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Note that the forms of the constants $h=\text{4}\text{.}\text{14}\times {\text{10}}^{\text{\u201315}}\phantom{\rule{0.25em}{0ex}}\text{eV}\cdot \text{s}$ and $\text{hc}=\text{1240 eV}\cdot \text{nm}$ may be particularly useful for this section’s Problems and Exercises.
The mass of a proton is 1.67 × 10 ^{–27} kg. If a proton has the same momentum as a photon with a wavelength of 325 nm, what is its speed?
(c)
A strip of metal foil with a mass of 5.00 × 10 ^{–7} kg is suspended in a vacuum and exposed to a pulse of light. The velocity of the foil changes from zero to 1.00 × 10 ^{–3} m/s in the same direction as the initial light pulse, and the light pulse is entirely reflected from the surface of the foil. Given that the wavelength of the light is 450 nm, and assuming that this wavelength is the same before and after the collision, how many photons in the pulse collide with the foil?
In an experiment in which the Compton effect is observed, a “gamma ray” photon with a wavelength of 5.00 × 10 ^{–13} m scatters from an electron. If the change in the electron energy is 1.60 × 10 ^{–15} J, what is the wavelength of the photon after the collision with the electron?
(c)
Consider two experiments involving a metal sphere with a radius of 2.00 μm that is suspended in a vacuum. In one experiment, a pulse of N photons reflects from the surface of the sphere, causing the sphere to acquire momentum. In a second experiment, an identical pulse of photons is completely absorbed by the sphere, so that the sphere acquires momentum. Identify each type of collision as either elastic or inelastic, and, assuming that the change in the photon wavelength can be ignored, use linear momentum conservation to derive the expression for the momentum of the sphere in each experiment.
Which formula may be used for the momentum of all particles, with or without mass?
Is there any measurable difference between the momentum of a photon and the momentum of matter?
Why don’t we feel the momentum of sunlight when we are on the beach?
(a) Find the momentum of a 4.00-cm-wavelength microwave photon. (b) Discuss why you expect the answer to (a) to be very small.
(a) $\text{1.66}\times {\text{10}}^{-\text{32}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$
(b) The wavelength of microwave photons is large, so the momentum they carry is very small.
(a) What is the momentum of a 0.0100-nm-wavelength photon that could detect details of an atom? (b) What is its energy in MeV?
(a) What is the wavelength of a photon that has a momentum of $5\text{.}\text{00}\times {\text{10}}^{-\text{29}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$ ? (b) Find its energy in eV.
(a) 13.3 μm
(b) $9\text{.}\text{38}\times {\text{10}}^{-2}$ eV
(a) A $\gamma $ -ray photon has a momentum of $8\text{.}\text{00}\times {\text{10}}^{-\text{21}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$ . What is its wavelength? (b) Calculate its energy in MeV.
(a) Calculate the momentum of a photon having a wavelength of $2\text{.}\text{50 \mu m}$ . (b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?
(a) $2\text{.}\text{65}\times {\text{10}}^{-\text{28}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$
(b) 291 m/s
(c) electron $3\text{.}\text{86}\times {\text{10}}^{-\text{26}}\phantom{\rule{0.25em}{0ex}}\text{J}$ , photon $7\text{.}\text{96}\times {\text{10}}^{-\text{20}}\phantom{\rule{0.25em}{0ex}}\text{J}$ , ratio $2\text{.}\text{06}\times {\text{10}}^{6}$
Repeat the previous problem for a 10.0-nm-wavelength photon.
(a) Calculate the wavelength of a photon that has the same momentum as a proton moving at 1.00% of the speed of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?
(a) $1\text{.}\text{32}\times {\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{m}$
(b) 9.39 MeV
(c) $4.70\times {\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{MeV}$
(a) Find the momentum of a 100-keV x-ray photon. (b) Find the equivalent velocity of a neutron with the same momentum. (c) What is the neutron’s kinetic energy in keV?
Take the ratio of relativistic rest energy, $E={\mathrm{\gamma mc}}^{2}$ , to relativistic momentum, $p=\gamma \text{mu}$ , and show that in the limit that mass approaches zero, you find $E/p=c$ .
$E={\mathrm{\gamma mc}}^{2}$ and $P=\mathrm{\gamma mu}$ , so
$\frac{E}{P}=\frac{{\text{\gamma mc}}^{2}}{\text{\gamma mu}}=\frac{{\text{c}}^{2}}{\text{u}}.$
As the mass of particle approaches zero, its velocity $u$ will approach $c$ , so that the ratio of energy to momentum in this limit is
${\mathrm{lim}}_{m\mathrm{\to 0}}\frac{E}{P}=\frac{{c}^{2}}{c}=c$
which is consistent with the equation for photon energy.
Construct Your Own Problem
Consider a space sail such as mentioned in [link] . Construct a problem in which you calculate the light pressure on the sail in ${\text{N/m}}^{2}$ produced by reflecting sunlight. Also calculate the force that could be produced and how much effect that would have on a spacecraft. Among the things to be considered are the intensity of sunlight, its average wavelength, the number of photons per square meter this implies, the area of the space sail, and the mass of the system being accelerated.
Unreasonable Results
A car feels a small force due to the light it sends out from its headlights, equal to the momentum of the light divided by the time in which it is emitted. (a) Calculate the power of each headlight, if they exert a total force of $2\text{.}\text{00}\times {\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{N}$ backward on the car. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
(a) $3\text{.}\text{00}\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{W}$
(b) Headlights are way too bright.
(c) Force is too large.
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