Chapter 38

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.

The human eye is most sensitive to green light of wavelength 505 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\(^{-12}\) g would move if it had that much energy.

A 75-W light source consumes 75 W of electrical power. Assume all this energy goes into emitted light of wavelength 600 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 laser used to weld detached retinas emits light with a wavelength of 652 nm in pulses that are 20.0 ms in duration. The average power during each pulse is 0.600 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?

A photon has momentum of magnitude 8.24 \(\times\) 10\(^{-28}\) kg \(\bullet\) m/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?

The photoelectric threshold wavelength of a tungsten surface is 272 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 (380-750 nm) to eject photoelectrons?

When ultraviolet light with a wavelength of 400.0 nm falls on a certain metal surface, the maximum kinetic energy of the emitted photoelectrons is measured to be 1.10 eV. What is the maximum kinetic energy of the photoelectrons when light of wavelength 300.0 nm falls on the same surface?

The photoelectric work function of potassium is 2.3 eV. If light that has a wavelength of 190 nm falls on potassium, find (a) the stopping potential in volts; (b) the kinetic energy, in electron volts, of the most energetic electrons ejected; (c) the speed of these electrons.

The cathode-ray tubes that generated the picture in early color televisions were sources of x rays. If the acceleration voltage in a television tube is 15.0 kV, what are the shortest-wavelength x rays produced by the television?

Protons are accelerated from rest by a potential difference of 4.00 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 wavelength if 4.00-keV electrons are used instead? Why do x-ray tubes 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 nm? (b) What is the shortest wavelength produced in an x-ray tube operated at 30.0 kV?

An x ray with a wavelength of 0.100 nm collides with an electron that is initially at rest. The x ray's final wavelength is 0.110 nm. What is the final kinetic energy of the electron?

X rays are produced in a tube operating at 24.0 kV. After emerging from the tube, x rays with the minimum wavelength produced strike a target and undergo Compton scattering 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 photon with wavelength \(\lambda\) = 0.1385 nm scatters from an electron that is initially at rest. What must be the angle between the direction of propagation of the incident and scattered photons if the speed of the electron immediately after the collision is 8.90 \(\times\) 10\(^6\) m/s?

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 wavelength 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 (\(\phi = 180^\circ\)) from a free proton that is initially at rest. What must the wavelength of the incident photon be if it is to undergo a 10.0\(\%\) change in wavelength as a result of the scattering?

X rays with an initial wavelength of 0.900 \(\times\) 10\(^{-10}\) m undergo Compton scattering. For what scattering angle is the wavelength of the scattered x rays greater by 1.0\(\%\) than that of the incident x rays?

An ultrashort pulse has a duration of 9.00 fs and produces light at a wavelength of 556 nm. What are the momentum and momentum uncertainty of a single photon in the pulse?

A horizontal beam of laser light of wavelength 585 nm passes through a narrow slit that has width 0.0620 mm. The intensity of the light is measured on a vertical screen that is 2.00 m from the slit. (a) What is the minimum uncertainty in the vertical component of the momentum of each photon in the beam after the photon has passed through the slit? (b) Use the result of part (a) to estimate the width of the central diffraction maximum that is observed on the screen.

A laser produces light of wavelength 625 nm in an ultrashort pulse. What is the minimum duration of the pulse if the minimum uncertainty in the energy of the photons is 1.0\(\%\)?

(a) If the average frequency emitted by a 120-W light bulb is 5.00 \(\times\) 10\(^{14}\) 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 cm\(^2\) per second if the light is emitted uniformly in all directions?

A pulsed dye laser emits light of wavelength 585 nm in 450-\(\mu\)s pulses. Because this wavelength 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 birthmarks. 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 (4190 J / kg \(\bullet\) K, 2.256 \(\times\) 10\(^6\) J / kg). Suppose that each pulse must remove 2.0 mg of blood by evaporating it, starting at 33\(^\circ\)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?

A 2.50-W beam of light of wavelength 124 nm falls on a metal surface. You observe that the maximum kinetic energy of the ejected electrons is 4.16 eV. Assume that each photon in the beam ejects a photoelectron. (a) What is the work function (in electron volts) of this metal? (b) How many photoelectrons 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)?

An incident x-ray photon of wavelength 0.0900 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 photon with wavelength \(\lambda\) = 0.0980 nm is incident on an electron that is initially at rest. If the photon scatters in the backward direction, what is the magnitude of the linear momentum of the electron just after the collision with the photon?

A photon of wavelength 4.50 pm scatters from a free electron that is initially at rest. (a) For \(\phi = 90.0^\circ\), what is the kinetic energy of the electron immediately after the collision with the photon? What is the ratio of this kinetic energy to the rest energy of the electron? (b) What is the speed of the electron immediately after the collision? (c) What is the magnitude of the momentum of the electron immediately after the collision? What is the ratio of this momentum value to the nonrelativistic expression \(mv\)?

Nuclear fusion reactions at the center of the sun produce gamma-ray photons with energies of about 1 MeV (10\(^6\) eV). By contrast, what we see emanating from the sun's surface are visiblelight photons with wavelengths of about 500 nm. A simple model that explains this difference in wavelength is that a photon undergoes Compton scattering many times-in fact, about 10\(^{26}\) times, as suggested by models of the solar interior-as it travels from the center of the sun to its surface. (a) Estimate the increase in wavelength of a photon in an average Compton-scattering event. (b) Find the angle in degrees through which the photon is scattered in the scattering event described in part (a). (\(Hint\): A useful approximation is cos \(\phi \approx 1 - \phi^2 /2\), which is valid for \(\phi \ll\) 1. Note that \(\phi\) is in radians in this expression.) (c) It is estimated that a photon takes about 10\(^6\) years to travel from the core to the surface of the sun. Find the average distance that light can travel within the interior of the sun without being scattered. (This distance is roughly equivalent to how far you could see if you were inside the sun and could survive the extreme temperatures there. As your answer shows, the interior of the sun is \(very\) opaque.)

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 x rays, at what rate is energy being delivered to the target? (b) If the target has mass \(m\) and specific heat \(c\) (in J/kg \(\bullet\) 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 kV and 60.0 mA that converts 1.0\(\%\) of the electric power into x rays. Assume that the 0.250-kg target is made of lead (\(c\) = 130 J/kg \(\bullet\) K). (d) What must the physical properties of a practical target material be? What would be some suitable target elements?

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 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?

An x-ray photon is scattered from a free electron (mass \(m\)) at rest. The wavelength of the scattered photon is \(\lambda'\), and the final speed of the struck electron is \(v\). (a) What was the initial wavelength \(\lambda\) of the photon? Express your answer in terms of \(\lambda\), \(v\), 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 \(\lambda\), \(\lambda'\), and \(m\). (c) Evaluate your results in parts (a) and (b) for a wavelength of 5.10 \(\times\) 10\(^{-3}\) nm for the scattered photon and a final electron speed of 1.80 \(\times\) 10\(^8\) m/s. Give \(\phi\) in degrees.

In developing night-vision equipment, you need to measure the work function for a metal surface, so you perform a photoelectric-effect experiment. You measure the stopping potential \(V_0\) as a function of the wavelength \(\lambda\) of the light that is incident on the surface. You get the results in the table. In your analysis, you use \(c\) = 2.998 \(\times\) 10\(^8\) m/s and \(e\) = 1.602 \(\times\) 10\(^{-19}\) C, which are values obtained in other experiments. (a) Select a way to plot your results so that the data points fall close to a straight line. Using that plot, find the slope and y-intercept of the best-fit straight line to the data. (b) Use the results of part (a) to calculate Planck's constant \(h\) (as a test of your data) and the work function (in eV) of the surface. (c) What is the longest wavelength of light that will produce photoelectrons from this surface? (d) What wavelength of light is required to produce photoelectrons with kinetic energy 10.0 eV?

Consider Compton scattering of a photon by a \(moving\) electron. Before the collision the photon has wavelength \(\lambda\) and is moving in the +\(x\)-direction, and the electron is moving in the -\(x\)-direction with total energy \(E\) (including its rest energy \(mc^2\)). The photon and electron collide head-on. After the collision, both are moving in the -\(x\)-direction (that is, the photon has been scattered by 180\(^\circ\)). (a) Derive an expression for the wavelength \(\lambda'\) of the scattered photon. Show that if \(E \gg mc^2\), where m is the rest mass of the electron, your result reduces to $$\lambda' = {hc \over E} (1 + {m^2c^4\lambda \over 4hcE}) $$ (b) A beam of infrared radiation from a CO\(_2\) laser (\(\lambda = 10.6 \mu{m}\)) collides head-on with a beam of electrons, each of total energy \(E\) = 10.0 GeV (1 GeV = 10\(^9\) eV). Calculate the wavelength \(\lambda'\) of the scattered photons, assuming a 180\(^\circ\) scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, etc.)? Can you think of an application of this effect?

Higher-energy photons might be desirable for the treatment of certain tumors. Which of these actions would generate higher-energy photons in this linear accelerator? (a) Increasing the number of electrons that hit the tungsten target; (b) accelerating the electrons through a higher potential difference; (c) both (a) and (b); (d) none of these.