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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 27

Quantum Physics

Educators


Problem 1

(a) What is the surface temperature of Betelgeuse, a red giant star in the constellation of Orion, which radiates with a peak wavelength of about 970 nm? (b) Rigel, a bluish-white star in Orion, radiates with a peak wavelength of 145 nm. Find the temperature of Rigel’s surface.

Katie M.
Numerade Educator

Problem 2

(a) Lightning produces a maximum air temperature on the order of $10^{4} \mathrm{K}$ , whereas (b) a nuclear explosion produces a temperature on the order of $10^{7} \mathrm{K}$ . Use Wien's displacement law to find the order of magnitude of the wavelength of the thermally produced photons radiated with greatest intensity by each of these sources. Name the part of the electromagnetic spectrum where you would expect each to radiate most strongly.

Katie M.
Numerade Educator

Problem 3

The temperature of a student's skin is $33.0^{\circ} \mathrm{C}$ . At what wavelength does the radiation emitted from the skin reach its peak?

Katie M.
Numerade Educator

Problem 4

The radius of our Sun is $6.96 \times 10^{8} \mathrm{m},$ and its total power output is $3.85 \times 10^{26} \mathrm{W} .$ (a) Assuming the Sun's surface emits as a black-body, calculate its surface temperature. (b) Using the result of part (a), find $\lambda_{\max }$ for the Sun.

Katie M.
Numerade Educator

Problem 5

Earth's average surface temperature is about 287 $\mathrm{K}$ . Assuming Earth radiates as a blackbody, calculate $\lambda_{\max }$ for the Earth.

Katie M.
Numerade Educator

Problem 6

Suppose a star with radius $8.50 \times 10^{8} \mathrm{m}$ has a peak wavelength of 685 $\mathrm{nm}$ in the spectrum of its emitted radiation. (a) Find the energy of a photon with this wavelength. (b) What is the surface temperature of the star? (c) At what rate is energy emitted from the star in the form of radiation? Assume the star is a blackbody $(e=1)$ . (d) Using the answer to part (a), estimate the rate at which photons leave the surface of the star.

Katie M.
Numerade Educator

Problem 7

Calculate the energy, in electron volts, of a photon whose frequency is (a) $6.20 \times 10^{2} \mathrm{THz},$ (b) 3.10 $\mathrm{GHz}$ , and (c) 46.0 $\mathrm{MHz}$

Katie M.
Numerade Educator

Problem 8

The threshold of dark-adapted (scotopic) vision is 4.0 $\times 10^{-11} \mathrm{W} / \mathrm{m}^{2}$ at a central wavelength of $5.00 \times 10^{2} \mathrm{nm}$ . If light with this intensity and wavelength enters the eye when the pupil is open to its maximum diameter of 8.5 $\mathrm{mm}$ , how many photons per second enter the eye?

Katie M.
Numerade Educator

Problem 9

When light of wavelength $3.50 \times 10^{2}$ nm falls on a potassium surface, electrons having a maximum kinetic energy of 1.31 eV are emitted. Find (a) the work function of potassium, (b) the cutoff wavelength, and (c) the frequency corresponding to the cutoff wavelength.

Katie M.
Numerade Educator

Problem 10

The work function for zinc is 4.31 $\mathrm{eV}$ (a) Find the cutoff wavelength for zinc. (b) What is the lowest frequency of light incident on zinc that releases photoelectrons from its surface? (c) If photons of energy 5.50 $\mathrm{eV}$ are incident on zinc, what is the maximum kinetic energy of the ejected photoelectrons?

Katie M.
Numerade Educator

Problem 11

The work function for platinum is 6.35 eV. (a) Convert the value of the work function from electron volts to joules. (b) Find the cutoff frequency for platinum. (c) What maximum wavelength of light incident on platinum releases photoelectrons from the platinum’s surface? (d) If light of energy 8.50 eV is incident on zinc, what is the maximum kinetic energy of the ejected photoelectrons? Give the answer in electron volts. (e) For photons of energy 8.50 eV, what stopping potential would be required to arrest the current of photoelectrons?

Katie M.
Numerade Educator

Problem 12

Lithium, beryllium, and mercury have work functions of 2.30 eV, $3.90 \mathrm{eV},$ and 4.50 $\mathrm{eV}$ , respectively. Light with a wavelength of $4.00 \times 10^{2} \mathrm{nm}$ is incident on each of these metals. (a) Which of these metals emit photoelectrons in response to the light? Why? (b) Find the maximum kinetic energy for the photoelectrons in each case.

Katie M.
Numerade Educator

Problem 13

When monochromatic light of an unknown wavelength falls on a sample of silver, a minimum potential of 2.50 $\mathrm{V}$ is required to stop all of the ejected photoelectrons. Determine the (a) maximum kinetic energy and (b) maximum speed of the ejected photoelectrons. (c) Determine the wavelength in $\mathrm{nm}$ of the incident light. (The work function for silver is 4.73 $\mathrm{eV.} )$

Katie M.
Numerade Educator

Problem 14

Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp $(\lambda=546.1 \mathrm{nm})$ is used, a stopping potential of 0.376 $\mathrm{V}$ reduces the photocurrent to zero. (a) Based on this measurement, what is the work function for this metal? (b) What stopping potential would be observed when using the yellow light from a helium discharge tube $(\lambda=587.5 \mathrm{nm}) ?$

Katie M.
Numerade Educator

Problem 15

The extremes of the x-ray portion of the electromagnetic spectrum range from approximately $1.0 \times 10^{-8} \mathrm{m}$ to $1.0 \times 10^{-13} \mathrm{m} .$ Find the minimum accelerating voltages required to produce wavelengths at these two extremes.

Katie M.
Numerade Educator

Problem 16

Calculate the minimum-wavelength $\mathrm{x}$ -ray that can be produced when a target is struck by an electron that has been accelerated through a potential difference of (a) 15.0 $\mathrm{kV}$ and (b) $1.00 \times 10^{2} \mathrm{kV}$ . (c) What happens to the minimum wavelength as the potential difference increases?

Katie M.
Numerade Educator

Problem 17

What minimum accelerating voltage is required to produce an $\mathrm{x}$ -ray with a wavelength of 70.0 $\mathrm{pm}$ ?

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Numerade Educator

Problem 18

When sodium is bombarded with electrons accelerated through a potential difference $\Delta V,$ its $x$ -ray spectrum contains emission peaks at 1.04 $\mathrm{keV}$ and 1.07 $\mathrm{keV}$ . Find the minimum value of $\Delta V$ required to produce both of these peaks.

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Numerade Educator

Problem 19

Lead has a prominent x-ray emission line at 75.0 keV. (a) What is the minimum speed of an incident electron that could produce this emission line? (Hint: Recall the expression for relativistic kinetic energy given in Topic $26 .$ ) (b) What is the wavelength of a 75.0 -keV x-ray photon?

Prashant B.
Numerade Educator

Problem 20

When x-rays of wavelength of 0.129 $\mathrm{nm}$ are incident on the surface of a crystal having a structure similar to that of NaCl, a first-order maximum is observed at $8.15^{\circ} .$ Calculate the interplanar spacing of the crystal based on this information.

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Numerade Educator

Problem 21

Potassium iodide has an interplanar spacing of $d=0.296 \mathrm{nm}$ . A monochromatic $\mathrm{x}$ -ray beam shows a first-order diffraction maximum when the grazing angle is $7.6^{\circ} .$ Calculate the x-ray wavelength.

Katie M.
Numerade Educator

Problem 22

The first-order diffraction maximum is observed at $12.6^{\circ}$ for a crystal having an interplanar spacing of 0.240 $\mathrm{nm}$ . How many other orders can be observed in the diffraction pattern, and at what angles do they appear? Why is there an upper limit to the number of observed orders?

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Numerade Educator

Problem 23

X-rays of wavelength 0.140 $\mathrm{nm}$ are reflected from a certain crystal, and the first-order maximum occurs at an angle of $14.4^{\circ} .$ What value does this give for the interplanar spacing of
the crystal?

Katie M.
Numerade Educator

Problem 24

X-rays are scattered from a target at an angle of $55.0^{\circ}$ with the direction of the incident beam. Find the wavelength shift of the scattered x-rays.

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Numerade Educator

Problem 25

A 0.001 60-nm photon scatters from a free electron.
For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered
photon?

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Numerade Educator

Problem 26

A $25.0-\mathrm{pm}$ x-ray photon scatters off a free electron at $A$ (Fig. $\mathrm{P} 27.26 ),$ producing a photon of wavelength $\lambda^{\prime}$ traveling at an angle $\theta=40.0^{\circ}$ relative to the first photon's direction. This second photon scatters off another free electron at $B,$ producing a photon with wavelength $\lambda^{\prime \prime}$ and moving in a direction directly opposite the first photon. Determine the wavelengths (a) $\lambda^{\prime}$ and ( b ) $\lambda^{\prime \prime}$

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Numerade Educator

Problem 27

A 0.110-nm photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backwards. Find (a) the momentum and (b) the kinetic energy of the electron.

Prashant B.
Numerade Educator

Problem 28

In a Compton scattering experiment, an x-ray photon scatters through an angle of $17.4^{\circ}$ from a free electron that is initially at rest. The electron recoils with a speed of 2180 $\mathrm{km} / \mathrm{s}$ . Calculate (a) the wavelength of the incident photon and (b) the angle through which the electron scatters.

Rashmi S.
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Problem 29

(a) If the wavelength of an electron is $5.00 \times 10^{-7} \mathrm{m},$ how fast is it moving? (b) If the electron has a speed equal to $1.00 \times$ $10^{7} \mathrm{m} / \mathrm{s},$ what is its wavelength?

Katie M.
Numerade Educator

Problem 30

Calculate the de Broglie wavelength of a proton moving at $(\text { a }) 2.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ and ( b ) $2.00 \times 10^{7} \mathrm{m} / \mathrm{s}$

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Problem 31

De Broglie postulated that the relationship $\lambda=h / p$ is valid for relativistic particles. What is the de Broglie wavelength for a (relativistic) electron having a kinetic energy of 3.00 $\mathrm{MeV}$ ?

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Numerade Educator

Problem 32

An electron and a $6.00-\mathrm{kg}$ bowling ball each have 4.50 $\mathrm{eV}$ of kinetic energy. Calculate (a) $\lambda_{e}$ and (b) $\lambda_{b}$ , the de Broglie wavelengths of the electron and the bowling ball, respectively. (c) Calculate the wavelength $\lambda_{p}$ of a $4.50-\mathrm{eV}$ photon.

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Numerade Educator

Problem 33

The resolving power of a microscope is proportional to the wavelength used. A resolution of $1.0 \times 10^{-11} \mathrm{m}(0.010 \mathrm{nm})$ would be required in order to see an atom. (a) If electrons were used (electron microscope), what minimum kinetic energy would be required of the electrons? (b) If photons were used, what minimum photon energy would be needed to obtain $1.0 \times 10^{-11} \mathrm{m}$ resolution?

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Numerade Educator

Problem 34

A nonrelativistic particle of mass $m$ and charge $q$ is accelerated from rest through a potential difference $\Delta V$ (a) Use conservation of energy to find a symbolic expression for the momentum of the particle in terms of $m, q,$ and $\Delta V$ . (b) Write a symbolic expression for the de Broglie wavelength using the result of part (a). (c) If an electron and proton go through the same potential difference but in opposite directions, which particle will have the shorter wavelength?

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Numerade Educator

Problem 35

In the ground state of hydrogen, the uncertainty in the position of the electron is roughly 0.10 nm. If the speed of the electron is approximately the same as the uncertainty in its speed, about how fast is it moving?

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Numerade Educator

Problem 36

An electron is located on a pinpoint having a diameter of 2.5$\mu \mathrm{m}$ . What is the minimum uncertainty in the speed of the electron?

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Numerade Educator

Problem 37

An electron and a 0.020 0-kg bullet each have a velocity of magnitude $5.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ , accurate to within 0.0100$\%$ . Within what lower limit could we determine the position of
each object along the direction of the velocity?

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Numerade Educator

Problem 38

In a nonrelativistic experiment, an electron and a proton are each located along the $x$ -axis to within an uncertainty of 2.50$\mu \mathrm{m}$ . Determine the minimum uncertainty in the $x$ -component of the velocity of (a) the electron, and (b) the proton.

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Numerade Educator

Problem 39

The average lifetime of a muon is about 2 \mus. Estimate the minimum uncertainty in the energy of a muon.

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Numerade Educator

Problem 40

(a) Show that the kinetic energy of a nonrelativistic particle can be written in terms of its momentum as $K E=$ $p^{2} / 2 m .$ (b) Use the results of part (a) to find the minimum kinetic energy of a proton confined within a nucleus having a diameter of $1.0 \times 10^{-15} \mathrm{m} .$

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Numerade Educator

Problem 41

A microwave photon in the x-band region has a wavelength of 3.00 $\mathrm{cm} .$ Find (a) the momentum, (b) the frequency, and (c) the energy of the photon in electron volts.

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Numerade Educator

Problem 42

Find the speed of an electron having a de Broglie wavelength equal to its Compton wavelength. Hint: This electron is relativistic.

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Numerade Educator

Problem 43

A 2.0 -kg object falls from a height of 5.0 $\mathrm{m}$ to the ground. If the change in the object's kinetic energy could be converted to visible light of wavelength $5.0 \times 10^{-7} \mathrm{m},$ how many photons would be produced?

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Numerade Educator

Problem 44

An x-ray tube is operated at $5.00 \times 10^{4} \mathrm{V}$ (a) Find the minimum wavelength of the radiation emitted by this tube. (b) If the radiation is directed at a crystal, the first-order maximum
in the reflected radiation occurs when the grazing angle is $2.5^{\circ} .$ What is the spacing between reflecting planes in the crystal?

Katie M.
Numerade Educator

Problem 45

Figure $P 27.45$ shows the spectrum of light emitted by a firefly. (a) Determine the temperature of a blackbody that would emit radiation peaked at the same frequency. (b) Based on your result, explain whether firefly radiation is blackbody radiation.

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Numerade Educator

Problem 46

Johnny Jumper's favorite trick is to step out of his 16 th-story window and fall 50.0 $\mathrm{m}$ into a pool. A news reporter takes a picture of $75.0-\mathrm{kg}$ Johnny just before he makes a splash, using an exposure time of 5.00 $\mathrm{ms}$ . Find (a) Johnny's de Broglie wavelength at this moment, (b) the uncertainty of his kinetic energy measurement during such a period of time, and
(c) the percent error caused by such an uncertainty.

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Numerade Educator

Problem 47

Photons of wavelength $4.50 \times 10^{2} \mathrm{nm}$ are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius 20.0 $\mathrm{cm}$ by a magnetic
field with a magnitude of $2.00 \times 10^{-5}$ T. What is the work function of the metal?

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Numerade Educator

Problem 48

An electron initially at rest recoils after a head-on collision with a 6.20 -keV photon. Determine the kinetic energy acquired by the electron.

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Problem 49

A light source of wavelength $\lambda$ illuminates a metal and ejects photoelectrons with a maximum kinetic energy of 1.00 eV. A second light source of wavelength $\lambda / 2$ ejects photoelectrons with a maximum kinetic energy of 4.00 $\mathrm{eV}$ . What is the work function of the metal?

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Numerade Educator

Problem 50

Red light of wavelength 670 . nm produces photoelectrons from a certain photoemissive material. Green light of wavelength 520 . nm produces photoelectrons from the same material with 1.50 times the maximum kinetic energy. What is the material's work function?

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Numerade Educator

Problem 51

How fast must an electron be moving if all its kinetic energy is lost to a single $\mathrm{x}$ -ray photon (a) at the high end of the $\mathrm{x}$ -ray electromagnetic spectrum with a wavelength of $1.00 \times 10^{-8} \mathrm{m}$ and (b) at the low end of the x-ray electromagnetic spectrum with a wavelength of $1.00 \times 10^{-13} \mathrm{m} ?$

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Problem 52

From the scattering of sunlight, J. J. Thomson calculated the classical radius of the electron as having the value $2.82 \times 10^{-15} \mathrm{m} .$ Sunlight with an intensity of $5.00 \times 10^{2} \mathrm{W} /$ $\mathrm{m}^{2}$ falls on a disk with this radius. Assume light is a classical wave and the light striking the disk is completely absorbed. (a) Calculate the time interval required to accumulate 1.00 $\mathrm{eV}$ of energy. (b) Explain how your result for part (a) compares with the observation that photoelectrons are emitted promptly (within $10^{-9} \mathrm{s} )$

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