Modern Physics

Kenneth S. Krane

Chapter 3

The Particle-Like Properties of Electromagnetic Radiation - all with Video Answers

Educators

Chapter Questions

Problem 1

A double-slit experiment is performed with sodium light $(\lambda=589.0 \mathrm{nm}) .$ The slits are separated by $1.25 \mathrm{mm},$ and the screen is $2.604 \mathrm{m}$ from the slits. Find the separation between adjacent maxima on the screen.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

In Example $3.1,$ what angle of incidence will produce the second-order Bragg peak?

Narayan Hari

Numerade Educator

Problem 3

Monochromatic X rays are incident on a crystal in the geometry of Figure $3.5 .$ The first-order Bragg peak is observed when the angle of incidence is $38.0^{\circ} .$ The crystal spacing is known to be $0.327 \mathrm{nm} .$ (a) What is the wavelength of the $X$ rays? $(b)$ Now consider a set of crystal planes that makes an angle of $45^{\circ}$ with the surface of the crystal (as shown in Figure 3.6 ). For $X$ rays of the same wavelength, find the angle of incidence measured from the surface of the crystal that produces the first-order Bragg peak. At what angle from the surface does the emerging beam appear in this case?

Suzanne W.

Numerade Educator

Problem 4

A certain device for analyzing electromagnetic radiation is based on the Bragg scattering of the radiation from a crystal. For radiation of wavelength $0.149 \mathrm{nm},$ the first-order Bragg peak appears centered at an angle of $15.15^{\circ} .$ The aperture of the analyzer passes radiation in the angular range of $0.015^{\circ} .$ What is the corresponding range of wavelengths passing through the analyzer?

Narayan Hari

Numerade Educator

Problem 5

Find the momentum of $(a)$ a $10.0-\mathrm{MeV}$ gamma ray; (b) a $25-\mathrm{keV} \times$ ray ;(c) a $1.0-\mu \mathrm{m}$ infrared photon; $(d)$ a 150-MHz radio-wave photon. Express the momentum in $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ and $\mathrm{eV} / \mathrm{c}$

Suzanne W.

Numerade Educator

Problem 6

What range of photon energies corresponds to radio-wave frequencies of 1 to $100 \mathrm{MHz} ?$ Our bodies are continuously bombarded by these photons. Why are they not dangerous to us?

Suzanne W.

Numerade Educator

Problem 7

(a) What is the wavelength of an X-ray photon of energy $10.0 \mathrm{keV} ?(b)$ What is the wavelength of a gamma-ray photon of energy $1.00 \mathrm{MeV} ?$ (c) What is the range of\begin{aligned} &\text { energies of photons of visible light with wavelengths }\\
&350-700 \mathrm{nm} ? \end{aligned}

Narayan Hari

Numerade Educator

Problem 8

What is the cutoff wavelength for the photoelectric effect using an aluminum surface?

Narayan Hari

Numerade Educator

Problem 9

Light of wavelength $304.2 \mathrm{nm}$ illuminates a metal surface whose cutoff wavelength is $352.8 \mathrm{nm} .$ What is the stopping potential?

Narayan Hari

Numerade Educator

Problem 10

When light of wavelength $\lambda$ illuminates a copper surface, the stopping potential is $V$. In terms of $V$, what will be the stopping potential if the same wavelength is used to illuminate a sodium surface?

Narayan Hari

Numerade Educator

Problem 11

The cutoff wavelength for the photoelectric effect in a certain metal is $254 \mathrm{nm}$. (a) What is the work function for that metal? (b) Will the photoelectric effect be observed for $\lambda>254 \mathrm{nm}$ or for $\lambda<254 \mathrm{nm} ?$

Narayan Hari

Numerade Educator

Problem 12

A surface of zinc is illuminated and photo electrons are observed. (a) What is the largest wavelength that will cause photo electrons to be emitted? (b) What is the stopping potential when light of wavelength $252.0 \mathrm{nm}$ is used?

Narayan Hari

Numerade Educator

Problem 13

(a) Show that in the classical result for the energy distribution of the cavity wall oscillators (Eq. 3.30 ), the total number of oscillators at all energies is $N .$ (b) Show that $E_{\mathrm{me}}=k T$ for the classical oscillators.

Suzanne W.

Numerade Educator

Problem 14

(a) Writing the discrete Maxwell-Boltzmann distribution for Planck's cavity wall oscillators as $N_{n}=$ $A e^{-E_{n} / k T}($ where $A$ is a constant to be determined), show that the condition $\sum_{n=0}^{\infty} N_{n}=N$ gives $A=N\left(1-e^{-\varepsilon / k T}\right)$ as in Eq. $3.36 .\left[\right.$ Hint: Use $\left.\sum_{n=0}^{\infty} e^{n x}=1 /\left(1-e^{x}\right) \cdot\right](b)$ By taking the derivative with respect to $x$ of the equation given in the hint, show that $\sum_{n=0}^{\infty} n e^{n x}=e^{x} /\left(1-e^{x}\right)^{2}$
(c) Use this result
to derive Eq. 3.38 from Eq. $3.37 .(d)$ Show that $E_{\mathrm{av}} \cong k T$ at large $\lambda$ and $E_{a v} \rightarrow 0$ for small $\lambda$.

Suzanne W.

Numerade Educator

Problem 15

By differentiating Eq. 3.39 show that $I(\lambda)$ has its maximum as expected according to Wien's displacement law, Eq. 3.25

Suzanne W.

Numerade Educator

Problem 16

Integrate Eq. 3.39 to obtain Eq. 3.24. Use the definite integral $\int_{0}^{\infty} x^{3} d x /\left(e^{x}-1\right)=\pi^{4} / 15$ to obtain Eq. 3.40 relating the Stefan-Boltzmann constant to Planck's constant.

Suzanne W.

Numerade Educator

Problem 17

Use the numerical value of the Stefan-Boltzmann constant to find the numerical value of Planck's constant from $\mathrm{Fa} .3 .40$

Narayan Hari

Numerade Educator

Problem 18

The surface of the Sun has a temperature of about $6000 \mathrm{K} .$ At what wavelength does the Sun emit its peak intensity? How does this compare with the peak sensitivity of the human eye?

Narayan Hari

Numerade Educator

Problem 19

The universe is filled with thermal radiation, which has a black body spectrum at an effective temperature of $2.7 \mathrm{K}$ (see Chapter 15). What is the peak wavelength of this radiation? What is the energy (in eV) of quanta at the peak wavelength? In what region of the electromagnetic spectrum is this peak wavelength?

Narayan Hari

Numerade Educator

Problem 20

(a) Assuming the human body (skin temperature $34^{\circ} \mathrm{C}$ ) to behave like an ideal thermal radiator, find the wavelength where the intensity from the body is a maximum. In what region of the electromagnetic spectrum is radiation with this wavelength? (b) Making whatever (reasonable) assumptions you may need, estimate the power radiated by a typical person isolated from the surroundings.
(c) Estimate the radiation power absorbed by a person in a room in which the temperature is $20^{\circ} \mathrm{C}$.

Suzanne W.

Numerade Educator

Problem 21

A cavity is maintained at a temperature of $1750 \mathrm{K}$. At what rate does energy escape from the interior of the cavity through a hole in its wall of diameter $1.24 \mathrm{mm} ?$

Narayan Hari

Numerade Educator

Problem 22

An analyzer for thermal radiation is set to accept wavelengths in an interval of $1.74 \mathrm{nm} .$ What is the intensity of the radiation in that interval at a wavelength of $932 \mathrm{nm}$ emitted from a glowing object whose temperature is $1546 \mathrm{K} ?$

Suzanne W.

Numerade Educator

Problem 23

(a) Assuming the Sun to radiate like an ideal thermal source at a temperature of $6000 \mathrm{K}$, what is the intensity of the solar radiation emitted in the range $530.0 \mathrm{nm}$ to $532.0 \mathrm{nm} ?$
(b) What fraction of the total solar radiation does this represent?

Suzanne W.

Numerade Educator

Problem 24

Show how Eq. 3.46 follows from Eq. 3.45.

Suzanne W.

Numerade Educator

Problem 25

Incident photons of energy $11.32 \mathrm{keV}$ are Compton scattered, and the scattered beam is observed at $62.9^{\circ}$ relative to the incident beam. (a) What is the energy of the scattered photons at that angle? (b) How much kinetic energy is given to the scattered electron?

Suzanne W.

Numerade Educator

Problem 26

X-ray photons of wavelength $0.02218 \mathrm{nm}$ are incident in a target and the Compton-scattered photons are observed at $90.0^{\circ} .(a)$ What is the wavelength of the scattered photons? (b) What is the momentum of the incident photons and the scattered photons? (c) What is the kinetic energy of the scattered electrons? $(d)$ What is the momentum (magnitude and direction) of the scattered electrons?

Suzanne W.

Numerade Educator

Problem 27

High-energy gamma rays can reach a radiation detector by Compton scattering from the surroundings, as shown in Figure $3.26 .$ This effect is known as back-scattering. Show that, when $E \gg m_{\mathrm{e}} c^{2},$ the back-scattered photon has an energy of approximately 0.25 MeV, independent of the energy of the original photon, when the scattering angle is nearly $180^{\circ}$.

Narayan Hari

Numerade Educator

Problem 28

Gamma rays of energy $0.662 \mathrm{MeV}$ are Compton scattered. (a) What is the energy of the scattered photon observed at a scattering angle of $52.2^{\circ} ?(b)$ What is the kinetic energy of the scattered electrons?

Suzanne W.

Numerade Educator

Problem 29

In Compton's original work he used a slightly different derivation to obtain Eq. $3.47 .(a)$ Conservation of momentum in Figure 3.18 means that the 3 vectors representing the momenta $p, p^{\prime},$ and $p_{\mathrm{e}}$ must form a closed triangle such that $\overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{p}}^{\prime}+\overrightarrow{\mathbf{p}}_{\mathrm{e}} .$ Draw this triangle and apply the law of cosines to the angle $\theta$ between $\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{p}}^{\prime}$ Express the photon momenta in terms of wavelength and the electron momentum in terms of its speed. (b) Write a second equation for conservation of energy, again expressing the photon energy in terms of wavelength and the electron energy in terms of its speed. (c) Eliminate the electron speed between the 2 equations to obtain $\mathrm{Eq} \cdot 3.47$

Lainey Roebuck

Numerade Educator

Problem 30

A photon enters a detector and undergoes Compton scattering. The scattered electron is captured within the detector and its kinetic energy is measured. The scattered photon then travels to a second detector where it is captured and its energy measured. In one particular experiment, the electron energy was determined to be $2.302 \mathrm{MeV}$ and the scattered photon energy $0.239 \mathrm{MeV}$ Determine the energy of the original photon and its direction relative to the scattered photon. This is the process used by the Compton Gamma-Ray Observatory (Figure 3.27 ) to determine the location in the sky from which energetic gamma rays reach the Earth.

Narayan Hari

Numerade Educator

Problem 31

Suppose an atom of gold at rest emits an X-ray photon of energy $69 \mathrm{keV}$. Calculate the "recoil" momentum and kinetic energy of the atom. (Hint: Do you expect to need classical or relativistic kinetic energy for the atom? Is the kinetic energy likely to be much smaller than the atom's rest energy?)

Suzanne W.

Numerade Educator

Problem 32

What is the minimum X-ray wavelength produced in bremsstrahlung by electrons that have been accelerated through $7.5 \times 10^{4}$ V?

Narayan Hari

Numerade Educator

Problem 33

An atom absorbs a photon of wavelength $425 \mathrm{nm}$ and immediately emits another photon of wavelength $643 \mathrm{nm}$. What is the net energy absorbed by the atom in this process?

Narayan Hari

Numerade Educator

Problem 34

A certain green light bulb emits at a single wavelength of $550 \mathrm{nm} .$ It consumes $55 \mathrm{W}$ of electrical power and is $75 \%$ efficient in converting electrical energy into light. $(a)$ How many photons does the bulb emit in one hour? (b) Assuming the emitted photons to be distributed uniformly in space, how many photons per second strike a $10 \mathrm{cm}$ by $10 \mathrm{cm}$ paper held facing the bulb at a distance of $1.0 \mathrm{m} ?$

Suzanne W.

Numerade Educator

Problem 35

When sodium metal is illuminated with light of wavelength $4.20 \times 10^{2} \mathrm{nm},$ the stopping potential is found to be $0.65 \mathrm{V} ;$ when the wavelength is changed to $3.10 \times 10^{2} \mathrm{nm},$ the stopping potential is $1.69 \mathrm{V}$. Using only these data and the values of the speed of light and the electronic charge, find the work function of sodium and a value of Planck's

Suzanne W.

Numerade Educator

Problem 36

A photon of wavelength $157 \mathrm{nm}$ strikes an aluminum surface along a line perpendicular to the surface and releases a photoelectron traveling in the opposite direction. Assume the recoil momentum is taken up by a single aluminum atom on the surface. Calculate the recoil kinetic energy of the atom. Would this recoil energy significantly affect the kinetic energy of the photoelectron?

Suzanne W.

Numerade Educator

Problem 37

A certain cavity has a temperature of $1325 \mathrm{K} .(a) \mathrm{At}$ what wavelength will the intensity of the radiation in the cavity have its maximum value? (b) As a fraction of the maximmm intensity, what is the intensity at twice the wavelength found in part ( $a$ )?

Suzanne W.

Numerade Educator

Problem 38

In Compton scattering, calculate the maximum kinetic energy given to the scattered electron for a given photon
energy.

Suzanne W.

Numerade Educator

Problem 39

The COBE satellite was launched in 1989 to study the cosmic background radiation and measure its temperature. By measuring at many different wavelengths, researchers were able to show that the background radiation exactly followed the spectral distribution expected for a blackbody. At a wavelength of $0.133 \mathrm{cm},$ the radiant intensity is $1.440 \times 10^{-7} \mathrm{W} / \mathrm{m}^{2}$ in a wavelength interval of $0.00833 \mathrm{cm} .$ What is the temperature of the radiation that would be deduced from these data?

Suzanne W.

Numerade Educator

Problem 40

The WMAP satellite launched in 2001 studied the cosmic
microwave background radiation and was able to chart small fluctuations in the temperature of different regions of the background radiation. These fluctuations in temperature correspond to regions of large and small density in the early universe. The satellite was able to measure differences in temperature of $2 \times 10^{-5} \mathrm{K}$ at a temperature of $2.7250 \mathrm{K} .$ At the peak wavelength, what is the difference in the radiation intensity per unit wavelength interval between the "hot" and "cold" regions of the background radiation?

Manik Pulyani

Numerade Educator

Problem 41

You have been hired as an engineer on a NASA project to design a microwave spectrometer for an orbital mission to measure the cosmic background radiation, which has a black body spectrum with an effective temperature of $2.725 \mathrm{K}$. (a) The spectrometer is to scan the sky between wavelengths of $0.50 \mathrm{mm}$ and $5.0 \mathrm{mm},$ and at each wavelength it accepts radiation in a wavelength range of $3.0 \times 10^{-4} \mathrm{mm} .$ What maximum and minimum radiation intensity do you expect to find in this region? $(b)$ The photon detector in the spectrometer is in the form of a disk of diameter $0.86 \mathrm{cm} .$ How many photons per second will the spectrometer record at its maximum and minimum intensities?

Ozenc Gungor

Numerade Educator

Problem 42

A photon of wavelength $6.13 \mathrm{pm}$ scatters from a free electron at rest. After the interaction, the electron is observed to be moving in the direction of the original photon. Find the momentum of the electron.

Susan Hallstrom

Susan Hallstrom

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Problem 43

A hydrogen atom is moving at a speed of $125.0 \mathrm{m} / \mathrm{s}$. It absorbs a photon of wavelength $97 \mathrm{nm}$ that is moving in the opposite direction. By how much does the speed of the atom change as a result of absorbing the photon?

Narayan Hari

Numerade Educator

Problem 44

Before a positron and an electron annihilate, they form a sort of "atom" in which each orbits about their common center of mass with identical speeds. As a result of this motion, the photons emitted in the annihilation show a small Doppler shift. In one experiment, the Doppler shift in energy of the photons was observed to be $2.41 \mathrm{keV}$.
(a) What would be the speed of the electron or positron before the annihilation to produce this Doppler shift?
(b) The positrons form these atom-like structures with the nearly "free" electrons in a solid. Assuming the positron and electron must have about the same speed to form this structure, find the kinetic energy of the electron. This technique, called "Doppler broadening," is an important method for learning about the energies of electrons in materials.

Suzanne W.

Numerade Educator

Problem 45

Prove that it is not possible to conserve both momentum and total relativistic energy in the following situation: A free electron moving at velocity $\overrightarrow{\mathbf{v}}$ emits a photon and then moves at a slower velocity $\overrightarrow{\mathbf{v}}^{\prime}$

Mayukh Banik

Numerade Educator

Problem 46

A photon of energy $E$ interacts with an electron at rest and undergoes pair production, producing a positive electron (positron) and an electron (in addition to the original electron): $$ \text { photon }+\mathrm{e}^{-} \rightarrow \mathrm{e}^{+}+\mathrm{e}^{-}+\mathrm{e}^{-} $$ The two electrons and the positron move off with identical momenta in the direction of the initial photon. Find the kinetic energy of the three final particles and find theenergy $E$ of the photon. (Hint: Conserve momentum and total relativistic energy.)

Suzanne W.

Numerade Educator

Problem 47

You have been hired by NASA to analyze a solar sail that uses the momentum of sunlight for interplanetary travel. A prototype sail of area $1 \mathrm{km}^{2}$ has been developed and is made from a thin lightweight polymer that has a highly reflective aluminum coating on one side. This material has thickness $2 \mu \mathrm{m}$ and density $0.29 \mathrm{g} / \mathrm{cm}^{2}$. You are limited by the design which requires that the supporting frame and the cargo weigh no more than the film itself. Make any necessary assumptions about the parameters of the calculation, and estimate the travel time of this spacecraft from Earth to Mars.

Suzanne W.

Numerade Educator

Problem 48

An electron is moving in the $+x$ direction with a speed of $0.46 c .$ It is struck from behind by a photon of energy $0.172 \mathrm{MeV}$ moving in the some direction. Find the energies of the scattered electron and photon. (Hint: Would you expect the scattered photon to be moving in the $+x$ or $-x$ direction?)

Zulfiqar Ali

Numerade Educator

Problem 49

A certain gamma-ray detector measures photon energies through the Compton interaction: the photon Compton scatters within the detector material, which then absorbs the kinetic energy of the scattered electron. The absorbed energy of the scattered electron is the response of the Suppose photons of energy $E$ are incident on this detector. (a) Find an expression for the maximum energy response $E_{\max }$ of this detector and show that $E_{\max }$ is less than the original energy of the photon. (b) Evaluate $E_{\max }$ for an incident photon energy of $1.5 \mathrm{MeV}$. (c) Occasionally the detector may report events with energy greater than $E_{\max }$ but less than $E .$ What other processes might be responsible for such events? $(d)$ What processes might contribute to the detector reporting the full energy $E$ of the photon?

Narayan Hari

Numerade Educator

Problem 50

An electron is moving in the negative $x$ direction with a speed of $0.95 c$ when it encounters a photon of energy $12.4 \mathrm{keV}$ moving in the positive $x$ direction. Find the energies of the photon and electron after the scattering. This process, in which photons gain energy by scattering from very energetic electrons, is called inverse Compton scattering and is thought to be the process that produces very high energy gamma rays ( $1 \mathrm{GeV}$ and above) that reach Earth from space. The Compton Gamma-Ray Observatory was launched by NASA in 1991 and operated until 2000 scanning the sky for gamma rays (Figure 3.27 ).

Suzanne W.

Numerade Educator