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CALTECH PH 136A - FINAL EXAM

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Physics 136a December 2004FINAL EXAMYou will be expected to answer 8 of the following questions in three hours, closed book.The questions might carry different weights. When you are ready to take the exam, pickup a list of question numbers and their weights from Shirley Hampton, Room 151 BridgeAnnex, or from JoAnn Boyd, Room 161 West Bridge. Please write your solutions in a bluebook, which can be purchased at the Caltech bookstore for a nominal cost. Completedexams must be placed in Kip Thorne’s mailbox at the west end of Bridge Annex before5:00 PM Friday, December 10. Late exams will not be accepted.Grades in this course will be determined as follows:(i) Students who score 60% or more on the homeworks will pass the course without havingto take the final. Those students who have petitioned for letter grades will receive agrade based on the homework, plus the exam if the exam is taken.(ii) Students who are failing on the basis of homeworks and those who wish to improvetheir letter grades should take this final exam. For such students, the grade will bedetermined by a combination of homework performance and exam performance, withthe proviso that the grade will be no lower than would be received based solely onhomework scores.1. Using a spacetime diagram, exhibit the Lorentz contraction of a moving, rigid rod, thetime dilation of a moving clock, and the fact that the concept of simultaneity is not frameinvariant.2. Explain the geometrical meanings of the direction of a volume vector~Σ and the sense of~Σ. What is the relationship between the direction and the sense for the case of a timelike~Σ, a spacelike~Σ, and a null~Σ?3. Without using any coordinate system or basis, explain the meaning of “stress-energytensor”, “electromagnetic field tensor”, and “Levi-Civita tensor”.4. Show that the number density of particles in phase space is Lorentz invariant (i.e. isthe same in all reference frames, independently of the frames’ speed of motion).5. Millions of asteroids move in orbits around the Sun, which has a Newtonian gravita-tional field Φ(x). Ignoring collisions between the asteroids, prove that their kinetic-theorydistribution function satisfies the collisionless Boltzman equation.6. A white dwarf star made of pure hydrogen gas has almost all its mass provided by thehydrogen’s protons and almost all its pressure by the hydrogen’s electrons. The star is inthermodynamic equilibrium with an electron chemical potential µ = ˜µ + methat varieswith radius but lies in the range 0 < ˜µ  me, and with a temperature T  ˜µ/k. Herek is Boltzmann’s constant and meis the electron rest mass. Sketch a derivation of theequation of state P = P (ρ) of the star’s matter7. For photons derive the relationship between the number density in phase space, themean occupation number, and the specific intensity.8. Explain what is meant by a “transport coefficient” and sketch the steps by which onecan compute transport coefficients from the Boltzmann transport equation.9. The shear of a (nonrelativistic) fluid is defined to be the trace-free part of the sym-metrized gradient of its velocity, σij= v(i,j)−13δijvk,k. Viscosity resists this shear byproducing a stress Tij= −2ησij. Here η is the coefficient of shear viscosity. Make a roughestimate of η for photons in a shearing hydrogen gas at a temperature T = 105K anddensity ρ = 10−16g cm−3.10. What is the statistical equilibrium distribution function that describes an ensembleof systems of the following type: Each system is a collection of photons in the interiorof a spherical container whose walls freely emit and absorb the photons. The containerhas volume V and temperature T , and it rotates with angular velocity Ω. What is thedistribution function for a subensemble of this ensemble in which each subsystem is a modeof the electromagnetic field that has angular frequency ω and spherical harmonic indicesl, m?11. How would one measure the chemical potential of the nitrogen molecules in a room fullof air?12. Explain what is meant by the ergodic hypothesis in statistical mechanics, and give oneconcrete example of how it could be applied.13. Explain what is meant by canonical ensemble, grand canonical ensemble, microcanon-ical ensemble, and Gibbs ensemble. Give an example of a system whose fluctuations awayfrom equilibrium are characterized by the canonical ensemble.14. Explain quantitatively the relationship between the second law of thermodynamics andthe principle that, when a system held at fixed pressure and temperature can exist in twophases, it will evolve toward the phase with minimum Gibbs potential.15. Explain the difference between a first-order phase transition and a second-order phasetransition, and give one example of each.16. A mixture of hydrogen and oxygen is inserted into a chamber with rigid, perfectlyinsulating walls. A spark ignites an explosion in which much of the hydrogen and oxygenforms water. Since no heat is added to or removed from the system through its walls,and one normally says that (heat added) = (temperature)×(change in entropy), one mightthink that the explosion leaves the system’s entropy unchanged. Explain why this is notso, and explain how you would go about computing how much water is formed and howmuch entropy increase accompanies that formation.17. Explain the meaning of correlation function and spectral density and the relationshipbetween them. If a correlation function has a relaxation time τr, what does that implyabout the form of its spectral density?18. A 100 gram pendulum with a 1Hz swinging frequency is set swinging with a largeamplitude in air at room temperature (300 K). The air molecules produce drag on thependulum, causing its swinging to die out with an amplitude e-folding time τ∗= 1 hour.The pendulum then is placed absolutely at rest and absolutely vertical, and its subsequentmotion is measured. Describe that subsequent motion in as much detail as you can, andas quantitatively as you can.19. A physics experiment produces a very weak voltage signal whose time dependence hasthe following form:tVVoτThis voltage signal is contaminated by noise with spectral densitySV(f) =constantf2The experimenter wants to know whether the signal is actually present or not, and ifpresent, what its amplitude is. To answer this question, the experimenter performs optimalsignal processing. Sketch the form of the optimal filter K(t) that the experimenter shoulduse, and


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CALTECH PH 136A - FINAL EXAM

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