### Problems

## 4.1 Single-Slit Diffraction

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width $1.00\mu \text{m}$ ? (b) Will there be a second minimum?

(a) Calculate the angle at which a $2.00\text{-}\mu \text{m}$-wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

(a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of $28.0\text{\xb0}$ ? (b) At what angle will the second minimum be?

(a) What is the width of a single slit that produces its first minimum at $60.0\text{\xb0}$ for 600-nm light? (b) Find the wavelength of light that has its first minimum at $62.0\text{\xb0}$.

Find the wavelength of light that has its third minimum at an angle of $48.6\text{\xb0}$ when it falls on a single slit of width $3.00\mu \text{m}$.

(a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width $7.50\mu \text{m}$. At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

Consider a single-slit diffraction pattern for $\lambda =589\phantom{\rule{0.2em}{0ex}}\text{nm}$, projected on a screen that is 1.00 m from a slit of width 0.25 mm. How far from the center of the pattern are the centers of the first and second dark fringes?

(a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width $2.00\mu \text{m}$. (b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit? (c) Discuss the ease or difficulty of measuring such a distance.

(a) What is the minimum width of a single slit (in multiples of $\lambda $) that will produce a first minimum for a wavelength $\lambda $ ? (b) What is its minimum width if it produces 50 minima? (c) 1000 minima?

(a) If a single slit produces a first minimum at $14.5\text{\xb0},$ at what angle is the second-order minimum? (b) What is the angle of the third-order minimum? (c) Is there a fourth-order minimum? (d) Use your answers to illustrate how the angular width of the central maximum is about twice the angular width of the next maximum (which is the angle between the first and second minima).

If the separation between the first and the second minima of a single-slit diffraction pattern is 6.0 mm, what is the distance between the screen and the slit? The light wavelength is 500 nm and the slit width is 0.16 mm.

A water break at the entrance to a harbor consists of a rock barrier with a 50.0-m-wide opening. Ocean waves of 20.0-m wavelength approach the opening straight on. At what angles to the incident direction are the boats inside the harbor most protected against wave action?

An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a 600-Hz sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is 0.800 m wide and the speed of sound is 340 m/s?

## 4.2 Intensity in Single-Slit Diffraction

A single slit of width $3.0\phantom{\rule{0.2em}{0ex}}\mu \text{m}$ is illuminated by a sodium yellow light of wavelength 589 nm. Find the intensity at a $15\text{\xb0}$ angle to the axis in terms of the intensity of the central maximum.

A single slit of width 0.1 mm is illuminated by a mercury light of wavelength 576 nm. Find the intensity at a $10\text{\xb0}$ angle to the axis in terms of the intensity of the central maximum.

The width of the central peak in a single-slit diffraction pattern is 5.0 mm. The wavelength of the light is 600 nm, and the screen is 2.0 m from the slit. (a) What is the width of the slit? (b) Determine the ratio of the intensity at 4.5 mm from the center of the pattern to the intensity at the center.

Consider the single-slit diffraction pattern for $\text{\lambda}=600\phantom{\rule{0.2em}{0ex}}\text{nm}$, $a=0.025\phantom{\rule{0.2em}{0ex}}\text{mm}$, and $x=2.0\phantom{\rule{0.2em}{0ex}}\text{m}$. Find the intensity in terms of ${I}_{o}$ at $\theta =0.5\text{\xb0}$, $1.0\text{\xb0}$, $1.5\text{\xb0}$, $3.0\text{\xb0}$, and $10.0\text{\xb0}$.

## 4.3 Double-Slit Diffraction

Two slits of width $2\phantom{\rule{0.2em}{0ex}}\mu \text{m,}$ each in an opaque material, are separated by a center-to-center distance of $6\phantom{\rule{0.2em}{0ex}}\mu \text{m}.$ A monochromatic light of wavelength 450 nm is incident on the double-slit. One ﬁnds a combined interference and diffraction pattern on the screen.

(a) How many peaks of the interference will be observed in the central maximum of the diffraction pattern?

(b) How many peaks of the interference will be observed if the slit width is doubled while keeping the distance between the slits same?

(c) How many peaks of interference will be observed if the slits are separated by twice the distance, that is, $12\phantom{\rule{0.2em}{0ex}}\mu \text{m,}$ while keeping the widths of the slits same?

(d) What will happen in (a) if instead of 450-nm light another light of wavelength 680 nm is used?

(e) What is the value of the ratio of the intensity of the central peak to the intensity of the next bright peak in (a)?

(f) Does this ratio depend on the wavelength of the light?

(g) Does this ratio depend on the width or separation of the slits?

A double slit produces a diffraction pattern that is a combination of single- and double-slit interference. Find the ratio of the width of the slits to the separation between them, if the first minimum of the single-slit pattern falls on the fifth maximum of the double-slit pattern. (This will greatly reduce the intensity of the fifth maximum.)

For a double-slit configuration where the slit separation is four times the slit width, how many interference fringes lie in the central peak of the diffraction pattern?

Light of wavelength 500 nm falls normally on 50 slits that are $2.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{mm}$ wide and spaced $5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{mm}$ apart. How many interference fringes lie in the central peak of the diffraction pattern?

A monochromatic light of wavelength 589 nm incident on a double slit with slit width $2.5\phantom{\rule{0.2em}{0ex}}\mu \text{m}$ and unknown separation results in a diffraction pattern containing nine interference peaks inside the central maximum. Find the separation of the slits.

When a monochromatic light of wavelength 430 nm incident on a double slit of slit separation $5\phantom{\rule{0.2em}{0ex}}\mu \text{m}$, there are 11 interference fringes in its central maximum. How many interference fringes will be in the central maximum of a light of the same wavelength and slit widths, but a new slit separation of $4\phantom{\rule{0.2em}{0ex}}\mu \text{m}$?

Determine the intensities of two interference peaks other than the central peak in the central maximum of the diffraction, if possible, when a light of wavelength 628 nm is incident on a double slit of width 500 nm and separation 1500 nm. Use the intensity of the central spot to be ${1\phantom{\rule{0.2em}{0ex}}\text{mW/cm}}^{2}$.

## 4.4 Diffraction Gratings

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?

Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a difraction grating having 1500 lines per centimeter.

How many lines per centimeter are there on a diffraction grating that gives a first-order maximum for 470-nm blue light at an angle of $25.0\text{\xb0}$ ?

What is the distance between lines on a diffraction grating that produces a second-order maximum for 760-nm red light at an angle of $60.0\text{\xb0}$ ?

Calculate the wavelength of light that has its second-order maximum at $45.0\text{\xb0}$ when falling on a diffraction grating that has 5000 lines per centimeter.

An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles $24.2\text{\xb0},\phantom{\rule{0.2em}{0ex}}25.7\text{\xb0},\phantom{\rule{0.2em}{0ex}}29.1\text{\xb0},$ and $41.0\text{\xb0}$ when projected on a diffraction grating having 10,000 lines per centimeter?

(a) What do the four angles in the preceding problem become if a 5000-line per centimeter diffraction grating is used? (b) Using this grating, what would the angles be for the second-order maxima? (c) Discuss the relationship between integral reductions in lines per centimeter and the new angles of various order maxima.

What is the spacing between structures in a feather that acts as a reflection grating, giving that they produce a first-order maximum for 525-nm light at a $30.0\text{\xb0}$ angle?

An opal such as that shown in Figure 4.15 acts like a reflection grating with rows separated by about $8\phantom{\rule{0.2em}{0ex}}\text{\mu m}.$ If the opal is illuminated normally, (a) at what angle will red light be seen and (b) at what angle will blue light be seen?

At what angle does a diffraction grating produce a second-order maximum for light having a first-order maximum at $20.0\text{\xb0}$ ?

(a) Find the maximum number of lines per centimeter a diffraction grating can have and produce a maximum for the smallest wavelength of visible light. (b) Would such a grating be useful for ultraviolet spectra? (c) For infrared spectra?

(a) Show that a 30,000 line per centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a first-order maximum? (c) What is the greatest number of line per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?

The analysis shown below also applies to diffraction gratings with lines separated by a distance *d*. What is the distance between fringes produced by a diffraction grating having 125 lines per centimeter for 600-nm light, if the screen is 1.50 m away? (*Hin*t*:* The distance between adjacent fringes is $\text{\Delta}y=x\lambda \text{/}d,$ assuming the slit separation *d* is comparable to $\text{\lambda}.$)

## 4.5 Circular Apertures and Resolution

The 305-m-diameter Arecibo radio telescope pictured in Figure 4.20 detected radio waves with a 4.00-cm average wavelength. (a) What is the angle between two just-resolvable point sources for this telescope? (b) How close together could these point sources be at the 2 million light-year distance of the Andromeda Galaxy?

Assuming the angular resolution found for the Hubble Telescope in Example 4.6, what is the smallest detail that could be observed on the moon?

Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally 5.00 cm in diameter with an average wavelength of 600 nm.

(a) What is the minimum angular spread of a 633-nm wavelength He-Ne laser beam that is originally 1.00 mm in diameter? (b) If this laser is aimed at a mountain cliff 15.0 km away, how big will the illuminated spot be? (c) How big a spot would be illuminated on the moon, neglecting atmospheric effects? (This might be done to hit a corner reflector to measure the round-trip time and, hence, distance.)

A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the moon. (a) If this is done with the Mount Wilson telescope, producing a 2.54-m-diameter beam of 633-nm light, what is the minimum angular spread of the beam? (b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the moon, assuming a lunar distance of $3.84\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m}$ ?

The limit to the eye’s acuity is actually related to diffraction by the pupil. (a) What is the angle between two just-resolvable points of light for a 3.00-mm-diameter pupil, assuming an average wavelength of 550 nm? (b) Take your result to be the practical limit for the eye. What is the greatest possible distance a car can be from you if you can resolve its two headlights, given they are 1.30 m apart? (c) What is the distance between two just-resolvable points held at an arm’s length (0.800 m) from your eye? (d) How does your answer to (c) compare to details you normally observe in everyday circumstances?

What is the minimum diameter mirror on a telescope that would allow you to see details as small as 5.00 km on the moon some 384,000 km away? Assume an average wavelength of 550 nm for the light received.

Find the radius of a star’s image on the retina of an eye if its pupil is open to 0.65 cm and the distance from the pupil to the retina is 2.8 cm. Assume $\text{\lambda}=550\phantom{\rule{0.2em}{0ex}}\text{nm}$.

(a) The dwarf planet Pluto and its moon, Charon, are separated by 19,600 km. Neglecting atmospheric effects, should the 5.08-m-diameter Palomar Mountain telescope be able to resolve these bodies when they are $4.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}\text{km}$ from Earth? Assume an average wavelength of 550 nm. (b) In actuality, it is just barely possible to discern that Pluto and Charon are separate bodies using a ground-based telescope. What are the reasons for this?

A spy satellite orbits Earth at a height of 180 km. What is the minimum diameter of the objective lens in a telescope that must be used to resolve columns of troops marching 2.0 m apart? Assume $\text{\lambda}=550\phantom{\rule{0.2em}{0ex}}\text{nm}.$

What is the minimum angular separation of two stars that are just-resolvable by the 8.1-m Gemini South telescope, if atmospheric effects do not limit resolution? Use 550 nm for the wavelength of the light from the stars.

The headlights of a car are 1.3 m apart. What is the maximum distance at which the eye can resolve these two headlights? Take the pupil diameter to be 0.40 cm.

When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh’s criterion. Take the pupil of the eye to be 3.0 mm and the distance from the paper to the eye of 35 cm; find the minimum separation of two dots such that they cannot be resolved. How many dots per inch (dpi) does this correspond to?

Suppose you are looking down at a highway from a jetliner flying at an altitude of 6.0 km. How far apart must two cars be if you are able to distinguish them? Assume that $\text{\lambda}=550\phantom{\rule{0.2em}{0ex}}\text{nm}$ and that the diameter of your pupils is 4.0 mm.

Can an astronaut orbiting Earth in a satellite at a distance of 180 km from the surface distinguish two skyscrapers that are 20 m apart? Assume that the pupils of the astronaut’s eyes have a diameter of 5.0 mm and that most of the light is centered around 500 nm.

The characters of a stadium scoreboard are formed with closely spaced lightbulbs that radiate primarily yellow light. (Use $\text{\lambda}=600\phantom{\rule{0.2em}{0ex}}\text{nm}.$) How closely must the bulbs be spaced so that an observer 80 m away sees a display of continuous lines rather than the individual bulbs? Assume that the pupil of the observer’s eye has a diameter of 5.0 mm.

If a microscope can accept light from objects at angles as large as $\alpha =70\text{\xb0}$, what is the smallest structure that can be resolved when illuminated with light of wavelength 500 nm and (a) the specimen is in air? (b) When the specimen is immersed in oil, with index of refraction of 1.52?

A camera uses a lens with aperture 2.0 cm. What is the angular resolution of a photograph taken at 700 nm wavelength? Can it resolve the millimeter markings of a ruler placed 35 m away?

## 4.6 X-Ray Diffraction

X-rays of wavelength 0.103 nm reflects off a crystal and a second-order maximum is recorded at a Bragg angle of $25.5\text{\xb0}$. What is the spacing between the scattering planes in this crystal?

A first-order Bragg reflection maximum is observed when a monochromatic X-ray falls on a crystal at a $32.3\text{\xb0}$ angle to a reflecting plane. What is the wavelength of this X-ray?

An X-ray scattering experiment is performed on a crystal whose atoms form planes separated by 0.440 nm. Using an X-ray source of wavelength 0.548 nm, what is the angle (with respect to the planes in question) at which the experimenter needs to illuminate the crystal in order to observe a first-order maximum?

The structure of the NaCl crystal forms reflecting planes 0.541 nm apart. What is the smallest angle, measured from these planes, at which X-ray diffraction can be observed, if X-rays of wavelength 0.085 nm are used?

On a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of $27.1\text{\xb0}$ relative to its surface, using an X-ray source of unknown wavelength. Additionally, when illuminated with a different, this time of known wavelength 0.137 nm, a second-order maximum is detected at $37.3\text{\xb0}.$ Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.

Calcite crystals contain scattering planes separated by 0.30 nm. What is the angular separation between first and second-order diffraction maxima when X-rays of 0.130 nm wavelength are used?

The first-order Bragg angle for a certain crystal is $12.1\text{\xb0}$. What is the second-order angle?