Department of Physics PRELIMINARY EXAMINATION 2016 Part I. Short Questions

Transcription

1 Department of Physics PRELIMINARY EXAMINATION 2016 Part I. Short Questions Thursday May 19th, 2016, 14-17h Examiners: Prof. J. Cline, Prof. H. Guo, Prof. G. Gervais (Chair), and Prof. D. Hanna INSTRUCTIONS Answer 10 questions out of the choice of 16. This is a closed book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid. If you attempt more than ten questions, you should clearly mark which ones should be graded. DO NOT WRITE YOUR NAME; write ONLY you student ID on the exam booklet. Clearly indicate the question number next to each answer. This exam has 7 pages, including this title page.

2 2016 Prelim Short Answers 2 1. Earths Consider the Earth as i) with a constant density of matter, ii) as a thin shell empty sphere and iii) with a constant linear density of matter ρ(r) = ρ 0 r. In all cases, calculate the gravitational potential and the gravitational field everywhere and make a sketch. 2. Hawking on a Motorcycle Per an old prediction of relativity, an accelerating observer with acceleration a should emit radiation at a temperature T = ha 2πck B. Assuming it is a blackbody radiator, provide a Fermi-style estimate (order of magnitude) for the rate of photon emission for Guillaume s motorcycle accelerating over 400 meters from rest to a final velocity of 250 km/h. You can use c 10 8 ; h ; k B and Stefan s constant σ 10 8, all in the relevant SI units. 3. Mass on Springs A mass m hangs from a combination of two springs, each with spring constant k, connected in series. If the mass is doubled to 2m the mass will hang lower by a distance h. If three such springs are arranged in parallel to support a mass of 5m what will be the frequency of small oscillations if the system is perturbed? 2

3 2016 Prelim Short Answers 3 4. Magnetron In the parallel-plate magnetron (see figure), the cathode and anode are flat parallel plates and a magnetic field is applied in a direction parallel to the plates. Electrons are emitted from the cathode at essentially zero velocity. If the anode is at a distance s from the cathode, and if it is held at potential V with respect to the cathode, show that for V eb 2 s 2 /2m no current will flow to the anode. (B is the magnetic induction and m is the mass of the electron). One way to do this problem is to use conservation of energy and the fact that that the magnetic field does no work. Why does it do no work? V X B e Figure 1: Magnetron. 5. Anharmonic Oscillator Consider a particle of mass m in the three-dimensional central potential V = λ r 4. Find the dependence of energy of a quantum state with angular momentum quantum number l 1 on the parameters of the problem. Use this method: imagine a trial wave function of width a; estimate the average kinetic and potential energies, and minimize the total energy to eliminate a. How does a depend on the parameters? Show that your result agrees with that from classical mechanics for circular motion in this potential. 3

4 2016 Prelim Short Answers 4 6. Collisions A high-energy photon collides with a proton at rest. A neutral pi meson is produced according to the reaction γ + p p + π 0. What is the minimum energy the photon must have for this reaction to occur? (The rest mass of a proton is 938 MeV/c 2 and the rest mass of a π 0 is 135 MeV/c 2.) 7. Rotation of the Moon The moon did not always rotate around its own axis at the same rate at which it orbits the earth. Estimate (in order of magnitude) the time scale for this tidal locking to set in, assuming that 1% of the energy of compression of the interior due to tidal forces is lost to heat (Call this fraction 1/Q, hence Q = 100). Relevant quantities are the mass and radius of the moon M m = kg, and R m = 1737 km, distance from earth d = 370, 000 km, mass of earth M e = kg, and Newton s constant G = Nm 2 /kg 2. Take the bulk modulus of moon rock to be K = dp/d ln V = Pa and assume the initial rotation frequency ω 0 is the same as that of the earth. 8. Thermal Contact Two identical objects, A and B, are thermally and mechanically isolated from the rest of the world. Their initial temperatures are τ A and τ B. Each object has heat capacity C (the same for both objects) which is independent of temperature. Suppose the objects are placed in thermal contact and allowed to come to thermal equilibrium. (a) What is their final temperature? (b) How much entropy is created in this process? (c) How much work is done on the outside world in this process? 4

5 2016 Prelim Short Answers 5 9. Quantum Isomers Isomers with the same molecular formula but different bonding structure are distinguished as Cis or Trans. Fig.2 shows a dichloroethene molecule in Cis and Trans: they only differ by exchanging the H and Cl atoms on the right hand side. The Cis is given by state 1 and Trans by state 2. The energies of Cis and Trans are taken as the same, E o. In addition, Cis can become Trans (and vice versa) by exchanging the H and Cl atoms on the right via an exchange coupling energy V. (a) Write down the Hamiltonian matrix in the Hilbert space of 1 and 2 for this molecule. (b) Calculate the energy eigenvalues and eigenstates of the molecule (assume E o and V positive numbers). For the ground state, what is the probability of finding the molecule in Cis? Figure 2: Isomers. 5

6 2016 Prelim Short Answers Escape from Prison A prisoner with mass 70kg escapes from a cell window by sliding down a rope attached at the top end to a hook which can support a pull of 600N. The window is 45m above ground and the rope is long enough to reach the ground. Assume the rope to be massless. What is the minimum speed the prisoner can land on the ground? 11. Dispersion of light Cauchy s formula for the index of refraction as a function of wavelength can be written as n = n 0 + (n 0 1)B/λ 2, where n 0 = 1.4 and B = m 2 for glass. Consider a wave packet whose initial width is σ 0 = 1 mm along the direction of propagation, and central wavelength is λ 0 = 600 nm. In order of magnitude, how far must it travel (in glass) for the width to double? Hint: a wave packet in a dispersive medium necessarily encompasses a range of group velocities. 12. Classical model of the electron Model the electron as a hollow sphere of radius a on which the charge is uniformly distributed, and which is rotating with angular velocity ω. Parametrically estimate the angular momentum in the fields (recall that (ɛ 0 E B) is the momentum density), and the value of aω needed to match the spin h/2 of the electron. Then estimate the total energy in the fields and equate it to the mass-energy of the electron to get separate estimates for ω and a. For simplicity ignore any fields inside the sphere for your estimates. 6

7 2016 Prelim Short Answers Ball Rolling Consider the motion of a balling ball of mass M and radius R. The ball starts at a speed v o. Initially it just slides - no rolling. Then it slows down due to friction between the ball and the ground, and the ball starts to roll. Determine the speed when it just starts to roll. (The moment of inertia of a ball is I = 2 5 MR2. When a ball only rolls, we have v = Rω where ω is the angular speed). 14. Relativistic Father A 30 year old man has a 10 year old daughter. The man travels away and back home at (essentially) constant speed. When he returns, both he and his daughter are 60 years old. How fast did the man travel? 15. Capacitance Two conductors are embedded in a material of conductivity 10 4 Ω/m and dielectric constant ɛ = 80ɛ 0. The resistance between the two conductors is measured to be 10 5 Ω. Derive an equation for the capacitance between the two conductors and calculate its value. 16. Carnot Cycle A Carnot cycle consists of four processes, two isothermal and two adiabatic. Illustrate the Carnot cycle on a pressure-volume and entropy-temperature diagram. Derive the efficiency of an engine using the Carnot cycle. 7