2011 Ph.D Preliminary Examination

Transcription

1 2011 Ph.D Preliminary Examination Part I. Short-questions (Answer 10 out of 16) Instructions: 1. This is a 3-hour close book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid. 2. This exam is graded on 10 questions; all questions are of equal value. It needs to be specified on the first page if more than 10 are attempted and which ones should be marked. 3. Write down your name and student ID on the exam booklet. Write down clearly the question number. 4. This paper has 5 pages including the title page. Examiners: Keshav Dasgupta Paul Francois Guillaume Gervais Hong Guo (Chair of Exam Committee) Shaun Lovejoy 1/5

2 1. Lagrangian Write down the Lagrangian for two bodies orbiting around each other in a plane, under a gravitational potential, and using the reduced mass μ. Show that angular momentum is conserved for this Lagrangian. 2. Space walkers Two space walkers, one with a mass of 120 kg and the other with a mass of 80 kg, hold on to the ends of a very light cable 10m long. The astronauts are originally at rest in deep space. (a) If the astronauts then start pulling themselves toward each other along the cable until they meet, roughly how far will the less massive person move? (b) Suppose one astronaut just holds on tight while the other does the pulling. Would that change your answer? Why or why not? (c) If the cable s mass is taken into account, how would this change your answers? 3. Crawling bug Consider a bug crawling on a horizontal turntable rotating with constant angular velocity ω. We call x, y its position in the rotating frame of reference and x', y' its position in the lab frame of reference. Assume the Lagrangian is i.e. the potential depends on the position in the rotating frame of reference. Compute the Lagrangian in the turntable coordinate, define p x, p y momenta associated to x, y and show that the Hamiltonian in the rotating frame of reference is 4. Magnetic field The figure shows a cross-sectional view of a hollow long cylindrical conductor with inner and outer radii of a = 2 cm and b = 4 cm. This conductor carries a current of 4.0A uniformly spread over its cross-section. Find the magnitude of the magnetic field at the following distances, measured from the centre of the hollow cylinder: (a) r = 1.5 cm; (b) r = 3 cm; (c) r = 8 cm. 2/5

3 5. Charged spheres A small uncharged metallic sphere is positioned exactly in the midpoint between two charges whose absolute values are the same but whose signs are opposite. Suppose the sphere is shifted somewhat. Will it remain in the new position or will it move in some direction? Give reasons. Now, a slight variant of the above problem is to replace the two charges by a parallel plate capacitor, with the metallic sphere now being suspended by a long nonconducting string in the region between the vertically positioned plates of the capacitor, closer to one plate than to the other. This is illustrated as: Discuss in details how the sphere will behave now, for example whether the sphere will touch the plates, and if yes, the subsequent behavior of the sphere. 6. Strange field Suppose that the electric field due to a point charge were square law. instead of the usual inverse (a) Find E. (b) Would Gauss s Law hold true for this field? Defend your answer. (c) Would the excess charge on a conductor lie on its surface? Defend your answer. 3/5

4 7. Ising chain Consider a chain of spin forming a 1D Ising model, H J i i 1. Assume spin 1 to j are up and spin j+1 to N are down. Compute energy (with respect to the ground state) and entropy, and deduce from your result that the 1D Ising model is always disordered. 8. Physics of cooking Butchers claim that you should cook in oven a roast beef 15 minutes per pound of flesh. Assuming a roast beef is cooked when temperature at its center reaches a given temperature T c, find out if your butcher's scaling law is plausible and if not, give the correct scaling law. 9. Free particles Consider a system of N>>1 non-interacting particles in which the energy of each particle can assume two and two distinct values: 0 and E (E>0). Denote by n 0 and n 1 the occupation numbers of the energy levels 0 and E, respectively. The fixed total energy of the system is U. (a) Find the entropy of the system. (b) Find the temperature as a function of U. For what range of values of n 0 it T < 0? (c) In which direction does heat flow when a system of negative temperature is brought into thermal contact with a system of positive temperature? Why? 10. Earthquake Some people claimed that they once read tweets about an earthquake just before actually feeling it themselves. Does that sound possible? 11. Polarizing light Discuss qualitatively a simple way to polarize light using simply two medium with different refraction index n 1 and n 2 and a beam of un-polarized light. 12. Interfering bodies Calculate the de Broglie wave associated with you running at 1 m/s and with a mass of 100 kg (you can use h=10-34 J.s). Redo the calculations now assuming that the Planck constant is given by h=1 J.s. Using again h=1 J.s, what do you conclude on the length scale required for you to interfere with yourself in a slit experiment? 13. Bosons Show that the symmetric combination of two single particle wavefunctions ab (r 1,r 2 )= a (r 1 ) b (r 2 )+ a (r 2 ) b (r 1 ) displays the exchange symmetry characteristic of bosons. Is it possible for two bosons to occupy the same quantum state? Explain your answer. N i 4/5

5 14. Spins An electron is in the spin state: (a) Determine the normalization constant A. (b) If you measure S z, what values might you get, and what is the probability of each? (c) Suppose that when you measured S z you obtained / 2. You then immediately measure S y. What values might you get, and what is the probability of each? (d) Finally, suppose that after measuring S y in part (c) you re-measure S z. For each of the possible outcomes of part (c), what is the probability of obtaining /2again? i 1 0 Note, =, =, =. x 1 0 y i 0 z Bart and Lisa Homer Simpsons two kids Bart and Lisa decide to test the special theory of relativity. Lisa goes on a trip on a spaceship moving very close to the speed of light, say 0.98c. When she meets her brother after the trip she finds that her brother has aged considerably. This puzzles her because she thought that in the spaceship she was in the rest frame and her brother Bart was in a frame moving at the speed 0.98c. So, if special theory is right, her brother shouldn't have aged so much, and therefore when they meet they should have had the same age difference! Is Lisa's conclusion right, or there is some way you can argue for the age difference between the two Simpsons' kids after the trip? Give detailed reasoning. 16. Speed Consider two particles having equal rest mass m o and both moving in the +x direction. In the lab frame, particle 1 has momentum 5m o c, particle 2 has momentum 10m o c. (a) In the lab frame, what is the speed of both particles? (b) In the rest frame of particle 1, what is the speed of particle 2? You may need the following formula: ux v u x' v u x' or u x vu x vu x' c c END /5