WeekProb_Mar_30 - D = 150 ∗ 106 km, and Ts = Quiz 10 5800...

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Unformatted text preview: D = 150 ∗ 106 km, and Ts = Quiz 10 5800 K. How does this compare to the observed value of about 300 K? Fall 2006 Due in section week of March 30, 2008 b) For what wavelength is the power output of the Sun maximum? What about for the Earth (assuming Te = 300 K)? c) You The formula in Tipler the Earth is a cold place with the Sun alone. The situation is Problem 1 should have found thatfor Compton scattering relates to the photon which scatters off significantly worsened by the fact that much of the an electron. Suppose, however Greenhouse Effect Sun’s light is about the electron. In 1. Blackbody Radiation and the that we are interested in information reflected at the surface particular, assume why aren’t is traveling down an ice age? In at 10 keV, and it collides While of the Earth, so an electron we in a permanent x-ray tube short, the atmosphere. with a it eV photon head-on. Thefor the is reflected for incoming T+y comes from the Sun, and 2 is good temperature to assume that most 90 to the e , direction (relative Sun Call theasurfaceapproximationphotonSun Ts , thatoff atthe ◦Earthheat the distance to the to the original path).the radius of the Earth Re , and the (after all are maximum see whybothdon’t from Earth D,of What angle is the electron reflected radius of the Sun Rs . to in the visibleas that most this gets through the atmosphere to? (You its suppose Modeling we and worry eyes answer theradiation dispersing cathode fact (Hint: you should understand the blackbodies, have evolved to take advantage of the rays).that visible light gets through), its our about ambient following: derivation of the Compton formula to answer this question See other problem. Earth (which is not a good approximation per area output for the the heat 4 , find a the One suggestion a)to boost to the rest frame of to assume that most ofSun is σTs leaving naive estimate Using is max at that the total power the electrongets then apply compton’s formula for the the for a much different wavelength) and through the atmosphere. If all of photon.) Sun’s the temperature of the Earth by doing the following steps: light gets through the atmosphere, what fraction f of the radiation power coming from (i.) Calculate the total power leaving the Sun. Earth must escape the atmosphere for the Earth to have its observed tempurature (again (ii.) Calculateall of the Sun’sthis that absorbed at the Earth’s surface)? assuming that the fraction of light is is incident on the Earth. (iii.) Calculate the total power leaving the Earth if its temperature is Te . (iv.) Equate the power incident on the Earth from the Sun to the power leaving the Earth to find Te . (They Effect 2. Derivation of Compton should be equal if the Earth is in equilibrium assuming the Earth is a perfect absorber of the Sun’s light.) Use the values Rs = 7∗105 km, Re = 6400 km, 6 D = wavelength i moving in the How does this compare to an electron at rest A photon with 150 ∗ 10 km,λand Ts = 5800 K. +ˆ-direction collides withthe observed value x of about 300 K? with mass m and scatters in the xy-plane. If the angle that the photon is scattered (above the x-axis) Forθwhat the photon’s final power output of fthe Sun maximum? for howabout for the ˆ b) is and wavelength is the wavelength is λ , derive a formula What the wavelength Earth changes T = 300 K)? of the photon(assuming ineterms of m and θ by following the steps below: c) You should have found that the Earth is a cold place with the Sun alone. The situation is a) Call φ the angle that the electron scatters below the x-axis, pi the magnitude of the ˆ significantly worsened by the fact that much of the Sun’s light is reflected at the surface photon’sEarth, so why aren’t we in a permanent ice age? In short,and atmosphere. While of initial momentum, pf the photon’s final momentum, the pe the momentum of the the electron after the collision.assume that formulas for momentum conservation (in both it is a good approximation to Write the most incoming heat comes from the Sun, and x-that most of this gets through the atmosphere (after all its maximum, ine the visible and ˆ and y -directions) and energy conservation in terms of θ, φ, pi , pf p , and m. ˆ our eyes have evolved to take advantage of the fact that visible light gets through), its not a good approximation to assume that most of the heat leaving the Earth (which is 1 max at a much different wavelength) gets through the atmosphere. If all of the Sun’s light gets through the atmosphere, what fraction f of the radiation power coming from Earth must escape the atmosphere for the Earth to have its observed tempurature (again assuming that all of the Sun’s light is absorbed at the Earth’s energy b) Use your formula for energy conservation to write the electronsurface)? after the collision in terms of pi , pf and m, and your formulas for momentum conservation to write the electron momentum components, p cos φ and pe sin φ, in terms of pi , pf and pe . 2. Derivation of Compton Effect e c) You should now have 3 equations so we can in principle use 2 to eliminate the intermediate A photon with pe in the λi moving in the +ˆ-direction collides with an electron prevent variables φ andwavelength third, which wouldxthen be the desired equation. Toat rest with mass algebra, the in the xy-plane. If the angle that use that formidablem and scatters following trick may be helpful: the photon is scattered (above the x-axis) is θ and the photon’s final wavelength is λf , derive a formula for how the wavelength ˆ of the photon changes)in = E 2 − (pe c)2 = E 2 followingcos φ)2 + (pe c sin φ)2 ) (mc2 2 terms of m and θ by − ((pe c the steps below: a) Call φ the angle that the electron scatters below the x-axis, pi the magnitude of the ˆ for the electron and substitute pyour formulas final E, pe cosφ, and p e sinφ from partof photon’s initial momentum, f the photon’s for momentum, and pe the momentum b). There will be many terms but most either formulas for momentum conservation (inyou are the electron after the collision. Write the cancel or simplify. Show that what both left with after the dust settles can conservationas terms of θ, φ, pi , pf , pe , and m. x- and y -directions) and energy be written in ˆ ˆ m(pi c − pf c) = pi pf (1 − cos θ) 11 . d) Finally, write the photon’s initial momemtum pi in terms of λi and its final momentum pf in terms of λf . Show that with these substitutions, Compton’s formula is obtained: λf − λi = h (1 − cos θ) mc Quiz 10 Fall 2006 1. Blackbody Radiation and the Greenhouse Effect Call the surface temperature for the Sun Ts , that for the Earth Te , the distance to the Sun from Earth D, the radius of the Earth Re , and the radius of the Sun Rs . Modeling both as blackbodies, answer the following: 4 a) Using that the total power per area output for the Sun is σTs , find a naive estimate for the temperature of the Earth by doing the following steps: (i.) (ii.) (iii.) (iv.) Calculate the total power leaving the Sun. Calculate the fraction of this that is incident on the Earth. Calculate the total power leaving the Earth if its temperature is Te . Equate the power incident on the Earth from the Sun to the power leaving the Earth to find Te . (They should be equal if the Earth is in equilibrium assuming the Earth is a perfect absorber of the Sun’s light.) Use the values Rs = 7∗105 km, Re = 6400 km, D = 150 ∗ 106 km, and Ts = 5800 K. How does this compare to the observed value of about 300 K? b) For what wavelength is the power output of the Sun maximum? What about for the Earth (assuming Te = 300 K)? c) You should have found that the Earth is a cold place with the Sun alone. The situation is significantly worsened by the fact that much of the Sun’s light is reflected at the surface of the Earth, so why aren’t we in a permanent ice age? In short, the atmosphere. While it is a good approximation to assume that most incoming heat comes from the Sun, and that most of this gets through the atmosphere (after all its maximum in the visible and our eyes have evolved to take advantage of the fact that visible light gets through), its not a good approximation to assume that most of the heat leaving the Earth (which is max at a much different wavelength) gets through the atmosphere. If all of the Sun’s light gets through the atmosphere, what fraction f of the radiation power coming from Earth must escape the atmosphere for the Earth to have its observed tempurature (again assuming that all of the Sun’s light is absorbed at the Earth’s surface)? 2. Derivation of Compton Effect A photon with wavelength λi moving in the +ˆ-direction collides with an electron at rest x with mass m and scatters in the xy-plane. If the angle that the photon is scattered (above the x-axis) is θ and the photon’s final wavelength is λf , derive a formula for how the wavelength ˆ of the photon changes in terms of m and θ by following the steps below: a) Call φ the angle that the electron scatters below the x-axis, pi the magnitude of the ˆ photon’s initial momentum, pf the photon’s final momentum, and pe the momentum of the electron after the collision. Write the formulas for momentum conservation (in both x- and y -directions) and energy conservation in terms of θ, φ, pi , pf , pe , and m. ˆ ˆ 1 ...
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This note was uploaded on 10/05/2009 for the course PHYSICS 7C taught by Professor Lin during the Spring '08 term at University of California, Berkeley.

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