Fall09_solutions

Fall09_solutions - Physics Qualifying Examination Problems...

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Physics Qualifying Examination Problems 1–6 Thursday, August 27, 2009 1–5 pm Problems 7–12 Friday, August 28, 2009 1-5 pm 1. Solve each problem. 2. Start each problem solution on a fresh page. You may use multiple pages per problem. 3. At the top of each solution page put the problem number (1–12) and your Social Security number, but not your name or any other information.
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Problem 1
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Problem 1 Solution
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Problem 2 Problem 2 Solution
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Problem 3 A photon with wavelength 24.8 fm strikes a proton at rest. The photon undergoes Compton scattering, and the scattered photon is seen by an observer in the lab to be emitted at 180 ° with respect to the direction of the incident photon. The mass of the proton is M p = 938 MeV/c 2 , and hc = 1240 MeV·fm. a) What is the energy of the incident photon? What name would typically be given to classify this “type” of photon? Give a very brief explanation for your choice. b) Using relativistic kinematics, find (i) the wavelength of the scattered photon and (ii) the de Broglie wavelength of the recoiling proton. c) If we could observe the reaction occurring in the center-of-mass frame instead of the lab frame, what would we then see as the difference between the wavelengths of the incoming and scattered photons? Explain your answer. Problem 3 Solution
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Problem 4 A thin fiber of length L is stretched between two supports. The speed of propagation of transverse waves on the fiber is c for both polarizations. (a) What is the contribution of these modes to the heat capacity of the fiber at low temperatures, assuming ħ c/L << k B T? (Don’t worry about solving the integral to find the dimensionless prefactor) (b) What is the heat capacity for ħ c/L >> k B T? Problem 4 Solution: (a) Boson modes: k = n π /L (n = 1, 2, 3, …) Let ω = kc = 1 2 / kT e E ω = = For ħ c/L << k B T, = = 0 2 0 / 0 / 1 2 1 2 1 2 x kT kT e x dx kT c L e d c L e dk L E = = = = = = π Letting x = ħω /k B T, and solving for the heat capacity: T c Lk T E C B = 2 4 ~ = Or, C ~ T as expected for 1D bosons. (b) For ħ c/L >> k B T, we have: = = LkT c LkT cn kT e L c n e n L c e E / / / 2 1 ) / ( 2 1 2 = = = = = = At low temperatures, only the n = 1 mode is occupied, so we have: / 2 cL kT c Ee L = = And the heat capacity is: LkT c B e T k L c T E C / 2 2 2 = = = =
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Problem 5 A classical particle of mass m moves in a circular orbit under the influence of a central force V = -mk/r n , where k is a constant. (a) Write down an effective potential function for this system (centrifugal potential + central potential). (b) Using the conditions for a circular orbit, solve for the condition on n under small perturbations of the radius. Problem 5 Solution: (a) 2 2 2 2 2 2 mr L r km r m r km V n n eff + = + = ω (b) Circular orbit: 0 0 3 2 1 = = + mr L r kmn r V n eff Stable orbit: 2 2 22 (1 ) 3 00 eff n V kmn n L rr m + + >→ + > 4 r Putting these two equations together, we have: 2 0 3 ) 1 ( 0 3 ) 1 ( 4 2 4 2 < > + + > + + n n mr L mr n L
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Problem 6 A parallel-plate capacitor is made of two plates of surface area A
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Fall09_solutions - Physics Qualifying Examination Problems...

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