UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART I Qualifying Examination August 22, 2017, 5:00 p.m. to 8:00 p.m. Instructions: The only material you are allowed in the examination room is a writing instrument and a calculator. You may not store any formulae in your calculator. Paper and question sheets are provided. Each student is assigned a capital English letter; this letter will identify your work on both parts (I and II) of this exam. In writing out your answers, use only one side of a page; use as many pages as necessary for each problem, and do not combine work for two different problems on the same page. Each page should be identified in the upper, right-hand corner according to the following scheme: A 4.3 i.e., student A, problem 4, page 3. Refer all questions to the exam proctor. In answering the examination questions, the following suggestions should be heeded: 1. Answer the exact question that is asked, not a similar question. 2. Use simple tests of correctness (such as a reasonable value, correct limiting values and dimensional analysis) in carrying out any derivation or calculation. 3. If there is any possibility of the grader being confused as to what your mathematical symbols mean, define them. You may leave when finished.
1. A bowling ball rolls without slipping up a ramp that slopes upward at an angle β to the horizontal. Treat the ball as a uniform solid sphere, ignoring the finger holes. The ball has radius R, mass M and momentum of inertia (about an axis that goes through its center) I = (2/5)MR +. (a) Draw the free-body diagram for the ball. Explain why the friction force must be directed uphill. (b) What is the acceleration of the center of mass of the ball? (c) What minimum coefficient of static friction is needed to prevent slipping? 2. A mass M is free to slide down a frictionless plane inclined at an angle β. A pendulum of length ll and mass m hangs from M, as shown in the figure. (Assume that M extends a short distance beyond the side of the plane, so the pendulum can hang down). (a) Find the Lagrangian of the system by employing the generalized coordinates x (distance of M from the top of the plane) and θ (angle formed by the pendulum with the vertical). (b) Derive the corresponding Lagrange equations of motion for x and θ. (c) In the limit θ 1 (small oscillations), combine the two equations of motion to obtain a closed equation of motion for θ, and find the frequency of the oscillations of the pendulum. l M m β - 2 -
3. A particle of mass m is confined in a one-dimensional box of width 2L, with potential energy V x = 0 for x < L otherwise (a) What are the energy eigenfunctions, ψ B (x), and eigenvalues, E B, of the particle? Express your answers in terms of L, m, and ħ. Next, assume that the particle has an energy that corresponds to the first excited state. In this state, the probability of finding the particle at the center of the box, x = 0, is zero. (b) Knowing that at time t F = 0 the particle is located in the left half of the box, is there a chance that at a later time, t > 0, we will find the particle in the right side of the box? Explain your answer. Finally, assume that we modify V(x) such that it has a value V F for x < L/4. All other values remain unchanged. Assume that V F equals the energy of the first excited state of the situation without the additional potential. (c) Sketch the wavefunctions with the 4 lowest energies corresponding to the solution of the Schrödinger equation in the presence of this modified potential. You are not required to perform any specific calculations. Simply sketch what the modified wavefunctions should look like, and explain (in words) why. 4. A two level system is described by the Hamiltonian H = ε 1 1 ε 2 2, where ε > 0 and the state vectors 1 and 2 are orthonormal. (a) Write the Hamiltonian as a 2 2 matrix that operates in the Hilbert space spanned by 1 and 2. What are the energy eigenvalues and eigenvectors of this system? We now switch on a perturbation Δ so that the Hamiltonian becomes Assume that 0 < Δ ε. H = ε 1 1 ε 2 2 + Δ 1 2 + Δ 2 1. (b) What are the energy eigenvalues and eigenvectors of this perturbed system? At t = 0, we prepare this (now perturbed) system to be in state 1. (c) What is the minimum time required for the state of the system to coincide with 2? Hint: recall that in a two-level system, only the difference in energy between the two levels determines the oscillation time between the levels. If you were unable to calculate the energy levels, you can still solve this problem (use the symbol δe for the level splitting in your answer). - 3 -
UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART II Qualifying Examination August 24, 2017, 5:00 p.m. to 8:00 p.m. Instructions: The only material you are allowed in the examination room is a writing instrument and a calculator. You may not store any formulae in your calculator. Paper and question sheets are provided. Each student is assigned a capital English letter; this letter will identify your work on both parts (I and II) of this exam. In writing out your answers, use only one side of a page; use as many pages as necessary for each problem, and do not combine work for two different problems on the same page. Each page should be identified in the upper, right-hand corner according to the following scheme: A 4.3 i.e., student A, problem 4, page 3. Refer all questions to the exam proctor. In answering the examination questions, the following suggestions should be heeded: 1. Answer the exact question that is asked, not a similar question. 2. Use simple tests of correctness (such as a reasonable value, correct limiting values and dimensional analysis) in carrying out any derivation or calculation. 3. If there is any possibility of the grader being confused as to what your mathematical symbols mean, define them. You may leave when finished.
1. An electronic circuit drives two linear microwave antennas in-phase at a frequency, f. The microwave radiation strikes a detector screen located at a distance, L, along the center-line between the antennas that are separated by a distance, D, as shown in the top-view diagram where the antennas are directed out of the page. For the purpose of this problem, assume that the electric field magnitude coming from each antenna individually does not vary significantly along the screen (the x-direction). The measured intensity at point A (x=0) is found to be I ". In the following, give all answers in terms of the given information. The speed of light is c. (a) Is the intensity at point A a minimum, a maximum or somewhere in between? Why? Credit will be based your answer to why. (b) Determine the minimum value of D for which the intensity is I " at point B (which is located at x = D/2). (c) With D having the value determined in part (b), if the electronic driving circuits are changed such that the two antennas operate 180 degrees out-of-phase from each other, what is the intensity at points A and B along the screen? Show your work. (d) With D having the value determined in part (b) and with the two antennas being 180 degrees out-of-phase, what is the minimum value of x where the intensity is I "? After setting up the equations that lead to the answer, you do not have to solve them. 2. A very long solenoid of n turns per unit length carries a current which increases uniformly with time, i(t) = Kt, where K is a constant. You may consider the solenoid to be sufficiently long so as to neglect edge effects in your calculations. (a) Calculate the magnetic field inside the solenoid at time t (neglect retardation effects). (b) Explain in a few words why there is also an electric field present inside the solenoid. Then calculate E(r), the electric field inside the solenoid. - 2 -
(c) Consider a cylinder of length l and radius equal to that of the solenoid and co-axial with the solenoid. Find the rate at which energy flows into the volume enclosed by this cylinder. The formula listed below for the curl of an arbitrary vector v in cylindrical coordinates may be useful in your calculations: 1 v v z j vr vz 1 vs Ñ v = [ - ] rˆ + [ - ] ˆ j + [ ( rv ) ] zˆ j - r j z z r r r j 3. Consider the same amount of n = 1 mole of two different monoatomic ideal gases that are isolated in separate thermally insulated containers at the same pressure, P ", and temperature, T ". (a) What is the total internal energy of the gases? At some moment, the partition separating the containers is removed and the gases are allowed to mix but not react. After the mixing is complete: (b) What is the final pressure and temperature of the gases? (c) What is the internal energy of the gases? (d) What is the entropy change of the system? (e) What is the maximum work that can be extracted through the mixing of the two gases? 4. (a) Consider an ideal gas of N particles of mass m confined to a volume V at a temperature T. Using classical statistical mechanics, and assuming that the particles are indistinguishable, show that the chemical potential of the gas is given by μ = k 6 T ln 9 : <=ħ? @A B C D/<. (b) A gas of N E particles, also of mass m, is absorbed on a surface of area A, forming a twodimensional (2D) ideal gas at temperature T on the surface. The energy of an absorbed particle is ε = p < /2m ε ", where p = (p K, p M ) is the 2D momentum vector of the particle (in the plane of - 3 -
the surface), and ε " is the surface binding energy per particle. Using the same approximations and assumptions as in part (a), show that the chemical potential μ E of the absorbed gas is given by μ E = k 6 T ln 9 N E <=ħ? @A B C ε ". (c) At temperature T, the particles on the surface and in the surrounding three-dimensional gas (vapors) are in equilibrium. This implies a relationship between the respective chemical potentials, μ and μ E. Using this condition, determine the mean number n of particles absorbed per unit area when the pressure of the vapor is P. - 4 -