UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT. PART I Qualifying Examination. August 20, 2005, 9:00 a.m. to 1:00 p.m.

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1 UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART I Qualifying Examination August, 5, 9: a.m. to : 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, mathematical handbooks 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:. Answer the exact question that is asked, not a similar question.. 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.

2 . A small solid marble of mass m and radius r rolls without slipping along the loop-theloop track shown in the figure after being released from rest somewhere on the straight section of the track. The radius of the loop-the-loop is R r. (a) By neglecting the momentum of inertia of the sphere, determine from what initial height H above the bottom of the track must the marble be released if it is to be on the verge of leaving the track at the top of the loop? H (b) How does your result at (a) change if you do not neglect R the momentum of inertia of the sphere I = mr? Does your 5 answer depend on the actual size of the radius r of the sphere?. Two identical point like particles of mass m are connected through an ideal (inextensible and massless) string of length. One of the particles can move only along the x-axis, while the other one can oscillate within a vertical x-y plane, as y shown in the figure. Are frictions are negligible. (a) Write down the Lagrangian of the system in terms of the generalized coordinates x and θ, and the corresponding x m generalized velocities x and θ. (b) Write down the corresponding Lagrange equations. (c) Show that the generalized momentum px = L/ x is θ conserved, and calculate the expression of p x. What is the physical interpretation of this result? (d) Assuming small θ amplitude oscillations, calculate the normal frequencies of the system. How many normal modes does the system have? Hint: In the Lagrange equations make the usual small angle approximations (i.e., sinθ θ i t and cosθ ) and assume harmonic oscillator solutions (i.e., x xe ω i = and θ = θe ωt ). m x 3. Answer all four parts: (a) Calculate the electrostatic energy of three charges q, q, -q located at the vertices of an equilateral triangle a. (b) A uniform electric field E in the x-direction is produced by an appropriate charge configuration. A thin sheet of charge σ per unit area is placed perpendicular to the x-direction at x =. If the initial charge configuration is assumed to be undisturbed by the presence of the sheet, what is the total electric field on each side of the sheet? (c) Consider two concentric spherical metal shells of radii r and r (r > r ). If the outer shell has a charge q and the potential of the inner shell is zero with respect to infinity, what is its charge? (d) Two identical iron toroids are wound with N and N turns of identical wire, respectively. Assume that the N turns requires exactly twice the wire length of the N turns. If the toroids Page of 5

3 are connected in series, what is the ratio of the potentials across the two windings when direct current flows in the windings? You can use the following: The magnitude of the magnetic field B due to a current carrying wire at a distance r is B= (µ /4π) I/r, and the electric field a distance r from a cylinder with uniform charge density ρ and cross- sectional area A is given by E = [/(πε )] ρa/r, and c = / ε µ. 4. The electro-magnetic force is considered to be one of the four fundamental forces in nature, instead of two separate forces. This problem demonstrates the full equivalence of the Coulomb and the Lorentz force. v Consider the laboratory frame S, in which a wire carries a current I in the negative x-direction (see figure). For the sake of simplicity, assume the laboratory is under vacuum. The wire is infinitely r long, and has a cross-sectional area A. Assume that the current carrying particles are electrons that move at speed v (about cm/s). Careful measurements in the laboratory reveal that there is no net-electric field associated with this current. a) What is the net charge per unit volume ρ on the wire in frame S, expressed in terms of the charge A I per unit volume ρ due to the electrons and the charge per unit volume ρ + due to the positive ions in S? Express the current magnitude I in terms of ρ+, v and A. b) Suppose an additional electron with charge e is moving with velocity v in the positive x-direction, a distance r from the center of the wire. What are the magnitude and direction of the Coulomb and Lorentz forces on this electron in the laboratory frame S. Show that the sum of the two can be expressed as F = (µ /4π) eρ + v A/r c) Next, look at this problem in the reference frame S which has its origin on the additional electron (hence v = in this frame). The observer in this moving reference frame measures a net electric field due to the wire. Why? Hint: this is where special relativity becomes important, even though the speed v is very small. According to classical physics, there is no net force on the electron in S. d) Show that the net charge per unit volume ρ on the wire in S is given by = ρ v /c / [(-v /c ] ρ + e) What are the magnitude and direction of the Coulomb and Lorentz forces on the additional electron measured by the observer in the moving reference frame S. Express the total force F in terms of the total force F in S. Comparing this net force to your answer in b) should make it clear why the electromagnetic force is a single force instead of two separate forces. 5. Suppose right-handed circularly polarized light (defined to be clockwise for an observer looking toward the on-coming wave) is incident on an absorbing slab. The slab is suspended by a vertical thread, and the light is directed upwards hitting the underside of the slab. Page 3 of 5

4 (a) If the circularly polarized light beam has watt of visible light of an average wavelength of 6 nm, and if all the light is absorbed by the slab, what is the torque, τa, exerted on the slab? [Give the answer to (a) in N-m and the remaining parts in units of τ.] (b) Suppose that instead of an absorbing slab you use an ordinary silvered mirror surface, so that the light is reflected back at 8± to its original direction. What is new torque? (c) Suppose that the slab is a transparent half-wave plate. The light goes through the plate and does not hit anything else. What is the torque now? (Neglect reflections at the surface of the slab.) (d) Now suppose that the slab is a transparent half-wave plate with the top surface silvered, so that the light goes through the half-wave plate, reflects from the mirror, and returns through the plate. What is the torque? 6. A diffraction grating has 8 lines uniformly spaced over 5.4 mm and is illuminated by light from a mercury vapor discharge. (a) What is the expected dispersion, in the third order, in the vicinity of the intense green line (λ =546 nm)? [Hint: the dispersion, D, of a grating is given by: D = m / d cosθ, where m is the order and d is the grating spacing] (b) For the fifth order, what is the minimum separation between two wavelengths ( λ) that can be resolved by the above grating when mean λ ~ 546 nm? 7. A 5 kg block of ice at - C is dropped into an insulated container holding kg of water at + C. (a) What is the final temperature of the system? (b) What phase(s) is(are) present? (c) Calculate the mass of liquid water, if any, and the mass of ice, if any, at equilibrium. (d) Calculate the total entropy change of the system. Not es: the specific heat of ice is.5 kcal/(kg K), the specific heat of water is. kcal/(kg K). The latent heat of fusion of water is 8 kcal/kg. [in SI units:.9 kj/(kg K), 4.8 kj/(kg K), and 335 kj/kg, respectively]. When calculating entropy change, you need to consider the temperature in K, i.e., T(K) = T(C) + 73 K. 8. A crude (very crude!) model for the unwrapping of a DNA double helix corresponds to the zipper model as show in the figure below. For N pairs we associate N zipper links (in what follows we will assume that we have N links). An open link has energy +ε, a closed link has energy. For simplicity we assume that the DNA strand can only unzip itself from one end (for definiteness, the left end, as shown), a Page 4 of 5

5 and that the n-th link can only be open if all the previous n- links are also open (i.e. the zipper is not defective ). You may take k B = for simplicity. (a) Calculate the partition function Z for the zipper. (b) What is the expectation number of open links at temperature T? (c) What is the fraction of open links at T = and T? Does it make sense? Page 5 of 5

6 UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART II Qualifying Examination August 7, 5, 9: a.m. to : 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, mathematical handbooks 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:. Answer the exact question that is asked, not a similar question.. 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.

7 . (a) Expand f ( x) = x for π x π in a Fourier series. (b) From the result in (a), show that n= n π = 6 Useful integral: π iax 4π π 4 x e dx= cos aπ + sin aπ π 3 a a a, for a.. (a) By employing the method of separation of variables, show that, in cylindrical coordinates, the general solution of the partial differential equation r Φ Φ + =, r r r r θ is given by m m ( m m ) Φ (, r θ ) = C + A r + B r e imθ m= m where the unknown coefficients Am, B m and C ought to be determined from the boundary conditions. Does the solution have the correct π periodicity in the angle θ? What is the condition that the solution is finite in the origin (i.e., for r = )? (b) Show that the particular solution Φ (, r θ ) in the region r r and θ < π that satisfies the following boundary conditions Φ= at r = r, θ < π, is given by Φ= at r = r, π θ < π, Φ= finite at r =, n+ ( n ) r sin + θ Φ (, r θ ) = +. π n= r n+ 3. (a) State the content of Pauli s exclusion principle. (b) In a simple metal, the electrons can be approximately described as a gas of non-interacting fermions in a box. The Fermi distribution function is defined as ft ( E) =, E EF + exp ( kt ) where T is temperature, kb is Boltzmann s constant, and E F is the Fermi energy. ft ( E ) is plotted below, for temperature T = and for finite T : B Page of 5

8 Fermi function.5. T= T finite.5 EF Energy. In the figure above, indicate the location of the Fermi energy, and explain its significance.. Explain how an electron gas at finite temperature, whose Fermi distribution is shown in the above figure, is different from a classical gas at the same temperature. (c) Explain the basic difference of a Fermi gas and a Bose gas at zero temperature. 4. An empirical expression which gives, with some accuracy, the potential energy of a bound state in a diatomic molecule for a given electronic configuration is the Morse potential, a( r r ) E () r = D e, p where the constants D, a, and r are adjustable parameters characteristic of each molecule. The graph of E () p r is as follows: Potential Energy r (a) Assume that the molecule can be described classically. From E () r, determine expressions for. the equilibrium separation of the molecule,. the dissociation energy, 3. the angular frequency of small vibrations, ω. y Hints: use the expansion e = + y+ y / +, and use the fact that a harmonic oscillator with the potential U( x ) = kx has the frequency ω = km /. For a diatomic molecule A-B, replace the mass m with the the reduced mass µ, which is defined as µ = mm/ ( m + m). A B A B (b) Now describe the same molecule quantum mechanically.. What are the vibrational energy levels? You may give your answer directly without p Page 3 of 5

9 detailed derivation.. What is the dissociation energy? 5. The first two eigenstates of an electron in a harmonic oscillator potential are given by λ λ x / ψ ( x) = e 4 / π 3 / λ ψ ( x) = xe λ x /, 4 / π Where λ is a constant. The energy eigenvalue of the n th level is given by En = ω( n + / ). Now consider an electron in this harmonic oscillator potential whose wave function at time t = is given by Ψ ( x, t = ) = ψ ( x) + ψ( x). (a) Write down the wave function Ψ (x, t) at time t (the Hamiltonian of the system is not timedependent). (b) Calculate the average energy E of the system at time t. Is E a constant or time-dependent? (c) Calculate the expectation value of position, x, at time t. Is it constant or time-dependent? A useful integral: λ x xe π dx= 3 λ 6. The Hamiltonian of a simple harmonic oscillator is given by: H p m = + mω x, where ω is the angular frequency of the oscillator, and m is the mass of the particle. The x and p are the canonical position and momentum, with the usual commutation relation [ x, p] = i. The energy of the n th state n (n ) is given by: En = ( n+ ) ω. If we define the operators mω p mω p a ( x+ i ), a ( x i ), mω mω it is easy to see that (don t prove it) they act as raising and lowering operators (or creation/annihilation): a n = n+ n+, and a n = n n a) Show that [a,a ]=. b) Prove that in terms of the raising and lowering operators the Hamiltonian can be written as H = a a+ ω. c) Consider adding to the Hamiltonian the additional interaction H = bx, where b is constant. Show that at first order in perturbation theory the energy of the n th state does not change. Page 4 of 5

10 d) Consider adding to H the mass renormalization perturbation H'' p c m =, where c is a constant. Use perturbation theory to calculate the first-order correction to the energy of the n th state. Hint for parts (c) and (d): express the perturbations H' and H'' in terms of the raising/lowering operators 7-8. Experiment ) and ) are experiments you will actually carry out in the lab, experimental question 3 asks you to design an experiment. You have been assigned a time slot to carry out parts ) and ). You cannot do these experiments outside your time slot. ) Provided are a flexible plastic ruler, a standard weight attached to a string and a wooden log and a pencil. Determine the mass (plus errorbars) of the log using only the items listed and your fingers. Specify your method, and include a full error analysis. ) An RLC circuit is plugged into a standard outlet (V rms, 6 Hz). Use the provided voltmeter to determine at which frequency this circuit would be at resonance (minimum impedance), and give the errorbars associated with your determination. Make sure your observations and method are clearly stated. 3) Design an experiment that would determine Avogadro s number. Make sure you indicate all the steps of your experimental procedure, as well as the equipment that is needed and how you would use it. What is the largest source of error in your setup? What is the expected accuracy of your measurement? Page 5 of 5

UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT. PART I Qualifying Examination. January 20, 2015, 5:00 p.m. to 8:00 p.m.

UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART I Qualifying Examination January 20, 2015, 5:00 p.m. to 8:00 p.m. Instructions: The only material you are allowed in the examination room is a writing