College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 26

Quantum Optics

Educators

Chapter Questions

Problem 1

Wavelength of radiation from a person If a person could be modeled as a black body, at what wavelength would his or her surface emit the maximum energy?

Sarah J.

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Problem 2

(a) A surface at $27^{\circ} \mathrm{C}$ emits radiation at a rate of 100 $\mathrm{W}$ . At what rate does an identical surface at $54^{\circ} \mathrm{C}$ emit radiation? (b) Determine the wavelength of the maximum amount of radiation emitted by each surface.

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Problem 3

Maximum radiation wavelength from star, Sun, and Earth Determine the wavelengths for the following black body radiation sources where they emit the most energy: (a) A bluewhite star at 40,000 K; (b) the Sun at 6000 K; and (c) Earth at about 300 K.

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Problem 4

Star colors and radiation frequency The colors of the stars in the sky range from red to blue. Assuming that the color indicates the frequency at which the star radiates the maximum amount of electromagnetic energy, estimate the surface temperature of red, yellow, white, and blue stars. What assumptions do you need to make about white stars to estimate the surface temperature?

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Problem 5

Estimate the surface area of a 60-watt lightbulb filament. Assume that the surface temperature of the filament when it is plugged into an outlet of 120 V is about 3000 K and the power rating of the bulb is the electric energy/s it consumes (not what it radiates). Incandescent lightbulbs usually radiate in visible
light about 10% of the electric energy that they consume.

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Problem 6

Photon emission rate from human skin Estimate the number of photons emitted per second from 1.0 $\mathrm{cm}^{2}$ of a person's skin if a typical emitted photon has a wavelength of $10,000 \mathrm{nm} .$

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Problem 7

Balancing Earth radiation absorption and emission Compare the average power that the surface of Earth facing the Sun receives from it to the energy that Earth emits over its entire surface due to it being a warm object. Assume that the average temperature of Earth's surface is about $15^{\circ} \mathrm{C}$ . The distance between Earth and the Sun is about $1.5 \times 10^{11} \mathrm{m} .$

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Problem 8

(a) Explain how you convert energy in joules into energy in electron volts. (b) The kinetic energy of an electron is 2.30 eV. What is its kinetic energy in joules?

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Problem 9

Draw a picture of a phototube and the electric circuit that you can build to study the photoelectric effect. Label all of the parts and explain the purpose of each part.

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Problem 10

(a) Describe the experimental findings for the photoelectric effect. (b) What findings could be explained by the wave model of light? (c) What experimental findings concerning the photoelectric effect could not be explained by the wave model of light?

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Problem 11

The stopping potential for an ejected photoelectron is -0.50 V. What is the maximum kinetic energy of the electron ejected by the light?

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Problem 12

Light shines on a cathode and ejects electrons. Draw an energy bar chart describing this process. Explain why the frequency of incident light determines whether the electrons will be ejected or not.

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Problem 13

What is the cutoff frequency of light if the cathode in a photoelectric tube is made of iron?

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Problem 14

The work function of cesium is 2.1 eV. (a) Determine the lowest frequency photon that can eject an electron from cesium. (b) Determine the maximum possible kinetic energy in electron volts of a photoelectron ejected from the metal that absorbs a 400-nm photon

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Problem 15

Visible light shines on the metal surface of a photocell having a work function of 1.30 eV. The maximum kinetic energy of the electrons leaving the surface is 0.92 eV. Determine the light’s wavelength.

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Problem 16

Equation Jeopardy 1 Solve for the unknown quantity in the equation below and write a problem for which the equation could be a solution.
$$
-3.9 \mathrm{eV}+\left(6.63 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\right) f\left(\frac{1 \mathrm{eV}}{1.6 \times 10^{-19} \mathrm{J}}\right)=(-e)(-1.0 \mathrm{V})
$$

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Problem 17

Camera film exposure In an old-fashioned camera, the film becomes exposed when light striking it initiates a complex chemical reaction. A particular type of film does not become exposed if struck by light of wavelength longer than 670 nm. Determine the minimum energy in electron volts needed to
initiate the chemical reaction.

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Problem 18

CO vibration A vibrating carbon monoxide (CO) molecule produces infrared photons of energy 0.26 eV. Determine the frequency of CO vibration, which is the same as the frequency of the infrared radiation the molecule emits

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Problem 19

Breaking a molecular bond Suppose the bond in a molecule is broken by photons of energy 5.0 eV. Determine the frequency and wavelength of these photons and the region of the electromagnetic spectrum in which they are located.

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Problem 20

A l.0-eV photon’s wavelength is 1240 nm. Use a ratio technique to determine the wavelength of a 5.0-eV photon.

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Problem 21

Tanning bed In a tanning bed, exposure to photons of wavelength 300 nm or less can do considerable damage. Determine the lowest energy in electron volts of such photons.

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Problem 22

Determine the number of 650 -nm photons that together have energy equal to the rest energy of an electron. [Hint: See Section $25.8 . ]$

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Problem 23

Laser surgery Scientists studying the use of lasers in various surgeries have found that very short $10^{-12}-$ -s laser pulses of power $10^{+12} \mathrm{W}$ with 65 pulses every $200 \times 10^{-6} \mathrm{s}$ produced much cleaner welds and ablations (removals of body tissues) than longer laser pulses. Determine the numberof $10.6-\mu \mathrm{m}$ photons in one pulse and the average power during the 65 pulses delivered in $200 \times 10^{-6} \mathrm{s}$ .

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Problem 24

A laser beam of power $P$ in watts consists of photons of wavelength $\lambda$ in nanometers. Determine in terms of these quantities the number of photons passing a cross section along the beam's path each second.

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Problem 25

What is the mass of the photons in the previous problem?

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Problem 26

Pulsed laser replaces dental drills A laser used for many applications of hard surface dental work emits 2780-nm wavelength pulses of variable energy (0–300 mJ) about 20 times per second. Determine the number of photons in one 100-mJ pulse and the average power of these photons during 1 s

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Problem 27

Light hitting Earth The intensity of light reaching Earth is about 1400 $\mathrm{W} / \mathrm{m}^{2} .$ Determine the number of photons reaching a $1.0-\mathrm{m}^{2}$ area each second. What assumptions did you make?

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Problem 28

Lightbulb Roughly 10% of the power of a 100-watt incandescent lightbulb is emitted as light, the rest being emitted as heat and longer-wavelength radiation. Estimate the number of photons of light coming from a bulb each second. What assumptions did you make? How will the answer change if the
assumptions are not valid?

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Problem 29

Human vision sensitivity To see an object with the unaided eye, the light intensity coming to the eye must be about $5 \times 10^{-12} \mathrm{J} / \mathrm{m}^{2} \cdot$ s or greater. Determine the minimum number of photons that must enter the eye's pupil each second in order for an object to be seen. Assume that the pupil's radius is 0.20 $\mathrm{cm}$ and the wavelength of the light is 550 $\mathrm{nm} .$

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Problem 30

Explain how a cathode ray tube works. Draw a picture and an electric circuit. Label the important elements and explain how they work together to produce cathode rays.

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Problem 31

Explain how we know that cathode rays are low-mass light negatively charged particles. Draw pictures and field diagrams to illustrate your explanation

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Problem 32

An X-ray tube emits photons of frequency $1.33 \times 10^{19} \mathrm{Hz}$ or less. (a) Explain how the tube creates the X-ray photons. (b) Determine the potential difference across the X-ray tube.

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Problem 33

Electrons are accelerated across a 40,000-V potential difference. (a) Explain why X-rays are created when the electrons crash into the anode of the X-ray tube. (b) Determine the frequency and wavelength of the maximum-energy photons created.

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Problem 34

An electron with kinetic energy $K$ moving horizontally to the right in a tube of length $L$ passes through a uniform electric field with an $\vec{E}$ field that points downward. The electron is initially moving toward the center of the screen. Develop an expression for the strength of the field so the electron hits the screen a vertical height $h$ above the center.

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Problem 35

An electron with kinetic energy K moving horizontally to the right in a tube of length L enters a uniform electric field that points upward. How strong and in what direction should a magnetic field be so that the electron moves straight ahead with no velocity change?

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Problem 36

A small $1.0 \times 10^{-5}$ -g piece of dust falls in Earth's gravitational field. Determine the distance it must fall so that the change in gravitational potential energy of the dust-Earth system equals the energy of a 0.10 -nm X-ray photon.

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Problem 37

X-ray exam While being $\mathrm{X}$ -rayed, a person absorbs $3.2 \times 10^{-3} \mathrm{J}$ of energy. Determine the number of $40,000$ -eV $\mathrm{X}$ -ray photons absorbed during the exam.

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Problem 38

Body cell $\mathrm{X}$ -ray $(\text { a })$ A body cell of $1.0 \times 10^{-5}$ -m radius absorbs $4.2 \times 10^{-14} \mathrm{J}$ of $\mathrm{X}$ -ray radiation. If the energy needed to produce one positively charged ion is $100 \mathrm{eV},$ how many positive ions are produced in the cell? (b) How many ions are formed in the $3.0 \times 10^{-6}$ -m-radius nucleus of that cell (the place where the genetic information is stored)?

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Problem 39

Equation Jeopardy 2 Solve for the unknown quantity in the equation below and write a problem for which the equation could be a solution.
$$
\lambda_{\mathrm{f}}-\left(100 \times 10^{-9} \mathrm{m}\right)=\frac{\left(6.63 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\right)\left(1-\cos 37^{\circ}\right)}{\left(9.1 \times 10^{-31} \mathrm{kg}\right)\left(3.0 \times 10^{8} \mathrm{m} / \mathrm{s}\right)}
$$

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Problem 40

In a Compton effect scattering experiment, an incident photon's frequency is $2.0 \times 10^{19} \mathrm{Hz} ;$ the scattered photon's frequency is $1.4 \times 10^{19} \mathrm{Hz}$ . Determine the kinetic energy increase of the electron, in units of electron volts, when the photon is scattered from it.

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Problem 41

An electron hit by an X-ray photon of energy $5.0 \times 10^{4} \mathrm{eV}$ gains $3.0 \times 10^{3} \mathrm{eV}$ of energy. Determine the wavelength of the scattered photon leaving the site of the collision.

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Problem 42

A laser produces a short pulse of light whose energy equals 0.20 $\mathrm{J}$ . The wavelength of the light is 694 $\mathrm{nm} .$ (a) How many photons are produced? (b) Determine the total momentum of
the emitted light pulse.

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Problem 43

Levitation with light Light from a relatively powerful laser can lift and support glass spheres that are $20.0 \times 10^{-6} \mathrm{m}$ in diameter (about the size of a body cell). Explain how that is possible.

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Problem 44

Light detection by human eye The dark-adapted eye can supposedly detect one photon of light of wavelength 500 nm. Suppose that 100 such photons enter the eye each second. Estimate the intensity of the light. Assume that the diameter of the eye’s pupil is 0.50 cm.

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Problem 45

Fireflies emit light of wavelengths from 510 nm to 670 nm. They are about 90% efficient at converting chemical energy into light (compared to about 10% for an incandescent lightbulb). Most living organisms, including fireflies, use adenosine triphosphate (ATP) as an energy molecule. Estimate the number of ATP molecules a firefly would use at 0.5 eV per molecule to produce one photon of 590-nm
wavelength if all the energy came from ATP

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Problem 46

Light of wavelength 430 nm strikes a metal surface, releasing electrons with kinetic energy equal to 0.58 eV or less. Determine the metal’s work function.

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Problem 47

Sail in laser wind 1 A powerful 0.50-W laser emitting 670-nm photons shines on the sail of a tiny 0.10-g cart that can coast on a horizontal frictionless track. (a) Determine the force of the light on the sail. Assume that the light is totally reflected. (b) What time interval is needed for the cart’s speed to increase from zero to 2.0 m/s?

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Problem 48

Sail in laser wind 2 A powerful 0.50-W laser emitting 670-nm photons shines on a black sail of a tiny 0.10-g cart that can coast on a frictionless track. (a) Determine the force of the light on the sail. Assume that the light is totally absorbed by the sail. (b) What time interval is needed for the cart’s speed to increase from zero to 2.0 m/s?

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Problem 49

Comet tails Comets are relatively small extraterrestrial objects that move around the Sun in highly elliptical orbits. The comet’s head is made primarily of ice with a small amount of dust. When the comet is near the Sun, gases and dust evaporated from the surface of the comet form a tail. Independent of the direction of motion of the comet, the tail always points away from the Sun. Use the photon model of light to explain why the comet’s tail points away from the Sun.

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Problem 50

Solar cell $\mathrm{A} 0.20-\mathrm{m} \times 0.20-\mathrm{m}$ photovoltaic solar cell is irradiated with 800 $\mathrm{W} / \mathrm{m}^{2}$ sunlight of wavelength 500 $\mathrm{nm} .$ (a) Determine the number of photons hitting the cell each second. (b) Determine the maximum possible electric current
that could be produced. (c) Explain how a solar cell converts the energy of sunlight into electric energy.

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Problem 51

The Sun is about 150 million km from Earth. The energy emitted by the Sun in all directions every second is about $4 \times 10^{26} \mathrm{J}$ J. Use this information to evaluate whether the value of the power per unit area provided in Problem 50 is reasonable.

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Problem 52

Sirius radiation power Sirius, a star in the constellation of Canis Major, is the second brightest star of the northern sky (the brightest is the Sun). Its surface temperature is 9880 $\mathrm{K}$ and its radius is 1.75 times greater than the radius of the Sun. Estimate the energy that Sirius emits every second from its surface and compare this energy to the energy that the Sun emits. The radius of the Sun is about $7.0 \times 10^{8} \mathrm{m}$ and the energy emitted per second is about $3.9 \times 10^{26} \mathrm{W} .$

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Problem 53

Owl night vision Owls can detect light of intensity $5 \times 10^{-13} \mathrm{W} / \mathrm{m}^{2} .$ Estimate the minimum number of photons an owl can detect. Indicate any assumptions you used in making the estimate.

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Problem 54

Photosynthesis efficiency During photosynthesis in a certain plant, eight photons of 670 -nm wavelength can cause the following reaction: $6 \mathrm{CO}_{2}+6 \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+6 \mathrm{O}_{2}$ During respiration, when the plant metabolizes sugar, the reverse reaction releases 4.9 eV of energy per $\mathrm{CO}_{2}$ molecule. Determine the ratio of the energy released (respiration) to the energy absorbed (photosynthesis), a measure of photosynthetic efficiency.

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Problem 55

Suppose that light of intensity $1.0 \times 10^{-2} \mathrm{W} / \mathrm{m}^{2}$ is made of waves rather than photons and that the waves strike a sodium surface with a work function of 2.2 $\mathrm{eV} .(\text { a })$ Determine the power in watts incident on the area of a single sodium atom
at the metal's surface (the radius of a sodium atom is approximately $1.7 \times 10^{-10} \mathrm{m}$ ). (b) How long will it take for an electron in the sodium to accumulate enough energy to escape the
surface, assuming it collects all light incident on the atom?

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Problem 56

Force of light on mirror A beam of light of wavelength 560 $\mathrm{nm}$ is reflected perpendicularly from a mirror. Determine the force that the light exerts on the mirror when $10^{20}$ photons hit the mirror each second. [Hint: Refer to the impulse momentum equation (Chapter 5 ). You may assume that the magnitude of the photons' momenta is unchanged by the collision, but their directions are reversed.]

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Problem 57

Force of sunlight on Earth We wish to determine the net force on Earth caused by the absorption of light from the Sun. (a) Determine the net area of the surface of Earth exposed to sunlight (Earth's radius is $6.38 \times 10^{6} \mathrm{m} ) .$ (b) The solar radiation intensity is 1400 $\mathrm{J} / \mathrm{s} \cdot \mathrm{m}^{2}$ . Determine the momentum of photons hitting Earth's surface each second. (c) Use the impulse-momentum equation to determine the average force of this radiation on Earth.

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Problem 58

Levitating a person Suppose that we wish to support a 70-kg person by levitating the person on a beam of light. (a) If all of the photons striking the person’s bottom surface are absorbed, what must be the power of the light beam, which is made of 500-nm-wavelength photons? (b) Estimate the person’s temperature change in 1 s.

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Problem 59

An electron that resides by itself in an open region of space is struck by a photon of light. Using nonrelativistic formulas, show that the electron cannot absorb the photon’s energy and simultaneously absorb its momentum. To conserve both energy and momentum, the photon must be absorbed by the electron near another mass, which carries away some of the momentum but little of the energy.

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Problem 60

Compton's original experiment involved scattering 0.0709 -nm X-rays off a graphite target (primarily composed of carbon atoms). He observed the scattered $X$ -rays at different angles using a spectrometer (a device that uses interference to determine the wavelength of the X-rays). What scattered the X-ray photons: the carbon nuclei or the electrons? To answer this question, determine the wavelength of the scattered photon when, after colliding with an electron or with a carbon atom, it travels at a $90^{\circ}$ angle relative to its initial momentum. The mass of an electron is $m_{e}=9.11 \times 10^{-31} \mathrm{kg}$ and the mass of a carbon nucleus is $m_{\mathrm{c}}=19.9 \times 10^{-27} \mathrm{kg} .$

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Problem 61

What is the number of antenna molecules that can absorb light in a photosynthetic unit?
(a)1
(b) About 10
(c) Over 100
(d) Over $10,000 $
(e) About $10^{8}$

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Problem 62

Suppose an antenna molecule absorbs a 430 -nm photon and that this energy is transferred directly to the acceptor molecule. Which answer below is closest to the energy that the photoelectron brings to the electron transport chain?
$$
\begin{array}{llll}{\text { (a) } 1 \mathrm{eV}} & {\text { (b) } 2 \mathrm{eV}} & {\text { (c) } 3 \mathrm{eV}} \\ {\text { (d) } 4 \mathrm{eV}} & {\text { (e) } 5 \mathrm{eV}}\end{array}
$$

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Problem 63

Suppose that the excited energy of one antenna molecule is transferred 100 times between neighboring antenna molecules. Which answer below is closest to the maximum time interval for the transfer between neighboring antenna molecules?
$$\begin{array}{llll}{\text { (a) } 10^{-3} \mathrm{s}} & {\text { (b) } 10^{-6} \mathrm{s}} & {\text { (c) } 10^{-8} \mathrm{s}}\end{array}$$
$$(d) 10^{-10} \mathrm{s} \quad (e) 10^{-14} \mathrm{s}$$

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Problem 64

Suppose that neighboring antenna molecules are separated by about $10^{-10} \mathrm{m}$ and that the excitation energy is transferred 100 times between neighboring antenna molecules before the
photoelectric transfer of an electron to the electron transport chain. Which answer below is closest to the minimum speed of the excitation energy through the photosynthetic unit?
$$\begin{array}{llll}{\text { (a) } 1 \mathrm{m} / \mathrm{s}} & {\text { (b) } 10^{2} \mathrm{m} / \mathrm{s}} & {\text { (c) } 10^{4} \mathrm{m} / \mathrm{s}}\end{array}$$
$$(d) 10^{-2} \mathrm{m} / \mathrm{s} \quad (e) 10^{-4} \mathrm{m} / \mathrm{s}$$

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Problem 65

The high-energy electron that transfers into an electron transport chain from a photosynthetic unit
(a) comes from the antenna molecule that absorbed the photon.
(b) comes from the acceptor molecule, which absorbed the photon.
(c) comes from the acceptor molecule that is excited by a nearby antenna molecule.
(d) is produced by the oxidation of a water molecule.
(e) is produced in the electron transport chain as other molecules react.

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Problem 66

The wavelength of maximum light emission from the body is closest to:
$$\begin{array}{llll}{\text { (a) } 500 \mathrm{nm}} & {\text { (b) } 700 \mathrm{nm}} & {\text { (c) } 1200 \mathrm{nm}}\end{array}$$
$$(d) 4500 \mathrm{nm} \quad (e) 9500 \mathrm{nm}$$

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Problem 67

During one day, the total radiative energy loss by a clothed person having a 2 $\mathrm{m}^{2}$ surface area in a $20^{\circ} \mathrm{C}$ room in kcal $(1 \mathrm{kcal}=4180 \mathrm{J})$ is closest to:
$$\begin{array}{lll}{\text { (a) } 0.5 \mathrm{kcal}} & {\text { (b) } 100 \mathrm{kcal}} & {\text { (c) } 400 \mathrm{kcal}}\end{array}$$
$$(d) 2000 kcal \quad (e) 3000 kcal$$

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Problem 68

Photographs taken with a regular camera and with an infrared camera are shown in Figure P26.68. The man’s arm is covered with a black plastic bag. Why is his arm visible in the infrared picture?
(a) Light does not pass through black plastic, but infrared radiation does.
(b) His arm is very warm under the black plastic bag and emits much more infrared radiation.
(c) The bag temperature is similar to his arm temperature.
(d) The black bag absorbs light and becomes warm and is a good thermal emitter.
(e) None of the above

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Problem 69

The man’s glasses appear clear with the regular camera photo and black with the infrared camera photo in Figure P26.68. Why?
(a) Light does not pass through glass, but infrared radiation does.
(b) Light passes through glass, but infrared radiation does not.
(c) The lenses of the glasses are cool compared to the man’s face, and thus they emit little infrared radiation.
(d) a and c
(e) b and c

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Problem 70

What is the ratio of the emitted radiative power from a 310 $\mathrm{K}$ surface and the same surface at 300 $\mathrm{K}$ closest to?
$$\begin{array}{llll}{\text { (a) } 0.86} & {\text { (b) } 0.97} & {\text { (c) } 1.03}\end{array}$$
$$(d) 1.07\quad (e) 1.14$$

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