Chapter 32, Interference and Diffraction Video Solutions, Essential University Physics: Volume 1 & 2 pack | Numerade (2024)

A prism bends blue light more than red. Is the same true of a diffraction grating?

Why does an oil slick show colored bands?

Why does a soap bubble turn colorless just before it dries up and pops?

Can optical interference be a viable technique to measure the refractive index of a liquid?

You can hear around corners, but you can't see around comers. Why?

. If you have a dual-color, coherent source that emits red and blue light, what sort of double-slit pattern would you see using it?

The primary maxima in multiple-slit interference are in the same angular positions as those in double-slit interference. Why, then. do diffraction gratings have thousands of slits instead of just two?

What changes do you expect to observe if you use white light instead of a monochromatic source in a double-slit experiment?

Sketch roughly the diffraction pattern you would expect for light passing through a square hole a few wavelengths wide.

A double-slit system is used to measure the wavelength of light. The system has slit spacing $d=15 \mu \mathrm{m}$ and slit-to-screen distance $L=2.2 \mathrm{~m}$. If the $m=1$ maximum in the interference pattern occurs $7.1 \mathrm{~cm}$ from screen center, what's the wavelength?

A double-slit experiment with $d=0.025 \mathrm{~mm}$ and $L=74 \mathrm{~cm}$ uses $550-\mathrm{nm}$ light. Find the spacing between adjacent bright fringes.

A double-slit experiment has slit spacing $0.12 \mathrm{~mm}$. (a) What should be the slit-to-screen distance $L$ if the bright fringes are to be $5.0 \mathrm{~mm}$ apart when the slits are illuminated with $633-\mathrm{nm}$ laser light? (b) What will be the fringe spacing with $480-\mathrm{nm}$ light?

The interference pattern from two slits separated by $0.37 \mathrm{~mm}$ has bright fringes with angular spacing $0.065^{\circ}$. Find the light's wavelength.

The 546 -nm green line of gaseous mercury falls on a double-slit apparatus. If the fifth dark fringe is at $0.113^{\circ}$ from the centerline, what's the slit separation?

In a five-slit system, how many minima lie between the zeroth-order and first-order maxima?

In a three-slit system, the first minimum occurs at angular position $5.40^{\circ}$. Where is the next maximum?

A five-slit system with $7.5-\mu \mathrm{m}$ slit spacing is illuminated with 633 -nm light. Find the angular positions of (a) the first two maxima and (b) the third and sixth minima.

Green light at $522 \mathrm{~nm}$ is diffracted by a grating with 3000 lines $/ \mathrm{cm}$. Through what angle is the light diffracted in (a) first and (b) fifth order?

Find the minimum thickness of a soap film $(n=1.333)$ in which 705 -nm light will undergo constructive interference.

Light of unknown wavelength shines on a precisely machined glass wedge with refractive index 1.52 . The closest point to the apex of the wedge where reflection is enhanced occurs where the wedge is $98 \mathrm{~nm}$ thick. Find the wavelength.

Monochromatic light shines on a glass wedge with refractive index 1.65, and enhanced reflection occurs where the wedge is $450 \mathrm{~nm}$ thick. Find all possible values for the wavelength in the visible range.

White light shines on a 98.0 -nm-thick sliver of fluorite $(n=1.43)$. What wavelength is most strongly reflected?

For the soap film described in Conceptual Example 32.1's "Making the Connection," what portion of the film will appear dark when it's illuminated with white light?

For what ratio of slit width to wavelength will the first minima of a single-slit diffraction pattern occur at $\pm 90^{\circ}$ ?

Light with wavelength $615 \mathrm{~nm}$ is incident on a $2.40-\mu \mathrm{m}$-wide slit. Find the angular width of the central peak in the diffraction pattern, taken as the angular separation between the first minima.

You're inside a metal building that blocks radio waves, but you're trying to make a call with your cellphone, which broadcasts at a frequency of $950 \mathrm{MHz}$. Down the hall from you is a narrow window measuring $35 \mathrm{~cm}$ wide. What's the horizontal angular width of the beam (i.e., the angle between the first minima) from your phone as it emerges from the window?

Find the intensity as a fraction of the central peak intensity for the second secondary maximum in single-slit diffraction, assuming the peak lies midway between the second and third minima.

. Find the minimum angular separation resolvable with $627-\mathrm{nm}$ laser light passing through a circular aperture of diameter $2.8 \mathrm{~cm}$.

Find the minimum telescope aperture that could resolve an object with angular diameter 0.41 arcsecond, observed at 515 -nm wavelength.

What's the longest wavelength of light you could use to resolve a structure with angular diameter 0.43 mrad, using a microscope with a 1.8 -mm-diameter objective lens?

In bright light, the human eye's pupil diameter is about $2 \mathrm{~mm}$. If Bio diffraction were the limiting factor, what's the eye's minimum angular resolution under these conditions, assuming 550 -nm light?

Example 32.2: Atomic hydrogen emits a total of four spectral lines in the visible part of the spectrum. In addition to the two used in Example 32.2, H $\gamma$ and $\mathrm{H} \delta$ are at $434.0 \mathrm{~nm}$ and $410.2 \mathrm{~nm}$, respectively. Find the angular separation of these two lines when observed with the spectrometer of Example 32.2.

Example 32.2; Two of the four visible spectral lines of atomic hydrogen (whose wavelengths are given either in Example 32.2 or in the preceding problem) exhibit an angular separation of $8.1^{\circ}$ when observed with the spectrometer of Example 32.2. Which two lines are they?

Example 32.2: Atomic sodium produces two prominent spectral lines at $588.995 \mathrm{~nm}$ and $589.592 \mathrm{~nm}$. Find the angular separation between these lines when observed in third order using a spectrometer with 2300 lines per $\mathrm{cm}$.

Example 32.2: Consider a grating spectrometer where the spacing $d$ between lines is large enough compared with the wavelength of light that you can apply the small-angle approximation $\sin \theta=\theta$ in Equation 32. 1a. Find an expression for the line spacing $d$ required for a given (small) angular separation $\Delta \theta$ between spectral lines with wavelengths $\lambda_1$ and $\lambda_2$, when observed in first order.

Example 32.5: Find the minimum asteroid size in Example 32.5 if it were imaged with the $6.5-\mathrm{m}$ James Webb Space Telescope observing at an infrared wavelength of $850 \mathrm{~nm}$.

. Example 32.5: An asteroid as small as $35 \mathrm{~m}$ in diameter could pose a threat to a city. What size telescope mirror would be needed to image such an asteroid when it's $1.20 \mathrm{Gm}$ from Earth (roughly three times the Moon's distance)? Assume it's imaged at an optical wavelength of $535 \mathrm{~nm}$.

Example 32.5: You've got binoculars with 28-mm-diameter lenses. You're observing two bees on a flower $840 \mathrm{~m}$ distant. What's the minimum separation between the bees for you to be able to resolve them as two distinct insects? Assume visible light with 550 -nm wavelength.

Example 32.5: WorldView-4 is a satellite that images Earth's surface for commercial purposes. It orbits Earth at $610 \mathrm{~km}$ altitude and achieves its highest resolution of $31 \mathrm{~cm}$ at a wavelength of $460 \mathrm{~nm}$. Find the diameter of the mirror that WorldView-4 uses for its imaging.

Find the angular position of the second-order bright fringe in a double-slit system whose slit spacing is $1.5 \mu \mathrm{m}$ for (a) red light at $640 \mathrm{~nm}$, (b) yellow light at $580 \mathrm{~nm}$, and (c) violet light at $410 \mathrm{~nm}$.

A double-slit experiment has slit spacing $0.035 \mathrm{~mm}$, slit-toscreen distance $1.5 \mathrm{~m}$, and wavelength $490 \mathrm{~nm}$. What's the phase difference between two waves arriving at a point $0.56 \mathrm{~cm}$ from the center line of the screen?

A tube of glowing gas emits light at $550 \mathrm{~nm}$ and $400 \mathrm{~nm}$. In a double-slit apparatus, what's the lowest-order 550 -nm bright fringe that will fall on a $400-\mathrm{nm}$ dark fringe, and what are the fringes' corresponding orders?

On the screen of a multiple-slit system, the interference pattern shows bright maxima separated by $0.80^{\circ}$ and seven minima between each bright maximum. (a) How many slits are there? (b) What's the slit separation if the incident light has wavelength $656.3 \mathrm{~nm}$ ?

You're designing a spectrometer whose specifications call for a minimum of $5^{\circ}$ separation between the red hydrogen- $\alpha$ line at $656 \mathrm{~nm}$ and the yellow sodium line at $589 \mathrm{~nm}$ when the two are observed in third order with a grating spectrometer. Available gratings have 2500 lines $/ \mathrm{cm}, 3500$ lines $/ \mathrm{cm}$, or 4500 lines $/ \mathrm{cm}$. What's the coarsest grating you can use?

What order is necessary to resolve $647.98-\mathrm{nm}$ and $648.07-\mathrm{nm}$ spectral lines using a 4500 -line grating?

A thin film of toluene $(n=1.49)$ floats on water. Find the minimum film thickness if the most strongly reflected light has wavelength $460 \mathrm{~nm}$.

NASA asks you to assess the feasibility of a single-mirror spacebased optical telescope that could resolve an Earth-sized planet 5 light-years away. What do you conclude?

In the second-order spectrum from a diffraction grating, yellow light at $588 \mathrm{~nm}$ overlaps violet light (wavelength range $390 \mathrm{~nm}$ $450 \mathrm{~nm}$ ) diffracted in a different order. What's the exact wavelength of the violet light, and what's the order of its diffraction?

X-ray diffraction in potassium chloride $(\mathrm{KCl})$ results in a firstorder maximum when 97 -pm-wavelength X-rays graze the crystal plane at $8.5^{\circ}$. Find the spacing between crystal planes.

As a soap bubble with $n=1.333$ evaporates and thins, reflected colors gradually disappear. What are (a) the bubble thickness just as the last vestige of color vanishes and (b) the last color seen?

An oil film with refractive index 1.25 floats on water. The film thickness varies from $0.80 \mu \mathrm{m}$ to $2.1 \mu \mathrm{m}$. If $630-\mathrm{nm}$ light is incident normally on the film, at how many locations will it undergo enhanced reflection?

The table below lists the angular positions of the bright fringes DATA that result when monochromatic laser light shines through a diffraction grating, as a function of order $m$. The spacing between lines of the grating is $d=3.2 \mu \mathrm{m}$. Determine a quantity that, when plotted against $m$, should give a straight line. Plot the data, determine a best-fit line, and use the result to find the wavelength of the light.

Two perfectly flat glass plates are separated at one end by a sheet of paper $0.065 \mathrm{~mm}$ thick. $535-\mathrm{nm}$ light illuminates the plates from above, as shown in Fig. 32.28. How many bright bands appear to an observer looking down on the plates?
(FIGURE CAN'T COPY)

An air wedge like that of Fig. 32.28 shows $N$ bright bands when illuminated from above. Find an expression for the number of bands if the air is replaced by a liquid of refractive index $n$ different from that of the glass.
(FIGURE CAN'T COPY)

A Michelson interferometer uses light from glowing hydrogen at $486.1 \mathrm{~nm}$. As you move one mirror, 527 bright fringes pass a fixed point in the viewer. How far did the mirror move?

Find the wavelength of light used in a Michelson interferometer if 570 bright fringes go by a fixed point when the mirror moves $0.155 \mathrm{~mm}$.

One arm of a Michelson interferometer is $42.5 \mathrm{~cm}$ long and is enclosed in a box that can be evacuated. The box initially contains air, which is gradually pumped out. In the process, 388 bright fringes pass a point in the viewer. If the interferometer uses light with wavelength $641.6 \mathrm{~nm}$, what's the air's refractive index?

You're driving over a bridge at $65.0 \mathrm{~km} / \mathrm{h}$, listening to an AM radio station broadcasting at $535 \mathrm{kHz}$. The station's transmitting antenna is directly behind you, and its signals are reaching your car both directly and after reflecting off a metal bridge structure ahead of you. The resulting interference means that you experience a varying signal strength. What's the time you perceive between the maximum signal strength and the following minimum signal?

A laser-based spacecraft accelerator (see Application Starshot! on page 597) requires that a laser spot $6.0 \mathrm{~m}$ in diameter be focused from Earth on a spacecraft that's $66 \mathrm{Mm}$ distant. Given a laser wavelength of $1.06 \mu \mathrm{m}$, what's the effective aperture diameter required for the optical system that shapes the laser beam?

Suppose one of the 10 -m-diameter Keck Telescopes in Hawaii is trained on San Francisco, $3400 \mathrm{~km}$ away. Assuming $550-\mathrm{nm}$ light, and ignoring atmospheric distortion, would it be possible to read (a) newspaper headlines or (b) a billboard sign at this distance? (c) Repeat for the case of the Keck optical interferometer, formed from the two $10-\mathrm{m}$ Keck Telescopes and several smaller ones, with a 50 -m effective aperture.

A camera has an $f / 1.4$ lens, meaning the ratio of focal length to lens diameter is 1.4. Find the smallest spot diameter (i.e., the diameter of the first diffraction minimum) to which this lens can focus parallel light with 610 -nm wavelength.

The CIA wants your help identifying individual terrorists in a photo of a training camp taken from a spy satellite at $100-\mathrm{km}$ altitude. You ask for details of the optical system used, but they're classified. However, they do tell you that the optics are diffraction limited and can resolve facial features as small as $5 \mathrm{~cm}$. Assuming a typical optical wavelength of $550 \mathrm{~nm}$, what do you conclude about the size of the mirror or lens in the satellite camera?

While driving at night, your eyes' irises dilate to $3.1-\mathrm{mm}$ diamman eter. If your vision were diffraction limited, what would be the greatest distance at which you could see the two headlights of an oncoming car, spaced $1.6 \mathrm{~m}$ apart, distinctly? Take $\lambda=550 \mathrm{~nm}$.

Under the best conditions, atmospheric turbulence limits groundbased telescopes' resolution to about 1 arcsecond ( $1 / 3600$ of a degree). For what apertures is this limitation more severe than that of diffraction at $550 \mathrm{~nm}$ ? (Your answer shows why large ground-based telescopes don't generally produce better images than small ones, although they do gather more light.)

You're a biologist studying rhinoviruses, which cause the comBio mon cold. These are among the smallest viruses, some $50 \mathrm{~nm}$ in diameter, and you can't image them with your optical microscope using visible light (average wavelength about $560 \mathrm{~nm}$ ). A sales rep tries to sell you an expensive microscope using 280-nm ultraviolet light, saying it will resolve structures half the size that's resolvable with your current microscope. Is the rep correct? Will the new microscope resolve your rhinoviruses?

An air wedge like that of Fig. 32.28 displays 10,003 bright bands when illuminated from above. If the region between the plates is then evacuated, the number of bands drops to 10,000 . Find the refractive index of the air.
(FIGURE CAN'T COPY)

A thin-walled glass tube of length $L$ containing a gas of unknown refractive index is placed in one arm of a Michelson interferometer using light of wavelength $\lambda$. The tube is then evacuated. During the process, $m$ bright fringes pass a fixed point in the viewer. Find an expression for the refractive index of the gas.

Light is incident on a diffraction grating at angle $\alpha$ to the normal. Show that the condition for maximum light intensity becomes

The Laser Interferometer Gravitational-Wave Observatory (LIGO) shown in Fig. 32.16 is essentially a Michelson interferometer as diagrammed in Fig. 32.15. Laser light with wavelength $1064 \mathrm{~nm}$ reflects 280 times as it traverses the $4-\mathrm{km}$ arms of the interferometer, making the effective one-way path length $1120 \mathrm{~km}$. Suppose that a gravitational wave causes the actual length of one $4-\mathrm{km}$ arm to change by a mere 1.2 attometers (am; $1.2 \times 10^{-18} \mathrm{~m}$ ). (a) Find the corresponding change in the longer effective path length that accounts for the multiple reflections. (b) By what fraction of a fringe spacing will the interference pattern shift, assuming the perpendicular arm is unaffected? It's this tiny effect that LIGO successfully measures.

The intensity of the single-slit diffraction pattern can be calculated
$\mathrm{CH}$ by summing the amplitudes of infinitely many field amplitudes corresponding to waves from every infinitesimal part of the slit. (a) Referring to Fig. 32.20, show that the field from an element of slit width $d y$, a distance $y$ from the bottom edge of the slit, is $d E=\left(E_{\mathrm{p}} d y / a\right) \sin (\omega t+\phi(y))$, where $\phi(y)=(2 \pi y / \lambda) \sin \theta$. (b) Integrate $d E$ over the entire slit from $y=0$ to $y=a$, and use trig identities from Appendix $\mathrm{A}$ to find the total amplitude and from there show that the average intensity is given by Equation 32.10 .

You're on an international panel charged with allocating "real estate" for communications satellites in geostationary orbit. You're to estimate how many satellites could fit in geostationary orbit without receivers on the ground picking up multiple signals. Satellites broadcast at $12 \mathrm{GHz}$ and receiver dishes are $47 \mathrm{~cm}$ in diameter. Use the Raleigh criterion to estimate the number of satellites allowed in geostationary orbit if each receiver dish is to "see" just one satellite. Make the simplifying assumption that all dishes are located on the equator and that each is pointed at the nearest satellite. (Hint: Consult Example 8.3).

You're investigating an oil spill for your state environmental EW protection agency. There's a thin film of oil on water, and you know its refractive index is $n_{\text {cil }}=1.39$. You shine white light vertically on the oil and use a spectrometer to determine that the most strongly reflected wavelength is $555 \mathrm{~nm}$. Assuming first order thin-film interference, what do you report for the thickness of the oil slick?

If the separation of two telescopes comprising an interferometer is doubled, the angular separation between two sources just barely resolvable by the interferometer will
(FIGURE CAN'T COPY)
a. not change.
b. decrease by a factor of $1 / \sqrt{2}$.
c. halve.
d. double.

If the separation of two telescopes comprising an interferometer is doubled, the instrument's light-collecting power will
(FIGURE CAN'T COPY)
a. not change.
b. increase by a factor of $\sqrt{2}$.
c. double.
d. quadruple.

If a point source is located directly above a two-telescope interferometer, on the perpendicular bisector of the line joining the telescopes (source 1 in Fig. 32.29), electromagnetic waves reaching the two will be
(FIGURE CAN'T COPY)
a. in phase.
b. out of phase by $45^{\circ}$.
c. out of phase by $90^{\circ}$.
d. you can't tell without further information

If a point source is located on a line at $45^{\circ}$ to the line joining the two telescopes (source 2 in Fig. 32.29), electromagnetic waves reaching the two will be
(FIGURE CAN'T COPY)
a. in phase.
b. out of phase by $45^{\circ}$.
c. out of phase by $90^{\circ}$.
d. you can't tell without further information