| A double slit is illuminated simultaneously with orange light of wavelength 620 nm and light of an unknown wavelength. The $m=4$ bright fringe of the unknown wavelength overlaps the $m=3$ bright orange fringe. What is the unknown wavelength?

| Two narrow slits $80\mu \text{m}$ apart are illuminated with light of wavelength 620 nm. What is the angle of the $m=3$ bright fringe in radians? In degrees?

| Light of wavelength 550 nm illuminates a double slit, and the interference pattern is observed on a screen behind the slit. The third maximum is measured to be 3.0 cm from the central maximum. The slits are then illuminated with light of wavelength 440 nm. How far is the fourth maximum from the central maximum?

| A double-slit experiment is performed with light of wavelength 630 nm. The bright interference fringes are spaced 1.8 mm apart on the viewing screen. What will the fringe spacing be if the light is changed to a wavelength of 420 nm?

|| Light of 630 nm wavelength illuminates two slits that are 0.25 mm apart. FIGURE EX33.5 shows the intensity pattern seen on a screen behind the slits. What is the distance to the screen?

FIGURE EX33.5

| Light from a sodium lamp $(\text{λ}=589\text{nm})$ illuminates two narrow slits. The fringe spacing on a screen 150 cm behind the slits is 4.0 mm. What is the spacing (in mm) between the two slits?

| In a double-slit experiment, the slit separation is 200 times the wavelength of the light. What is the angular separation (in degrees) between two adjacent bright fringes?

|| A double-slit interference pattern is created by two narrow slits spaced 0.25 mm apart. The distance between the first and the fifth minimum on a screen 60 cm behind the slits is 5.5 mm. What is the wavelength (in nm) of the light used in this experiment?

|| FIGURE EX33.9 shows the light intensity on a screen behind a double slit. The slit spacing is 0.20 mm and the wavelength of the light is 620 nm. What is the distance from the slits to the screen?

FIGURE EX33.9

|| Figure EX33.9 shows the light intensity on a screen behind a double slit. The slit spacing is 0.20 mm and the screen is 2.0 m behind the slits. What is the wavelength (in nm) of the light?

| Light of wavelength 620 nm illuminates a diffraction grating. The second-order maximum is at angle $39.5\text{°}$. How many lines per millimeter does this grating have?

| A 4.0-cm-wide diffraction grating has 2000 slits. It is illuminated by light of wavelength 550 nm. What are the angles (in degrees) of the first two diffraction orders?

| A diffraction grating produces a first-order maximum at an angle of $20.0\text{°}$. What is the angle of the second-order maximum?

| Excited potassium atoms emit deep-violet light at wavelengths 404.4 nm and 404.7 nm. Suppose light from a potassium lamp passes through a diffraction grating that has 800 slits per millimeter. How far behind the grating must a viewing screen be placed so that the two first-order fringes are separated by 1.0 mm?

|| The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656 nm (red) and 486 nm (blue). Light from a hydrogen lamp illuminates a diffraction grating with $500\text{lines}/\text{m}\text{m}$, and the light is observed on a screen 1.50 m behind the grating. What is the distance between the first-order red and blue fringes?

|| A helium-neon laser $(\text{λ}=633\text{nm})$ illuminates a diffraction grating. The distance between the two $m=1$ bright fringes is 32 cm on a screen 2.0 m behind the grating. What is the spacing between slits of the grating?

|| FIGURE EX33.17 shows the interference pattern on a screen 1.0 m behind an $800\text{lines}/\text{m}\text{m}$ diffraction grating. What is the wavelength (in nm) of the light?

FIGURE EX33.17

|| Figure EX33.17 shows the interference pattern on a screen 1.0 m behind a diffraction grating. The wavelength of the light is 620 nm. How many lines per millimeter does the grating have?

|| A helium-neon laser $(\text{λ}=633\text{nm})$ illuminates a single slit and is observed on a screen 1.5 m behind the slit. The distance between the first and second minima in the diffraction pattern is 4.75 mm. What is the width (in mm) of the slit?

|| In a single-slit experiment, the slit width is 200 times the wavelength of the light. What is the width (in mm) of the central maximum on a screen 2.0 m behind the slit?

|| A 0.50-mm-wide slit is illuminated by light of wavelength 500 nm. What is the width (in mm) of the central maximum on a screen 2.0 m behind the slit?

|| Light of 630 nm wavelength illuminates a single slit of width 0.15 mm. FIGURE EX33.22 shows the intensity pattern seen on a screen behind the slit. What is the distance to the screen?

FIGURE EX33.22

| Light from a helium-neon laser $(\text{λ}=633\text{nm})$ is incident on a single slit. What is the largest slit width for which there are no minima in the diffraction pattern?

|| Light of 600 nm wavelength passes through a single slit and creates a 2.0-cm-wide central maximum on a screen behind the slit. What wavelength of light will create a 3.0-cm-wide central maximum on a screen twice as far away?

|| You need to use your cell phone, which broadcasts an 800 MHz signal, but you’re behind two massive, radio-wave-absorbing buildings that have only a 15 m space between them. What is the angular width, in degrees, of the electromagnetic wave after it emerges from between the buildings?

|| FIGURE EX33.26 shows the light intensity on a screen behind a single slit. The slit width is 0.20 mm and the screen is 1.5 m behind the slit. What is the wavelength (in nm) of the light?

FIGURE EX33.26

|| Figure EX33.26 shows the light intensity on a screen behind a single slit. The wavelength of the light is 600 nm and the slit width is 0.15 mm. What is the distance from the slit to the screen?

|| A laser beam illuminates a single, narrow slit, and the diffraction pattern is observed on a screen behind the slit. The first secondary maximum is 26 mm from the center of the diffraction pattern. How far is the first minimum from the center of the diffraction pattern?

||| Two $50\text{-}\mu \text{m}\text{-}\text{wide}$ slits spaced 0.25 mm apart are illuminated by blue laser light with a wavelength of 450 nm. The interference pattern is observed on a screen 2.0 m behind the slits. How many bright fringes are seen in the central maximum that spans the distance between the first missing order on one side and the first missing order on the other side?

||| A laser beam with a wavelength of 480 nm illuminates two 0.12-mm-wide slits separated by 0.30 mm. The interference pattern is observed on a screen 2.3 m behind the slits. What is the light intensity, as a fraction of the maximum intensity $I_0$, at a point halfway between the center and the first minimum?

| Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength 670 nm, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength 410 nm. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

|| A 0.50-mm-diameter hole is illuminated by light of wavelength 550 nm. What is the width (in mm) of the central maximum on a screen 2.0 m behind the slit?

|| Infrared light of wavelength $2.5\mu \text{m}$ illuminates a 0.20-mm-diameter hole. What is the angle of the first dark fringe in radians? In degrees?

|| Light from a helium-neon laser $(\text{λ}=633\text{nm})$ passes through a circular aperture and is observed on a screen 4.0 m behind the aperture. The width of the central maximum is 2.5 cm. What is the diameter (in mm) of the hole?

|| You want to photograph a circular diffraction pattern whose central maximum has a diameter of 1.0 cm. You have a helium-neon laser $(\text{λ}=633\text{nm})$ and a 0.12-mm-diameter pinhole. How far behind the pinhole should you place the screen that’s to be photographed?

| Moving mirror $\text{M}{}_2$ of a Michelson interferometer a distance of $100\mu \text{m}$ causes 500 bright-dark-bright fringe shifts. What is the wavelength of the light?

| A Michelson interferometer uses red light with a wavelength of 656.45 nm from a hydrogen discharge lamp. How many bright-dark-bright fringe shifts are observed if mirror $\text{M}{}_2$ is moved exactly 1 cm?

|| A Michelson interferometer uses a diode laser with a wavelength of 620 nm. The interferometer is adjusted so that the central spot is a bright fringe. A glass wedge with index of refraction 1.50 is slowly lowered into the laser beam in one arm of the interferometer. The edge that enters first is razor sharp, then the thickness of the glass gradually increases. How thick, in $\mu \text{m}$, is the portion of the wedge that the laser beam passes through after you have counted 650 bright-dark-bright fringe shifts?

| FIGURE P33.39 shows the light intensity on a screen 2.5 m behind an aperture. The aperture is illuminated with light of wavelength 620 nm.

1. Is the aperture a single slit or a double slit? Explain.

2. If the aperture is a single slit, what is its width? If it is a double slit, what is the spacing between the slits?

FIGURE P33.39

| FIGURE P33.40 shows the light intensity on a screen 2.5 m behind an aperture. The aperture is illuminated with light of wavelength 620 nm.

1. Is the aperture a single slit or a double slit? Explain.

2. If the aperture is a single slit, what is its width? If it is a double slit, what is the spacing between the slits?

FIGURE P33.40

|| Light from a helium-neon laser $(\text{λ}=633\text{nm})$ is used to illuminate two narrow slits. The interference pattern is observed on a screen 3.0 m behind the slits. Twelve bright fringes are seen, spanning a distance of 52 mm. What is the spacing (in mm) between the slits?

| The intensity at the central maximum of a double-slit interference pattern is $4I_1$. The intensity at the first minimum is zero. At what fraction of the distance from the central maximum to the first minimum is the intensity $I_1?$ Assume an ideal double slit.

| Two vertical, high-frequency radio antennas are 20 m apart. 2.0 km away, in a plane parallel to the plane of the antennas, “bright” spots of radio intensity are spaced 5.0 m apart, separated by spots with almost no radio intensity. What is the radio frequency?

|| A diffraction grating having $500\text{lines}/\text{mm}$ diffracts visible light at $30\text{°}$. What is the light’s wavelength?

|| Helium atoms emit light at several wavelengths. Light from a helium lamp illuminates a diffraction grating and is observed on a screen 50.00 cm behind the grating. The emission at wavelength 501.5 nm creates a first-order bright fringe 21.90 cm from the central maximum. What is the wavelength of the bright fringe that is 31.60 cm from the central maximum?

|| X-ray diffraction is a technique used to determine the structure of crystals. X rays are light waves that have a wavelength of only a few nanometers or less. For wavelengths this short, the parallel planes of atoms in a crystal, spaced less than a nanometer apart, act very much like the slits of a diffraction grating. If x rays are sent through a crystal, a careful analysis finds that the condition for bright fringes of x-ray diffraction is $2d\text{sin}({\theta }_m\text{/}2)=m\text{λ}$, where $d$ is the spacing between the planes of the crystal—called the lattice spacing. This agrees with the diffraction-grating equation at small angles, where $\text{sin}({\theta }_m\text{/}2)\approx {\theta }_m\text{/}2$, but it differs slightly at larger angles.

1. If x rays with a wavelength of 0.190 nm are sent through a crystal of tungsten, the first-order diffraction is observed at 34.5°. What is the lattice spacing of tungsten?

2. At what angle is second-order diffraction observed?

|| A diffraction grating with $600\text{lines}/\text{mm}$ is illuminated with light of wavelength 510 nm. A very wide viewing screen is 2.0 m behind the grating.

1. What is the distance between the two $m=1$ bright fringes?

2. How many bright fringes can be seen on the screen?

|| INT Because sound is a wave, it’s possible to make a diffraction grating for sound from a large board of sound-absorbing material with several parallel slits cut for sound to go through. When 10 kHz sound waves pass through such a grating, listeners 10 m from the grating report “loud spots” 1.4 m on both sides of center. What is the spacing between the slits? Use 340 m/s for the speed of sound.

| White light (400–700 nm) incident on a $600\text{lines}/\text{mm}$ diffraction grating produces rainbows of diffracted light. What is the width of the first-order rainbow on a screen 2.0 m behind the grating?

|| A chemist identifies compounds by identifying bright lines in their spectra. She does so by heating the compounds until they glow, sending the light through a diffraction grating, and measuring the positions of first-order spectral lines on a detector 15.0 cm behind the grating. Unfortunately, she has lost the card that gives the specifications of the grating. Fortunately, she has a known compound that she can use to calibrate the grating. She heats the known compound, which emits light at a wavelength of 461 nm, and observes a spectral line 9.95 cm from the center of the diffraction pattern. What are the wavelengths emitted by compounds A and B that have spectral lines detected at positions 8.55 cm and 12.15 cm, respectively?

|||

1. Find an expression for the positions $y_1$ of the first-order fringes of a diffraction grating if the line spacing is large enough for the small-angle approximation $\text{tan}\theta \approx \text{sin}\theta \approx \theta$ to be valid. Your expression should be in terms of $d$, $L$, and $\text{λ}$.

2. Use your expression from part a to find an expression for the separation $\text{Δ}y$ on the screen of two fringes that differ in wavelength by $\text{Δ}\text{λ}$.

3. Rather than a viewing screen, modern spectrometers use detectors—similar to the one in your digital camera—that are divided into pixels. Consider a spectrometer with a 333 lines/mm grating and a detector with 100 pixels/mm located 12 cm behind the grating. The resolution of a spectrometer is the smallest wavelength separation $\text{Δ}{\text{λ}}_{\text{min}}$ that can be measured reliably. What is the resolution of this spectrometer for wavelengths near 550 nm, in the center of the visible spectrum? You can assume that the fringe due to one specific wavelength is narrow enough to illuminate only one column of pixels.

|| What is the longest wavelength for which a third-order fringe can be observed with a diffraction grating that has 800 lines/mm?

| BIO The wings of some beetles have closely spaced parallel lines of melanin, causing the wing to act as a reflection grating. Suppose sunlight shines straight onto a beetle wing. If the melanin lines on the wing are spaced $2.0\mu \text{m}$ apart, what is the first-order diffraction angle for green light $(\text{λ}=550\text{nm})?$

|| Light from a sodium lamp $(\text{λ}=589\text{nm})$ illuminates a narrow slit and is observed on a screen 75 cm behind the slit. The distance between the first and third dark fringes is 7.5 mm. What is the width (in mm) of the slit?

| BIO If sunlight shines straight onto a peacock feather, the feather appears bright blue when viewed from $15\text{°}$ on either side of the incident beam of light. The blue color is due to diffraction from parallel rods of melanin in the feather barbules, as was described . Other wavelengths in the incident light are diffracted at different angles, leaving only the blue light to be seen. The average wavelength of blue light is 470 nm. Assuming this to be the first-order diffraction, what is the spacing of the melanin rods in the feather?

|| You’ve found an unlabeled diffraction grating. Before you can use it, you need to know how many lines per mm it has. To find out, you illuminate the grating with light of several different wavelengths and then measure the distance between the two first-order bright fringes on a viewing screen 150 cm behind the grating. Your data are as follows:

Use the best-fit line of an appropriate graph to determine the number of lines per mm.

|| A diffraction grating has slit spacing $d$. Fringes are viewed on a screen at distance $L$. Find an expression for the wavelength of light that produces a first-order fringe on the viewing screen at distance $L$ from the center of the screen.

|| For light that strikes a reflection grating perpendicular to the grating, the condition for bright fringes in the reflected light is exactly the same as the bright-fringe condition for a transmission grating. A DVD makes a useful reflection grating because it has many narrow, mirror-like surfaces between the regularly spaced circular tracks of “pits” where the data are encoded. If a green laser pointer with a wavelength of 532 nm illuminates a DVD, a first-order diffraction spot is seen 1.00 m from the laser beam on a screen that is 1.00 m from the DVD and parallel to it. What is the track spacing on the DVD?

||| A student performing a double-slit experiment is using a green laser with a wavelength of 530 nm. She is confused when the $m=5$ maximum does not appear. She had predicted that this bright fringe would be 1.6 cm from the central maximum on a screen 1.5 m behind the slits.

1. Explain what prevented the fifth maximum from being observed.

2. What is the width of her slits?

|| Scientists shine a laser beam on a $35\text{-}\mu \text{m}\text{-}\text{wide}$ slit and produce a diffraction pattern on a screen 70 cm behind the slit. Careful measurements show that the intensity first falls to 25% of maximum at a distance of 7.2 mm from the center of the diffraction pattern. What is the wavelength of the laser light?

Hint: Use the trial-and-error technique demonstrated in Example 33.5 to solve the transcendental equation.

| Light from a helium-neon laser $(\text{λ}=633\text{nm})$ illuminates a circular aperture. It is noted that the diameter of the central maximum on a screen 50 cm behind the aperture matches the diameter of the geometric image. What is the aperture’s diameter (in mm)?

|| A helium-neon laser $(\text{λ}=633\text{nm})$ is built with a glass tube of inside diameter 1.0 mm, as shown in FIGURE P33.62. One mirror is partially transmitting to allow the laser beam out. An electrical discharge in the tube causes it to glow like a neon light. From an optical perspective, the laser beam is a light wave that diffracts out through a 1.0-mm-diameter circular opening.

1. Can a laser beam be perfectly parallel, with no spreading? Why or why not?

2. The angle ${\theta }_1$ to the first minimum is called the divergence angle of a laser beam. What is the divergence angle of this laser beam?

3. What is the diameter (in mm) of the laser beam after it travels 3.0 m? Note that the wave model is appropriate because the spreading, at this distance, is significantly larger than the size of the opening.

4. What is the diameter of the laser beam after it travels 1.0 km?

FIGURE P33.62

|| One day, after pulling down your window shade, you notice that sunlight is passing through a pinhole in the shade and making a small patch of light on the far wall. Having recently studied optics in your physics class, you’re not too surprised to see that the patch of light seems to be a circular diffraction pattern. It appears that the central maximum is about 1 cm across, and you estimate that the distance from the window shade to the wall is about 3 m. Estimate (a) the average wavelength of the sunlight (in nm) and (b) the diameter of the pinhole (in mm).

| A radar for tracking aircraft broadcasts a 12 GHz microwave beam from a 2.0-m-diameter circular radar antenna. From a wave perspective, the antenna is a circular aperture through which the microwaves diffract.

1. What is the diameter of the radar beam at a distance of 30 km?

2. If the antenna emits 100 kW of power, what is the average microwave intensity at 30 km?

| Scientists use laser range-finding to measure the distance to the moon with great accuracy. A brief laser pulse is fired at the moon, then the time interval is measured until the “echo” is seen by a telescope. A laser beam spreads out as it travels because it diffracts through a circular exit as it leaves the laser. In order for the reflected light to be bright enough to detect, the laser spot on the moon must be no more than 1.0 km in diameter. Staying within this diameter is accomplished by using a special large-diameter laser. If $\text{λ}=532\text{nm,}$ what is the minimum diameter of the circular opening from which the laser beam emerges? The earth-moon distance is 384,000 km.

| Suppose two distant sources of light are separated by angle $\varphi$ when viewed from the center of a circular aperture. Each source creates a diffraction pattern, and the two central maxima are separated by the same angle $\varphi$. If $\varphi$ is large, there will be two distinct bull’s-eye patterns and you can tell, just by looking at the screen, that there are two sources. If $\varphi$ is very small, the diffraction patterns will overlap so much that you cannot observe two distinct sources. The two sources will be marginally resolved, which means that you can just barely detect two sources, if the central maximum of one diffraction pattern falls on the first minimum of the other diffraction pattern—that is, if $\varphi ={\theta }_1=1.22\text{λ}\text{/}D$. The Arecibo Observatory in Puerto Rico was a 310-m-diameter radio telescope that collapsed in 2020. Incoming radio waves reflected from the receiving dish rather than passing through it, but the laws of diffraction work the same in both cases. What was the angular resolution—the smallest angle by which two sources could be separated and still be recognized as two distinct sources—when the radio telescope was looking for signals at a frequency of 1.5 GHz? Give your answer in minutes of arc, where 1° is divided into 60 minutes.

|| Light of wavelength 600 nm passes though two slits separated by 0.20 mm and is observed on a screen 1.0 m behind the slits. The location of the central maximum is marked on the screen and labeled $y=0$.

1. At what distance, on either side of $y=0$, are the $m=1$ bright fringes?

2. A very thin piece of glass is then placed in one slit. Because light travels slower in glass than in air, the wave passing through the glass is delayed by $5.0\times {10}^{-16}\text{s}$ in comparison to the wave going through the other slit. What fraction of the period of the light wave is this delay?

3. With the glass in place, what is the phase difference $\text{Δ}{\varphi }_0$ between the two waves as they leave the slits?

4. The glass causes the interference fringe pattern on the screen to shift sideways. Which way does the central maximum move (toward or away from the slit with the glass) and by how far?

|| A Michelson interferometer operating at a 600 nm wavelength has a 2.00-cm-long glass cell in one arm. To begin, the air is pumped out of the cell and mirror $\text{M}{}_2$ is adjusted to produce a bright spot at the center of the interference pattern. Then a valve is opened and air is slowly admitted into the cell. The index of refraction of air at 1.00 atm pressure is 1.00028. How many bright-dark-bright fringe shifts are observed as the cell fills with air?

|| Optical computers require microscopic optical switches to turn signals on and off. One device for doing so, which can be implemented in an integrated circuit, is the Mach-Zender interferometer seen in FIGURE P33.69. Light from an on-chip infrared laser $(\text{λ}=1.000\mu \text{m})$ is split into two waves that travel equal distances around the arms of the interferometer. One arm passes through an electro-optic crystal, a transparent material that can change its index of refraction in response to an applied voltage. Suppose both arms are exactly the same length and the crystal’s index of refraction with no applied voltage is 1.522.

1. With no voltage applied, is the output bright (switch closed, optical signal passing through) or dark (switch open, no signal)? Explain.

2. What is the first index of refraction of the electro-optic crystal larger than 1.522 that changes the optical switch to the state opposite the state you found in part a?

FIGURE P33.69

|| To illustrate one of the ideas of holography in a simple way, consider a diffraction grating with slit spacing $d$. The small-angle approximation is usually not valid for diffraction gratings, because $d$ is only slightly larger than $\text{λ}$, but assume that the $\text{λ}/d$ ratio of this grating is small enough to make the small-angle approximation valid.

1. Use the small-angle approximation to find an expression for the fringe spacing on a screen at distance $L$ behind the grating.

2. Rather than a screen, suppose you place a piece of film at distance $L$ behind the grating. The bright fringes will expose the film, but the dark spaces in between will leave the film unexposed. After being developed, the film will be a series of alternating light and dark stripes. What if you were to now “play” the film by using it as a diffraction grating? In other words, what happens if you shine the same laser through the film and look at the film’s diffraction pattern on a screen at the same distance $L$? Demonstrate that the film’s diffraction pattern is a reproduction of the original diffraction grating.

||| A 500 lines/mm diffraction grating is illuminated by light of wavelength 510 nm. How many bright fringes are seen on a 2.0-m-wide screen located 2.0 m behind the grating?

||| A double-slit experiment is set up using a helium-neon laser $(\text{λ}=633\text{nm})$. Then a very thin piece of glass $(n=1.50)$ is placed over one of the slits. Afterward, the central point on the screen is occupied by what had been the $m=10$ dark fringe. How thick is the glass?

||| FIGURE CP33.73 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure 33.9b). As a practical matter, two peaks can just barely be resolved if their spacing $\text{Δ}y$ equals the width $w$ of each peak, where $w$ is measured at half of the peak’s height. Two peaks closer together than $w$ will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating.

1. In the small-angle approximation, the position of the $m=1$ peak of a diffraction grating falls at the same location as the $m=1$ fringe of a double slit: $y_1=\text{λ}L/d$. Suppose two wavelengths differing by $\text{Δ}\text{λ}$ pass through a grating at the same time. Find an expression for $\text{Δ}y$, the separation of their first-order peaks.

2. We noted that the widths of the bright fringes are proportional to $1/N$, where $N$ is the number of slits in the grating. Let’s hypothesize that the fringe width is $w=y_1/N$. Show that this is true for the double-slit pattern. We’ll then assume it to be true as $N$ increases.

3. Use your results from parts a and b together with the idea that $\text{Δ}{y}_{\text{min}}=w$ to find an expression for $\text{Δ}{\text{λ}}_{\text{min}}$, the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved.

4. Ordinary hydrogen atoms emit red light with a wavelength of 656.45 nm. In deuterium, which is a “heavy” isotope of hydrogen, the wavelength is 656.27 nm. What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern?

FIGURE CP33.73

||| FIGURE CP33.74 shows light of wavelength $\text{λ}$ incident at angle $\varphi$ on a reflection grating of spacing $d$. We want to find the angles ${\theta }_m$ at which constructive interference occurs.

1. The figure shows paths 1 and 2 along which two waves travel and interfere. Find an expression for the path-length difference $\text{Δ}r=r_2-r_1$.

2. Using your result from part a, find an equation (analogous to Equation 33.15) for the angles ${\theta }_m$ at which diffraction occurs when the light is incident at angle $\varphi$. Notice that $m$ can be a negative integer in your expression, indicating that path 2 is shorter than path 1.

3. Show that the zeroth-order diffraction is simply a “reflection.” That is, ${\theta }_0=\varphi$.

4. Light of wavelength 500 nm is incident at $\varphi =40\text{°}$ on a reflection grating having 700 reflection $\text{lines}/\text{mm}$. Find all angles ${\theta }_m$ at which light is diffracted. Negative values of ${\theta }_m$ are interpreted as an angle left of the vertical.

5. Draw a picture showing a single 500 nm light ray incident at $\varphi =40\text{°}$ and showing all the diffracted waves at the correct angles.

FIGURE CP33.74

||| The pinhole camera of FIGURE CP33.75 images distant objects by allowing only a narrow bundle of light rays to pass through the hole and strike the film. If light consisted of particles, you could make the image sharper and sharper (at the expense of getting dimmer and dimmer) by making the aperture smaller and smaller. In practice, diffraction of light by the circular aperture limits the maximum sharpness that can be obtained. Consider two distant points of light, such as two distant streetlights. Each will produce a circular diffraction pattern on the film. The two images can just barely be resolved if the central maximum of one image falls on the first dark fringe of the other image. (This is called Rayleigh’s criterion, and we will explore its implication for optical instruments in Chapter 35.)

1. Optimum sharpness of one image occurs when the diameter of the central maximum equals the diameter of the pinhole. What is the optimum hole size for a pinhole camera in which the film is 20 cm behind the hole? Assume $\text{λ}=550\text{nm,}$ an average value for visible light.

2. For this hole size, what is the angle $\alpha$ (in degrees) between two distant sources that can barely be resolved?

3. What is the distance between two street lights 1 km away that can barely be resolved?

FIGURE CP33.75