1. Home
  2. College physics
  3. Introduction to quantum physics
  4. Photon momentum

29.4 Photon momentum  (Page 3/5)

<< Chapter < Page
  College physics     Page 3 / 5
Chapter >> Page >
(a) A payload having an umbrella-shaped solar sail attached to it is shown. The direction of movement of payload and direction of incident photons are shown using arrows. (b) A photograph of the top view of a silvery space sail.
(a) Space sails have been proposed that use the momentum of sunlight reflecting from gigantic low-mass sails to propel spacecraft about the solar system. A Russian test model of this (the Cosmos 1) was launched in 2005, but did not make it into orbit due to a rocket failure. (b) A U.S. version of this, labeled LightSail-1, is scheduled for trial launches in the first part of this decade. It will have a 40-m 2 sail. (credit: Kim Newton/NASA)

Relativistic photon momentum

There is a relationship between photon momentum p size 12{p} {} and photon energy E size 12{E} {} that is consistent with the relation given previously for the relativistic total energy of a particle as E 2 = ( pc ) 2 + ( mc ) 2 size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {} . We know m size 12{m} {} is zero for a photon, but p size 12{p} {} is not, so that E 2 = ( pc ) 2 + ( mc ) 2 size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {} becomes

E = pc , size 12{E = ital "pc"} {}

or

p = E c (photons). size 12{p = { {E} over {c} } } {}

To check the validity of this relation, note that E = hc / λ size 12{E = ital "hc"/λ} {} for a photon. Substituting this into p = E / c size 12{p = E"/c"} {} yields

p = hc / λ / c = h λ , size 12{p = left ( ital "hc"/λ right )/c = { {h} over {λ} } } {}

as determined experimentally and discussed above. Thus, p = E / c size 12{p = E"/c"} {} is equivalent to Compton’s result p = h / λ size 12{p = h/λ} {} . For a further verification of the relationship between photon energy and momentum, see [link] .

Photon detectors

Almost all detection systems talked about thus far—eyes, photographic plates, photomultiplier tubes in microscopes, and CCD cameras—rely on particle-like properties of photons interacting with a sensitive area. A change is caused and either the change is cascaded or zillions of points are recorded to form an image we detect. These detectors are used in biomedical imaging systems, and there is ongoing research into improving the efficiency of receiving photons, particularly by cooling detection systems and reducing thermal effects.

Photon energy and momentum

Show that p = E / c size 12{p = E"/c"} {} for the photon considered in the [link] .

Strategy

We will take the energy E size 12{E} {} found in [link] , divide it by the speed of light, and see if the same momentum is obtained as before.

Solution

Given that the energy of the photon is 2.48 eV and converting this to joules, we get

p = E c = ( 2.48 eV ) ( 1 . 60 × 10 –19 J/eV ) 3 . 00 × 10 8 m/s = 1 . 33 × 10 –27 kg ⋅ m/s . size 12{p = { {E} over {c} } = { { \( 2 "." "48 eV" \) \( 1 "." "60 " times " 10" rSup { size 8{"–19"} } " J/eV" \) } over {3 "." "00 " times " 10" rSup { size 8{8} } " m/s"} } =" 1" "." "33 " times " 10" rSup { size 8{"–27"} } " kg " cdot " m/s"} {}

Discussion

This value for momentum is the same as found before (note that unrounded values are used in all calculations to avoid even small rounding errors), an expected verification of the relationship p = E / c size 12{p = E"/c"} {} . This also means the relationship between energy, momentum, and mass given by E 2 = ( pc ) 2 + ( mc ) 2 size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {} applies to both matter and photons. Once again, note that p size 12{p} {} is not zero, even when m size 12{m} {} is.

Got questions? Get instant answers now!

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.

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

(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?

Got questions? Get instant answers now!

(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

Got questions? Get instant answers now!

(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.

Got questions? Get instant answers now!

(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} } } {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

(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"} {}

Got questions? Get instant answers now!

(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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask