Relate the linear momentum of a photon to its energy or wavelength, and also apply linear momentum conservation to an easy processes involving the emission, absorption, or have fun of photons.Account qualitatively for the boost of photon wavelength that is observed, and explain the definition of the Compton wavelength.

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Measuring Photon Momentum

The quantum of EM radiation we speak to a photon has actually properties analogous come those of corpuscle we can see, such as grains the sand. A photon interacts as a unit in collisions or as soon as absorbed, quite than as comprehensive wave. Enormous quanta, favor electrons, additionally act like macroscopic particles—something us expect, since they room the smallest systems of matter. Particles lug momentum as well as energy. Despite photons having actually no mass, there has actually long been evidence that EM radiation tote momentum. (Maxwell and also others who studied EM waves predicted the they would lug momentum.) that is now a well-established truth that photons do have momentum. In fact, photon momentum is said by the photoelectric effect, whereby photons hit electrons the end of a substance. Number 1 shows macroscopic proof of photon momentum.


Figure 1. The tails of the Hale-Bopp comet suggest away native the Sun, proof that light has actually momentum. Dust create from the human body of the comet develops this tail. Particles of dust are pushed away from the sunlight by light mirroring from them. The blue ionized gas tail is also produced through photons interacting with atoms in the comet material. (credit: Geoff Chester, U.S. Navy, via Wikimedia Commons)


Figure 1 shows a comet v two prominent tails. What most civilization do not know around the tails is that they always suggest away native the Sun rather than trailing behind the comet (like the tail that Bo Peep’s sheep). Comet tails space composed the gases and dust evaporated native the human body of the comet and also ionized gas. The dust corpuscle recoil away from the Sun as soon as photons scatter native them. Evidently, photons bring momentum in the direction the their movement (away native the Sun), and also some the this momentum is transferred to dust particles in collisions. Gas atoms and also molecules in the blue tail room most affected by various other particles of radiation, such together protons and also electrons emanating from the Sun, quite than by the momentum of photons.


Making Connections: conservation of Momentum

Not just is inert conserved in all realms of physics, however all varieties of corpuscle are found to have actually momentum. We suppose particles with mass to have actually momentum, however now we view that massless particles including photons also carry momentum.


Figure 2. The Compton effect is the name offered to the scattering that a photon by an electron. Energy and momentum space conserved, bring about a palliation of both for the scattered photon. Studying this effect, Compton proved that photons have momentum.


Momentum is conserved in quantum mechanics just as it is in relativity and classical physics. Few of the earliest straight experimental evidence of this came from scattering that x-ray photons by electron in substances, named Compton scattering after ~ the American physicist Arthur H. Compton (1892–1962). About 1923, Compton observed that x rays scattered from materials had a decreased energy and correctly analyzed this together being because of the scattering that photons from electrons. This phenomenon could be tackled as a collision between two particles—a photon and an electron at rest in the material. Energy and momentum room conserved in the collision. (See figure 2) He won a Nobel prize in 1929 because that the exploration of this scattering, now dubbed the Compton effect, since it aided prove the photon momentum is offered by p=frachlambda\, where h is Planck’s continuous and λ is the photon wavelength. (Note the relativistic momentum given as p = γmu is valid only for particles having mass.)

We have the right to see that photon momentum is small, due to the fact that p=frachlambda\ and h is really small. It is for this reason that we execute not ordinarily observe photon momentum. Our mirrors carry out not recoil as soon as light reflects from them (except possibly in cartoons). Compton saw the impacts of photon momentum because he was observing x rays, which have actually a tiny wavelength and a relatively large momentum, connecting with the lightest of particles, the electron.


Example 1. Electron and Photon momentum Compared

Calculate the momentum of a clearly shows photon that has actually a wavelength of 500 nm.Find the velocity of an electron having actually the very same momentum.What is the energy of the electron, and also how does the compare with the power of the photon?Strategy

Finding the photon inert is a straightforward application of its definition: p=frachlambda\. If we uncover the photon momentum is small, climate we deserve to assume that an electron with the same momentum will be nonrelativistic, making it straightforward to find its velocity and also kinetic energy from the classic formulas.

Solution for component 1

Photon momentum is given by the equation: p=frachlambda\.

Entering the given photon wavelength yields

displaystylep=frac6.63 imes10^-34 ext J cdot ext s500 imes10^-9 ext m=1.33 imes10^-27 ext kgcdot ext m/s\

Solution for Part 2

Since this momentum is without doubt small, we will use the timeless expression pmv to find the velocity of one electron v this momentum. Solving for v and also using the well-known value for the massive of one electron gives

displaystylev=fracpm=frac1.33 imes10^-27 ext kgcdot ext m/s9.11 imes10^-31 ext kg=1460 ext m/sapprox1460 ext m/s\

Solution for Part 3

The electron has actually kinetic energy, which is classically provided by extKE_e=frac12mv^2\.

Thus, extKE_e=frac12left(9.11 imes10^-3 ext kg ight)left(1455 ext m/s ight)^2=9.64 imes10^-25 ext J\.

Converting this come eV by multiply by frac1 ext eV1.602 imes10^-19 ext J\ yields KEe = 6.02 × 10−6 eV.

The photon power E is

E=frachclambda=frac1240 ext eVcdot ext nm500 ext nm=2.48 ext eV\,

which is about five order of magnitude greater.

Discussion

Photon momentum is undoubtedly small. Even if we have huge numbers of them, the complete momentum they lug is small. An electron through the exact same momentum has actually a 1460 m/s velocity, i m sorry is clearly nonrelativistic. A an ext massive bit with the very same momentum would have actually an also smaller velocity. This is borne the end by the fact that that takes far less energy to give an electron the very same momentum as a photon. Yet on a quantum-mechanical scale, especially for high-energy photons interacting with little masses, photon momentum is significant. Also on a huge scale, photon momentum can have an impact if there are sufficient of them and if over there is naught to prevent the slow-moving recoil the matter. Comet tails are one example, but there are additionally proposals come build space sails the use large low-mass mirrors (made that aluminized Mylar) come reflect sunlight. In the vacuum that space, the mirrors would slowly recoil and also could actually take spacecraft from place to place in the solar system. (See number 3.)


Figure 3. (a) space sails have been proposed that usage the momentum of sunlight mirroring from large low-mass sails come propel spacecraft around the solar system. A Russian test design of this (the Cosmos 1) was launched in 2005, however did not make it into orbit due to a rocket failure. (b) A U.S. Variation of this, labeled LightSail-1, is booked for psychological launches in the very first part the this decade. The will have actually a 40-m2 sail. (credit: Kim Newton/NASA)


Relativistic Photon Momentum

There is a relationship in between photon momentum p and also photon energy E the is continual with the relation offered previously for the relativistic complete energy the a particle as E2 = (pc)2 + (mc)2. We know m is zero for a photon, yet p is not, so that E2 = (pc)2 + (mc)2 becomes E = pc, or p=fracEc\ (photons).

To examine the validity that this relation, note that E=frachclambda\ for a photon. Substituting this into p=fracEc\ yields

displaystylep=fracfrachclambdac=frachlambda\,

as determined experimentally and also discussed above. Thus, p=E/c is tantamount to Compton’s result p=h/λ. For a additional verification the the relationship between photon energy and also momentum, see example 2.


Photon Detectors

Almost every detection systems talked around thus far—eyes, photographic plates, photomultiplier tube in microscopes, and also CCD cameras—rely top top particle-like nature of photons interacting with a sensitive area. A readjust is caused and either the readjust is cascaded or zillions of points are tape-recorded to type an image we detect. These detectors are used in biomedical imaging systems, and there is continuous research into boosting the effectiveness of receiving photons, an especially by cooling detection systems and reducing heat effects.


Example 2. Photon Energy and also Momentum

Show that p=fracEc\ for the photon considered in the instance 1.

Strategy

We will certainly take the power E discovered in instance 1, division it through the speed of light, and see if the very same momentum is acquired as before.

Solution

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

p=fracEc=fracleft(2.48 ext eV ight)left(1.60 imes10^-16 ext J/eV ight)3.00 imes10^8 ext m/s=1.33 imes10^-27 ext kg cdot ext m/s\

Discussion

This value for momentum is the exact same as found prior to (note that unrounded values are supplied in every calculations to stop even tiny rounding errors), an supposed verification the the connection p=fracEc\. This also way the relationship between energy, momentum, and also mass provided by E2 = (pc)2 + (mc)2 applies to both matter and photons. As soon as again, note that p is no zero, even when m is.


Problem-Solving Suggestion

Note the the forms of the constants h = 4.14 × 10–15 eV ⋅ s and also hc = 1240 eV ⋅ nm might be particularly useful because that this section’s Problems and Exercises.


Section Summary

Photons have actually momentum, provided by p=frachlambda\, where λ is the photon wavelength.Photon energy and also momentum are connected by p=fracEc\, where E=hf=frachclambda\ because that a photon.

Conceptual Questions

Which formula might be offered for the inert of every particles, through or there is no mass?Is there any type of measurable difference between the momentum of a photon and also the momentum of matter?Why don’t us feel the momentum of sunlight once we are on the beach?

Problems & Exercises

(a) uncover the inert of a 4.00-cm-wavelength microwave photon. (b) talk about why you mean the answer come (a) to be an extremely small.(a) What is the inert of a 0.0100-nm-wavelength photon that can detect details of an atom? (b) What is its energy in MeV?(a) What is the wavelength the a photon that has actually a inert of 5.00 × 10−29 kg · m/s? (b) discover its power in eV.(a) A γ-ray photon has a inert of 8.00 × 10−21 kg · m/s. What is that wavelength? (b) calculate its energy in MeV.(a) calculation the momentum of a photon having a wavelength of 2.50 μm. (b) discover the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and also how does that compare with that that the photon?Repeat the previous trouble for a 10.0-nm-wavelength photon.(a) calculate the wavelength that a photon that has actually the same momentum together a proton relocating at 1.00% that the rate of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?(a) discover the momentum of a 100-keV x-ray photon. (b) uncover the tantamount velocity of a neutron through the very same momentum. (c) What is the neutron’s kinetic power in keV?Take the ratio of relativistic remainder energy, = γmc2, come relativistic momentum, = γmu, and show the in the limit the mass philosophies zero, you discover fracEp=c\.Construct Your own Problem. Consider a an are sail together as mentioned in instance 1. Build a difficulty in which you calculate the light push on the sail in N/m2 created by mirroring sunlight. Also calculate the pressure that might be produced and also how much result that would have on a spacecraft. Among the things to be considered are the strongness of sunlight, its average wavelength, the variety of photons per square meter this implies, the area that the room sail, and also the massive of the mechanism being accelerated.Unreasonable Results. A vehicle feels a little force because of the irradiate it sends out indigenous its headlights, equal to the inert of the light split by the moment in which the is emitted. (a) calculation the power of each headlight, if they exert a full force of 2.00 × 10−2 N backward on the car. (b) What is unreasonable about this result? (c) Which presumptions are insignificant or inconsistent?

Glossary

photon momentum: the lot of inert a photon has, calculation by p=frachlambda =fracEc\

Compton effect: the phenomenon by which x light ray scattered from products have lessened energy


Selected solutions to Problems & Exercises

1. (a) 1.66 × 10−32 kg ⋅ m/s; (b) The wavelength the microwave photons is large, for this reason the momentum they lug is an extremely small.

3. (a) 13.3 μm; (b) 9.38 × 10−2 eV

5. (a) 2.65 × 10−28 kg · m/s; (b) 291 m/s; (c) electron 3.86 × 10−26 J, photon 7.96 × 10−20 J, ratio 2.06 × 106

7. (a) 1.32 × 10−13 m; (b) 9.39 MeV; (c) 4.70 × 10−2 MeV

9. = γmc2 and = γmu, so

displaystylefracEP=fracgammamc^2gammamu=fracc^2u\

As the mass of particle approaches zero, that velocity will strategy c , so that the ratio of energy to momentum in this border is

displaystylelim_m o0fracEP=fracc^2c=c\

which is continuous with the equation because that photon energy.

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11. (a) 3.00 × 106 W; (b) Headlights are way too bright; (c) force is as well large.