Chapter 28

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 eV) 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.

An excited nucleus emits a gamma-ray photon with an energy of \(2.45 \mathrm{MeV}\). (a) What is the photon's energy in joules? (b) What is the photon's frequency? (c) What is the photon's wavelength? (d) How does this wavelength compare with a typical nuclear diameter of \(10^{-14} \mathrm{~m} ?\)

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 expended during each pulse is \(0.600 \mathrm{~W}\). (a) How much energy is in each pulse, in joules? In electronvolts? (b) What is the energy of one photon in joules? In electronvolts? (c) How many photons are in each pulse?

A radio station broadcasts at a frequency of \(92.0 \mathrm{MHz}\) with a power output of \(50.0 \mathrm{~kW}\). (a) What is the energy of each emitted photon, in joules and in electronvolts? (b) How many photons are emitted per second?

The predominant wavelength emitted by an ultraviolet lamp is \(248 \mathrm{nm} .\) If the total power emitted at this wavelength is \(12.0 \mathrm{~W}\), how many photons are emitted per second?

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 electronvolts. (b) What is the wavelength of this photon? In what region of the electromagnetic spectrum does it lie?

In the photoelectric effect, what is the relationship between the threshold frequency \(f_{0}\) and the work function \(\phi ?\)

The photoelectric threshold wavelength of a tungsten surface is \(272 \mathrm{nm} .\) (a) What are the threshold frequency and work function (in eV) of this tungsten? (b) Calculate the maximum kinetic energy (in eV) of the electrons ejected from this tungsten surface by ultraviolet radiation of frequency \(1.45 \times 10^{15} \mathrm{~Hz}\)

What is the maximum work function that a metal can have and still produce photoelectrons with visible light (having wavelengths between \(400 \mathrm{nm}\) and \(700 \mathrm{nm}\) )?

When ultraviolet light with a wavelength of \(254 \mathrm{nm}\) falls upon a clean metal surface, the stopping potential necessary to terminate the emission of photoelectrons is \(0.181 \mathrm{~V}\). (a) What is the photoelectric threshold wavelength for this metal? (b) What is the work function for the metal?

In a photoelectric-effect experiment, it is found that no current flows unless the incident light has a wavelength shorter than \(289 \mathrm{nm}\). (a) What is the work function of the metal surface? (b) What stopping potential will be needed to halt the current if light of \(225 \mathrm{nm}\) falls on the surface?

Light with a wavelength range of \(145-295 \mathrm{nm}\) shines on a silicon surface in a photoelectric effect apparatus, and a reversing potential of \(3.50 \mathrm{~V}\) is applied to the resulting photoelectrons. (a) What is the longest wavelength of the light that will eject electrons from the silicon surface? (b) With what maximum kinetic energy will electrons reach the anode?

(a) How much energy is needed to ionize a hydrogen atom that is in the \(n=4\) state? (b) What would be the wavelength of a photon emitted by a hydrogen atom in a transition from the \(n=4\) state to the \(n=2\) state?

For a hydrogen atom in the ground state, determine (a) the circumference of the electron orbit, (b) the speed of the electron, (c) the total energy of the electron, and (d) the minimum energy required to remove the electron completely from the atom.

Use the Bohr model for the following calculations: (a) What is the speed of the electron in a hydrogen atom in the \(n=1,2,\) and 3 levels? (b) Calculate the radii of each of these levels. (c) Find the total energy (in eV) of the atom in each of these levels.

An electron in an excited state of hydrogen makes a transition from the \(n=5\) level to the \(n=2\) level. (a) Does the atom emit or absorb a photon during this process? How do you know? (b) Calculate the wavelength of the photon involved in the transition.

A hydrogen atom initially in the ground state absorbs a photon, which excites it to the \(n=4\) state. Determine the wavelength and frequency of the photon.

Light of wavelength \(59 \mathrm{nm}\) ionizes a hydrogen atom that was originally in its ground state. What is the kinetic energy of the ejected electron?

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 \(\mathrm{Be}^{3+}\) ? How does this compare with the ground-level energy of the hydrogen atom? (b) What is the ionization energy of \(\mathrm{Be}^{3+}\) ? How does this compare with the ionization energy of the hydrogen atom? (c) For the hydrogen atom, the wavelength of the photon emitted in the transition \(n=2\) to \(n=1\) is \(122 \mathrm{nm} .\) (See Example \(28.6 .)\) What is the wavelength of the photon emitted when a \(\mathrm{Be}^{3+}\) ion undergoes this transition? (d) For a given value of \(n\), how does the radius of an orbit in \(\mathrm{Be}^{3+}\) compare with that for hydrogen?

The diode laser keychain you use to entertain your cat has a wavelength of \(645 \mathrm{nm} .\) If the laser emits \(4.50 \times 10^{17}\) photons during a \(30.0 \mathrm{~s}\) feline play session, what is its average power output?

Laser surgery. Using a mixture of \(\mathrm{CO}_{2}, \mathrm{~N}_{2},\) and sometimes He, \(\mathrm{CO}_{2}\) lasers emit a wavelength of \(10.6 \mu \mathrm{m}\). At power outputs of \(0.100 \mathrm{~kW},\) such lasers are used for surgery. How many photons per second does a \(\mathrm{CO}_{2}\) laser deliver to the tissue during its use in an operation?

PRK surgery. Photorefractive keratectomy (PRK) is a laserbased surgery process that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers \(0.25 \mu \mathrm{m}\) thick in pulses lasting \(12.0 \mathrm{~ns}\) with 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 lie? (b) What is the energy of a single photon? (c) If a \(1.50 \mathrm{~mW}\) beam is used, how many photons are delivered to the lens in each pulse?

Removing birthmarks. Pulsed dye lasers emit light of wavelength \(585 \mathrm{nm}\) in \(0.45 \mathrm{~ms}\) pulses to remove skin blemishes such as birthmarks. The beam is usually focused onto a circular spot \(5.0 \mathrm{~mm}\) in diameter. Suppose that the output of one such laser is \(20.0 \mathrm{~W}\). (a) What is the energy of each photon, in eV? (b) How many photons per square millimeter are delivered to the blemish during each pulse?

(a) What is the minimum potential difference between the filament and the target of an X-ray tube if the tube is to accelerate electrons 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 \(30.0 \mathrm{kV} ?\) (c) Would the answers to parts (a) and (b) be different if the tube accelerated protons instead of electrons? Why or why not?

An X-ray tube is operated at \(50 \mathrm{kV}\). The shortest wavelength photons from this tube are used in a Compton scattering experiment. One of these photons strikes a free electron and is scattered directly back at an angle of \(180^{\circ} .\) (a) What is the wavelength of the incident photon that comes from the X-ray tube? (b) What is the wavelength of the scattered photon?

An X-ray with a wavelength of \(0.100 \mathrm{nm}\) collides with an electron that is initially at rest. The X-ray's final wavelength is \(0.110 \mathrm{nm}\). What is the final kinetic energy of the electron?

If a photon of wavelength \(0.04250 \mathrm{nm}\) 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?), and (d) the energy gained by the electron.

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?

An incident X-ray photon is scattered from a free electron that is initially at rest. The photon is scattered straight back at an angle of \(180^{\circ}\) from its initial direction. The wavelength of the scattered photon is \(0.0830 \mathrm{nm}\). (a) What is the wavelength of the incident photon? (b) What is the magnitude of the momentum of the electron after the collision? (c) What is the kinetic energy of the electron after the collision?

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 wavelength if \(4.00 \mathrm{keV}\) electrons are used instead? Why do X-ray tubes use electrons rather than protons to produce X-rays?

(a) An electron moves with a speed of \(4.70 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

I How fast would an electron have to move so that its de Broglie wavelength would be \(1.00 \mathrm{~mm} ?\)

A proton and an electron are each given a kinetic energy of \(1000 \mathrm{eV}\) by accelerating them through a potential difference of \(1000 \mathrm{~V}\). Find the ratio of the proton's de Broglie wavelength to that of the electron.

In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) the \(n=1\) level and (b) the \(n=4\) level? In each case, compare the de Broglie wavelength to the circumference \(2 \pi r_{n}\) of the orbit.

(a) What is the de Broglie wavelength of an electron accelerated through \(800 \mathrm{~V} ?\) (b) What is the de Broglie wavelength of a proton accelerated through the same potential difference?

Find the wavelengths of a photon and an electron that have the same energy of \(25 \mathrm{eV}\). (The energy of the electron is its kinetic energy.)

(a) The uncertainty in the \(x\) component of the position of a proton is \(2.0 \times 10^{-12} \mathrm{~m} .\) What is the minimum uncertainty in the \(x\) component of the velocity of the proton? (b) The uncertainty in the \(x\) component of the velocity of an electron is \(0.250 \mathrm{~m} / \mathrm{s}\). What is the minimum uncertainty in the \(x\) coordinate of the electron?

A certain atom has an energy level \(3.50 \mathrm{eV}\) above the ground state. When excited to this state, it remains there for \(4.0 \mu \mathrm{s},\) on the average, before emitting a photon and returning to the ground state. (a) What is the energy of the photon? What is its wavelength? (b) What is the smallest possible uncertainty in energy of the photon?

A pesky \(1.5 \mathrm{mg}\) mosquito is annoying you as you attempt to study physics in your room, which is \(5.0 \mathrm{~m}\) wide and \(2.5 \mathrm{~m}\) high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit. (a) What is the maximum uncertainty in the horizontal position of the mosquito? (b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

Suppose that the uncertainty in position of an electron is equal to the radius of the \(n=1\) Bohr orbit, about \(0.5 \times 10^{-10} \mathrm{~m}\). Calculate the minimum uncertainty in the corresponding momentum component, and compare this with the magnitude of the momentum of the electron in the \(n=1\) Bohr orbit.

(a) What accelerating potential is needed to produce electrons of wavelength \(5.00 \mathrm{nm} ?\) (b) What would be the energy of photons having the same wavelength as these electrons? (c) What would be the wavelength of photons having the same energy as the electrons in part (a)?

(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength \(0.0600 \mathrm{nm} ?\) (b) If protons are used instead of electrons, what accelerating voltage is needed to produce protons with wavelength \(0.0600 \mathrm{nm} ?\) (Hint: In each case, the initial kinetic energy is negligible.)

Structure of a virus. To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but \(50 \mathrm{nm}\) is not unusual. Suppose you want to study such a virus, using a wave of wavelength \(5.00 \mathrm{nm} .\) (a) If you use light of this wavelength, what would be the energy (in eV) of a single photon? (b) If you use an electron of this wavelength, what would be its kinetic energy (in eV)? Is it now clear why matter waves (such as in the electron microscope) are often preferable to electromagnetic waves for studying microscopic objects?

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 an electron. (a) What is the work function (in electronvolts) 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 unknown element has a spectrum for absorption from its ground level with lines at \(2.0,5.0,\) and \(9.0 \mathrm{eV}\). Its ionization energy is \(10.0 \mathrm{eV}\). (a) Draw an energy-level diagram for this element. (b) If a \(9.0 \mathrm{eV}\) photon is absorbed, what energies can the subsequently emitted photons have?

(a) What is the least amount of energy, in electronvolts, that must be given to a hydrogen atom that is initially in its ground level so that it can emit the \(\mathrm{H}_{\alpha}\) 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 emitted photon in each case.

A specimen of the microorganism Gastropus hyptopus measures \(0.0020 \mathrm{~cm}\) in length and can swim at a speed of 2.9 times its body length per second. The tiny animal has a mass of roughly \(8.0 \times 10^{-12} \mathrm{~kg} .\) (a) Calculate the de Broglie wavelength of this organism when it is swimming at top speed. (b) Calculate the kinetic energy of the organism (in eV) when it is swimming at top speed.

A photon with a wavelength of \(0.1800 \mathrm{nm}\) is Compton scattered through an angle of \(180^{\circ} .\) (a) What is the wavelength of the scattered photon? (b) How much energy is given to the electron? (c) What is the recoil speed of the electron? Is it necessary to use the relativistic kinetic-energy relationship?

(a) Calculate the maximum increase in photon wavelength that can occur during Compton scattering. (b) What is the energy (in electronvolts) of the smallest- energy X-ray photon for which Compton scattering could result in doubling the original wavelength?

A photon with wavelength \(\lambda\) collides with a free electron. The scattered photon has a wavelength of \(\lambda^{\prime}=2 \lambda .\) If the incident photon has a wavelength of \(\lambda=1.0 \times 10^{-12} \mathrm{~m},\) through what angle is it Compton scattered?