UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT. PART I Qualifying Examination. August 20, 2013, 5:00 p.m. to 8:00 p.m.

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1 UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART I Qualifying Examination August 20, 2013, 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

2 1. A spool of wire of mass M and radius R is unwound under a constant force F as shown in the figure. Assuming that the spool is a uniform solid cylinder that does not slip, determine: (a) The acceleration a of the center of mass of the spool. (b) The static friction force (both magnitude and orientation) acting on the spool. (c) If the cylinder starts from rest, what is the speed v of its center of mass after it has rolled through a distance d. 2. A particle of mass m = 0.1 kg slides without friction on a half circular track of radius R = 0.1 m in the presence of a uniform gravitational field [ g 10 m/s 2 ] directed along the negative z axis. z (a) Write the Lagrangian in terms of the generalized coordinate θ. (b) Obtain the equation of motion for θ. (c) For a particle starting from rest at θ = π / 2 calculate the period of oscillation in seconds [Useful integral: π /2 dθ = 2K(1/ 2) ] 0 cosθ (d) d) For the same initial condition as in (c) calculate the force exerted by the track on the particle as a function of θ. θ 3. A particle of mass m is contained in a one-dimensional impenetrable box extending from x = L / 2 to x = +L / 2. The particle is in its ground state. (a) Find the normalized eigenfunctions of the ground state and the first excited state. 1/2 [Useful integrals: cos 2 (π x)dx = 1/ 2; sin 2 (2π x)dx = 1/ 2 ] 1/2 1/2 1/2 (b) The walls of the box are moved outward instantaneously to form a box extending from L to +L. Calculate the probability that the particle stays in the ground state during this sudden expansion. [Useful integral: L/2 cos π x 2L cos π x L dx = 4 2 3π L ]. L/2 (c) Calculate the probability that the particle jump from the initial ground state to the first final excited state. 2

3 4. The Rashba effect describes the momentum-dependent spin splitting of the energy of an electron moving in an electric field and is a contemporary area of research in physics. The effect originates from the spin-orbit interaction and may be described by the phenomenological Hamiltonian: H = α ( p σ ) ẑ, where p is the momentum of the electron, the σ x, σ y, and σ z are the Pauli matrices, ẑ is the direction of the electric field, and α is a constant. (a) Write down the above Hamiltonian in a 2 2 matrix form using the Pauli matrices. (b) Diagonalize the Hamiltonian to obtain the energy eigenvalues and show that the eigenvalues have a linear dependence on the momentum magnitude. (c) If the electron is moving along the x direction, so that p x = p and p y = 0, obtain the two eigenfunctions for the Hamiltonian corresponding to the energy eigenvalues you found in part (b). (d) For a certain momentum p, Ψ = i was found to be an eigenstate of the Hamiltonian. What would be the expectation value for this state if the x component of the spin is measured? Recall: The three Pauli matrices for a spin ½ particle are σ x = , σ y = 0 i i 0, and σ z =

4 UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART II Qualifying Examination August 22, 2013, 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.

5 1. In the circuit, shown in the figure, the switch is initially open and the capacitor is charged to 50 V. The large circuit on the left is rectangular with dimensions 8 m by 2 m. The smaller circuit on the right is a S square loop having dimensions of 20 cm by 20 cm with turns of wire that has a resistance per length of 1 Ω /m. Note that the diameter of all wires can be neglected and that the diagram is not drawn to scale. Both circuits are stationary and the small loop is positioned 5 cm away from the larger circuit, as indicated in the diagram. Assume that the wire of the large circuit, which is nearest to the small circuit, is the only wire that produces an appreciable magnetic field for the purpose of this problem. (a) Find the current in the large circuit 400 µ s after the switch, S, is closed. (b) Find the current in the small circuit 400 µ s after the switch, S, is closed. (c) What is the direction of the current in the small circuit? 40µF + 5cm (d) Explain why you can assume that the wire of the large circuit which is nearest to the small circuit is the only wire that produces an appreciable magnetic field for the purpose of this problem. Give a numerical estimate of the effect of also including the left vertical wire of the large circuit. 2. (a) A thin cylindrical ferromagnet has radius a and height, h, with h a. Its axis is oriented along ẑ and its magnetization is M ẑ. The ends of the cylinder are at z = 0 and z = h, as shown in the diagram. What is the magnetic dipole moment, m, of the cylinder? (b) What condition must be fulfilled in order to introduce a magnetic scalar potential, M, that describes the magnetizing field, H? (c) For a given magnetization, M r ( ), show that the magnetic scalar potential is: M ( r ) = " 1 M ( r& ) $ 4# r %' d 3 r& r " r&

6 (d) Using the result from (c), show that far from the ferromagnet the magnetic scalar potential has the form of a dipole: m# r M ( r ) " 4$r 3 3. One mole of a monoatomic ideal gas undergoes a quasistatic transformation 1 2 described by the equation P = P 0 [7 (V / V 0 ) 2 ] / 3, where the volume V increases from V 1 = V 0 to V 2 = 2V 0. Express all your answers in terms of P 0, V 0 and R (the ideal gas constant). Calculate: (a) the work, W, done by the gas in the process; (b) the highest temperature, T max, reached during the process; (c) the heat, Q 1, absorbed during the process; (d) the heat, Q 2, released during the process. 4. Consider a system of N non-interacting (distinguishable) particles ( N 1) in which the energy of each particle can assume only two distinct values: 1 = 0 and 2 = > 0. The system is in equilibrium at temperature T. As a function of T, find: (a) the partition function, Z(T ), and the free energy, F(T ) of the system (b) the entropy, S(T ), and the internal energy, E(T ) (c) the specific heat per particle, c(t ). Also, make a qualitative plot of c(t ) vs T /.

PART I Qualifying Examination. August 22, 2017, 5:00 p.m. to 8:00 p.m.

PART I Qualifying Examination. August 22, 2017, 5:00 p.m. to 8:00 p.m. 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