College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 27

Quantum Physics - all with Video Answers

Educators

Chapter Questions

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 $970 \mathrm{~nm} ?$ (b) Rigel, a bluish-white star in Orion, radiates with a peak wavelength of $145 \mathrm{~nm}$. Find the temperature of Rigel's surface.

Guilherme Barros

Guilherme Barros

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 Mcalpine

Numerade Educator

Problem 3

The human eye is most sensitive to 560 -nm (green) light. What is the temperature of a blackbody that would radiate most intensely at this wavelength?

Sheh Lit Chang

University of Washington

Problem 4

A tungsten filament of a glowing lightbulb has a temperature of approximately $2000 \mathrm{~K}$. (a) Assuming the operating filament is a blackbody, determine the peak wavelength of its emitted radiation at this temperature. (b) Why does your answer to part (a) suggest that more energy from a lightbulb goes into heat than into light?

Sheh Lit Chang

University of Washington

Problem 5

Calculate the energy in electron volts of a photon having a wavelength (a) in the microwave range, $5.00 \mathrm{~cm}$, (b) in the visible light range, $500 \mathrm{~nm}$, and $(\mathrm{c})$ in the $\mathrm{x}$ -ray range, $5.00 \mathrm{~nm}$.

Sheh Lit Chang

University of Washington

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 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 Mcalpine

Numerade Educator

Problem 7

An FM radio transmitter has a power output of $150 \mathrm{~kW}$ and operates at a frequency of $99.7 \mathrm{MHz}$. How many photons per second does the transmitter emit?

Guilherme Barros

Guilherme Barros

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 $500 \mathrm{~nm}$. 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?

Sheh Lit Chang

University of Washington

Problem 9

When light of wavelength $350 \mathrm{~nm}$ falls on a potassium surface, electrons having a maximum kinetic energy of $1.31 \mathrm{eV}$ are emitted. Find (a) the work function of potassium, (b) the cutoff wavelength, and (c) the frequency corresponding to the cutoff wavelength.

Sheh Lit Chang

University of Washington

Problem 10

Electrons are ejected from a certain metallic surface with speeds ranging up to $4.6 \times 10^{5} \mathrm{~m} / \mathrm{s}$ when light with a wavelength of $\lambda=625 \mathrm{~nm}$ is used. (a) What is the work function of the metal? (b) What is the cutoff frequency for this metal?

Sheh Lit Chang

University of Washington

Problem 11

The work function for platinum is $6.35 \mathrm{eV}$. (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 $8.50 \mathrm{eV}$ 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 $8.50 \mathrm{eV}$, what stopping potential would be required to arrest the current of photoelectrons?

Sheh Lit Chang

University of Washington

Problem 12

Lithium, beryllium, and mercury have work functions of $2.30 \mathrm{eV}, 3.90 \mathrm{eV}$, 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 Mcalpine

Numerade Educator

Problem 13

When light of wavelength $254 \mathrm{~nm}$ falls on cesium. the required stopping potential is $3.00 \mathrm{~V}$. If light of wavelength $436 \mathrm{~nm}$ is used, the stopping potential is $0.900 \mathrm{~V}$. Use this information to plot a graph like that shown in Figure $27.6$, and from the graph determine the cutoff frequency for cesium and its work function.

Linda Winkler

Numerade Educator

Problem 14

Ultraviolet light is incident normally on the surface of a certain substance. The binding energy of the electrons in this substance is $3.44 \mathrm{eV}$. The incident light has an intensity of $0.055 \mathrm{~W} / \mathrm{m}^{2}$. The electrons are photoelectrically emitted with a maximum speed of $4.2 \times 10^{5} \mathrm{~m} / \mathrm{s}$. How many electrons are emitted from a square centimeter of the surface each second? Assume the absorption of every photon ejects an electron.

Sheh Lit Chang

University of Washington

Problem 15

The extremes of the $\mathrm{x}$ -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 Mcalpine

Numerade Educator

Problem 16

Calculate the minimum-wavelength 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 Mcalpine

Numerade Educator

Problem 17

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

Katie Mcalpine

Numerade Educator

Problem 18

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.

Katie Mcalpine

Numerade Educator

Problem 19

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

Katie Mcalpine

Numerade Educator

Problem 20

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?

Katie Mcalpine

Numerade Educator

Problem 21

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 Mcalpine

Numerade Educator

Problem 22

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.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 23

A $0.0016-\mathrm{nm}$ photon scatters from a free electron. For what (photon) scattering angle will the recoiling electron and scattered photon have the same kinetic energy?

Sheh Lit Chang

University of Washington

Problem 24

A beam of $0.68-n m$ photons undergoes Compton scattering from free electrons. What are the energy and momentum of the photons that emerge at a $45^{\circ}$ angle with respect to the incident beam?

Sheh Lit Chang

University of Washington

Problem 25

A $0.110-\mathrm{nm}$ photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backwards. Find the momentum and kinetic energy of the electron.

Sheh Lit Chang

University of Washington

Problem 26

X-rays with an energy of $300 \mathrm{keV}$ undergo Compton scattering from a target. If the scattered rays are deflected at $37.0^{\circ}$ relative to the direction of the incident rays, find
(a) the Compton shift at this angle, (b) the energy of the scattered $\mathrm{x}$ -ray, and (c) the kinetic energy of the recoiling electron.

Sheh Lit Chang

University of Washington

Problem 27

(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 Mcalpine

Numerade Educator

Problem 28

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

Katie Mcalpine

Numerade Educator

Problem 29

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 $3.00 \mathrm{MeV}$

Prashant Bana

Numerade Educator

Problem 30

(a) Calculate the momentum of a photon having a wavelength of $4.00 \times 10^{2} \mathrm{~nm} .$ (b) Find the speed of an electron having the same momentum as the photon in part (a).

Sheh Lit Chang

University of Washington

Problem 31

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?

Katie Mcalpine

Numerade Educator

Problem 32

A 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 $\Delta V$. Assume the particle's speed isn't relativistic. (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?

Sheh Lit Chang

University of Washington

Problem 33

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

Katie Mcalpine

Numerade Educator

Problem 34

A $0.50-\mathrm{kg}$ block rests on the icy surface of a frozen pond, which you can assume to be frictionless. If the location of the block is measured to a precision of $0.50 \mathrm{~cm}$, what is the minimum uncertainty in the block's speed, assuming the mass is known exactly?

Sheh Lit Chang

University of Washington

Problem 35

Suppose optical radiation $\left(\lambda=5.00 \times 10^{-7} \mathrm{~m}\right)$ is used to determine the position of an electron to within the wavelength of the light. What will be the resulting uncertainty in the electron's velocity?

Sheh Lit Chang

University of Washington

Problem 36

Suppose Fuzzy, a quantum mechanical duck, lives in a world in which $h=2 \pi J \cdot \mathrm{s}$. Fuzzy has a mass of $2.00 \mathrm{~kg}$ and is initially known to be within a pond $1.00 \mathrm{~m}$ wide.
(a) What is the minimum uncertainty in the duck's speed? (b) Assuming this uncertainty in speed to prevail for $5.00 \mathrm{~s}$, determine the uncertainty in Fuzzy's position after this time.

Sheh Lit Chang

University of Washington

Problem 37

The average lifetime of a muon is about $2 \mu \mathrm{s}$. Estimate the minimum uncertainty in the energy of a muon.

Sheh Lit Chang

University of Washington

Problem 38

(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 $1.0 \times 10^{-15} \mathrm{~m}$.

Katie Mcalpine

Numerade Educator

Problem 39

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.

Katie Mcalpine

Numerade Educator

Problem 40

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

Katie Mcalpine

Numerade Educator

Problem 41

A $2.0$ -kg object falls from a height of $5.0 \mathrm{~m}$ to the ground. If all the gravitational potential energy of this mass could be converted to visible light of wavelength $5.0 \times 10^{-7} \mathrm{~m}$, how many photons would be produced?

Sheh Lit Chang

University of Washington

Problem 42

An x-ray tube is operated at $50000 \mathrm{~V}$. (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?

Sheh Lit Chang

University of Washington

Problem 43

Figure P27.43 shows the spectrum of light emitted by a firefly. Determine the temperature of a blackbody that would emit radiation peaked at the same frequency. Based on your result, would you say that firefly radiation is blackbody radiation?

Sheh Lit Chang

University of Washington

Problem 44

Johnny Jumper's favorite trick is to step out of his 16 thstory window and fall $50.0 \mathrm{~m}$ into a pool. A news reporter takes a picture of $75.0$ -kg Johnny just before he makes ? 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.

Sheh Lit Chang

University of Washington

Problem 45

Photons of wavelength $450 \mathrm{~nm}$ 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 $2.00 \times 10^{-5} \mathrm{~T}$. What is the work function of the metal?

Sheh Lit Chang

University of Washington

Problem 46

A $200-\mathrm{MeV}$ photon is scattered at $40.0^{\circ}$ by a free proton that is initially at rest. Find the energy (in MeV) of the scattered photon.

Sheh Lit Chang

University of Washington

Problem 47

A light source of wavelength $\lambda$ illuminates a metal and ejects photoelectrons with a maximum kinetic energy of $1.00 \mathrm{eV} .$ 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?

Katie Mcalpine

Numerade Educator

Problem 48

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

Katie Mcalpine

Numerade Educator

Problem 49

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

Sheh Lit Chang

University of Washington

Problem 50

Show that if an electron were confined inside an atomic nucleus of diameter $2.0 \times 10^{-15} \mathrm{~m}$, it would have to be moving relativistically, whereas a proton confined to the same nucleus can be moving at less than one-tenth the speed of light.

Sheh Lit Chang

University of Washington