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PHOTONS (Answers)We have been looking at the wave properties of electromagnetic radiation. As with other types of waves, the rate an electromagnetic wave transmits energy is proportional to the square of its amplitude. But (strangely) electromagnetic radiation also behaves like a stream of particles called photons. Each photon is a “packet” of electromagnetic energy. The energy of each photon is directly proportional to its frequency and inversely proportional to its wavelength and can be calculated using the formula: Ephoton = hf = hc/?. The constant of proportionality, h, is called “Planck’s Constant” and is equal to 6.63x10-34 J s. Problem 1. At the atomic level it is often useful to work with nanometers instead of meters, and with electron-volts instead of Joules. Use factor label conversion to show that Planck’s constant times the speed of light is equal to 1240 eV nm. (c=3x108 m/s, 1eV=1.6x10-19 J)hc = 6.63x10-34 J s x 3.0 x 108 m/s x (1 eV/1.6x10-19 J) x (1nm/10-9m) = 1243 eV nm Problem 2. What is the energy of each photon for the wavelengths and frequencies shown? Give answers in both Joules and electron-volts.?=240 nm (ultraviolet)8.3x10-19 J5.2 eVF=104.3 MHz (radio)6.9x10-26 J4.3x10-7 eV?=21 cm (microwave)9.5x10-25 J5.9x10-6 eV?=650 nm (red light)3.1x10-19 J1.9 eVProblem 3. Find the wavelength and frequency of photons with the following energies:Ephoton=1200 eV (x-ray)1.0 nm3.0x10-17 HzEphoton=4.2x10-19 J (blue light)476 nm6.3x1014 HzEphoton=0.51 MeV (gamma)2.4 pm1.25x1020 HzMASS-ENERGY EQUIVALENCE AND PAIR PRODUCTIONEinstein’s famous equation E=mc2 describes how much energy is associated with the mass of a particle. If enough energy is available new particles can “pop” into existence. These new particles are always produced in particle/anti-particle pairs because or other conservation laws. So, a photon with sufficient energy could vanish and be replaced by an electron and a positron, or a proton and anti-proton.Problem 4. Use Einstein’s equation to find out how much energy is associated with the following particles. Give answers in both Joules and electron-volts.Electron (m=9.1x10-31 kg)8.2x10-14 J0.51 MeVPositron (m=0.51 MeV/c2 )8.2x10-14 J0.51 MeVProton (1.67x10-27 kg)1.5x10-10 J0.94 GeVMuon (m=106 MeV/c2)1.7x10-11 J106 MeVHigg’s Boson (m=2.3x10-25 kg)2.1x10-9 J130 GeVProblem 5. Find the minimum frequency and corresponding wavelength of photon that could produce an electron/positron pair. (Ephoton = mc2particle + mc2anti-particle)0.51 MeV + 0.51 MeV = hc/? –> ?=1.2x10-12 mf=c/?=2.5x1020 HzProblem 6. Less than 1% of the time a Higg’s Boson decays into 2 photons. Find the minimum frequency and corresponding wavelength of photons produced by this process. (Each photon would get half of the energy associated with the particle’s mass)(mc2)Higgs=130 GeVEphoton=65 GeV = hf = hc/?f=6.5x1010/6.63x10-34=1.0x1044 Hz?=1243 eVnm/6.5x1010eV = 1.91x10-8 nm = 1.91x10-17mVOLTAGE AND ELECTRIC ENERGYElectric charge is a fundamental property of particles. In the SI system it is measured in units called Coulombs and is represented in formulas by the letter q. Voltage (a.k.a. electric potential difference) describes the change in potential energy per unit charge. The units of potential difference are volts, where 1 V = 1 J/C. As a falling object is pulled down by gravity it loses gravitational potential energy, but gains kinetic energy. Similarly, a charge accelerated through a potential difference increases its kinetic energy by an amount equal to the potential energy it loses (?KE=-?U).Problem 7. Describe the change in gravitational potential energy and kinetic energy when a 3.0 kg rock falls 42 m. What is the final speed of the rock if it was initially at rest? (?Ugravitational=mg?h, K=1/2 mv2)(mgh)i + (1/2 mv2)i = (mgh)f + (1/2 mv2)f v=(2gh)1/2 =29 m/sProblem 8. Describe the change in electric potential energy and kinetic energy when an electron is accelerated through a 12 V potential difference. Give your answers in both Joules and electron-volts. What is the final speed of the electron if it was initially at rest? (?Uelectric=q?V)?KE = -q?V = 12 eV = 1.9x10-18 Jv=(2K/m)1/2=2x106 m/s(v=velocity, V=Voltage)Problem 9. The radiation produced by x-ray machines is a result of accelerating electrons through a potential difference towards a metal target. What is the maximum frequency of x-ray photons if the electrons are accelerated through a 25 kV potential difference? hf=q?V f=q?V/h=6.0x1014 HzDeBROGLIE WAVESAlthough it is difficult to imagine how electromagnetic radiation can have a dual wave-particle nature, the photon hypothesis has successfully explained many observed phenomena such as the photo-electric effect, blackbody radiation, and Compton scattering. So, if a wave can also be particle-like, can a particle be wave-like? This was the question posed by Louis deBroglie in 1924. The answer to the question is yes, particles can and do exhibit wave behavior. All particles have a “deBroglie wavelength” which is given by the formula: ?=h/p (where p is the particle’s momentum).Problem 10. What is the deBroglie wavelength of a proton moving at 3000 m/s??=h/(mv) = 1.3x10-10 mProblem 11. The electrons in an electron microscope are accelerated through a 1.2 kV potential difference before striking the sample. What is the deBroglie wavelength of the electrons?(Hint: Use what you have learned to find the kinetic energy and the speed of the electrons).p=mvK=1/2 mv2 = p2/2m = qV p=(2qVm)1/2?=h/p=h(2qVm)-1/2=3.5x10-11 mReview questions:What is a photon? How is the energy of photons related to their frequency and wavelength? Photons are ‘packets’ or particles of electromagnetic energy. The energy of each photon is directly proportional to the frequency and inversely proportional to the wavelength. (It is really weird to think about a particle having a wavelength but you have to get used to this sort of weirdness at the atomic level.) What are three observable phenomenon which can be explained by the photon hypothesis? Photo-electric effect, black body radiation, Compton scatteringWhat is pair-production? Is mass conserved in pair-production? Is energy? Pair production is when a photon has enough energy for a particle and an anti-particle to pop into existence. Mass is not conserved since the photon has no mass, but the particle and anti-particle do. Energy is conserved because there is energy associated with the mass of the particles. What are the SI units of electric charge? Of voltage? In the SI system electric charge is measured in Coulombs and voltage is measured in Volts.What is the relationship between voltage and energy? Voltage (a.k.a. electric potential) describes how much energy per unit charge.How do you calculate the kinetic energy gained by a charged particle that is accelerated through a potential difference? Multiply the charge of the particle by the potential difference to determine how much the potential energy has decreased. This is equal to the kinetic energy gained.A particle with charge qo and mass mo is initially at rest. After being accelerated through a potential difference Vo the particle has kinetic energy Ko, speed vo, momentum po, and wavelength ?o. What would happen to each of the following quantities if the particle was accelerated through a potential difference of 4Vo?electric charge The charge would still be qo since it is the same particle.MassThe mass would still be mo since it is the same particle.kinetic energyKE would equal 4Vo since voltage and energy are directly proportional.SpeedSpeed would equal 2vo since speed is proportional to the square root of KEMomentumMomentum would equal 2po since momentum is directly proportional to velocity )Wavelength Wavelength would equal ? ?o since deBroglie wavelength is inversely proportional to momentum ................
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