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Problem-solving suggestion

Note that the forms of the constants h = 4 . 14 × 10 –15 eV s size 12{h =" 4" "." "14 " times " 10" rSup { size 8{"–15"} } " eV " cdot " s"} {} and hc = 1240 eV nm size 12{ ital "hc" =" 1240 eV " cdot " nm"} {} may be particularly useful for this section’s Problems and Exercises.

Test prep for ap courses

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?

  1. 2.73 × 10 –3 m/s
  2. 0.819 m/s
  3. 1.22 m/s
  4. 2.71 × 10 4 m/s

(c)

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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?

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Test prep for ap courses

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?

  1. 4.95 × 10 –13 m
  2. 4.98 × 10 –13 m
  3. 5.02 × 10 –13 m
  4. 5.05 × 10 –13 m

(c)

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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.

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Section summary

  • Photons have momentum, given by p = h λ size 12{p = { {h} over {λ} } } {} , where λ size 12{λ} {} is the photon wavelength.
  • Photon energy and momentum are related by p = E c size 12{p = { {E} over {c} } } {} , where E = hf = hc / λ size 12{E = ital "hf"= ital "hc"/λ } {} for a photon.

Conceptual questions

Which formula may be used for the momentum of all particles, with or without mass?

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Is there any measurable difference between the momentum of a photon and the momentum of matter?

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Why don’t we feel the momentum of sunlight when we are on the beach?

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Problems&Exercises

(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) 1.66 × 10 32 kg m/s size 12{1 "." "66" times "10" rSup { size 8{ - "32"} } `"kg" cdot "m/s"} {}

(b) The wavelength of microwave photons is large, so the momentum they carry is very small.

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(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?

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(a) What is the wavelength of a photon that has a momentum of 5 . 00 × 10 29 kg m/s size 12{5 "." "00" times "10" rSup { size 8{ - "29"} } `"kg" cdot "m/s"} {} ? (b) Find its energy in eV.

(a) 13.3 μm

(b) 9 . 38 × 10 -2 eV

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(a) A γ size 12{γ} {} -ray photon has a momentum of 8 . 00 × 10 21 kg m/s size 12{8 "." "00" times "10" rSup { size 8{ - "21"} } `"kg" cdot "m/s"} {} . What is its wavelength? (b) Calculate its energy in MeV.

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(a) Calculate the momentum of a photon having a wavelength of 2 . 50 μm size 12{2 "." "50"" μm"} {} . (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 . 65 × 10 28 kg m/s size 12{2 "." "65" times "10" rSup { size 8{ - "28"} } `"kg" cdot "m/s"} {}

(b) 291 m/s

(c) electron 3 . 86 × 10 26 J size 12{3 "." "86" times "10" rSup { size 8{ - "26"} } " J"} {} , photon 7 . 96 × 10 20 J size 12{7 "." "96" times "10" rSup { size 8{ - "20"} } " J"} {} , ratio 2 . 06 × 10 6 size 12{2 "." "06" times "10" rSup { size 8{6} } } {}

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Repeat the previous problem for a 10.0-nm-wavelength photon.

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(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 . 32 × 10 13 m size 12{1 "." "32" times "10" rSup { size 8{ - "13"} } " m"} {}

(b) 9.39 MeV

(c) 4.70 × 10 2 MeV size 12{4 "." "70" times "10" rSup { size 8{ - 2} } " MeV"} {}

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(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?

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Take the ratio of relativistic rest energy, E = γmc 2 mc 2 , to relativistic momentum, p = γ mu size 12{p=γ ital "mu"} {} , and show that in the limit that mass approaches zero, you find E / p = c size 12{E/p=c} {} .

E = γmc 2 mc 2 and P = γmu , so

E P = γmc 2 γmu = c 2 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

lim m →0 E P = c 2 c = c

which is consistent with the equation for photon energy.

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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 N/m 2 size 12{"N/m" rSup { size 8{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.

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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 . 00 × 10 2 N size 12{2 "." "00" times "10" rSup { size 8{ - 2} } " N"} {} backward on the car. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) 3 . 00 × 10 6 W size 12{3 "." "00" times "10" rSup { size 8{6} } " W"} {}

(b) Headlights are way too bright.

(c) Force is too large.

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Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
room?
Abrar
What is position?
Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 2

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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