2008 Ph.D Preliminary Examination

Transcription

1 008 Ph.D Preliminary Examination Part I. Short-questions (Answer 0 out of 5) Instructions:. This is a 3-hour close book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid.. This exam is graded on 0 questions; all questions are of equal value. It needs to be specified on the first page if more than 0 are attempted. 3. Write down your name and student ID on the exam booklet. Write down clearly the question number. 4. This paper has 5 pages. Examiners: Hong Guo (Chair), Roland Bennewitz, Keshav Dasgupta, Gil Holder, Sangyong Jeon, Brigitte Vachon /5

2 . Ratio of mass density One wishes to estimate the ratio of mass density between the Sun and the Earth. Assuming both are spherical objects with uniform density. The following parameters are given: From the Earth, o the angular diameter of the Sun is measured to be θ =/ ; the Earth s radius is R e 6000km ; 7 one year of time 3 0 s ; g 0ms. (Hint: the gravitational force causes a circular motion whose frequency you know; the same force also gives g).. Wireless charger One kind of electric toothbrush has a base that holds the toothbrush handle when not in use, as shown in the figure below. The base is plugged into the wall electric outlet. The toothbrush handle has a cylindrical hole that fits loosely over a matching cylinder on the base, all surfaces are plastics. When one finishes using the toothbrush, it is returned to sit on the base. Somehow, the rechargeable battery inside the toothbrush gets charged. Write down your guess on how this wireless charger works (obviously, the trick is of E&M origin). 3. Sound velocity in copper The elastic modulus of copper is E=0GPa. Give a rough estimate for the sound velocity in copper using dimensional analysis. In order to do so, make some reasonable assumptions about the distance between atoms in a copper crystal and about the mass of one copper atom. One atomic mass unit is kg. 4. Dropping ball If you drop a rubber ball and a metal ball of the same size and shape into a fluid, what will be the ratio of the magnitude of the terminal velocities? The density of the rubber is 000kg/m 3 and the density of the metal is 8000kg/m 3. /5

3 5. Murphy s Law Murphy's Law: A slice of toast falling onto the floor will always land with the buttered side down. Some people claim that the Murphy's Law can be explained easily by classical mechanics. Assume that you support the toast with your finger (with buttered side up) at one of its edges while it starts falling down. (a) Estimate the angular velocity of the toast as it leaves your finger; (b) Estimate from what height does the toast need to fall in order to violate Murphy s Law. Neglect air friction, life is hard enough. The moment of inertia of a slice of toast is I=(ML )/3 with respect to rotation around the edge of the toast. The mass of the slice is M, the length of one edge L. 6. Kepler s nd law Starting from the Lagrangian, L = m r κ + r ϕ + r prove Kepler's second law, namely, The line joining the planet to the Sun sweeps out equal area in equal times. 7. Entropy For each of the following irreversible processes, briefly explain how you can tell that the total entropy of the universe has increased (no need to write too much). (a) Stirring salt into a pot of soup; (b) Scrambling an egg; (c) Cutting down a tree; (d) Burning gasoline in a car engine (e) A water wave washing down a sand castle on beach. 8. Optical fibre An optical fibre is made of a cylindrical core surrounded by a thin layer called cladding''. Can the index of refraction of the cladding layer be greater than the index of refraction of the core? Why? 3/5

4 9. Electron interference A two-slit electron diffraction experiment is done with slits of unequal widths. When only slit- is open, the number of electrons reaching the screen per second is 5 times the number of electrons reaching the screen per second when only slit- is open. When both slits are open, an interference pattern results in which the destructive interference is not complete. Find the ratio of the probability of an electron arriving at an interference maximum to the probability of an electron arriving at an adjacent interference minimum. (Hint: Use the superposition principle of wave functions) 0. Neutrinos Two neutrino states ν a and ν b are mixtures of pure eigenstates: with energies E and E ν a = ( ν + ν ) ν b = ( ν ν ) for the two eigenstates. The system is initially prepared to be entirely in the ν a state. After a time t has evolved, what is the probability that the system will be observed to be in the ν b state?.. Coin toss If I toss two unbiased coins, then the probability of getting two heads is /4. How would this probability change if the coins are quantum-mechanical and satisfy a Bose-Einstein statistics? Give a short answer.. Polarization A laser pointer is handed to you and you are told that it is strongly linearly polarized. How would you determine the angle of polarization? You can use any device that you want, provided that you can fully explain the physics of how that device responds to various polarization states. 4/5

5 3. Pendulum A ball of mass m and charge q is tied to an insulating cord of length L, so that it behaves as a pendulum. The system is hung between two infinite walls with charge densities -Q each (see figure below). The distance between the two infinite walls is d and assuming that the connecting rod to be completely insulating, determine how the time period of the pendulum would change in this setup. Again ignore any radiation from the system due to the motion of the pendulum. 4. Photon emission Prove that it is not possible to conserve both momentum and total relativistic energy in the following situation: A free electron moving at velocity v emits a photon and then continues to move but at a slower velocity v'. 5. Doppler effect In a stationary lab, the frequency of the light coming from a sodium lamp is measured to be 509 THz. If you are in the frame that is receding from the lab frame with a constant velocity, will you read a different frequency? Explain your answer. END 5/5