A daredevils favorite trick is to step out of a 16th-story

Rank the wavelengths of the following quantum particles from the largest to the smallest. If any have equal wavelengths, display the equality in your ranking. (a) a photon with energy 3 eV (b) an electron with kinetic energy 3 eV (c) a proton with kinetic energy 3 eV (d) a photon with energy 0.3 eV (e) an electron with momentum 3 eV/c

An x-ray photon is scattered by an originally stationary electron. Relative to the frequency of the incident photon, is the frequency of the scattered photon (a) lower, (b) higher, or (c) unchanged?

In a Compton scattering experiment, a photon of energy E is scattered from an electron at rest. After the scattering event occurs, which of the following statements is true? (a) The frequency of the photon is greater than E/h. (b) The energy of the photon is less than E. (c) The wavelength of the photon is less than hc/E. (d) The momentum of the photon increases. (e) None of those statements is true.

In a certain experiment, a filament in an evacuated lightbulb carries a current I1 and you measure the spectrum of light emitted by the filament, which behaves as a black body at temperature T1. The wavelength emitted with highest intensity (symbolized by lmax) has the value l1. You then increase the potential difference across the filament by a factor of 8, and the current increases by a factor of 2. (i) After this change, what is the new value of the temperature of the filament? (a) 16T1 (b) 8T1 (c) 4T1 (d) 2T1 (e) still T1 (ii) What is the new value of the wavelength emitted with highest intensity? (a) 4l1 (b) 2l1 (c) l1 (d) 1 2l1 (e) 1 4l1

Which of the following statements are true according to the uncertainty principle? More than one statement may be correct. (a) It is impossible to simultaneously determine both the position and the momentum of a particle along the same axis with arbitrary accuracy. (b) It is impossible to simultaneously determine both the energy and momentum of a particle with arbitrary accuracy. (c) It is impossible to determine a particles energy with arbitrary accuracy in a finite amount of time. (d) It is impossible to measure the position of a particle with arbitrary accuracy in a finite amount of time. (e) It is impossible to simultaneously measure both the energy and position of a particle with arbitrary accuracy.

A monochromatic light beam is incident on a barium target that has a work function of 2.50 eV. If a potential difference of 1.00 V is required to turn back all the ejected electrons, what is the wavelength of the light beam? (a)355nm (b)497 nm (c) 744 nm (d) 1.42 pm (e) none of those answers

Which of the following is most likely to cause sunburn by delivering more energy to individual molecules in skin cells? (a) infrared light (b) visible light (c) ultraviolet light (d) microwaves (e) Choices (a) through (d) are equally likely

Which of the following phenomena most clearly demonstrates the wave nature of electrons? (a) the photoelectric effect (b) blackbody radiation (c) the Compton effect (d) diffraction of electrons by crystals (e) none of those answers

What is the de Broglie wavelength of an electron accelerated from rest through a potential difference of 50.0 V? (a) 0.100 nm (b) 0.139 nm (c) 0.174 nm (d) 0.834 nm (e) none of those answers

A proton, an electron, and a helium nucleus all move at speed v. Rank their de Broglie wavelengths from largest to smallest.

Consider (a) an electron, (b) a photon, and (c) a proton, all moving in vacuum. Choose all correct answers for each question. (i) Which of the three possess rest energy? (ii) Which have charge? (iii) Which carry energy? (iv) Which carry momentum? (v) Which move at the speed of light? (vi) Which have a wavelength characterizing their motion?

An electron and a proton, moving in opposite directions, are accelerated from rest through the same potential difference. Which particle has the longer wavelength? (a) The electron does. (b) The proton does. (c) Both are the same. (d) Neither has a wavelength.

Which of the following phenomena most clearly demonstrates the particle nature of light? (a) diffraction (b) the photoelectric effect (c) polarization (d) interference (e) refraction

Both an electron and a proton are accelerated to the same speed, and the experimental uncertainty in the speed is the same for the two particles. The positions of the two particles are also measured. Is the minimum possible uncertainty in the electrons position (a) less than the minimum possible uncertainty in the protons position, (b) the same as that for the proton, (c) more than that for the proton, or (d) impossible to tell from the given information?

Iridescence is the phenomenon that gives shining colors to the feathers of peacocks, hummingbirds (see page 1134), resplendent quetzals, and even ducks and grackles. Without pigments, it colors Morpho butterflies (Fig. CQ40.15), Urania moths, some beetles and flies, rainbow trout, and mother-of-pearl in abalone shells. Iridescent colors change as you turn an object. They are produced by a wide variety of intricate structures in different species. Problem 64 in Chapter 38 describes the structures that produce iridescence in a peacock feather. These structures were all unknown until the invention of the electron microscope. Explain why light microscopes cannot reveal them.

In describing the passage of electrons through a slit and arriving at a screen, physicist Richard Feynman said that electrons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of the waves would be. It is in this sense that the electron behaves sometimes like a particle and sometimes like a wave. Elaborate on this point in your own words. For further discussion, see R. Feynman, The Character of Physical Law (Cambridge, MA: MIT Press, 1980), chap. 6.

The classical model of blackbody radiation given by the RayleighJeans law has two major flaws. (a) Identify the flaws and (b) explain how Plancks law deals with them.

The work function for zinc is 4.31 eV. (a) Find the cutoff wavelength for zinc. (b) What is the lowest frequency of light incident on zinc that releases photoelectrons from its surface? (c) If photons of energy 5.50 eV are incident on zinc, what is the maximum kinetic energy of the ejected photoelectrons?

Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp (l 5 546.1 nm) is used, a stopping potential of 0.376 V reduces the photocurrent to zero. (a) Based on this measurement, what is the work function for this metal? (b) What stopping potential would be observed when using the yellow light from a helium discharge tube (l 5 587.5 nm)?

Lithium, beryllium, and mercury have work functions of 2.30 eV, 3.90 eV, and 4.50 eV, respectively. Light with a wavelength of 400 nm is incident on each of these metals. (a) Determine which of these metals exhibit the photoelectric effect for this incident light. Explain your reasoning. (b) Find the maximum kinetic energy for the photoelectrons in each case.

Electrons are ejected from a metallic surface with speeds of up to 4.60 3 105 m/s when light with a wavelength of 625 nm is used. (a) What is the work function of the surface? (b) What is the cutoff frequency for this surface?

From the scattering of sunlight, J. J. Thomson calculated the classical radius of the electron as having the value 2.82 3 10215 m. Sunlight with an intensity of 500 W/m2 falls on a disk with this radius. Assume light is a classical wave and the light striking the disk is completely absorbed. (a) Calculate the time interval required to accumulate 1.00 eV of energy. (b) Explain how your result for part (a) compares with the observation that photoelectrons are emitted promptly (within 1029 s).

Review. An isolated copper sphere of radius 5.00 cm, initially uncharged, is illuminated by ultraviolet light of wavelength 200 nm. The work function for copper is 4.70 eV. What charge does the photoelectric effect induce on the sphere?

The work function for platinum is 6.35 eV. Ultraviolet light of wavelength 150 nm is incident on the clean surface of a platinum sample. We wish to predict the stopping voltage we will need for electrons ejected from the surface. (a) What is the photon energy of the ultraviolet light? (b) How do you know that these photons will eject electrons from platinum? (c) What is the maximum kinetic energy of the ejected photoelectrons? (d) What stopping voltage would be required to arrest the current of photoelectrons?

X-rays are scattered from a target at an angle of 55.0 with the direction of the incident beam. Find the wavelength shift of the scattered x-rays.

A photon having wavelength l scatters off a free electron at A (Fig. P40.26), producing a second photon having wavelength l9. This photon then scatters off another free electron at B, producing a third photon having wavelength l0 and moving in a direction directly opposite the original photon as shown in the figure. Determine the value of Dl 5 l0 2 l.

A 0.110-nm photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backward. Find the momentum and the kinetic energy of the electron.

X-rays with a wavelength of 120.0 pm undergo Compton scattering. (a) Find the wavelengths of the photons scattered at angles of 30.0, 60.0, 90.0, 120, 150, and 180. (b) Find the energy of the scattered electron in each case. (c) Which of the scattering angles provides the electron with the greatest energy? Explain whether you could answer this question without doing any calculations

A 0.001 60-nm photon scatters from a free electron. For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered photon?

After a 0.800-nm x-ray photon scatters from a free electron, the electron recoils at 1.40 3 106 m/s. (a) What is the Compton shift in the photons wavelength? (b) Through what angle is the photon scattered?

A photon having energy E0 5 0.880 MeV is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered photon as shown in Figure P40.31. (a) Determine the scattering angle of the photon and the electron. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron.

A photon having energy E0 is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered photon as shown in Figure P40.31. (a) Determine the angle u. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron.

X-rays having an energy of 300 keV undergo Compton scattering from a target. The scattered rays are detected at 37.0 relative to the incident rays. Find (a) the Compton shift at this angle, (b) the energy of the scattered x-ray, and (c) the energy of the recoiling electron.

In a Compton scattering experiment, a photon is scattered through an angle of 90.0 and the electron is set into motion in a direction at an angle of 20.0 to the original direction of the photon. (a) Explain how this information is sufficient to determine uniquely the wavelength of the scattered photon and (b) find this wavelength.

In a Compton scattering experiment, an x-ray photon scatters through an angle of 17.4 from a free electron that is initially at rest. The electron recoils with a speed of 2 180 km/s. Calculate (a) the wavelength of the incident photon and (b) the angle through which the electron scatters.

Find the maximum fractional energy loss for a 0.511-MeV gamma ray that is Compton scattered from (a) a free electron and (b) a free proton.

An electromagnetic wave is called ionizing radiation if its photon energy is larger than, say, 10.0 eV so that a single photon has enough energy to break apart an atom. With reference to Figure P40.37 (page 1264), explain what region or regions of the electromagnetic spectrum fit this definition of ionizing radiation and what do not. (If you wish to consult a larger version of Fig. P40.37, see Fig. 34.13.)

Review. A heliumneon laser produces a beam of diameter 1.75 mm, delivering 2.00 3 1018 photons/s. Each photon has a wavelength of 633 nm. Calculate the amplitudes of (a) the electric fields and (b) the magnetic fields inside the beam. (c) If the beam shines perpendicularly onto a perfectly reflecting surface, what force does it exert on the surface? (d) If the beam is absorbed by a block of ice at 0C for 1.50 h, what mass of ice is melted?

(a) Calculate the momentum of a photon whose wavelength is 4.00 3 1027 m. (b) Find the speed of an electron having the same momentum as the photon in part (a).

(a) An electron has a kinetic energy of 3.00 eV. Find its wavelength. (b) What If? A photon has energy 3.00 eV. Find its wavelength.

The resolving power of a microscope depends on the wavelength used. If you wanted to see an atom, a wavelength of approximately 1.00 3 10211 m would be required. (a) If electrons are used (in an electron microscope), what minimum kinetic energy is required for the electrons? (b) What If? If photons are used, what minimum photon energy is needed to obtain the required resolution?

Calculate the de Broglie wavelength for a proton moving with a speed of 1.00 3 106 m/s.

In the DavissonGermer experiment, 54.0-eV electrons were diffracted from a nickel lattice. If the first maximum in the diffraction pattern was observed at f 5 50.0 (Fig. P40.43), what was the lattice spacing a between the vertical columns of atoms in the figure?

The nucleus of an atom is on the order of 10214 m in diameter. For an electron to be confined to a nucleus, its de Broglie wavelength would have to be on this order of magnitude or smaller. (a) What would be the kinetic energy of an electron confined to this region? (b) Make an order-of-magnitude estimate of the electric potential energy of a system of an electron inside an atomic nucleus. (c) Would you expect to find an electron in a nucleus? Explain.

Robert Hofstadter won the 1961 Nobel Prize in Physics for his pioneering work in studying the scattering of 20-GeV electrons from nuclei. (a) What is the g factor for an electron with total energy 20.0 GeV, defined by g 5 1/!1 2 u2 /c 2 ? (b) Find the momentum of the electron. (c) Find the wavelength of the electron. (d) State how the wavelength compares with the diameter of an atomic nucleus, typically on the order of 10214 m.

Why is the following situation impossible? After learning about de Broglies hypothesis that material particles of momentum p move as waves with wavelength l 5 h/p, an 80-kg student has grown concerned about being diffracted when passing through a doorway of width w 5 75 cm. Assume significant diffraction occurs when the width of the diffraction aperture is less than ten times the wavelength of the wave being diffracted. Together with his classmates, the student performs precision experiments and finds that he does indeed experience measurable diffraction.

A photon has an energy equal to the kinetic energy of an electron with speed u, which may be close to the speed of light c. (a) Calculate the ratio of the wavelength of the photon to the wavelength of the electron. (b) Evaluate the ratio for the particle speed u 5 0.900c. (c) What If? What would happen to the answer to part (b) if the material particle were a proton instead of an electron? (d) Evaluate the ratio for the particle speed u 5 0.001 00c. (e) What value does the ratio of the wavelengths approach at high particle speeds? (f) At low particle speeds?

(a) Show that the frequency f and wavelength l of a freely moving quantum particle with mass are related by the expression a f c b 2 5 1 l2 1 1 lC 2 where lC 5 h/mc is the Compton wavelength of the particle. (b) Is it ever possible for a particle having nonzero mass to have the same wavelength and frequency as a photon? Explain.

Consider a freely moving quantum particle with mass m and speed u. Its energy is E 5 K 5 1 2mu2 . (a) Determine the phase speed of the quantum wave representing the particle and (b) show that it is different from the speed at which the particle transports mass and energy

For a free relativistic quantum particle moving with speed u, the total energy of the particle is E 5 hf 5 Uv 5 !p 2 c 2 1 m2 c 4 and the momentum is p 5 h/l 5 Uk 5 gmu. For the quantum wave representing the particle, the group speed is vg 5 dv/dk. Prove that the group speed of the wave is the same as the speed of the particle.

Neutrons traveling at 0.400 m/s are directed through a pair of slits separated by 1.00 mm. An array of detectors is placed 10.0 m from the slits. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zero-intensity point on the detector array? (c) When a neutron reaches a detector, can we say which slit the neutron passed through? Explain.

In a certain vacuum tube, electrons evaporate from a hot cathode at a slow, steady rate and accelerate from rest through a potential difference of 45.0 V. Then they travel 28.0 cm as they pass through an array of slits and fall on a screen to produce an interference pattern. If the beam current is below a certain value, only one electron at a time will be in flight in the tube. In this situation, the interference pattern still appears, showing that each individual electron can interfere with itself. What is the maximum value for the beam current that will result in only one electron at a time in flight in the tube?

A modified oscilloscope is used to perform an electron interference experiment. Electrons are incident on a pair of narrow slits 0.060 0 mm apart. The bright bands in the interference pattern are separated by 0.400 mm on a screen 20.0 cm from the slits. Determine the potential difference through which the electrons were accelerated to give this pattern.

Suppose a duck lives in a universe in which h 5 2p J ? s. The duck has a mass of 2.00 kg and is initially known to be within a pond 1.00 m wide. (a) What is the minimum uncertainty in the component of the ducks velocity parallel to the ponds width? (b) Assuming this uncertainty in speed prevails for 5.00 s, determine the uncertainty in the ducks position after this time interval.

An electron and a 0.020 0-kg bullet each have a velocity of magnitude 500 m/s, accurate to within 0.010 0%. Within what lower limit could we determine the position of each object along the direction of the velocity?

A 0.500-kg block rests on the frictionless, icy surface of a frozen pond. If the location of the block is measured to a precision of 0.150 cm and its mass is known exactly, what is the minimum uncertainty in the blocks speed?

The average lifetime of a muon is about 2 ms. Estimate the minimum uncertainty in the rest energy of a muon.

Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of vx 5 100 m/s. The rifles barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Use the uncertainty principle to show that if an electron were confined inside an atomic nucleus of diameter on the order of 10214 m, it would have to be moving relativistically, whereas a proton confined to the same nucleus can be moving nonrelativistically.

The accompanying table shows data obtained in a photoelectric experiment. (a) Using these data, make a graph similar to Figure 40.11 that plots as a straight line. From the graph, determine (b) an experimental value for Plancks constant (in joule-seconds) and (c) the work function (in electron volts) for the surface. (Two significant figures for each answer are sufficient.) Wavelength Maximum Kinetic Energy (nm) of Photoelectrons (eV) 588 0.67 505 0.98 445 1.35 399 1.63

Photons of wavelength 450 nm are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius 20.0 cm by a magnetic field with a magnitude of 2.00 3 1025 T. What is the work function of the metal?

Review. Photons of wavelength l are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius R by a magnetic field having a magnitude B. What is the work function of the metal?

Review. Design an incandescent lamp filament. A tungsten wire radiates electromagnetic waves with power 75.0 W when its ends are connected across a 120-V power supply. Assume its constant operating temperature is 2 900 K and its emissivity is 0.450. Also assume it takes in energy only by electric transmission and emits energy only by electromagnetic radiation. You may take the resistivity of tungsten at 2 900 K as 7.13 3 1027 V ? m. Specify (a) the radius and (b) the length of the filament.

Derive the equation for the Compton shift (Eq. 40.13) from Equations 40.14 through 40.16.

Figure P40.65 shows the stopping potential versus the incident photon frequency for the photoelectric effect for sodium. Use the graph to find (a) the work function of sodium, (b) the ratio h/e, and (c) the cutoff wavelength. The data are taken from R. A. Millikan, Physical Review 7:362 (1916).

A photon of initial energy E0 undergoes Compton scattering at an angle u from a free electron (mass me) initially at rest. Derive the following relationship for the final energy E9 of the scattered photon: E r 5 E0 1 1 a E0 mec 2 b 11 2 cos u 2

A daredevils favorite trick is to step out of a 16th-story window and fall 50.0 m into a pool. A news reporter takes a picture of the 75.0-kg daredevil just before he makes a splash, using an exposure time of 5.00 ms. Find (a) the daredevils de Broglie wavelength at this moment, (b) the uncertainty of his kinetic energy measurement during the 5.00-ms time interval, and (c) the percent error caused by such an uncertainty.

Show that the ratio of the Compton wavelength lC to the de Broglie wavelength l 5 h/p for a relativistic electron is lC l 5 c a E mec 2 b 2 2 1d 1/2 where E is the total energy of the electron and me is its mass.

Monochromatic ultraviolet light with intensity 550 W/m2 is incident normally on the surface of a metal that has a work function of 3.44 eV. Photoelectrons are emitted with a maximum speed of 420 km/s. (a) Find the maximum possible rate of photoelectron emission from 1.00 cm2 of the surface by imagining that every photon produces one photoelectron. (b) Find the electric current these electrons constitute. (c) How do you suppose the actual current compares with this maximum possible current?

A p0 meson is an unstable particle produced in highenergy particle collisions. Its rest energy is approximately 135 MeV, and it exists for a lifetime of only 8.70 3 10217 s before decaying into two gamma rays. Using the uncertainty principle, estimate the fractional uncertainty Dm/m in its mass determination.

The neutron has a mass of 1.67 3 10227 kg. Neutrons emitted in nuclear reactions can be slowed down by collisions with matter. They are referred to as thermal neutrons after they come into thermal equilibrium with the environment. The average kinetic energy 1 3 2k BT2 of a thermal neutron is approximately 0.04 eV. (a) Calculate the de Broglie wavelength of a neutron with a kinetic energy of 0.040 0 eV. (b) How does your answer compare with the characteristic atomic spacing in a crystal? (c) Explain whether you expect thermal neutrons to exhibit diffraction effects when scattered by a crystal.

A woman on a ladder drops small pellets toward a point target on the floor. (a) Show that, according to the uncertainty principle, the average miss distance must be at least Dxf 5 a 2U m b 1/2 a 2H g b 1/4 where H is the initial height of each pellet above the floor and m is the mass of each pellet. Assume that the spread in impact points is given by Dxf 5 Dxi 1 (Dvx)t. (b) If H 5 2.00 m and m = 0.500 g, what is Dxf?

Review. A light source emitting radiation at frequency 7.00 3 1014 Hz is incapable of ejecting photoelectrons from a certain metal. In an attempt to use this source to eject photoelectrons from the metal, the source is given a velocity toward the metal. (a) Explain how this procedure can produce photoelectrons. (b) When the speed of the light source is equal to 0.280c, photoelectrons just begin to be ejected from the metal. What is the work function of the metal? (c) When the speed of the light source is increased to 0.900c, determine the maximum kinetic energy of the photoelectrons.

Using conservation principles, prove that a photon cannot transfer all its energy to a free electron.

The total power per unit area radiated by a black body at a temperature T is the area under the I(l,T )-versusl curve as shown in Figure 40.3. (a) Show that this power per unit area is 3 ` 0 I 1l,T 2 dl 5 sT 4 where I(l,T ) is given by Plancks radiation law and s is a constant independent of T. This result is known as Stefans law. (See Section 20.7.) To carry out the integration, you should make the change of variable x 5 hc/lkBT and use 3 ` 0 x 3 dx e x 2 1 5 p4 15 (b) Show that the StefanBoltzmann constant s has the value s 5 2p5 k B 4 15c 2 h3 5 5.67 3 1028 W/m2 # K4

(a) Derive Wiens displacement law from Plancks law. Proceed as follows. In Figure 40.3, notice that the wavelength at which a black body radiates with greatest intensity is the wavelength for which the graph of I(l,T ) versus l has a horizontal tangent. From Equation 40.6, evaluate the derivative dI/dl. Set it equal to zero. Solve the resulting transcendental equation numerically to prove that hc/lmaxkBT 5 4.965 . . . or lmaxT 5 hc/4.965kB. (b) Evaluate the constant as precisely as possible and compare it with Wiens experimental value.