Chapter 38

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.

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.

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?

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 \(10^{-14} \mathrm{m} ?\)

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.

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

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.

(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 \(E_{1} ?\)

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.

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?

(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?

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?

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?

(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).

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?

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.

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?

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.)

(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?

(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?

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

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?

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?

(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 \(\mathrm{kV}\) ?

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.)

\(X\) 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)?

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?

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) \(0.0521 \mathrm{nm} ;(\mathrm{c}) 0.0500 \mathrm{nm} ?\)

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.

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?

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

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

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?

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?

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.

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.

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.

(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?

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)?

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?

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 \((400-700 \mathrm{nm}) ?\)

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?

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?

(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.

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?

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?

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?

(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).

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.

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?