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Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 40

All About Atoms - all with Video Answers

Educators


Chapter Questions

09:01

Problem 1

An electron in a hydrogen atom is in a state with $\ell=5$. What is the minimum possible value of the semiclassical angle between $\vec{L}$ and $L_z$ ?

Kyle Godbey
Kyle Godbey
Numerade Educator
03:52

Problem 2

How many electron states are there in a shell defined by the quantum number $n=5$ ?

Guilherme Barros
Guilherme Barros
Numerade Educator
04:28

Problem 3

(a) What is the magnitude of the orbital angular momentum in a state with $\ell=3$ ? (b) What is the magnitude of its largest projection on an imposed $z$ axis?

Alia Hamdan
Alia Hamdan
Numerade Educator
01:53

Problem 4

How many electron states are there in the following shells: (a) $n=4$, (b) $n=1$, (c) $n=3$, (d) $n=2$ ?

Salamat Ali
Salamat Ali
Numerade Educator
01:33

Problem 5

(a) How many $\ell$ values are associated with $n=3$ ? (b) How many $m_{\ell}$ values are associated with $\ell=1$ ?

Mayank Tripathi
Mayank Tripathi
Numerade Educator
02:53

Problem 6

How many electron states are in these subshells: (a) $n=4$, $\ell=3$; (b) $n=3, t=1$; (c) $n=4, \ell=1$; (d) $n=2, t=0$ ?

Salamat Ali
Salamat Ali
Numerade Educator
03:20

Problem 7

An electron in a multielectron atom has $m_{\ell}=+4$. For this electron, what are (a) the value of $\ell$, (b) the smallest possible value of $n$, and (c) the number of possible values of $m_s$ ?

Alia Hamdan
Alia Hamdan
Numerade Educator
02:14

Problem 8

In the subshell $\ell=3$, (a) what is the greatest (most positive) $m_{\ell}$ value, (b) how many states are available with the greatest $m_{\ell}$ value, and (c) what is the total number of states available in the subshell?

Salamat Ali
Salamat Ali
Numerade Educator
04:59

Problem 9

An electron is in a state with $\ell=3$. (a) What multiple of $\hbar$ gives the magnitude of $\vec{L}$ ? (b) What multiple of $\mu_{\mathrm{B}}$ gives the magnitude of $\vec{\mu}$ ? (c) What is the largest possible value of $m_e$,
(d) what multiple of $\hbar$ gives the corresponding value of $L_z$, and
(e) what multiple of $\mu_{\mathrm{B}}$ gives the corresponding value of $\mu_{\mathrm{orb}, z}$ ?
(f) What is the value of the semiclassical angle $\theta$ between the directions of $L_z$ and $\vec{L}$ ? What is the value of angle $\theta$ for $(\mathrm{g})$ the second largest possible value of $m_{\ell}$ and (h) the smallest (that is, most negative) possible value of $m_{\ell}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
03:02

Problem 10

An electron is in a state with $n=3$. What are (a) the number of possible values of $\ell$, (b) the number of possible values of $m_{\ell}$, (c) the number of possible values of $m_s$, (d) the number of states in the $n=3$ shell, and (e) the number of subshells in the $n=3$ shell?

Salamat Ali
Salamat Ali
Numerade Educator
05:44

Problem 11

If orbital angular momentum $\vec{L}$ is measured along, say, a $z$ axis to obtain a value for $L_z$, show that
$$
\left(L_x^2+L_y^2\right)^{1 / 2}=\left[t(\ell+1)-m_l^2\right]^{1 / 2} h
$$
is the most that can be said about the other two components of the orbital angular momentum.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:41

Problem 12

A magnetic field is applied to a freely floating uniform iron sphere with radius $R=2.00 \mathrm{~mm}$. The sphere initially had no net magnetic moment, but the field aligns $12 \%$ of the magnetic moments of the atoms (that is, $12 \%$ of the magnetic moments of the loosely bound electrons in the sphere, with one such electron per atom). The magnetic moment of those aligned electrons is the sphere's intrinsic magnetic moment $\vec{\mu}_s$. What is the sphere's resulting angular speed $\omega$ ?

Salamat Ali
Salamat Ali
Numerade Educator
03:37

Problem 13

What is the acceleration of a silver atom as it passes through the deflecting magnet in the Stern-Gerlach experiment of Fig. 40.2 .1 if the magnetic field gradient is $1.4 \mathrm{~T} / \mathrm{mm}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:41

Problem 14

Suppose that a hydrogen atom in its ground state moves $80 \mathrm{~cm}$ through and perpendicular to a vertical magnetic field that has a magnetic field gradient $d B / d z=1.6 \times 10^2 \mathrm{~T} / \mathrm{m}$. (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atom's electron, which we take to be 1 Bohr magneton? (b) What is the vertical displacement of the atom in the $80 \mathrm{~cm}$ of travel if its speed is $1.2 \times 10^5 \mathrm{~m} / \mathrm{s}$ ?

Salamat Ali
Salamat Ali
Numerade Educator
06:11

Problem 15

Calculate the (a) smaller and (b) larger value of the semiclassical angle between the electron spin angular momentum vector and the magnetic field in a Stern-Gerlach experiment. Bear in mind that the orbital angular momentum of the valence electron in the silver atom is zero.

Alia Hamdan
Alia Hamdan
Numerade Educator
03:07

Problem 16

Assume that in the Stern-Gerlach experiment as described for neutral silver atoms, the magnetic field $\vec{B}$ has a magnitude of $0.50 \mathrm{~T}$. (a) What is the energy difference between the magnetic moment orientations of the silver atoms in the two subbeams?
(b) What is the frequency of the radiation that would induce a transition between these two states? (c) What is the wavelength of this radiation, and (d) to what part of the electromagnetic spectrum does it belong?

Salamat Ali
Salamat Ali
Numerade Educator
09:40

Problem 17

In an NMR experiment, the RF source oscillates at $34 \mathrm{MHz}$ and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field $\vec{B}_{\text {ext }}$ has magnitude $0.78 \mathrm{~T}$. Assume that $\vec{B}_{\text {int }}$ and $\vec{B}_{\text {ext }}$ are in the same direction and take the proton magnetic moment component $\mu_z$ to be $1.41 \times 10^{-26} \mathrm{~J} / \mathrm{T}$. What is the magnitude of $\vec{B}_{\text {int }}$ ?

Alia Hamdan
Alia Hamdan
Numerade Educator
01:23

Problem 18

A hydrogen atom in its ground state actually has two possible, closely spaced energy levels because the electron is in the magnetic field $\vec{B}$ of the proton (the nucleus). Accordingly, an energy is associated with the orientation of the electron's
magnetic moment $\vec{\mu}$ relative to $\vec{B}$, and the electron is said to be either spin up (higher energy) or spin down (lower energy) in that field. If the electron is excited to the higher-energy level, it can de-excite by spin-flipping and emitting a photon. The wavelength associated with that photon is $21 \mathrm{~cm}$. (Such a process occurs extensively in the Milky Way Galaxy, and reception of the $21 \mathrm{~cm}$ radiation by radio telescopes reveals where hydrogen gas lies between stars.) What is the effective magnitude of $\vec{B}$ as experienced by the electron in the ground-state hydrogen atom?

Salamat Ali
Salamat Ali
Numerade Educator
09:54

Problem 19

What is the wavelength associated with a photon that will induce a transition of an electron spin from parallel to antiparallel orientation in a magnetic field of magnitude $0.200 \mathrm{~T}$ ? Assume that $\ell=0$.

Alia Hamdan
Alia Hamdan
Numerade Educator
04:16

Problem 20

A rectangular corral of widths $L_x=L$ and $L_y=2 L$ contains seven electrons. What multiple of $h^2 / 8 m L^2$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Salamat Ali
Salamat Ali
Numerade Educator
05:28

Problem 21

Seven electrons are trapped in a one-dimensional infinite potential well of width $L$. What multiple of $h^2 / 8 m L^2$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Alia Hamdan
Alia Hamdan
Numerade Educator
04:19

Problem 22

(e. Figure 40.2 is an energy-level diagram for a fictitious infinite potential well that contains one electron. The number of degenerate states of the levels are indicated: "non" means nondegenerate (which includes the ground state of the electron), "double" means 2 states, and "triple" means 3 states. We put a total of 11 electrons in the well. If the electrostatic forces between the electrons can be neglected, what multiple of $h^2 / 8 m L^2$ gives the energy of the first excited state of the 11 -electron system?
( FIGURE CAN'T COPY )

Keshav Singh
Keshav Singh
Numerade Educator
08:49

Problem 23

A cubical box of widths $L_x=L_y=L_z=L$ contains eight electrons. What multiple of $h^2 / 8 m L^2$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Alia Hamdan
Alia Hamdan
Numerade Educator
07:51

Problem 24

For Problem 20, what multiple of $h^2 / 8 m L^2$ gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons? (d) Construct an energy-level diagram for the lowest four energy levels.

Salamat Ali
Salamat Ali
Numerade Educator
09:12

Problem 25

For the situation of Problem 21, what multiple of $h^2 / 8 m L^2$ gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons? (d) Construct an energy-level diagram for the lowest four energy levels of the system.

Alia Hamdan
Alia Hamdan
Numerade Educator
08:18

Problem 26

For the situation of Problem 23 , what multiple of $h^2 / 8 m L^2$ gives the energy of (a) the first excited state, (b) the
second excited state, and (c) the third excited state of the system of eight electrons? (d) Construct an energy-level diagram for the lowest four energy levels of the system.

Salamat Ali
Salamat Ali
Numerade Educator
03:24

Problem 27

Two of the three electrons in a lithium atom have quantum numbers $\left(n, \ell, m_{\ell}, m_s\right)$ of $\left(1,0,0,+\frac{1}{2}\right)$ and $\left(1,0,0,-\frac{1}{2}\right)$ What quantum numbers are possible for the third electron if the atom is (a) in the ground state and (b) in the first excited state?

Keshav Singh
Keshav Singh
Numerade Educator
04:12

Problem 28

Show that the number of states with the same quantum number $n$ is $2 n^2$.

Mayank Tripathi
Mayank Tripathi
Numerade Educator
02:07

Problem 29

A recently named element is darmstadtium (Ds), which has 110 electrons. Assume that you can put the 110 electrons into the atomic shells one by one and can neglect any electron-electron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number $\ell$ for the last electron?

Keshav Singh
Keshav Singh
Numerade Educator
02:48

Problem 30

For a helium atom in its ground state, what are quantum numbers ( $n, \ell, m_{\ell}$, and $m_s$ ) for the (a) spin-up electron and (b) spin-down electron?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:20

Problem 31

Eonsider the elements selenium $(Z=34)$, bromine $(Z=35)$, and krypton $(Z=36)$. In their part of the periodic table, the subshells of the electronic states are filled in the sequence
$$
1 s 2 s 2 p 3 s 3 p 3 d 4 s 4 p \ldots .
$$

What are (a) the highest occupied subshell for selenium and (b) the number of electrons in it, (c) the highest occupied subshell for bromine and (d) the number of electrons in it, and (c) the highest occupied subshell for krypton and (f) the number of electrons in it?

Keshav Singh
Keshav Singh
Numerade Educator
06:47

Problem 32

Suppose two electrons in an atom have quantum numbers $n=2$ and $\ell=1$. (a) How many states are possible for those two electrons? (Keep in mind that the electrons are indistinguishable.) (b) If the Pauli exclusion principle did not apply to the electrons, how many states would be possible?

Salamat Ali
Salamat Ali
Numerade Educator
01:38

Problem 33

Ehrough what minimum potential difference must an electron in an $x$-ray tube be accelerated so that it can produce $x$ rays with a wavelength of $0.100 \mathrm{~nm}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:59

Problem 34

The wavelength of the $K_a$ line from iron is $193 \mathrm{pm}$. What is the energy difference between the two states of the iron atom that give rise to this transition?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:27

Problem 35

In Fig. 40.6.1, the $x$ rays shown are produced when $35.0 \mathrm{keV}$ electrons strike a molybdenum $(Z=42)$ target. If the accelerating potential is maintained at this value but a silver $(Z=47)$ target is used instead, what values of (a) $\lambda_{\min }$, (b) the wavelength of the $K_a$ line, and (c) the wavelength of the $K_\beta$ line result? The $K, L$, and $M$ atomic x-ray levels for silver (compare Fig. 40.6.3) are 25.51, 3.56, and $0.53 \mathrm{keV}$.

Keshav Singh
Keshav Singh
Numerade Educator
02:22

Problem 36

When electrons bombard a molybdenum target, they produce both continuous and characteristic $x$ rays as shown in Fig. 40.6.1. In that figure the kinetic energy of the incident electrons is $35.0 \mathrm{keV}$. If the accelerating potential is increased to $50.0 \mathrm{keV}$, (a) what is the value of $\lambda_{\min }$, and (b) do the wavelengths of the $K_u$ and $K_\beta$ lines increase, decrease, or remain the same?

Salamat Ali
Salamat Ali
Numerade Educator
01:51

Problem 37

Show that a moving electron cannot spontaneously change into an $x$-ray photon in free space. A third body (atom or nucleus) must be present. Why is it needed?

Keshav Singh
Keshav Singh
Numerade Educator
02:16

Problem 38

Here are the $K_a$ wavelengths of a few elements:
$$
\begin{array}{lccc}
\hline \text { Element } & \lambda(\mathrm{pm}) & \text { Element } & \lambda(\mathrm{pm}) \\
\hline \mathrm{Ti} & 275 & \mathrm{Co} & 179 \\
\mathrm{~V} & 250 & \mathrm{Ni} & 166 \\
\mathrm{Cr} & 229 & \mathrm{Cu} & 154 \\
\mathrm{Mn} & 210 & \mathrm{Zn} & 143 \\
\mathrm{Fe} & 193 & \mathrm{Ga} & 134 \\
\hline
\end{array}
$$
Make a Moseley plot (like that in Fig. 40.6.4) from these data and verify that its slope agrees with the value given for $C$ in Module 40.6.

Salamat Ali
Salamat Ali
Numerade Educator
04:12

Problem 39

Calculate the ratio of the wavelength of the $K_a$ line for niobium $(\mathrm{Nb})$ to that for gallium (Ga). Take needed data from the periodic table of Appendix G.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:19

Problem 40

(a) From Eq. 40.6.4, what is the ratio of the photon energies due to $K_a$ transitions in two atoms whose atomic numbers are $Z$ and $Z^{\prime \prime}$ ? (b) What is this ratio for uranium and aluminum? (c) For uranium and lithium?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:51

Problem 41

The binding energies of $K$-shell and $L$-shell electrons in copper are 8.979 and $0.951 \mathrm{keV}$, respectively. If a $K_a$ x ray from copper is incident on a sodium chloride crystal and gives a firstorder Bragg reflection at an angle of $74.1^{\circ}$ measured relative to parallel planes of sodium atoms, what is the spacing between these parallel planes?

Keshav Singh
Keshav Singh
Numerade Educator
03:10

Problem 42

Fig. 40.6.1, calculate approximately the energy difference $E_L-E_M$ for molybdenum. Compare it with the value that may be obtained from Fig. 40.6.3.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:11

Problem 43

A tungsten ( $Z=74)$ target is bombarded by electrons in an x-ray tube. The $K, L$, and $M$ energy levels for tungsten (compare Fig. 40.6.3) have the energies $69.5,11.3$, and $2.30 \mathrm{keV}$, respectively. (a) What is the minimum value of the accelerating potential that will permit the production of the characteristic $K_a$ and $K_\rho$ lines of tungsten? (b) For this same accelerating potential, what is $\lambda_{\min }$ ? What are the (c) $K_\alpha$ and (d) $K_\beta$ wavelengths?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:11

Problem 44

A $20 \mathrm{keV}$ electron is brought to rest by colliding twice with target nuclei as in Fig. 40.6.2. (Assume the nuclei remain stationary.) The wavelength associated with the photon emitted in the second collision is $130 \mathrm{pm}$ greater than that associated with the photon emitted in the first collision. (a) What is the kinetic energy of the electron after the first collision? What are (b) the wavelength $\lambda_1$ and (c) the energy $E_1$ associated with the first photon? What are (d) $\lambda_2$ and (c) $E_2$ associated with the second photon?

Salamat Ali
Salamat Ali
Numerade Educator
05:44

Problem 45

X rays are produced in an $\mathrm{x}$-ray tube by electrons accelerated through an electric potential difference of $50.0 \mathrm{kV}$. Let $K_0$ be the kinetic energy of an electron at the end of the acceleration. The electron collides with a target nucleus (assume the nucleus remains stationary) and then has kinetic energy $K_1=0.500 K_0$. (a) What wavelength is associated with the photon that is emitted? The electron collides with another target nucleus (assume it, too, remains stationary) and then has kinetic
energy $K_2=0.500 K_1$. (b) What wavelength is associated with the photon that is emitted?

Alia Hamdan
Alia Hamdan
Numerade Educator
06:15

Problem 46

Determine the constant $C$ in Eq. 40.6.5 to five significant figures by finding $C$ in terms of the fundamental constants in Eq. 40.6.2 and then using data from Appendix B to evaluate those constants. Using this value of $C$ in Eq. 40.6.5, determine the theoretical energy $E_{\text {theory }}$ of the $K_a$ photon for the low-mass elements listed in the following table. The table includes the value $(\mathrm{eV})$ of the measured energy $E_{\exp }$ of the $K_{a r}$ photon for each listed element. The percentage deviation between $E_{\text {theory }}$ and $E_{\exp }$ can be calculated as
$$
\text { percentage deviation }=\frac{E_{\text {theory }}-E_{\text {exp }}}{E_{\text {exp }}} 100 .
$$

What is the percentage deviation for (a) $\mathrm{Li}$, (b) $\mathrm{Be}$, (c) $\mathrm{B}$, (d) $\mathrm{C}$, (e) $\mathrm{N}$, (f) $\mathrm{O}$, (g) $\mathrm{F}$, (h) $\mathrm{Ne}$, (i) $\mathrm{Na}$, and (j) $\mathrm{Mg}$ ?
$$
\begin{array}{lclc}
\hline \mathrm{Li} & 54.3 & \mathrm{O} & 524.9 \\
\mathrm{Be} & 108.5 & \mathrm{~F} & 676.8 \\
\mathrm{~B} & 183.3 & \mathrm{Ne} & 848.6 \\
\mathrm{C} & 277 & \mathrm{Na} & 1041 \\
\mathrm{~N} & 392.4 & \mathrm{Mg} & 1254 \\
\hline
\end{array}
$$
(There is actually more than one $K_a$ ray because of the splitting of the $L$ energy level, but that effect is negligible for the elements listed here.)

Keshav Singh
Keshav Singh
Numerade Educator
03:39

Problem 47

The active volume of a laser constructed of the semiconductor GaAlAs is only $200 \mu \mathrm{m}^3$ (smaller than a grain of sand), and yet the laser can continuously deliver $5.0 \mathrm{~mW}$ of power at a wavelength of $0.80 \mu \mathrm{m}$. At what rate does it generate photons?

Alia Hamdan
Alia Hamdan
Numerade Educator
04:07

Problem 48

A high-powered laser beam $(\lambda=600 \mathrm{~nm})$ with a beam diameter of $12 \mathrm{~cm}$ is aimed at the Moon, $3.8 \times 10^5 \mathrm{~km}$ distant. The beam spreads only because of diffraction. The angular location of the edge of the central diffraction disk (see Eq. 36.3.1) is given by
$$
\sin \theta=\frac{1.22 \lambda}{d}
$$
where $d$ is the diameter of the beam aperture. What is the diameter of the central diffraction disk on the Moon's surface?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:15

Problem 49

Assume that lasers are available whose wavelengths can be precisely "tuned" to anywhere in the visible range-that is, in the range $450 \mathrm{~nm}<\lambda<650 \mathrm{~nm}$. If every television channel occupies a bandwidth of $10 \mathrm{MHz}$, how many channels can be accommodated within this wavelength range?

Alia Hamdan
Alia Hamdan
Numerade Educator
02:00

Problem 50

A hypothetical atom has only two atomic energy levels, separated by $3.2 \mathrm{eV}$. Suppose that at a certain altitude in the atmosphere of a star there are $6.1 \times 10^{13} / \mathrm{cm}^3$ of these atoms in the higher-energy state and $2.5 \times 10^{15} / \mathrm{cm}^3$ in the lower-energy state. What is the temperature of the star's atmosphere at that altitude?

Salamat Ali
Salamat Ali
Numerade Educator
04:44

Problem 51

A hypothetical atom has energy levels uniformly separated by $1.2 \mathrm{eV}$. At a temperature of $2000 \mathrm{~K}$, what is the ratio of the number of atoms in the 13 th excited state to the number in the 11 th excited state?

Alia Hamdan
Alia Hamdan
Numerade Educator
02:54

Problem 52

(c) A laser emits at $424 \mathrm{~nm}$ in a single pulse that lasts $0.500 \mu \mathrm{s}$. The power of the pulse is $2.80 \mathrm{MW}$. If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the $0.500 \mu \mathrm{s}$, how many atoms contributed?

Salamat Ali
Salamat Ali
Numerade Educator
04:58

Problem 53

A helium-neon laser emits laser light at a wavelength of $632.8 \mathrm{~nm}$ and a power of $2.3 \mathrm{~mW}$. At what rate are photons emitted by this device?

Alia Hamdan
Alia Hamdan
Numerade Educator
02:17

Problem 54

A certain gas laser can emit light at wavelength $550 \mathrm{~nm}$, which involves population inversion between ground state and an excited state. At room temperature, how many moles of neon are needed to put 10 atoms in that excited state by thermal agitation?

Salamat Ali
Salamat Ali
Numerade Educator
05:39

Problem 55

A pulsed laser emits light at a wavelength of $694.4 \mathrm{~nm}$. The pulse duration is $12 \mathrm{ps}$, and the energy per pulse is $0.150 \mathrm{~J}$. (a) What is the length of the pulse? (b) How many photons are emitted in each pulse?

Alia Hamdan
Alia Hamdan
Numerade Educator
04:53

Problem 56

A population inversion for two energy levels can be described by assigning a negative Kelvin temperature to the system. What negative temperature would describe a system in which the population of the upper energy level exceeds that of the lower level by $10 \%$ and the energy difference between the two levels is $2.26 \mathrm{eV}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
07:07

Problem 57

A hypothetical atom has two energy levels, with a transition wavelength between them of $580 \mathrm{~nm}$. In a particular sample at $300 \mathrm{~K}, 4.0 \times 10^{20}$ such atoms are in the state of lower energy. (a) How many atoms are in the upper state, assuming conditions of thermal equilibrium? (b) Suppose, instead, that $3.0 \times 10^{20}$ of these atoms are "pumped" into the upper state by an external process, with $1.0 \times 10^{20}$ atoms remaining in the lower state. What is the maximum energy that could be released by the atoms in a single laser pulse if each atom jumps once between those two states (either via absorption or via stimulated emission)?

Alia Hamdan
Alia Hamdan
Numerade Educator
03:00

Problem 58

The mirrors in the laser of Fig. 40.7.4, which are separated by $8.0 \mathrm{~cm}$, form an optical cavity in which standing waves of laser light can be set up. Each standing wave has an integral number $n$ of half wavelengths in the $8.0 \mathrm{~cm}$ length, where $n$ is large and the waves differ slightly in wavelength. Near $\lambda=533 \mathrm{~nm}$, how far apart in wavelength are the standing waves?

Salamat Ali
Salamat Ali
Numerade Educator
01:39

Problem 59

(c. Figure 40.3 shows the energy levels of two types of atoms. Atoms $A$ are in one tube, and atoms $B$ are in another tube. The energies (relative to a ground-state energy of zero) are indicated; the average lifetime of atoms in each level is also indicated. All the atoms are initially pumped to levels higher than the levels shown in the figure. The atoms then drop down through the levels, and many become "stuck" on certain levels, leading to population inversion and lasing. The light emitted by $A$ illuminates $B$ and can cause stimulated emission of $B$. What is the energy per photon of that stimulated emission of $B$ ?
( FIGURE CAN'T COPY )

Keshav Singh
Keshav Singh
Numerade Educator
02:50

Problem 60

The beam from an argon laser (of wavelength $515 \mathrm{~nm}$ ) has a diameter $d$ of $3.00 \mathrm{~mm}$ and a continuous energy output rate of $5.00 \mathrm{~W}$. The beam is focused onto a diffuse surface by a lens whose focal length $f$ is $3.50 \mathrm{~cm}$. A diffraction pattern such as that of Fig. 36.3.1 is formed, the radius of the central disk being given by
$$
R=\frac{1.22 \mathrm{f \lambda}}{d}
$$
(see Eq. 36.3.1 and Fig. 36.3.5). The central disk can be shown to contain $84 \%$ of the incident power. (a) What is the radius of the central disk? (b) What is the average intensity (power per unit area) in the incident beam? (c) What is the average intensity in the central disk?

Salamat Ali
Salamat Ali
Numerade Educator
11:14

Problem 61

The active medium in a particular laser that generates laser light at a wavelength of $694 \mathrm{~nm}$ is $6.00 \mathrm{~cm}$ long and $1.00 \mathrm{~cm}$ in diameter. (a) Treat the medium as an optical resonance cavity analogous to a closed organ pipe. How many standing-wave nodes are there along the laser axis? (b) By what amount $\Delta f$ would the beam frequency have to shift to increase this number by one? (c) Show that $\Delta f$ is just the inverse of the travel time of laser light for one round trip back and forth along the laser axis. (d) What is the corresponding fractional frequency shift $\Delta f / f$ ? The appropriate index of refraction of the lasing medium (a ruby crystal) is 1.75 .

Alia Hamdan
Alia Hamdan
Numerade Educator
04:59

Problem 62

(6. Ruby lases at a wavelength of $694 \mathrm{~nm}$. A certain ruby crystal has $4.00 \times 10^{19} \mathrm{Cr}$ ions (which are the atoms that lase). The lasing transition is between the first excited state and the ground state, and the output is a light pulse lasting $2.00 \mu \mathrm{s}$. As the pulse begins, $60.0 \%$ of the $\mathrm{Cr}$ ions are in the first excited state and the rest are in the ground state. What is the average power emitted during the pulse? (Hint: Don't just ignore the ground-state ions.)

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:44

Problem 63

Figure 40.4 is an energy-level diagram for a fictitious three-dimensional infinite potential well that contains one electron. The number of degenerate states of the levels are indicated: "non" means nondegenerate (which includes the ground state) and "triple" means 3 states. If we put a total of 22 electrons in the well, what multiple of $h^2 / 8 m L^2$ gives the energy of the ground state of the 22 -electron system? Assume that the electrostatic forces between the electrons are negligible.
( FIGURE CAN'T COPY )

Alia Hamdan
Alia Hamdan
Numerade Educator
03:11

Problem 64

Martian $\mathrm{CO}_2$ laser. Where sunlight shines on the atmosphere of Mars, carbon dioxide molecules at an altitude of about $75 \mathrm{~km}$ undergo natural laser action. The energy levels involved in the action are shown in Fig. 40.5; population inversion occurs between energy levels $E_2$ and $E_1$. (a) What wavelength of sunlight excites the molecules in the las-
ing action? (b) At what wavelength does lasing occur? (c) In what region of the electromagnetic spectrum do the excitation and lasing wavelengths lie?
( FIGURE CAN'T COPY )

Keshav Singh
Keshav Singh
Numerade Educator
03:04

Problem 65

(co) Excited sodium atoms emit two closely spaced spectrum lines called the sodium doublet (Fig. 40.6) with wavelengths $588.995 \mathrm{~nm}$ and $589.592 \mathrm{~nm}$. (a) What is the difference in energy between the two upper energy levels $(n=3, \ell=1)$ ? (b) This energy difference occurs because the electron's spin magnetic moment can be oriented either parallel or antiparallel to the internal magnetic field associated with
the electron's orbital motion. Use your result in (a) to find the magnitude of this internal magnetic field.
( FIGURE CAN'T COPY )

Keshav Singh
Keshav Singh
Numerade Educator
01:48

Problem 66

Comet stimulated emission. When a comet approaches the Sun, the increased warmth evaporates water from the ice on the surface of the comet nucleus, producing a thin atmosphere of water vapor around the nucleus. Sunlight can then dissociate $\mathrm{H}_2 \mathrm{O}$ molecules in the vapor to $\mathrm{H}$ atoms and $\mathrm{OH}$ molecules. The sunlight can also excite the $\mathrm{OH}$ molecules to higher energy levels.

When the comet is still relatively far from the Sun, the sunlight causes equal excitation to the $E_2$ and $E_1$ levels (Fig. 40.7a). Hence, there is no population inversion between the two levels. However, as the comet approaches the Sun, the excitation to the $E_1$ level decreases and population inversion occurs. The reason has to do with one of the many wavelengths-said to be Fraunhofer lines-that are missing in sunlight because, as the light travels outward through the Sun's atmosphere, those particular wavelengths are absorbed by the atmosphere.

As a comet approaches the Sun, the Doppler effect due to the comet's speed relative to the Sun shifts the Fraunhofer lines in wavelength, apparently overlapping one of them with the wavelength required for excitation to the $E_1$ level in $\mathrm{OH}$ molecules. Population inversion then occurs in those molecules, and they radiate stimulated emission (Fig. 40.7b). For example, as comet Kouhoutek approached the Sun in December 1973 and January 1974, it radiated stimulated emission at about $1666 \mathrm{MHz}$ during mid-January. (a) What was the energy difference $E_2-E_1$ for that emission? (b) In what region of the electromagnetic spectrum was the emission?
A ( FIGURE CAN'T COPY )
B( FIGURE CAN'T COPY )

Keshav Singh
Keshav Singh
Numerade Educator
04:15

Problem 67

Show that the cutoff wavelength (in picometers) in the continuous $\mathrm{x}$-ray spectrum from any target is given by $\lambda_{\min }=1240 / \mathrm{V}$, where $V$ is the potential difference (in kilovolts) through which the electrons are accelerated before they strike the target.

Mayank Tripathi
Mayank Tripathi
Numerade Educator
03:20

Problem 68

By measuring the go-and-return time for a laser pulse to travel from an Earth-bound observatory to a reflector on the Moon, it is possible to measure the separation between these bodies. (a) What is the predicted value of this time? (b) The separation can be measured to a precision of about $15 \mathrm{~cm}$. To what uncertainty in travel time does this correspond? (c) If the laser beam forms a spot on the Moon $3 \mathrm{~km}$ in diameter, what is the angular divergence of the beam?

Salamat Ali
Salamat Ali
Numerade Educator
04:01

Problem 69

Can an incoming intercontinental ballistic missile be
destroyed by an intense laser beam? A beam of intensity $10^8$ $\mathrm{W} / \mathrm{m}^2$ would probably burn into and destroy a nonspinning missile in $1 \mathrm{~s}$. (a) If the laser had $5.0 \mathrm{MW}$ power, $3.0 \mu \mathrm{m}$ wavelength, and a $4.0 \mathrm{~m}$ beam diameter (a very powerful laser indeed), would it destroy a missile at a distance of $3000 \mathrm{~km}$ ? (b) If the wavelength could be changed, what maximum value would work? Use the equation for the central diffraction maximum as given by Eq. 36.3.1 ( $\sin \theta=1.22 \mathrm{\lambda} / d)$.

Keshav Singh
Keshav Singh
Numerade Educator
03:15

Problem 70

A molybdenum $(Z=42)$ target is bombarded with $35.0 \mathrm{keV}$ electrons and the x-ray spectrum of Fig. 40.6 .1 results. The $K_\beta$ and $K_a$ wavelengths are 63.0 and $71.0 \mathrm{pm}$, respectively. What photon energy corresponds to the (a) $K_\beta$ and (b) $K_\alpha$ radiation? The two radiations are to be filtered through one of the substances in the following table such that the substance absorbs the $K_p$ line more strongly than the $K_{a t}$ line. A substance will absorb radiation $x_1$ more strongly than it absorbs radiation $x_2$ if a photon of $x_1$ has enough energy to eject a $K$ electron from an atom of the substance but a photon of $x_2$ does not. The table gives the ionization energy of the $K$ electron in molybdenum and four other substances. Which substance in the table will serve (c) best and (d) second best as the filter?
$$
\begin{array}{llllll}
\hline & \mathrm{Zr} & \mathrm{Nb} & \mathrm{Mo} & \mathrm{Tc} & \mathrm{Ru} \\
\hline Z & 40 & 40 & 42 & 43 & 44 \\
E_K(\mathrm{keV}) & 18.00 & 18.99 & 20.00 & 21.04 & 22.12 \\
\hline
\end{array}
$$

Keshav Singh
Keshav Singh
Numerade Educator
01:50

Problem 71

An electron in a multielectron atom is known to have the quantum number $\ell=3$. What are its possible $n, m_{\ell}$, and $m_s$ quantum numbers?

Alia Hamdan
Alia Hamdan
Numerade Educator
01:18

Problem 72

Show that if the 63 electrons in an atom of europium were assigned to shells according to the "logical" sequence of quantum numbers, this element would be chemically similar to sodium.

Keshav Singh
Keshav Singh
Numerade Educator
06:04

Problem 73

Lasers can be used to generate pulses of light whose durations are as short as $10 \mathrm{fs}$. (a) How many wavelengths of light $(\lambda=500 \mathrm{~nm})$ are contained in such a pulse? (b) In
$$
\frac{10 \mathrm{fs}}{1 \mathrm{~s}}=\frac{1 \mathrm{~s}}{X}
$$
what is the missing quantity $X$ (in years)?

Alia Hamdan
Alia Hamdan
Numerade Educator
02:07

Problem 74

Show that $h=1.06 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=6.59 \times 10^{-16} \mathrm{eV} \cdot \mathrm{s}$.

Salamat Ali
Salamat Ali
Numerade Educator
02:16

Problem 75

Suppose that the electron had no spin and that the Pauli exclusion principle still held. Which, if any, of the present noble gases would remain in that category?

Alia Hamdan
Alia Hamdan
Numerade Educator
03:23

Problem 76

(A correspondence principle problem.) Estimate (a) the quantum number $\ell$ for the orbital motion of Earth around the Sun and (b) the number of allowed orientations of the plane of Earth's orbit. (c) Find $\theta_{\min }$, the half-angle of the smallest cone
that can be swept out by a perpendicular to Earth's orbit as Earth revolves around the Sun.

Keshav Singh
Keshav Singh
Numerade Educator
02:39

Problem 77

Knowing that the minimum $x$-ray wavelength produced by $40.0 \mathrm{keV}$ electrons striking a target is $31.1 \mathrm{pm}$, determine the Planck constant $h$.

Alia Hamdan
Alia Hamdan
Numerade Educator
01:37

Problem 78

Consider an atom with two closely spaced excited states $A$ and $B$. If the atom jumps to ground state from $A$ or from $B$, it emits a wavelength of $500 \mathrm{~nm}$ or $510 \mathrm{~nm}$, respectively. What is the energy difference between states $A$ and $B$ ?

Salamat Ali
Salamat Ali
Numerade Educator
08:36

Problem 79

In 1911, Ernest Rutherford modeled an atom as being a point of positive charge $Z e$ surrounded by a negative charge - $Z e$ uniformly distributed in a sphere of radius $R$ centered at the point. At distance $r$ within the sphere, the electric potential is
$$
V=\frac{Z e}{4 \pi \varepsilon_0}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^2}{2 R^3}\right)
$$
(a) From this formula, determine the magnitude of electric field for $0 \leq r \leq R$. What are the (b) electric field and (c) potential for $r \geq R$ ?

Alia Hamdan
Alia Hamdan
Numerade Educator