machine oil on the water. Darken the room and shine a ashlight from an angle, as in Figure A24.1. Note the interference pattern of various colors you observe under the white light. How does the pattern change if you cover the ashlight with a sheet of red, blue, or green cellophane, which acts as a lter? As an extension of the preceding experiment, observe the colors appearing to swirl on the surface of a soap bubble. What color do you see just before a bubble bursts?

Stand a couple of meters from a lightbulb. Facing away from the light, hold a compact disc about 10 cm from your eye and tilt it until the reection of the bulb is located in the hole at the discs center. You should see spectra radiating out from the center, with violet on the inside and red on the outside. Now move the disc away from your eye until the violet band is at the outer edge. Carefully measure the distance from your eye to the center of the disc, and also determine the radius of the disc. Use this information to nd the angle  to the rst-order maximum for violet light. Now use the relationship d sin   m  to determine the spacing between the grooves of the disc. The industry standard is 1.6 m. How close did you come? While you are observing the spectrum from a CD, note that the color of the light from a given point changes with the viewing angle. Explain this effect in terms of changes in the path length. It is of interest that the blues and bluegreens in hummingbird feathers and butteries are caused by diffraction off nely aligned structures in feathers and wings. (See chapter opener photo.)

(a) Devise a way to use a protractor, a desk lamp, and polarizing sunglasses to measure Brewsters angle for the glass in a window. From this, determine the index of refraction of the glass. (b) Put on a pair of polarizing sunglasses and close one eye. Hold up a lens of a second pair of polarizing glasses in front of your open eye so that light must pass through a lens of each pair before entering your eye. Now rotate the second pair of glasses around. You will note that the light reaching your eye is considerably reduced at some orientations and will pass freely at others. (c) On a sunny day, rotate your polarizing sunglasses in front of your eye and observe how light reects from a window or the surface of water. Note the change in the amount of light entering your eye for various orientations of the glasses. (d) For a nal observation concerning polarized light, rotate a pair of polarizing sunglasses while looking at various areas of the sky. From what direction do you nd the light to be most highly polarized?