Questions and Answers of Modern Physics

The wave function describing a state of an electron confined to move along the x axis is given at time zero byFind the probability of finding the electron in a region dx centered at (a) x =
A radar transmitter used to measure the speed of pitched baseballs emits pulses of 2.0 cm wavelength that are 0.25 μs in duration.(a) What is the length of the wave packet produced? (b) To what
A certain standard tuning fork vibrates at 880 Hz. If the tuning fork is tapped, causing it to vibrate, then stopped a quarter of a second later, what is the approximate range of frequencies
If a phone line is capable of transmitting a range of frequencies Δf = 5000 Hz, what is the approximate duration of the shortest pulse that can be transmitted over the line?
(a) You are given the task of constructing a double-slit experiment for 5 eV electrons. If you want the first minimum of the diffraction pattern to occur at 5º, what must be the separation of the
A 100 g rigid sphere of radius 1 cm has a kinetic energy of 2 J and is confined to move in a force-free region between two rigid walls separated by 50 cm. (a) What is the probability of finding
A particle moving in one dimension between rigid walls separated by a distance L has the wave function Ψ (x) = A sin(πx/L). Since the particle must always be located between the walls, what must be
A ladybug 5 mm in diameter with a mass of 1.0 mg being viewed through a low-power magnifier with a calibrated reticule is observed to be stationary with an uncertainty of 10-2 mm. How fast might the
222Rn decays by the emission of an a particle with a lifetime of 3.823 days. The kinetic energy of the a particle is measured to be 5.490 MeV. What is the uncertainty in this energy? Describe in one
If the uncertainty in the position of a wave packet representing the state of a quantum-system particle is equal to its de Broglie wavelength, how does the uncertainty in momentum compare with the
An excited state of a certain nucleus has a half-life of 0.85 ns. Taking this to be the uncertainty Δt for emission of a photon, calculate the uncertainty in the frequency Δf, using Equation 5-25.
The lifetimes of so-called resonance particles cannot be measured directly but is computed from the energy width (or uncertainty) of the scattering cross section versus energy graph. For example, the
In order to locate a particle, for example, an electron, to within 5 x 10-12 m using electromagnetic waves (“light”), the wavelength must be at least this small. Calculate the momentum and energy
Laser pulses of femtosecond duration can be produced, but for such brief pulses it makes no sense to speak of the pulse’s color. To demonstrate this, compute the time duration of a laser pulse
Show that Equation 5-11 is satisfied by y = f (φ), where φ = x - vt for any function f. ay 1 a²y əx² var 5-11
Protons and neutrons in nuclei are bound to the nucleus by exchanging pions (π mesons) with each other. This is possible to do without violating energy conservation provided the pion is re-absorbed
Show that the relation Δps Δs > ћ can be written ΔL Δφ > ћ for a particle moving in a circle about the z axis, where ps is the linear momentum tangential to the circle, s is the
De Broglie developed Equation 5-2 initially for photons, assuming that they had a small but finite mass. His assumption was that RF waves with λ = 30 m traveled at a speed of at least 99 percent of
A particle with a mass of 4 g is moving at 100 m/s. What size aperture would be needed in order to observe diffraction of this particle wave? Explain why no common object could pass through such an
Neutrons and protons in atomic nuclei are confined within a region whose diameter is about 10-15 m.(a) At any given instant, how fast might an individual proton or neutron be moving? (b) What is
A particle of mass m moves in a one-dimensional box of length L. (Take the potential energy of the particle in the box to be zero so that its total energy is its kinetic energy p2/2m.) Its energy is
(a) Suppose that a particle of mass m is constrained to move in a one-dimensional space between two infinitely high barriers located A apart. Using the uncertainty principle, find an expression for
A proton and a bullet each move with a speed of 500 m/s, measured with an uncertainty of 0.01 percent. If measurements of their respective positions are made simultaneous with the speed measurements,
An electron and a positron are moving toward each other with equal speeds of 3 x 106 m/s. The two particles annihilate each other and produce two photons of equal energy.(a) What were the de Broglie
It is possible for some fundamental particles to “violate” conservation of energy by creating and quickly re-absorbing another particle. For example, a proton can emit a π+ according to p = n +
Suppose that you drop BBs onto a bull’s-eye marked on the floor. According to the uncertainty principle, the BBs do not necessarily fall straight down from the release point to the center of the
Using the first-order Doppler-shift formula f= f0 (1 + v/)c2, calculate the energy shift of a 1 eV photon emitted from an iron atom moving toward you with energy (3/2)kT at T = 300 K. Compare this
Calculate the order of magnitude of the shift in energy of a (a) 1 eV photon and(b) 1 MeV photon resulting from the recoil of an iron nucleus. Do this by first calculating the momentum of the
If a particle is deflected by 0.01º in each collision, about how many collisions would be necessary to produce an rms deflection of 10º? (Use the result from the one-dimensional random walk problem
Consider the foil and a particle energy in Problem 4-6. Suppose that 1000 of those particles suffer a deflection of more than 25º. (a) How many of these are deflected by more than
The radius of the n = 1 orbit in the hydrogen atom is a0 = 0.053 nm. (a) Compute the radius of the n = 6 orbit. (b) Compute the radius of the n = 6 orbit in singly ionized helium (He+),
Calculate the three longest wavelengths in the Lyman series (nf = 1) in nm, and indicate their position on a horizontal linear scale. Indicate the series limit (shortest wavelength) on this scale.
An atom in an excited state will on the average undergo a transition to a state of lower energy in about 10-8 seconds. If the electron in a doubly ionized lithium atom (Li+2, which is hydrogenlike)
It is possible for a muon to be captured by a proton to form a muonic atom. A muon is identical to an electron except for its mass, which is 105.7 MeV/c2. (a) Calculate the radius of the first
In the lithium atom (Z = 3) two electrons are in the n = 1 orbit and the third is in the n = 2 orbit. (Only two are allowed in the n = 1 orbit because of the exclusion principle, which will be
Draw to careful scale an energy-level diagram for hydrogen for levels with n = 1, 2, 3, 4, ∞. Show the following on the diagram: (a) The limit of the Lyman series, (b) The Hβ
A hydrogen atom at rest in the laboratory emits the Lyman a radiation. (a) Compute the recoil kinetic energy of the atom. (b) What fraction of the excitation energy of the n = 2 state is
(a) Draw accurately to scale and label completely a partial energy-level diagram for C5+ including at minimum the energy levels for n = 1, 2, 3, 4, 5, and ∞. (b) Compute the wavelength of the
The electron-positron pair that was discussed in Chapter 2 can form a hydrogenlike system called positronium. Calculate(a) The energies of the three lowest states and (b) The wavelength of the
(a) Compute the energy of an electron in the n = 1 (K shell) of tungsten, using Z - 1 for the effective nuclear charge. (b) The experimental result for this energy is 69.5 keV. Assume that the
Suppose that, in a Franck-Hertz experiment, electrons of energy up to 13.0 eV can be produced in the tube. If the tube contained atomic hydrogen,(a) what is the shortest-wavelength spectral line that
If we could somehow fill a Franck-Hertz tube with positronium, what cathode-grid voltage would be needed to reach the second current decrease in the positronium equivalent of Figure 4-23? Figure
The Li2+ ion is essentially identical to the H atom in Bohr’s theory, aside from the effect of the different nuclear charges and masses. (a) What transitions in Li2+ will yield emission lines
If electric charge did not exist and electrons were bound to protons by the gravitational force to form hydrogen, derive the corresponding expressions for a0 and En and compute the energy and
(a) Write a computer program for your personal computer or programmable calculator that will provide you with the spectral series of H-like atoms. Inputs to be included are ni, nf, Z, and the nuclear
In this problem you are to obtain the Bohr results for the energy levels in hydrogen without using the quantization condition of Equation 4-17. In order to relate Equation 4-14to the Balmer-Ritz
Listed in the table are the La x-ray wavelengths for several elements. Construct a Moseley plot from these data. Compare the slope with the appropriate one in Figure 4-19. Determine and interpret the
A small shot of negligible radius hits a stationary smooth, hard sphere of radius R, making an angle β with the normal to the sphere, as shown in Figure 4-25. It is reflected at an equal angle to
Figure 3-15b shows the Ka and Kβ characteristic x rays emitted by a molybdenum (Mo) target in an x-ray tube whose accelerating potential is 35 kV. The wavelengths are Ka = 0.071 nm and Kβ = 0.063
The Ka, La, and Ma x rays are emitted in the n = 2 → n = 1, n = 3 → n = 2, and n = 4 → n = 3 transitions respectively. For calcium (Z = 20) the energies of these transitions are 3.69 keV, 0.341
In an a scattering experiment, the area of the a particle detector is 0.50 cm2. The detector is located 10 cm from a 1.0-μm-thick silver foil. The incident beam carries a current of 1.0 nA, and the
(a) The current i due to a charge q moving in a circle with frequency frev is q frev. Find the current due to the electron in the first Bohr orbit. (b) The magnetic moment of a current loop is
Geiger and Marsden used a particles with 7.7 MeV kinetic energy and found that when they were scattered from thin gold foil, the number observed to be scattered at all angles agreed with
Three isotopes of hydrogen occur in nature; ordinary hydrogen, deuterium, and tritium. Their nuclei consist of, respectively, 1 proton, 1 proton and 1 neutron (deuteron), and 1 proton and 2 neutrons
A Rydberg hydrogen atom is in the n = 45 energy state. (a) What is the energy difference (in eV) between this state and the n = 46 level? (b) What is the ionization energy of the atom in
Electrons in the Franck-Hertz tube can also have elastic collisions with the Hg atoms. If such a collision is a head-on, what fraction of its initial kinetic energy will an electron lose, assuming
Show that a small change in the reduced mass of the electron produces a small change in a spectral line given by Δλ/λ = Δμ/μ. Use this to calculate the difference Δλ in the Balmer red line λ
Consider the Franck-Hertz experiment with Hg vapor in the tube and the voltage between the cathode and the grid equal to 4.0 V, that is, not enough to for the electrons to excite the Hg atom’s
A particular radiating cavity has the maximum of its spectral distribution of radiated power at a wavelength of 27.0 μm (in the infrared region of the spectrum). The temperature is then changed so
Equation 3-3 suggests how a velocity selector for particles or mixtures of different particles all having the same charge can be made. Suppose you wish to make a velocity selector that allows
On drop #16, Millikan measured the following total charges, among others, at different times:What value of the fundamental quantized charge e do these numbers imply? 25.41 X 10-19 C 20.64 × 10-19
If electrons have kinetic energy of 2000 eV, find (a) Their speed,(b) The time needed to traverse a distance of 5cm between plates D and E in Figure 3-1, and (c) The vertical component of
One variation of the Millikan oil-drop apparatus arranges the electric field horizontal, rather than vertical, giving charged droplets acceleration in the horizontal direction. The result is that the
A beam of charged particles consisting of protons, electrons, deuterons, and singly ionized helium atoms and H2 molecules all pass through a velocity selector, all emerging with speeds of 2.5 x 106
Consider Thomson’s experiment with the electric field turned “off.” If the electrons enter a region of uniform magnetic field B and length ℓ, show that the electrons are deflected through an
Show that the electric field needed to make the rise time of the oil drop equal to its field-free fall time is 8 = 2mg/q.
A cosmic-ray proton approaches Earth vertically at the equator, where the horizontal component of Earth’s magnetic field is 3.5 x 10-5 T. If the proton is moving at 3.0 x106 m/s, what is the ratio
In J. J. Thomson’s first method (see Problem 3-46), the heat capacity of the beam stopper was about 5 x10-3 cal/°C and the temperature increase was about 2°C. How many 2000 eV electrons struck
Use the result of Example 3-4 and Equations 3-4 and 3-6 to express Stefan’s constant in terms of h, c, and k. Using the known values of these constants, calculate Stefan’s constant. R = GT 3-4
Show that Planck’s law, Equation 3-18, expressed in terms of the frequency f, is u(f) 8πf² hf c³ ehf/kT - 1 C
A charged oil droplet falls 5.0 mm in 20.0 s at terminal speed in the absence of an electric field. The specific gravity of air is 1.35 x10-3 and that of oil is 0.75. The viscosity of air is 1.80
Find λm for blackbody radiation at (a) T = 3 K, (b) T = 300 K, and (c) T = 3000 K.
Use Planck’s law, Equation 3-18, to derive the constant in Wien’s law, Equation 3-5. XT = constant = m 2.898 x 10³ m. K 3-5
As noted in the chapter, the cosmic microwave background radiation fits the Planck equation for a blackbody at 2.7 K. (a) What is the wavelength at the maximum intensity of the spectrum of the
Find the temperature of a blackbody if its spectrum has its peak at (a) λm = 700 nm (visible), (b) λm = 3 cm (microwave region), and (c) λm = 3 m (FM radio waves).
If the absolute temperature of a blackbody is doubled, by what factor is the total emitted power increased?
A certain very bright star has an effective surface temperature of 20,000 K.(a) Assuming that it radiates as a blackbody, what is the wavelength at which u(λ) is maximum? (b) In what part of
The energy reaching Earth from the Sun at the top of the atmosphere is 1.36 x 103 W/m2, called the solar constant. Assuming that Earth radiates like a blackbody at uniform temperature, what do you
A 40 W incandescent bulb radiates from a tungsten filament operating at 3300 K. Assuming that the bulb radiates like a blackbody, (a) what are the frequency fm and the wavelength λm at the
Use Compton’s equation (Equation 3-25) to compute the value of Δλ in Figure 3-17d. To what percent shift in the wavelength does this correspond?Figure 3-17(d) λε - λι h muc (1 - cost) 3-25
Using apparatus similar to that in Figure 3-8, the photoelectric effect data below were measured.(a) From a graph of Ek,max versus f, find a value for Planck’s constant. (b) By what percentage
The wavelengths of visible light range from about 380 nm to about 750 nm.(a) What is the range of photon energies (in eV) in visible light? (b) A typical FM radio station’s broadcast frequency
The orbiting space shuttle moved around Earth well above 99 percent of the atmosphere, yet it still accumulated an electric charge on its skin due, in part, to the loss of electrons caused by the
(a) If 5 percent of the power of a 100 W bulb is radiated in the visible spectrum, how many visible photons are radiated per second? (b) If the bulb is a point source radiating equally in all
A Compton scattering experiment yielded the data in the table below.(a) Using Equation 3-25 as a guide, construct an appropriate graph that enables you to obtain a value for the Compton wavelength of
The work function of molybdenum is 4.22 eV. (a) What is the threshold frequency for the photoelectric effect in molybdenum? (b) Will yellow light of wavelength 560 nm cause ejection of
The NaCl molecule has a bond energy of 4.26 eV; that is, this energy must be supplied in order to dissociate the molecule into neutral Na and Cl atoms.(a) What are the minimum frequency and maximum
Under optimum conditions, the eye will perceive a flash if about 60 photons arrive at the cornea. How much energy is this in joules if the wavelength of the light is 550 nm?
The longest wavelength of light that will cause emission of electrons from cesium is 653 nm. (a) Compute the work function for cesium. (b) If light of 300 nm (ultraviolet) were to shine on
x-ray tubes currently used by dentists often have accelerating voltages of 80 kV. What is the minimum wavelength of the x rays they produce?
Compton’s equation (Equation 3-25) indicates that a graph of λ2 versus (1- cos θ) should be a straight line whose slope h/mc allows a determination of h. Given that the wavelength of λ1 in
In the Compton scattering of a photon with energy E1 from an electron at rest, show that the energy of the scattered photon E2 is given by E₂ = E₁ (E₁/mc²) (1 cos ) + 1
Find the momentum of a photon in eV/c and in kg·m/s if the wavelength is (a) 400 nm, (b) 1 Å = 0.1 nm, (c) 3 cm, and (d) 2 nm.
Gamma rays emitted by radioactive nuclei also exhibit measurable Compton scattering. Suppose a 0.511 MeV photon from a positron-electron annihilation scatters at 110° from a free electron. What are
Assuming that the difference between Thomson’s calculated e/m in his second experiment (see Figure 3-19) and the currently accepted value was due entirely to his neglecting the horizontal component
Compute Δλ for photons scattered at 120° from (a) Free protons, (b) Free electrons, and (c) N2 molecules in air.
Millikan’s data for the photoelectric effect in lithium are shown in the table.(a) Graph the data and determine the work function for lithium. (b) Find the value of Planck’s constant
When light of wavelength 450 nm is incident on potassium, photoelectrons with stopping potential of 0.52 V are emitted. If the wavelength of the incident light is changed to 300 nm, the stopping
Show that the maximum kinetic energy Ek, called the Compton edge, that a recoiling electron can carry away from a Compton scattering event is given by Ex = 2E² hf 1+ mc²/2hf 2E + mc²
In his first e/m experiment Thomson determined the speed of electrons accelerated through a potential ΔV by collecting them in an insulated beam stopper and measuring both the total collected charge

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