Chapter 38

(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 \(E_{1} ?\)

BIO Response of the Eve. The human eve 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 smount 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 photon of green light has a wavelength of 520 \(\mathrm{nm} .\) Find the photon's frequency, magnitude of momentum, and energy. Express the energy in both joules and electron volts.

BIO 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? (b) How many photons are in each pulse?

A 75 -W light source consumes 75 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 photon has momentum of magnitude \(8.24 \times\) \(10^{-28} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) (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?

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 \(1.45 \times 10^{15} \mathrm{Hz}\) . Express the answer in electron volts.

What would the minimum work function for a metal have to be for visible light \((380-750 \mathrm{nm})\) to eject photoelectrons?

When ultraviolet light with a wavelength of 400.0 \(\mathrm{nm}\) 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 \(\mathrm{eV}\) If light having a wavelength of 250 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.

When ultraviolet light with a wavelength of 254 nm falls on a clean copper surface, the stopping potential necessary to stop emission of photoelectrons is 0.181 \(\mathrm{V}\) (a) What is the photoelectric threshold wavelength for this copper surface? (b) What is the work function for this surface, and how does your calculated value compare with that given in Table 38.1\(?\)

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 \(\mathrm{kV}\) , what are the shortest-wavelength \(\mathrm{x}\) rays produced by the television? (Modern televisions contain shielding to stop these \(x\) rays.)

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?

(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 x-ray tube operated at 30.0 \(\mathrm{kV}\) ?

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 18.0 \(\mathrm{kV}\) . After emerging 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)?

\(\mathrm{X}\) rays with initial wavelength 0.0665 \(\mathrm{nm}\) 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 nm is Comptonscattered 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 \(\mathrm{nm}\) ; ( ) 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 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 \(\left(\phi=180^{\circ}\right)\) 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 \(0.900 \times 10^{-10} \mathrm{m}\) 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?

A photon with wavelength \(\lambda=0.1385 \mathrm{nm}\) 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 \(8.90 \times 10^{6} \mathrm{m} / \mathrm{s} ?\)

An electron and a positron are moving toward each other and each has speed 0.500\(c\) in the lab frame. (a) What is the kinetic energy of each particle? (b) The e' and e^ - meet head-on and andhilate. What is the energy of each photon that is produced? (c) What is the wavelength of each photon? How does the wavelength compare to the photon wavelength when the initial kinetic energy of the \(\mathrm{e}^{+}\) and \(\mathrm{e}^{-}\) is negligibly small (see Example 38.6\() ?\)

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

A horizontal beam of laser light of wavelength 585 \(\mathrm{nm}\) passes through a narrow slit that has width 0.0620 \(\mathrm{mm}\) . The intensity of the light is measured on a vertical screen that is 2.00 \(\mathrm{m}\) . 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.

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 MHz? (e) Light from a firelly 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 -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 directions?

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 \(\mathrm{cV}\) . 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 eachsecond 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)?

Cp Blo Removing Vascular Lesions. 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 birthmark. 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 \(\left(4190 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, 256 \times 10^{6} \mathrm{J} / \mathrm{kg}\right) .\) Suppose that each pulse must remove 2.0\(\mu \mathrm{g}\) 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 zinc, 4.3 \(\mathrm{eV}\) . (a) What is the threshold wavelength for each metal surface? (b) Which of these metals could not emit photoelectrons when irradiated with visible light \((380-750 \mathrm{nm}) ?\)

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?

CP A photon with wavelength \(\lambda=0.1050 \mathrm{nm}\) is incident on an electron that is initially at rest. If the photon scatters at an angle of \(60.0^{\circ}\) from its original direction, what are the magnitude and direction of the linear momentum of the electron just after the collision with the photon?

CP 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 \(J / k g \cdot K ),\) 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 wavelength 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 wavelength 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 \(\theta=0^{\circ}\) , for which the total shifts are the same? (c) Use the result of part (a) to calculate the total wavelength 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).

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?

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 \(5.10 \times 10^{-3}\) 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) Calculate the maximum increase in photon wavelength that can occur during Compton scattering, (b) What is the energy (in electron volts) of the lowest- energy \(x\) -ray photon for which Compton scattering could result in doubling the original wavelength?

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 \(m c^{2} )\) 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 \(\lambda^{\prime}\) of the scattered photon. Show that if \(E>m c^{2},\) where \(m\) is the rest mass of the electron, your result reduces to $$\lambda^{\prime}=\frac{h c}{E}\left(1+\frac{m^{2} c^{4} \lambda}{4 h c E}\right)$$ (b) A beam of infrared radiation from a \(\mathrm{CO}_{2}\) laser \((\lambda=10.6 \mu \mathrm{m})\) collides head-on with a beam of electrons, each of total energy \(E=10.0 \mathrm{GeV}\left(1 \mathrm{GeV}=10^{9} \mathrm{eV}\right) .\) Calculate the wavelength \(\lambda^{\prime}\) of the scattered photons, assuming a \(180^{\circ}\) scattering angle. (c) What kind of scattered photons are these (infrared, microwave, ultraviolet, ett.. \(.\) Can you think of an application of this effect?