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(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.

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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.

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

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
David Reply
hour glass, pendulum clock, atomic clock?
S.M
tnks
David
how did they solve for "t" after getting 67.6=.5(Voy + 0)t
Martin Reply
Find the following for path D in [link] : (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
David Reply
the topic is kinematics
David
can i get notes of solid state physics
Lohitha
just check the chpt. 13 kinetic theory of matter it's there
David
is acceleration a fundamental unit.
David Reply
no it is derived
Abdul
no
Nisha
K thanks
David
no it's not its derived
Emmanuel
hi
Gift
Hello
Gift
hello gift
Emmanuel
hello
David
Hello Emmanuel
Gift
how are you gift
Emmanuel
I'm good
Gift
that's good
Emmanuel
how are you too
Gift
am cool
Emmanuel
spending time summarizing
Emmanuel
broadening my horizon
Emmanuel
I am fin
Longwar
ok
Gift
hi guys can you teach me how to solve a logarithm?
Villaflor Reply
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
Sagcurse Reply
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
Lila Reply
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
Emmanuel Reply
who can help me with my problem about acceleration?
Vann Reply
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
adefenwa Reply
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
Okwethu Reply
v=d/t
Emeka
his about the speed?
Villaflor
how about speed
Villaflor
v=d/t
Nisha
hello bro hw is life with you
Jacob Reply
Mine is good. How about you?
Chase
Hi room of engineers
lawan Reply
yes,hi sir
Okwethu
hello
akinmeji
Hello
Mishael
hello
Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
u are all wlc just ask your question anybody. can answer
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
Riaz Reply
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
Badmus
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
Olatunji Reply
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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