1. Assume that a human body emits blackbody radiation at the
standard body temperature.(a) Estimate how much energy is radiated by
the body in one hour.(b) At what wavelength does this radiation have its
maximum intensity? Get solution
2. A distant red star is observed to have a blackbody spectrum with a maximum at a wavelength of 3500 Å [1 Å = 10−10 m]. What is the temperature of the star? Get solution
3. The universe is filled with blackbody radiation at a temperature of 2.7 K left over from the Big Bang. [This radiation was discovered in 1965 by Bell Laboratory scientists, who thought at one point that they were seeing interference from pigeon droppings on their microwave receiver.](a) What is the total energy density of this radiation?(b) What is the total energy density with wavelengths between 1 mm and 1.01 mm? Is the Rayleigh-Jeans formula a good approximation at these wavelengths? Get solution
4. Over what range in frequencies does the Rayleigh-Jeans formula give a result within 10% of the Planck blackbody spectrum? Get solution
5. Let ρ(υ0) be the total energy density of blackbody radiation in all frequencies less than υ0, where hυ0 ≪ kT. Derive an expression for ρ(υ0). Get solution
6. Suppose we want to measure the total energy density in blackbody radiation above some cutoff frequency υ0. Let ρ(> υ0) be the total radiation density in all frequencies greater than υ0. Using the Planck blackbody spectrum show that ...... is a good approximation when hυ0 is much larger than kT. Get solution
7. (a) Express the Planck spectrum (Equation 1.7) as a function of the wavelength λ of the radiation, rather than the frequency υ.(b) Use this expression to derive the wavelength λmax at which the spectrum is a maximum.(c) Does λmaxυmax = c? Get solution
8. In a photoelectric experiment, electrons are emitted from a surface illuminated by light of wavelength 4000 Å, and the stopping potential for these electrons is found to be Φ0 = 0.5 V. What is the longest wavelength of light that can illuminate this surface and still produce a photoelectric current? Get solution
9. A lightbulb emits 40 W of power at a wavelength of 6 × 10−7 m.(a) What is the total number of photons emitted per second?(b) What is the energy of each photon? Get solution
10. (a) Using the Planck blackbody spectrum, and the fact that a photon with a frequency υ has an energy of hυ, derive an expression for n(υ)dυ, the total number density of photons with frequencies between υ and υ + dυ in blackbody radiation.(b) Using the expression from part (a), show that the total number density of photons in blackbody radiation is given by...where ß is a constant given by ß ≈ 60. [Note that the integral ... cannot be done analytically, so use the numerical result that ... 2.4.] Get solution
11. A gamma ray with energy 1 MeV is scattered off of an unknown particle which is at rest. The gamma ray is reflected directly backward with a final energy of 0.98 MeV. What is m0c2 for the unknown particle? (Express your answer in MeV.) Get solution
12. Calculate the de Broglie wavelength of a proton (mc2 = 938 MeV) with(a) a kinetic energy of 0.1 MeV(b) a total energy of 3 GeV. Get solution
13. The Balmer series (the m = 2 case in Equation 1.18) was discovered before the other series of spectral lines (m = 1, m = 3, etc.). Why? (Hint: Plug in some numbers and calculate wavelengths for m = 1, m = 2, and m = 3.) Get solution
14. Verify that ħ has units of angular momentum. Get solution
15. Beginning with the Bohr energy levels (Equation 1.24), derive the expression for the wavelengths of the spectral lines in hydrogen (Equation 1.18) and use this result to express R as a function of m, e, ħ, c, and ϵ0. Plug in values for these constants and verify that the correct result for R is obtained. Get solution
16. Suppose that the attractive force between the electron and proton in the hydrogen atom as given by some power law other than the inverse square law, i.e., assume that the force is given by F = krβ, where k is a constant, and ß is an arbitrary number with ß ≠ 1. [For example, the ordinary Coulomb law corresponds to the case ß = −2. The harmonic oscillator corresponds to ß = 1.] Use the Bohr quantization rule to show that for ß ≠ −1, the energy levels of the atom are give by...This formula gives an absurd answer when ß = −3; why? Get solution
2. A distant red star is observed to have a blackbody spectrum with a maximum at a wavelength of 3500 Å [1 Å = 10−10 m]. What is the temperature of the star? Get solution
3. The universe is filled with blackbody radiation at a temperature of 2.7 K left over from the Big Bang. [This radiation was discovered in 1965 by Bell Laboratory scientists, who thought at one point that they were seeing interference from pigeon droppings on their microwave receiver.](a) What is the total energy density of this radiation?(b) What is the total energy density with wavelengths between 1 mm and 1.01 mm? Is the Rayleigh-Jeans formula a good approximation at these wavelengths? Get solution
4. Over what range in frequencies does the Rayleigh-Jeans formula give a result within 10% of the Planck blackbody spectrum? Get solution
5. Let ρ(υ0) be the total energy density of blackbody radiation in all frequencies less than υ0, where hυ0 ≪ kT. Derive an expression for ρ(υ0). Get solution
6. Suppose we want to measure the total energy density in blackbody radiation above some cutoff frequency υ0. Let ρ(> υ0) be the total radiation density in all frequencies greater than υ0. Using the Planck blackbody spectrum show that ...... is a good approximation when hυ0 is much larger than kT. Get solution
7. (a) Express the Planck spectrum (Equation 1.7) as a function of the wavelength λ of the radiation, rather than the frequency υ.(b) Use this expression to derive the wavelength λmax at which the spectrum is a maximum.(c) Does λmaxυmax = c? Get solution
8. In a photoelectric experiment, electrons are emitted from a surface illuminated by light of wavelength 4000 Å, and the stopping potential for these electrons is found to be Φ0 = 0.5 V. What is the longest wavelength of light that can illuminate this surface and still produce a photoelectric current? Get solution
9. A lightbulb emits 40 W of power at a wavelength of 6 × 10−7 m.(a) What is the total number of photons emitted per second?(b) What is the energy of each photon? Get solution
10. (a) Using the Planck blackbody spectrum, and the fact that a photon with a frequency υ has an energy of hυ, derive an expression for n(υ)dυ, the total number density of photons with frequencies between υ and υ + dυ in blackbody radiation.(b) Using the expression from part (a), show that the total number density of photons in blackbody radiation is given by...where ß is a constant given by ß ≈ 60. [Note that the integral ... cannot be done analytically, so use the numerical result that ... 2.4.] Get solution
11. A gamma ray with energy 1 MeV is scattered off of an unknown particle which is at rest. The gamma ray is reflected directly backward with a final energy of 0.98 MeV. What is m0c2 for the unknown particle? (Express your answer in MeV.) Get solution
12. Calculate the de Broglie wavelength of a proton (mc2 = 938 MeV) with(a) a kinetic energy of 0.1 MeV(b) a total energy of 3 GeV. Get solution
13. The Balmer series (the m = 2 case in Equation 1.18) was discovered before the other series of spectral lines (m = 1, m = 3, etc.). Why? (Hint: Plug in some numbers and calculate wavelengths for m = 1, m = 2, and m = 3.) Get solution
14. Verify that ħ has units of angular momentum. Get solution
15. Beginning with the Bohr energy levels (Equation 1.24), derive the expression for the wavelengths of the spectral lines in hydrogen (Equation 1.18) and use this result to express R as a function of m, e, ħ, c, and ϵ0. Plug in values for these constants and verify that the correct result for R is obtained. Get solution
16. Suppose that the attractive force between the electron and proton in the hydrogen atom as given by some power law other than the inverse square law, i.e., assume that the force is given by F = krβ, where k is a constant, and ß is an arbitrary number with ß ≠ 1. [For example, the ordinary Coulomb law corresponds to the case ß = −2. The harmonic oscillator corresponds to ß = 1.] Use the Bohr quantization rule to show that for ß ≠ −1, the energy levels of the atom are give by...This formula gives an absurd answer when ß = −3; why? Get solution
