Modern Physics

Final Exam

Name: Student I.D.:

Short Answer (10 pts each)

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference?

2. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

3. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

4. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

A Partial List of Fundamental Constants

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer Key for Exam A

Short Answer (10 pts each)

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference? Answer:

\frac{δν}{ν} = \frac{gH}{c^{2}} = \frac{(9.8m/s^{2})(300m)}{(3 × 10^{8}m/s)^{2}} = 3.27 × 10^{−14}

2. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

Answer: The rest mass of the proton is 1.672649×10^{−27}kg; that of the electron is 9.109534×10^{−31}kg. When the electron and proton combine to form a hydrogen atom, the ionization energy of 13.6 eV is released as ultraviolet radiation. The rest mass of the hydrogen atom is therefore smaller than the sum of the electron and proton rest masses by the amount

ΔM = \frac{(13.6eV)(1.6 × 10^{−19}J/eV)}{(3 × 10^{8}m/s)^{2}} = 2.42 × 10^{−35}kg

3. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

Answer: The radiance

4. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

Answer: The mass of the sun is 2 × 10^{30}kg. The universal gravitational constant is 6.67 × 10^{−11}N · m^{2}/kg^{2}. The Schwarzschild radius of one solar mass is

R_{c} = \frac{2(2 × 10^{30}lg)(6.67 × 10^{−11}N · m62/kg^{2})}{(3 × 10^{8}m/s)^{2}}
= 2.96 × 10^{3}m ≈ 3km

If the sun collapsed to a sphere of 3 km radius without loss of mass, it would then be a black hole. The mass density would be

ρ = \frac{M}{V} = \frac{2 × 10^{30}kg}{(\frac{4π}{3} )(2.96 × 10^{3}m)^{3}} = 1.84 × 10^{19}kg/m^{3}

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

2. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference?

3. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

4. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer Key for Exam B

Short Answer (10 pts each)

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

Answer: The mass of the sun is 2 × 10^{30}kg. The universal gravitational constant is 6.67 × 10^{−11}N · m^{2}/kg^{2}. The Schwarzschild radius of one solar mass is

R_{c} = \frac{2(2 × 10^{30}lg)(6.67 × 10^{−11}N · m62/kg^{2})}{(3 × 10^{8}m/s)^{2}}
= 2.96 × 10^{3}m ≈ 3km

If the sun collapsed to a sphere of 3 km radius without loss of mass, it would then be a black hole. The mass density would be

ρ = \frac{M}{V} = \frac{2 × 10^{30}kg}{(\frac{4π}{3} )(2.96 × 10^{3}m)^{3}} = 1.84 × 10^{19}kg/m^{3}

2. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference? Answer:

\frac{δν}{ν} = \frac{gH}{c^{2}} = \frac{(9.8m/s^{2})(300m)}{(3 × 10^{8}m/s)^{2}} = 3.27 × 10^{−14}

3. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

Answer: The rest mass of the proton is 1.672649×10^{−27}kg; that of the electron is 9.109534×10^{−31}kg. When the electron and proton combine to form a hydrogen atom, the ionization energy of 13.6 eV is released as ultraviolet radiation. The rest mass of the hydrogen atom is therefore smaller than the sum of the electron and proton rest masses by the amount

ΔM = \frac{(13.6eV)(1.6 × 10^{−19}J/eV)}{(3 × 10^{8}m/s)^{2}} = 2.42 × 10^{−35}kg

4. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

Answer: The radiance

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference?

2. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

3. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

4. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer Key for Exam C

Short Answer (10 pts each)

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference? Answer:

\frac{δν}{ν} = \frac{gH}{c^{2}} = \frac{(9.8m/s^{2})(300m)}{(3 × 10^{8}m/s)^{2}} = 3.27 × 10^{−14}

2. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

Answer: The radiance

3. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

Answer: The rest mass of the proton is 1.672649×10^{−27}kg; that of the electron is 9.109534×10^{−31}kg. When the electron and proton combine to form a hydrogen atom, the ionization energy of 13.6 eV is released as ultraviolet radiation. The rest mass of the hydrogen atom is therefore smaller than the sum of the electron and proton rest masses by the amount

ΔM = \frac{(13.6eV)(1.6 × 10^{−19}J/eV)}{(3 × 10^{8}m/s)^{2}} = 2.42 × 10^{−35}kg

4. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

Answer: The mass of the sun is 2 × 10^{30}kg. The universal gravitational constant is 6.67 × 10^{−11}N · m^{2}/kg^{2}. The Schwarzschild radius of one solar mass is

R_{c} = \frac{2(2 × 10^{30}lg)(6.67 × 10^{−11}N · m62/kg^{2})}{(3 × 10^{8}m/s)^{2}}
= 2.96 × 10^{3}m ≈ 3km

If the sun collapsed to a sphere of 3 km radius without loss of mass, it would then be a black hole. The mass density would be

ρ = \frac{M}{V} = \frac{2 × 10^{30}kg}{(\frac{4π}{3} )(2.96 × 10^{3}m)^{3}} = 1.84 × 10^{19}kg/m^{3}

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference?

2. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

3. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

4. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836

Name: Student I.D.:

Answer Key for Exam D

Short Answer (10 pts each)

Answer all problems in as thorough detail as possible. Be sure to include all your work. Partial credit will be given even if the answer is not fully correct.

1. The Eiffel Tower in Paris is 300m tall. What is the fractional gravitational red shift due to this elevation difference? Answer:

\frac{δν}{ν} = \frac{gH}{c^{2}} = \frac{(9.8m/s^{2})(300m)}{(3 × 10^{8}m/s)^{2}} = 3.27 × 10^{−14}

2. Derive Stefan’s law from Planck’s law. Using that result, obtain an expression for the Stefan-Boltzmann constant σ in terms of known physical constants and determine its value.

Answer: The radiance

3. Calculate the density of an object of one solar mass whose radius is the critical Schwarzschild radius. Compare this density with the nuclear density of approximately 2.3 × 10^{17}kg/m^{3}.

Answer: The mass of the sun is 2 × 10^{30}kg. The universal gravitational constant is 6.67 × 10^{−11}N · m^{2}/kg^{2}. The Schwarzschild radius of one solar mass is

R_{c} = \frac{2(2 × 10^{30}lg)(6.67 × 10^{−11}N · m62/kg^{2})}{(3 × 10^{8}m/s)^{2}}
= 2.96 × 10^{3}m ≈ 3km

If the sun collapsed to a sphere of 3 km radius without loss of mass, it would then be a black hole. The mass density would be

ρ = \frac{M}{V} = \frac{2 × 10^{30}kg}{(\frac{4π}{3} )(2.96 × 10^{3}m)^{3}} = 1.84 × 10^{19}kg/m^{3}

4. Determine the charge in the rest mass of a system consisting of a proton and an electron as the two particles combine to form a hydrogen atom. The ionization energy of hydrogen is 13.6 eV.

Answer: The rest mass of the proton is 1.672649×10^{−27}kg; that of the electron is 9.109534×10^{−31}kg. When the electron and proton combine to form a hydrogen atom, the ionization energy of 13.6 eV is released as ultraviolet radiation. The rest mass of the hydrogen atom is therefore smaller than the sum of the electron and proton rest masses by the amount

ΔM = \frac{(13.6eV)(1.6 × 10^{−19}J/eV)}{(3 × 10^{8}m/s)^{2}} = 2.42 × 10^{−35}kg

Constant  Symbol Approximate Value
Speed of light in vacuum  c  3.00 × 10^{8}m/s
Permeability of vacuum  μ_{0}  12.6 × 10^{−7}H/m
Permittivity of vacuum  ϵ_{0}  8.85 × 10^{−12}F/m
Magnetic flux quantum  ϕ_{0} = \frac{h}{2e} 2.07 × 10^{−15}Wb
Electron mass  m_{e}  9.11 × 10^{−31}kg
Proton mass  m_{p}  1.673 × 10^{−27}kg
Neutron mass  m_{n}  1.675 × 10^{−27}kg
Proton-electron mass ratio  \frac{m_{p}}{m_{e}}  1836