University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 38

Photons, Flectrons, and Atoms - all with Video Answers

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DB

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Chapter Questions

Problem 1

The graph in Figure 38.34 shows the stopping potential as a function of the frequency of the incident light falling on a metal surface, (a) Find the photoelectric work function for this metal. (b) What value of Planck's constant does the graph yield? (c) Why does the graph not extend below the $x$ -axis? (d) If a different metal were used, what characteristics of the graph would you expect to be the same and which ones to be different?

Ren Jie Tuieng

Ren Jie Tuieng

Numerade Educator

Problem 2

Response of the Eye. The human eye is most sensitive to green light of wavelength 505 $\mathrm{nm}$ . Experiments have found that when people are kept in a dark room until their eyes adapt to the darkness, a single photon of green light will trigger receptor cells in the rods of the retina. (a) What is the frequency of this photon? (b) How much energy (in joules and electron volts) does it deliver
to the receptor cells? (c) to appreciate what a small amount of energy this is, calculate how fast a typical bacterium of mass $9.5 \times 10^{-12} \mathrm{g}$ would move if it had that much energy.

Ren Jie Tuieng

Numerade Educator

Problem 3

A photon of green light has a wavelength of 520 nm. Find the photon's frequency, magnitude of momentum, and energy.Express the energy in both joules and electron volts.

Ren Jie Tuieng

Numerade Educator

Problem 4

A laser used to weld detached retinas emits light with a wavelength of 652 $\mathrm{nm}$ in pulses that are 20.0 $\mathrm{ms}$ in duration. The average power during each pulse is 0.600 $\mathrm{W}$ . (a) How much energy is in each pulse in joules? In electron volts? (b) What is the energy of one photon in joules? In electron volts? (c) How many photons are in each pulse?

Ren Jie Tuieng

Numerade Educator

Problem 5

An excited nucleus emits a gamma-ray photon with an energy of 2.45 $\mathrm{MeV}$ . (a) What is the photon frequency?(b) What is the photon wavelength? (c) How does the wavelength compare with a typical nuclear diameter of $10^{-14} \mathrm{m} ?$

Ren Jie Tuieng

Numerade Educator

Problem 6

The photoelectric threshold wavelength of a tungsten surface is 272 $\mathrm{nm}$ . Calculate the maximum kinetic energy of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency $1.45 \times 10^{15}$ Hz. Express the answer in electron volts.

Ren Jie Tuieng

Numerade Educator

Problem 7

A clean nickel surface is exposed to light of wavelength $235 \mathrm{nm} .$ What is the maximum speed of the photoelectrons emitted from this surface? Use Table $38.1 .$

Elan Stopnitzky

Elan Stopnitzky

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Problem 8

What would the minimum work function for a metal have to be for visible light $(400-700 \mathrm{mn})$ to eject photoelectrons?

Ren Jie Tuieng

Numerade Educator

Problem 9

A $75-\mathrm{W}$ light source consumes 75 $\mathrm{W}$ of electrical power. Assume all this energy goes into emitted light of wavelength 600 $\mathrm{nm}$ (a) Calculate the frequency of the emitted light. (b) How many photons per second does the source emit?(c) Are the answers to parts (a) and (b) the same? Is the frequency of the light the same thing as the number of photons emitted per second? Explain.

DB

Daniel Bregar

Numerade Educator

Problem 10

In a photoelectric-effect experiment, the maximum kinetic energy of the ejected photoelectrons is measured for various wave-lengths of the incident light. Figure 38.35 shows a graph of this maximum kinetic energy, $K_{\max }$ as a function of the wavelength $\lambda$ of the light falling on the surface of the metal. What are (a) the threshold frequency and (b) the work function (in electron volts) for this metal? (c) Data from experiments like this are often graphed showing $K_{\max }$ as a function of 1$/ \lambda$ . Make a qualitative (no numbers) sketch of what this graph would look like. Identify the threshold wavelength $\left(\lambda_{0}\right)$ on your sketch. What advantages are there to graphing the data this way?

Ren Jie Tuieng

Numerade Educator

Problem 11

(a) A proton is moving at a speed much slower than the speed of light. It has kinetic energy $K_{1}$ and momentum $p_{1}$ . If the momentum of the proton is doubled, so $p_{2}=2 p_{1},$ how is its new kinetic energy $K_{2}$ related to $K_{1} ?(b)$ A photon with energy $E_{1}$ has momentum $p_{1} .$ If another photon has momentum $p_{2}$ that is twice $p_{1},$ how is the energy $E_{2}$ of the second photon related to $E_{1} ?$

Ren Jie Tuieng

Numerade Educator

Problem 12

The photoelectric work function of potassium is 2.3 eV. If light having a wavelength of 250 $\mathrm{nm}$ falls on potassium, find (a) the stopping potential in volts; (b) the kinetic energy in electron volts of the most energetic electrons ciected; (c) the speed of these electrons.

Manish Kumar

Numerade Educator

Problem 13

When ultraviolet light with a wavelength of 254 nm falls on a clean copper surface, the stopping potential necessary to stop emission of photoelectrons is 0.181 $\mathrm{V}$ . (a) What is the photoelectric threshold wavelength for this copper surface? (b) What is the work function for this surface, and how does your calculated value compare with that given in Table 38.1$?$

Ren Jie Tuieng

Numerade Educator

Problem 14

A photon has momentum of magnitude $8.24 \times 10^{-28} \mathrm{kg}$ . $\mathrm{m} / \mathrm{s}$ (a) What is the energy of this photon? Give your answer in joules and in electron volts. (b) What is the wavelength of this photon? In what region of the electromagnetic spectrum does it lie?

Ren Jie Tuieng

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Problem 15

Use Balmer's formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the $\mathrm{H}_{y}$ line of the Balmer series for hydrogen.

Ren Jie Tuieng

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Problem 16

Find the longest and shortest wavelengths in the Lyman and Paschen series for hydrogen. In what region of the electromagnetic spectrum does each series lie?

Ren Jie Tuieng

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Problem 17

(a) An atom initially in an energy level with $E=-6.52$ eV absorbs a photon that has wavelength 860 $\mathrm{nm}$ . What is the internal energy of the atom after it absorbs the photon? (b) An atom initially in an energy level with $E=-2.68 \mathrm{eV}$ emits a photon that has wavelength 420 $\mathrm{nm}$ . What is the internal energy of the atom after it emits the photon?

Ren Jie Tuieng

Numerade Educator

Problem 18

The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. 38.36 . The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. (a) How much energy (in electron volts) does it take to ionize an electron from the ground level? (b) An 18 -eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels. (c) What will happen if a photon with an energy of 8 eV strikes a Searsium atom in its ground level? Why? (d) Photons emitted in the Searsium transitions $n=3 \rightarrow n=2$ and $n=3 \rightarrow n=1$ will eject photoclectrons from an unknown metal, but the photon emitted from the transition $n=4 \rightarrow n=3$ will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

Ren Jie Tuieng

Numerade Educator

Problem 19

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground state $(n=1),$ as shown in the energy-level diagram in Fig. 38.37 . You also observe that it takes 17.50 eV to ionize this atom. (a) What is the energy of the atom in each of the levels $(n=1, n=2, \text { etc. })$ shown in the figure? (b) If an electron made a transition from the $n=4$ to the $n=2$ level what wavelength of light would it emit?

Ren Jie Tuieng

Numerade Educator

Problem 20

A $4.78-\mathrm{MeV}$ alpha particle from a 226 $\mathrm{Ra}$ decay makes a head-on collision with a uranium nucleus. A uranium nucleus has 92 protons. (a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uramium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus.
(b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

Susan Hallstrom

Susan Hallstrom

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Problem 21

A beam of alpha particles is incident on a target of lead. A particular alpha particle comes in "head-on" to a particular lead nucleus and stops $6.50 \times 10^{-14} \mathrm{m}$ away from the center of the
nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has 82 protons, remains at rest. The mass of the alpha particle is $6.64 \times 10^{-27} \mathrm{kg} .$ (a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in MeV. (b) What initial kinetic energy (in joules and in MeV) did the alpha particle have? (c) What was the initial speed of the alpha particle?

Ren Jie Tuieng

Numerade Educator

Problem 22

(a) What is the angular momentum $L$ of the electron in a hydrogen atom, with respect to an origin at the nucleus, when the atom is in its lowest energy level? (b) Repeat part (a) for the ground level of He'. Compare to the answer in part (a).

Ren Jie Tuieng

Numerade Educator

Problem 23

A hydrogen atom is in a state with energy $-1.51$ eV. In the Bohr model, what is the angular momentum of the electron in the atom, with respect to an axis at the nucleus?

Ren Jie Tuieng

Numerade Educator

Problem 24

A hydrogen atom initially in the ground level absorbs a photon, which excites it to the $n=4$ level. Determine the wave- length and frequency of the photon.

Ren Jie Tuieng

Numerade Educator

Problem 25

A triply ionized beryllium ion, $\mathrm{Be}^{3+}$ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. (a) What is the ground-level energy of Be"t? How does this compare to the ground-level energy of the hydrogen atom? (b) What is the ionization energy of $\mathrm{Be}^{3+} ?$ How does this compare to the ionization energy of the hydrogen atom?(c) For the hydrogen atom the wave-length of the photon emitted in the $n=2$ to $n=1$ transition is 122 $\mathrm{nm}$ (see Example $38.6 ) .$ What is the wavelength of the photon emitted when a $B e^{3+}$ ion undergoes this transition? (d) For a given value of $n$ , how does the radius of an orbit in $B e^{3+}$ compare to that for hydrogen?

Ren Jie Tuieng

Numerade Educator

Problem 26

A hydrogen atom undergoes a transition from the $n=5$ to the $n=2$ state. (a) What are the energy and wavelength of the photon that is emitted? (b) If the angular momentum is conserved and if the Bohr model is used to describe the atom, what must the angular momentum be of the photon that is emitted? (As we will see in Chapter $41,$ the modern quantum-mechanical description of the hydrogen atom gives a different result.)

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 27

(a) Using the Bohr model, calculate the speed of the electron in a hydrogen atom in the $n=1,2$ and 3 levels. (b) Calculate- the orbital period in each of these levels. (c) The average lifetime of the first excited level of a hydrogen atom is $1.0 \times 10^{-8} \mathrm{s}$ . In the Bohr model, how many orbits does an electron in the $n=2$ level complete before reurning to the ground level?

Ren Jie Tuieng

Numerade Educator

Problem 28

(a) Show that, as $n$ gets very large, the energy levels of the hydrogen atom get closer and closer together in energy. (b) Do the radii of these energy levels also get closer together?

Ren Jie Tuieng

Numerade Educator

Problem 29

How many photons per second are emitted by a $7.50-\mathrm{mW}$ $\mathrm{CO}_{2}$ laser that has a wavelength of 10.6$\mu \mathrm{m} ?$

Ren Jie Tuieng

Numerade Educator

Problem 30

PRK Surgery. Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part on the lens of the eye to change its curvature and hence focal length. This procedure can remove layers 0.25$\mu \mathrm{m}$ thick using pulses lasting 12.0 $\mathrm{ns}$ from a laser beam of wavelength 193 $\mathrm{nm}$ . Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. (a) In what part of the electromagnetic spectrum does this light he? (b) What is the energy of a single photon? (c) If
1.50-mW beam is used, how many photons are delivered to the lens in each pulse?

Ren Jie Tuieng

Numerade Educator

Problem 31

A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5$s$ state to the number in a 3$p$ state at $(a) 300 \mathrm{K} ;(\mathrm{b}) 600 \mathrm{K} ;(\mathrm{c}) 1200 \mathrm{K} ?$ The energies of these states are shown in Fig. 38.24 $\mathrm{a}$ . (d) At any of these temperatures, the rate at which a neon gas will spontaneously emit 632.8 -nm radiation is quite low. Explain why.

Ren Jie Tuieng

Numerade Educator

Problem 32

Figure 38.10 a shows the energy levels of the sodium atom. The two lowest excited levels are shown in columns labeled $^{2} \mathrm{P}_{3 / 2}$ and $^{2} P_{1 / 2}$ . Find the ratio of the number of atoms in a $^{2} P_{3 / 2}$ state to the number in $a^{2} P_{1 / 2}$ state for a sodium gas in thermal equilibrium at 500 K. In which state are more atoms found?

Ren Jie Tuieng

Numerade Educator

Problem 33

Protons are accelerated from rest by a potential difference of 4.00 $\mathrm{kV}$ and strike a metal target. If a proton produces one photon on impact, what is the minimum wavelength of the resulting
x rays? How does your answer compare to the minimum wave-length if $4.00-\mathrm{keV}$ electrons are used instead? Why do x-ray tubes use electrons rather than protons to produce x rays? use electrons rather than protons to produce x rays?

Ren Jie Tuieng

Numerade Educator

Problem 34

(a) What is the minimum potential difference between the filament and the target of an x-ray tube if the tube is to produce x rays with a wavelength of 0.150 $\mathrm{nm}$ ? (b) What is the shortest
wavelength produced in an $\mathrm{x}$ -ray tube operated at 30.0 $\mathrm{kV}$ ?

Kayla Gephart

Numerade Educator

Problem 35

X Rays from Television Screens. Accelerating voltages in cathode-ray-tube (CRT) TVs are about 25.0 $\mathrm{kV}$ . What are (a) the highest frequency and (b) the shortest wavelength (in nm) of the
x rays that such a TV screen could produce? (c) What assumptions did you need to make? (CRT televisions contain shielding to absorb these x rays.)

Ren Jie Tuieng

Numerade Educator

Problem 36

$X$ rays are produced in a tube operating at 18.0 $\mathrm{kV}$ . After cmerging from the tube, $x$ rays with the minimum wavelength produced strike a target and are Compton-scattered through an angle of $45.0^{\circ} .$ (a) What is the original $x$ -ray wavelength? (b) What is the wavelength of the scattered $x$ rays? (c) What is the energy of the scattered $x$ rays (in electron volts)?

Ren Jie Tuieng

Numerade Educator

Problem 37

X rays with initial wavelength 0.0665 nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed?

Ren Jie Tuieng

Numerade Educator

Problem 38

A beam of $x$ rays with wavelength 0.0500 $\mathrm{nm}$ is Compton- scattered by the electrons in a sample. At what angle from the incident beam should you look to find $x$ rays with a wavelength of (a) $0.0542 \mathrm{nm} ;$ (b) $0.0521 \mathrm{nm} ;(\mathrm{c}) 0.0500 \mathrm{nm} ?$

Ren Jie Tuieng

Numerade Educator

Problem 39

If a photon of wavelength 0.04250 nm strikes a free electron and is scattered at an angle of $35.0^{\circ}$ from its original direction. find (a) the change in the wavelength of this photon; (b) the wave-length of the scattered light; (c) the change in energy of the photon (is it a loss or a gain?); (d) the energy gained by the electron.

Ren Jie Tuieng

Numerade Educator

Problem 40

A photon scatters in the backward direction $\left(\theta=180^{\circ}\right)$ from a free proton that is initially at rest. What must the wave- length of the incident photon be if it is to undergo a 10.0$\%$ change in wavelength as a result of the scattering?

Ren Jie Tuieng

Numerade Educator

Problem 41

Complete the derivation of the Compton-scautering formula, Eq. $(38.23),$ following the outline given in Eqs. $(38.24)$ through $(38.27) .$

Ren Jie Tuieng

Numerade Educator

Problem 42

Determine $\lambda_{\mathrm{m}}$ , the wavelength at the peak of the Planck distribution, and the corresponding frequency $f,$ at these temperatures: (a) $3.00 \mathrm{K} ;(\mathrm{b}) 300 \mathrm{K} ;(\mathrm{c}) 3000 \mathrm{K}$

Ren Jie Tuieng

Numerade Educator

Problem 43

A $100-W$ incandescent light bulb has a cylindrical tungsten filament 30.0 $\mathrm{cm}$ long, 0.40 $\mathrm{mm}$ in diameter, and with an emissivity of 0.26 . (a) What is the temperature of the filament? (b) For what wavelength does the spectral emittance of the bulb peak? (c) Incandescent light bulbs are not very efficient sources of visible light. Explain why this is so.

Ren Jie Tuieng

Numerade Educator

Problem 44

The shortest visible wavelength is about 400 $\mathrm{nm}$ . What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

Ren Jie Tuieng

Numerade Educator

Problem 45

Radiation has been detected from space that is characteristic of an ideal radiator at $T=2.728 \mathrm{K}$ . (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?

Ren Jie Tuieng

Numerade Educator

Problem 46

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature $T$ and 3.0 times the diameter of the hotter star. (a) What
is the temperature of the hotter star in terms of $T$ ? (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

Dading Chen

Numerade Educator

Problem 47

(a) Show that the maximum in the Planck distribution, Eq. $(38.32),$ occurs at a wavelength $\lambda_{\mathrm{m}}$ given by $\lambda_{\mathrm{m}}=h c / 4.965 \mathrm{kT}$ (Eq. 38.33$)$ . As discussed in the text, 4.965 is the root of Eq. $(38.34)$ . (b) Evaluate the constants in the expression derived in part (a) to show that $\lambda_{m} T$ has the numerical value given in the Wien displacement law, Eq ( 38.30 ).

Ren Jie Tuieng

Numerade Educator

Problem 48

Sirius B. The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is $24,000 \mathrm{K}$ and that it radiates energy at a total rate of $1.0 \times 10^{25} \mathrm{W}$ . Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius $\mathrm{B}$ (b) What is the peak-intensity wavelength? Is this wave-length visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's
radius. (d) Which star radiates more total energy per second, the hot Sirius or the (relatively) cool sun with a surface temperature of 5800 $\mathrm{K}$ ? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

Dading Chen

Numerade Educator

Problem 49

Show that for large values of $\lambda$ the Planck distribution, Eq. $(38.32),$ agrees with the Rayleigh distribution, Eq. $(38.31)$ .

Ren Jie Tuieng

Numerade Educator

Problem 50

Blue Supergiants. A typical blue supergiant star (the type that explode and leave behind black holes) has a surface temperature of $30,000 \mathrm{K}$ and a visual luminosity $100,000$ times that of our sun. Our sun radiates at the rate of $3.86 \times 10^{26} \mathrm{W}$ . (Visual) luminosity is the total power radiated at visible wavelengths. (a) Assuming that this star behaves like an ideal blackbody, what is, the principal wavelength it radiates? Is this light visible? Use your answer to explain why these stars are blue. (b) If we assume that the power radiated by the star is also $100,000$ times that of our sun, what is the radius of this star? Compare its size to that of our sun, which has a radius of $6.96 \times 10^{5} \mathrm{km}$ . (c) Is it really correct to say, that the visual luminosity is proportional to the total power radiated? Explain.

Ren Jie Tuieng

Numerade Educator

Problem 51

Exposing Photographic Film. The light-sensitive compound on most photographic films is silver bromide, AgBr. A film is "exposed" when the light energy absorbed dissociates this molecule into its atoms. (The actual process is more complex, but the quantitative result does not differ greatly.) The energy of dissociation of AgBr is $1.00 \times 10^{5} \mathrm{J} / \mathrm{mol}$ . For a photon that is just able to dissociate a molecule of silver bromide, find (a) the photon energy in electron volts; (b) the wavelength of the photon; (c) the frequency of the photon. (d) What is the energy in electron volts of a photon having a frequency of 100 $\mathrm{MHz}$ (e) Light from a firefly can expose photographic film, but the radiation from an FM station broadcasting $50,000 \mathrm{W}$ at 100 $\mathrm{MHz}$ cannot. Explain why this is so.

Ren Jie Tuieng

Numerade Educator

Problem 52

An atom with mass $m$ emits a photon of wavelength $\lambda$ . (a) What is the recoil speed of the atom? (b) What is the kinetic energy $K$ of the recoiling atom? (c) Find the ratio $K / E,$ where $E$ is the energy of the emitted photon. If this ratio is much less than unity, the recoil of the atom can be neglected in the emission process. Is the recoil of the atom more important for small or large atomic masses? For long or short wavelengths? (d) Calculate $K$ (in electron volts) and $K / E$ for a hydrogen atom (mass $1.67 \times 10^{-27} \mathrm{kg} )$ that emits an ultraviolet photon of energy 10.2 $\mathrm{eV}$ . Is recoil an important consideration in this emission process?

Ren Jie Tuieng

Numerade Educator

Problem 53

When a certain photoclectric surface is illuminated with light of different wavelengths, the following stopping potentials are observed: Plot the stopping potential on the vertical axis against the frequency of the light on the horizontal axis. Determine (a) the threshold frequency; (b) the threshold wavelength; (c) the photo-electric work function of the material (in electron volts); (d) the value of Planck's constant $h$ (assuming that the value of $e$ is known).

Ren Jie Tuieng

Numerade Educator

Problem 54

(a) If the average frequency emitted by a $200-\mathrm{W}$ light bulb is $5.00 \times 10^{14} \mathrm{Hz},$ and 10.0$\%$ of the input power is emitted as visible light, approximately how many visible-light photons are emitted per second? (b) At what distance would this correspond to $1.00 \times 10^{11}$ visible-light photons per square centimeter per second if the light is emitted uniformly in all directlons?

Ren Jie Tuieng

Numerade Educator

Problem 55

(a) The wavelength of light incident on a metal surface is reduced from $\lambda_{1}$ to $\Lambda_{2}$ . (Both $\lambda_{1}$ and $\lambda_{2}$ are less than the threshold wavelength for the surface.) When the wavelength is reduced in this way, what is the change in the stopping potential for photo-electrons emitted from the surface? (b) Evaluate the change in stopping potential for $\lambda_{1}=295 \mathrm{nm}$ and $\lambda_{2}=265 \mathrm{nm} .$

Ren Jie Tuieng

Numerade Educator

Problem 56

A $2.50-\mathrm{W}$ beam of light of wavelength 124 $\mathrm{nm}$ falls on a metal surface. You observe that the maximum kinetic energy of the ejected electrons is 4.16 $\mathrm{eV}$ . Assume that each photon in the beam ejects a photoclectron. (a) What is the work function (in electron volts of this metal? (b) How many photoclectrons are ejected each second from this metal?(c) If the power of the light beam, but not its wavelength, were reduced by half, what would be the answer to part (b)? (d) If the wavelength of the beam, but not its power, were reduced by half, what would be the answer to part (b)?

Ren Jie Tuieng

Numerade Educator

Problem 57

Removing Vascular Lesions. A pulsed dye laser emits light of wavelength 585 $\mathrm{nm}$ in $450-\mu$ s pulses. Because this wave- length is strongly absorbed by the hemoglobin in the blood, the method is especially effective for removing various types of blemishes due to blood, such as port-wine- colored birth-marks. To get a reasonable estimate of the power required for such laser surgery, we can model the blood as having the same specific heat and heat of vaporization as water $\left(4190 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, 2.256 \times 10^{6} \mathrm{J} / \mathrm{kg}\right) .$ Suppose that each pulse must remove 2.0$\mu g$ of blood by evaporating it, starting at $33^{\circ} \mathrm{C}$ (a) How much energy must each pulse deliver to the blemish? (b) What must be the power output of this laser? (c) How many photons does each pulse deliver to the blemish?

Ren Jie Tuieng

Numerade Educator

Problem 58

The photoelectric work functions for particular samples of certain metals are as follows: cesium, 2.1 eV; copper, 4.7 eV; potassium, $2.3 \mathrm{eV} ;$ and $\mathrm{zinc}, 4.3$ eV. (a) What is the threshold wavelength for each metal surface? (b) Which of these metals could not emit photoelectrons when irradiated with visible light $(400-700 \mathrm{nm}) ?$

Ren Jie Tuieng

Numerade Educator

Problem 59

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the $n=2$ level to the $n=1$ level?

Ren Jie Tuieng

Numerade Educator

Problem 60

An incident $x$ -ray photon is scattered from a free electron that is initially at rest. The photon is scattered straight back at an angle of $180^{\circ}$ from its initial direction. The wavelength of the scattered photon is $0.0830 \mathrm{nm} .$ (a) What is the wavelength of the incident photon? (b) What is the magnitude of the momentum of the electron after the collision? (c) What is the kinetic energy of the electron after the collision?

Ren Jie Tuieng

Numerade Educator

Problem 61

An incident x-ray photon of wavelength 0.0900 $\mathrm{nm}$ is scattered in the backward direction from a free electron that is initially at rest. (a) What is the magnitude of the momentum of the scattered photon? (b) What is the kinetic energy of the electron after the photon is scattered?

Ren Jie Tuieng

Numerade Educator

Problem 62

Bohr Orbits of a Satellite. A $20.0-\mathrm{kg}$ satellite circles the carth once every 2.00 $\mathrm{h}$ in an orbit having a radius of 8060 $\mathrm{km}$ . (a) Assuming that Bohr's angular-momentum result $(L=n h / 2 \pi)$ . applies to satellites just as it does to an electron in the hydrogen atom, find the quantum number $n$ of the orbit of the satellite. (b) Show from Bohr's angular momentum result and Newton's law of gravitation that the radius of an earth-satellite orbit is directly proportional to the square of the quantum number, $r=k n^{2},$ where $k$ is the constant of proportionality. (c) Using the result from part (b), find the distance between the orbit of the satellite in this problem and its next "allowed" orbit. (Calculate a numerical value.) (d) Comment on the possibility of observing the separation of the two adjacent orbits. (e) Do quantized and classical orbits correspond for this satellite? Which is the "correct" method for calculating the orbits?

Ren Jie Tuieng

Numerade Educator

Problem 63

(a) What is the smallest amount of energy in electron volts that must be given to a hydrogen atom initially in its ground level so that it can emit the $\mathrm{H}_{a}$ line in the Balmer series? (b) How many different possibilities of spectral-line emissions are there for this atom when the electron starts in the $n=3$ level and eventually ends up in the ground level? Calculate the wavelength of the emilted photon in each case.

Ren Jie Tuieng

Numerade Educator

Problem 64

A large number of hydrogen atoms are in thermal equilibrium. Let $n_{2} / n_{1}$ be the ratio of the number of atoms in an $n=2$ excited state to the number of atoms in an $n=1$ ground state. At what temperature is $n_{2} / n_{1}$ equal to (a) $10^{-12}$ (b) $10^{-8}$ , (c) $10^{-4}$ ? (d) Like the sun, other stars have continuous spectra with dark absorption lines (see Fig. 38.12$)$ . The absorption takes place in the star's atmosphere, which in all stars is composed primarily of hydrogen. Explain why the Balmer absorption lines are relatively weak in stars with low atmospheric temperatures such as the sun (atmosphere temperature 5800 $\mathrm{K} )$ but strong in stars with higher atmospheric temperatures.

Ren Jie Tuieng

Numerade Educator

Problem 65

A sample of hydrogen atoms is irradiated with light with wavelength 85.5 $\mathrm{nm}$ , and electrons are observed leaving the gas. (a) If each hydrogen atom were initially in its ground level, what would be the maximum kinetic energy in electron volts of these photoelectrons? (b) A few electrons are detected with energies as much as 10.2 eV greater than the maximum kinetic energy calculated in part (a). How can this be?

Ren Jie Tuieng

Numerade Educator

Problem 66

Light from an ideal spherical blackbody $15.0 \mathrm{~cm}$ in diameter is analyzed using a diffraction grating having 3850 lines/cm. When you shine this light through the grating, you observe that the peak-intensity wavelength forms a first-order bright fringe at $\pm 11.6^{\circ}$ from the central bright fringe.
(a) What is the temperature of the blackbody?
(b) How long will it take this sphere to radiate $12.0 \mathrm{MJ}$ of energy?

Ren Jie Tuieng

Numerade Educator

Problem 67

The Red Giant Betelgeuse. The red giant Betelgeuse has a surface temperature of 3000 $\mathrm{K}$ and is 600 times the diameter of our sun. (If our sun were that large, we would be inside it) Assume that it radiates like an ideal blackbody. (a) If Betelgeuse were to radiate all of its energy at the peak-intensity wavelength, how many photons per second would it radiate? (b) Find the ratio of the power radiated by Betelgeuse to the power radiated by our sun (at 5800 $\mathrm{K}$ ).

Ren Jie Tuieng

Numerade Educator

Problem 68

An ideal spherical blackbody 24.0 $\mathrm{cm}$ in diameter is maintained at $225^{\circ} \mathrm{C}$ by an internal electrical heater and is immersed in a very large open-faced tank of water that is kept boiling by the energy radiated by the sphere. You can neglect any heat transferred by conduction and convection. Consult Table 17.4 as needed. (a) At what rate, in $\mathrm{g} / \mathrm{s}$ , is water evaporating from the $\operatorname{tank} 7$ (b) If a physics-wise thermophile organism living in the hot water is observing this process, what will it measure for the peak-intensity (i) wavelength and (ii) frequency of the electromagnetic waves emitted by the sphere?

Ren Jie Tuieng

Numerade Educator

Problem 69

What must be temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wavelength can excite the electron in the Bohr-model hydrogen atom from the ground state to the third excited state?

Ren Jie Tuieng

Numerade Educator

Problem 70

An x-ray tube is operating at voltage $V$ and current $I$ . (a) If only a fraction $p$ of the electric power supplied is converted into $\mathbf{x}$ . rays, at what rate is energy being delivered to the target? (b) If the target has mass $m$ and specific heat capacity $c(\text { in } \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}),$ at what average rate would its temperature rise if there were no thermal losses?(c) Evaluate your results from parts (a) and (b) for an x-ray tube operating at 18.0 $\mathrm{kV}$ and 60.0 $\mathrm{mA}$ that converts 1.0$\%$ of the electric power into $\mathrm{x}$ rays. Assume that the $0.250-\mathrm{kg}$ target is made of lead $(c=130 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}) .$ (d) What must the physical properties of a practical target material be? What would be some suitable target elements?

Ren Jie Tuieng

Numerade Educator

Problem 71

When a photon is emitted by an atom, the atom must recoil to conserve momentum. This means that the photon and the recoiling atom share the transition energy. (a) For an atom with mass $m$ . calculate the correction $\Delta \lambda$ due to recoil to the wavelength of an emitted photon. Let $\lambda$ be the wavelength of the photon if recoil is not taken into consideration. (Hint. The correction is very small, as Problem 38.52 suggests, so $|\Delta \lambda| / \lambda \ll 1 .$ Use this fact to obtain an approximate but very accurate expression for $\Delta \lambda . )(\text { b) Evaluate the }$ correction for a hydrogen atom in which an electron in the $n$ th level returns to the ground level. How does the answer depend on $n ?$

Ren Jie Tuieng

Numerade Educator

Problem 72

(a) Derive an expression for the total shift in photon wave- length after two successive Compton scatterings from electrons at rest. The photon is scattered by an angle $\theta_{1}$ in the first scattering and by $\theta_{2}$ in the second. (b) In general, is the total shift in wave-length produced by two successive scatterings of an angle $\theta / 2$ the same as by a single scattering of $\theta ?$ If not, are there any specific values of $\theta,$ other than $\theta=0$ , for which the total shifts are the same? (c) Use the result of part (a) to calculate the total wave- length shift produced by two successive Compton scatterings of $30.0^{\circ}$ each. Express your answer in terms of $h / m c .$ (d) What is the wavelength shift produced by a single Compton scattering of $60.0^{\circ} ?$ Compare to the answer in part (c).

Ren Jie Tuieng

Numerade Educator

Problem 73

Nuclear fusion reactions at the center of the sun produce gamma-ray photons with energies of order 1 MeV $\left(10^{5} \mathrm{eV}\right)$ . By contrast, what we see emanating from the sun's surface are visible-light photons with wavelengths of order 500 $\mathrm{nm}$ . A simple model that explains this difference in wavelength is that a photon undergoes Compton scattering many times- in fact, about $10^{26}$ times, as suggested by models of the solar interior- as it travels from the center of the sun to its surface. (a) Estimate the increase in wave- length of a photon in an average Compton-scattering event. (b) Find the angle in degrees through which the photon is scattered in the scattering event described in part (a). (Hint: A useful approximation is $\cos \phi \approx 1-\phi^{2} / 2,$ which is valid for $\phi \ll 1 .$ Note that $\phi$ is in radians in this expression.) (c) It is estimated that a photon takes about $10^{6}$ years to travel from the core to the surface of the sun. Find the average distance that light can travel within the interior of the sun without being scattered. (This distance is roughly equivalent to how far you could see if you were inside the sun and could survive the extreme temperatures there. As your answer shows, the interior of the sun is very opaque.)

Ren Jie Tuieng

Numerade Educator

Problem 74

An x-ray photon is scattered from a free electron (mass m) at rest. The wavelength of the scattered photon is $\lambda^{\prime},$ and the final speed of the struck electron is $v$ . (a) What was the initial wave-length $\lambda$ of the photon? Express your answer in terms of $\lambda^{\prime}, v,$ and $m$ . (Hint: Use the relativistic expression for the electron kinetic energy.) (b) Through what angle $\phi$ is the photon scattered? Express your answer in terms of $\lambda, \lambda^{\prime},$ and $m .$ (c) Evaluate your results in parts $(a)$ and $(b)$ for a wavelength of $5.10 \times 10^{-3} \mathrm{nm}$ for the scattered photon and a final electron speed of $1.80 \times 10^{8} \mathrm{m} / \mathrm{s}$ . Give $\phi$ in degrees.

Ren Jie Tuieng

Numerade Educator

Problem 75

A photon with wavelength 0.1100 nm collides with a free electron that is initially at rest. After the collision the wavelength is $0.1132 \mathrm{nm} .$ (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid target), all of its kinetic energy is used to create a photon. What is the wavelength of this photon?

Ren Jie Tuieng

Numerade Educator

Problem 76

(a) Calculate the maximum increase in photon wavelength that can occur during Compton scattering. (b) What is the energy (in electron volts) of the lowest-energy $x$ -ray photon for which Compton scattering could result in doubling the original wave- length?

Ren Jie Tuieng

Numerade Educator

Problem 77

(a) Write the Planck distribution law in terms of the frequency $f$ rather than the wavelength $\lambda$ , to obtain $I(f)$ . Show that
$$
\int_{0}^{\infty} I(\lambda) d \lambda=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}} T^{4}
$$
where $I(\lambda)$ is the Planck distribution formula of Eq. $(38.32)$ (Hint: Change the integration variable from $\lambda$ to $f . )$ You will need to use the following tabulated integral:
$$
\int_{0}^{\infty} \frac{x^{3}}{e^{a x}-1} d x=\frac{1}{240}\left(\frac{2 \pi}{\alpha}\right)^{4}
$$
(c) The result of $(b)$ is $I$ and has the form of the Stefan-Boltzmann law, $I=\sigma T^{4}(\text { Eq. } 38.28)$ . Evaluate the constants in (b) to show that $\sigma$ has the value given in Section 38.8 .

Ren Jie Tuieng

Numerade Educator

Problem 78

An Ideal Blackbody. A large cavity with a very small hole and maintained at a temperature $T$ is a good approximation to an ideal radiator or blackbody. Radiation can pass into or out of the cavity only through the hole. The cavity is a perfect absorber, since any radiation incident on the hole becomes trapped inside the cavity. Such a cavity at $200^{\circ} \mathrm{C}$ has a hole with area 4.00 $\mathrm{mm}^{2}$ . How long does it take for the cavity to radiate 100 $\mathrm{J}$ of energy through
the hole?

JL

Jacob Leuquire

Numerade Educator

Problem 79

(a) Show that in the Bohr model, the frequency of revolution of an electron in its circular orbit around a stationary hydrogen nucleus is $f=m e^{4} / 4 \epsilon_{0}^{2} n^{3} h^{3}$ . (b) In classical physics, the frequency of revolution of the electron is equal to the frequency of the radiation that it emits. Show that when $n$ is very large, the frequency of revolution does indeed equal the radiated frequency calculated from Eq. $(38.6)$ for a transition from $n_{1}=n+1$ to $n_{2}=n .$ (This illustrates Bohr's correspondence principle, which is often used as a check on quantum calculations. When $n$ is small, quantum physics gives results that are very different from those of classical physics. When $n$ is large, the differences are not significant, and the two methods then "correspond." In fact, when Bohr first tackled the hydrogen atom problem, he sought to determine $f$ as a function of $n$ such that it would correspond to classical results for large $n$ .)

Ren Jie Tuieng

Numerade Educator

Problem 80

Consider a beam of monochromatic light with intensity $I$ incident on a perfectly absorbing surface oriented perpendicular to the beam. Use the photon concept to show that the radiation pressure exerted by the light on the surface is given by $I / c$ .

Ren Jie Tuieng

Numerade Educator

Problem 81

Consider Compton scattering of a photon by a moving electron. Before the collision the photon has wavelength $\lambda$ and is moving in the $+x$ -direction, and the electron is moving in the - $x$ -direction with total energy $E$ (including its rest energy $m c^{2} )$ . The photon and electron collide head on. After the collision, both are moving in the $-x$ -direction (that is, the photon has been scattered by $180^{\circ}$ ). (a) Derive an expression for the wavelength $\lambda^{\prime}$ of the scattered photon. Show that if $E \gg m c^{2}$ , where $m$ is the rest mass of the electron, your result reduces to
$$
\Lambda^{\prime}=\frac{h c}{E}\left(1+\frac{m^{2} c^{4} \lambda}{4 h c E}\right)
$$
(b) A beam of infrared radiation from a $\mathrm{CO}_{2}$ laser $(\lambda=10.6 \mu \mathrm{m})$
collides head-on with a beam of electrons, each of total energy $E=10.0 \mathrm{GeV}\left(1 \mathrm{GeV}=10^{9} \mathrm{eV}\right) .$ Calculate the wavelength $\lambda^{\prime}$ of the scattered photons, assuming a $180^{\circ}$ scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, etc.)? Can you think of an application of this effect?

Ren Jie Tuieng

Numerade Educator