The third-order bright fringe of 610 nm light is

Derive the law of reflection namely, that the angle of incidence equals the angle of reflection from a flat surface using Huygens principle for waves.

Monochromatic light falling on two slits 0.018 mm apart produces the fifth-order bright fringe at a 9.8 angle. What is the wavelength of the light used?

The third-order bright fringe of 610 nm light is observed at an angle of 28 when the light falls on two narrow slits. How far apart are the slits?

Monochromatic light falls on two very narrow slits 0.048 mm apart. Successive fringes on a screen 6.00 m away are 8.5 cm apart near the center of the pattern. Determine the wavelength and frequency of the light.

I f 720-nm and 660-nm light passes through two slits 0.68 mm apart, how far apart are the second-order fringes for these two wavelengths on a screen 1.0 m away?

red laser from the physics lab is marked as producing 632.8-nm light. When light from this laser falls on two closely spaced slits, an interference pattern formed on a wall several meters away has bright fringes spaced 5.00 mm apart near the center of the pattern. When the laser is replaced by a small laser pointer, the fringes are 5.14 mm apart. What is the wavelength of light produced by the pointer?

Light of wavelength A passes through a pair of slits separated by 0.17 mm, forming a double-slit interference pattern on a screen located a distance 35 cm away. Suppose that the image in Fig. 34-9a is an actual-size reproduction of this interference pattern. Use a ruler to measure a pertinent distance on this image; then utilize this measured value to determine A (nm).

Light of wavelength 680 nm falls on two slits and produces an interference pattern in which the third-order bright fringe is 38 mm from the central fringe on a screen 2.6 m away. What is the separation of the two slits?

A parallel beam of light from a He-Ne laser, with a wavelength 633 nm, falls on two very narrow slits 0.068 mm apart. How far apart are the fringes in the center of the pattern on a screen 3.8 m away?

A physics professor wants to perform a lecture demonstration o f Youngs double-slit experiment for her class using the 633-nm light from a He-Ne laser. Because the lecture hall is very large, the interference pattern w ill be projected on a wall that is 5.0 m from the slits. For easy viewing by all students in the class, the professor wants the distance between the m = 0 and m = 1 maxima to be 25 cm. What slit separation is required in order to produce the desired interference pattern?

Suppose a thin piece of glass is placed in front of the 3 4 lower slit in Fig. 34-7 so that the two waves enter the slits 23. 180 out of phase (Fig. 34-25). Describe in detail the interference pattern on the screen.FIGURE 34-25 Problem 11.

In a double-slit experiment it is found that blue light of wavelength 480 nm gives a second-order maximum at a 26. certain location on the screen. What wavelength of visible light would have a minimum at the same location?

Two narrow slits separated by 1.0 mm are illuminated by 544 nm light. Find the distance between adjacent bright 27. fringes on a screen 5.0 m from the slits.

In a double-slit experiment, the third-order maximum for light of wavelength 500 nm is located 12 mm from the central bright spot on a screen 1.6 m from the slits. Light of wavelength 650 nm is then projected through the same slits. How far from the central bright spot w ill the second-order maximum of this light be located?

Light of wavelength 470 nm in air falls on two slits 6.00 X 10 2 mm apart. The slits are immersed in water, as is a viewing screen 50.0 cm away. How far apart are the fringes on the screen?

A very thin sheet of plastic (n = 1.60) covers one slit of a double-slit apparatus illuminated by 680-nm light. The center point on the screen, instead of being a maximum, is dark. What is the (minimum) thickness of the plastic?

I f one slit in Fig. 34-12 is covered, by what factor does the intensity at the center of the screen change?

Derive an expression similar to Eq. 34-2 which gives the angles for which the double-slit intensity is one-half its maximum value, I6 = \Iq .

Show that the angular fu ll width at half maximum of the central peak in a double-slit interference pattern is given by A 6 = A/2d if A d.

(II) In a two-slit interference experiment, the path length to a certain point P on the screen differs for one slit in compar- ison with the other by 1.25A. (a) What is the phase difference between the two waves arriving at point P? (b ) Determine the intensity at P, expressed as a fraction of the maximum intensity I0 on the screen.

( I ll) Suppose that one slit of a double-slit apparatus is wider than the other so that the intensity of light passing through it is twice as great. Determine the intensity I as a function of position (0 ) on the screen for coherent light.

( I ll) (a) Consider three equally spaced and equal-intensity coherent sources of light (such as adding a third slit to the two slits of Fig. 34-12). Use the phasor method to obtain the intensity as a function of the phase difference 8 (Eq. 34-4). (b) Determine the positions of maxima and minima.

(I) I f a soap bubble is 120 nm thick, what wavelength is most strongly reflected at the center of the outer surface when illuminated normally by white light? Assume that n = 1.32.

(I) How far apart are the dark fringes in Example 34-6 if the glass plates are each 28.5 cm long?

(II) (a) What is the smallest thickness of a soap film (n = 1.33) that would appear black if illuminated with 480-nm light? Assume there is air on both sides of the soap film, (b) What are two other possible thicknesses for the film to appear black? (c) I f the thickness t was much less than A, why would the film also appear black?

(II) A lens appears greenish yellow ( A = 570 nm is strongest) when white light reflects from it. What minimum thickness of coating (n = 1.25) do you think is used on such a glass (n = 1.52) lens, and why?

(II) A thin film of o il (nQ = 1.50) with varying thickness floats on water (nw = 1.33). When it is illuminated from above by white light, the reflected colors are as shown in Fig. 34-26. In air, the wavelength of yellow light is 580 nm. (a) Why are there no reflected colors at point A? (b) What is the o ils thickness t at point B? FIGURE 34-26 Problem 27

(II) A thin oil slick (nQ = 1.50) floats on water (nw = 1.33). When a beam of white light strikes this film at normal incidence from air, the only enhanced reflected colors are red (650 nm) and violet (390 nm). From this information, deduce the (minimum) thickness t of the o il slick.

(II) A total of 31 bright and 31 dark Newtons rings (not counting the dark spot at the center) are observed when 560-nm light falls normally on a planoconvex lens resting on a fla t glass surface (Fig. 34-18). How much thicker is the center than the edges?

(II) A fine metal fo il separates one end of two pieces of optically fla t glass, as in Fig. 34-20. When light of wavelength 670 nm is incident normally, 28 dark lines are observed (with one at each end). How thick is the foil?

(II) How thick (minimum) should the air layer be between two fla t glass surfaces if the glass is to appear bright when 450-nm light is incident normally? What if the glass is to appear dark?

(II) A uniform thin film of alcohol (n = 1.36) lies on a flat glass plate (n = 1.56). When monochromatic light, whose wavelength can be changed, is incident normally, the reflected light is a minimum for A = 512 nm and a maximum for A = 635 nm. What is the minimum thickness of the film?

Show that the radius r o f the mth dark Newtons ring, as viewed from directly above (Fig. 34-18), is given by r = V'mXR where R is the radius of curvature of the curved glass surface and A is the wavelength of light used. Assume that the thickness of the air gap is much less than R at all points and that r <SC R. [Hint: Use the binomial expansion.]

Use the result of Problem 33 to show that the distance between adjacent dark Newtons rings is Ar for the mth ring, assuming ra 1.

When a Newtons ring apparatus (Fig. 34-18) is immersed in a liquid, the diameter of the eighth dark ring decreases from 2.92 cm to 2.54 cm. What is the refractive index of the liquid? [Hint: see Problem 33.]

A planoconvex lucite lens 3.4 cm in diameter is placed on a fla t piece of glass as in Fig. 34-18. When 580-nm light is incident normally, 44 bright rings are observed, the last one right at the edge. What is the radius of curvature of the lens surface, and the focal length of the lens? [Hint: see Problem 33.]

Lets explore why only th in layers exhibit thin-film interference. Assume a layer of water, sitting atop a fla t glass surface, is illuminated from the air above by white light (all wavelengths from 400 nm to 700 nm). Further, assume that the water layers thickness t is much greater than a micron (= 1 0 0 0nm); in particular, let t = 200 ftm. Take the index of refraction for water to be n = 1.33 for all visible wavelengths, (a) Show that a visible color w ill be reflected from the water layer if its wavelength is A = 2nt/m, where ra is an integer. (b) Show that the two extremes in wavelengths (400 nm and 700 nm) of the incident light are both reflected from the water layer and determine the m-value associated with each, (c) How many other visible wavelengths, besides A = 400 nm and 700 nm, are reflected from the thick layer of water? (d ) How does this explain why such a thick layer does not reflect colorfully, but is white or grey?

A single optical coating reduces reflection to zero for A = 550 nm. By what factor is the intensity reduced by the coating for A = 430 nm and A = 670 nm as compared to no coating? Assume normal incidence.

How far must the mirror M i in a Michelson interferometer be moved if 650 fringes of 589-nm light are to pass by a reference line?

What is the wavelength of the light entering an interferometer if 384 bright fringes are counted when the movable mirror moves 0.125 mm?

A micrometer is connected to the movable mirror of an interferometer. When the micrometer is tightened down on a thin metal fo il, the net number of bright fringes that move, compared to the empty micrometer, is 272. What is the thickness of the foil? The wavelength of light used is 589 nm.

One of the beams of an interferometer (Fig. 34-27) passes through a small evacuated glass container 1.155 cm deep. When a gas is allowed to slowly fill the container, a total of 176 dark fringes are counted to move past a reference line. The light used has a wavelength of 632.8 nm. Calculate the index of refraction of the gas at its final density, assuming that the interferometer is in vacuum.FIGURE 34-27 Problem 42.

The yellow sodium D lines have wavelengths of 589.0 and 589.6 nm. When they are used to illuminate a Michelson interferometer, it is noted that the interference fringes disappear and reappear periodically as the mirror M i is moved. Why does this happen? How far must the mirror move between one disappearance and the next?

The illuminance of direct sunlight on Earth is about 105 lm/m2. Estimate the luminous flux and luminous intensity of the Sun.

The luminous efficiency of a lightbulb is the ratio of luminous flux to electric power input, (a) What is the luminous efficiency (%) of a 100-W, 1700-lm bulb? (b ) How many 40-W, 60-lm/W fluorescent lamps would be needed to provide an illuminance of 250 lm/m2 on a factory floor of area 25 m X 30 m? Assume the lights are 10 m above the floor and that half their flux reaches the floor.

Light of wavelength 5.0 X 10_7m passes through two parallel slits and falls on a screen 4.0 m away. Adjacent bright bands of the interference pattern are 2.0 cm apart. (a) Find the distance between the slits, (b) The same two slits are next illuminated by light of a different wavelength, and the fifth-order minimum for this light occurs at the same point on the screen as the fourth-order minimum for the previous light. What is the wavelength of the second source of light?

Television and radio waves reflecting from mountains or airplanes can interfere with the direct signal from the station, (a) What kind of interference w ill occur when 75-MHz television signals arrive at a receiver directly from a distant station, and are reflected from a nearby airplane 122 m directly above the receiver? Assume \ \ change in phase of the signal upon reflection, (b) What kind of interference w ill occur if the plane is 22 m closer to the receiver?

A radio station operating at 88.5 MHz broadcasts from two identical antennas at the same elevation but separated by a 9.0-m horizontal distance d, Fig. 34-28. A maximum signal is found along the midline, perpendicular to d at its midpoint and extending horizontally in both directions. I f the midline is taken as 0 , at what other angle(s) 6 is a maximum signal detected? A minimum signal? Assume all measurements are made much farther than 9.0 m from the antenna towers.FIGURE 34-28 Problem 48.

Light of wavelength 690 nm passes through two narrow slits 0.66 mm apart. The screen is 1.60 m away. A second source of unknown wavelength produces its second-order fringe 1.23 mm closer to the central maximum than the 690-nm light. What is the wavelength of the unknown light?

Monochromatic light of variable wavelength is incident normally on a thin sheet o f plastic film in air. The reflected light is a minimum only for A = 491.4 nm and A = 688.0 nm in the visible spectrum. What is the thickness of the film (n = 1.58)? [Hint Assume successive values of ra.]

Suppose the mirrors in a Michelson interferometer are perfectly aligned and the path lengths to mirrors M i and M 2 are identical. With these in itia l conditions, an observer sees a bright maximum at the center of the viewing area. Now one of the mirrors is moved a distance x. Determine a formula for the intensity at the center of the viewing area as a function of x, the distance the movable mirror is moved from the initia l position.

A highly reflective mirror can be made for a particular wavelength at normal incidence by using two thin layers of transparent materials of indices of refraction ni and n2 ( l < < n2) on the surface of the glass (n > n2). What should be the minimum thicknesses d \ and d2 in Fig. 34-29 in terms of the incident wavelength A, to maximize reflection?FIGURE 34-29 Problem 52.

Calculate the minimum thickness needed for an antireflective coating (n = 1.38) applied to a glass lens in order to eliminate (a) blue (450 nm), or (b) red (720 nm) reflections for light at normal incidence.

Stealth aircraft are designed to not reflect radar, whose wavelength is typically 2 cm, by using an antireflecting coating. Ignoring any change in wavelength in the coating, estimate its thickness

Light of wavelength A strikes a screen containing two slits a distance d apart at an angle 6\ to the normal. Determine the angle 6m at which the mth-order maximum occurs.

Consider two antennas radiating 6.0-MHz radio waves in phase with each other. They are located at points S1 and S2, separated by a distance d = 175 m, Fig. 34-30. Determine the points on the y axis where the signals from the two sources w ill be out of phase (crests of one meet troughs of the other). FIGURE 34-30 Problem 56.

What is the minimum (non-zero) thickness for the air layer between two fla t glass surfaces if the glass is to appear dark when 680-nm light is incident normally? What if the glass is to appear bright?

L lo yd s mirror provides one way of obtaining a double-slit interference pattern from a single source so the light is coherent. As shown in Fig. 34-31, the light that reflects from the plane mirror appears to come from the virtual image of the slit. Describe in detail the interference pattern on the screen.FIGURE 34-31 Problem 58.

Consider the antenna array of Example 34-5, Fig. 34-15. Let d = A/2, and suppose that the two antennas are now 180 out of phase with each other. Find the directions for constructive and destructive interference, and compare with the case when the sources are in phase. (These results illustrate the basis for directional antennas.)

Suppose you viewed the light transmitted through a thin film layered on a flat piece of glass. Draw a diagram, similar to Fig. 34-17 or 34-23, and describe the conditions required for maxima and minima. Consider all possible values of index of refraction. Discuss the relative size of the minima compared to the maxima and to zero.

A thin film of soap (n = 1.34) coats a piece of fla t glass (n = 1.52). How thick is the film if it reflects 643-nm red light most strongly when illuminated normally by white light?

Two identical sources Si and S2, separated by distance d, coherently emit light of wavelength A uniformly in all directions. Defining the x axis with its origin at Si as shown in Fig. 34-32, find the locations (expressed as multiples of A) where the signals from the two sources are out of phase along this axis for x > 0, if d = 3A. FIGURE 34-32 Problem 62.

A two-slit interference set-up with slit separation d = 0.10 mm produces interference fringes at a particular set of angles 0m (where m = 0, 1, 2 ,...) for red light of frequency / = 4.6 X 1014 Hz. If one wishes to construct an analogous two-slit interference set-up that produces interference fringes at the same set of angles 6m for room-temperature sound of middle-C frequency / s = 262 Hz, what should the slit separation ds be for this analogous set-up?

A radio telescope, whose two antennas are separated by 55 m, is designed to receive 3.0-MHz radio waves produced by astronomical objects. The received radio waves create 3.0-MHz electronic signals in the telescopes left and right antennas. These signals then travel by equal-length cables to a centrally located amplifier, where they are added together. The telescope can be pointed to a certain region of the sky by adding the instantaneous signal from the right antenna to a time-delayed signal received by the left antenna a time At ago. (This time delay of the left signal can be easily accomplished with the proper electronic circuit.) If a radio astronomer wishes to view radio signals arriving from an object oriented at a 12 angle to the vertical as in Fig. 34-33, what time delay A? is necessary?

In a compact disc (CD), digital information is stored as a sequence of raised surfaces called pits and recessed surfaces called lands. Both pits and lands are highly reflective and are embedded in a thick plastic material with index of refraction n = 1.55 (Fig. 34-34). As a 780-nm wavelength (in air) laser scans across the pit-land sequence, the transition between a neighboring pit and land is sensed by monitoring the intensity of reflected laser light from the CD. A t the moment when half the width of the laser beam is reflected from the pit and the other half from the land, we want the two reflected halves of the beam to be 180 out of phase with each other. What should be the (minimum) height difference t between a pit and land? [When this light enters a detector, cancellation of the two out-of-phase halves of the beam produces a minimum detector output.] FIGURE 34-34 Problem 65.

(II) A Michelson interferometer can be used to determine the index of refraction of a glass plate. A glass plate (thickness t) is placed on a platform that can rotate. The plate is placed in the lights path between the beam splitter and either the fixed or movable mirror, so that its thickness is in the direction of the laser beam. The platform is rotated to various angles, and the number of fringes shifted is counted. It can be shown that if N is the number of fringes shifted when the angle of rotation changes by 0, the index of refraction is n = (21 - NX)(1 - cos0)/[2(l - cos0) - NX] where t is the thickness of the plate. The accompanying Table shows the data collected by a student in determining the index of refraction of a transparent plate by a Michelson interferometer. N 25 50 75 100 125 150 0 (degree) 5.5 6.9 8.6 10.0 11.3 12.5 In the experiment A = 632.8 nm and t = 4.0 mm. Determine n for each 0 and find the average n.