UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT. PART I Qualifying Examination. August 19, 2006, 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 9, 006, 9:00 a.m. to :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, 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 bowler throws a bowling ball of radius R = cm along a lane. The ball slides on the lane, with initial speed v 0 = 8.5 m/s and initial angular speed ω 0 = 0. The coefficient of kinetic friction between the ball and the lane is μ k = 0.. The kinetic frictional force F k acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When linear speed v has decreased enough and angular speed ω has increased enough, the ball stops sliding and then rolls smoothly. (a) During the sliding, what are the ball s linear acceleration and angular acceleration? (b) How long (how much time) does the ball slide (before the nonslide condition is met)? (c) Find the speed of the bowling ball when it begins to roll without slipping. (d) How far does the ball slide before it rolls without slipping/sliding? (e) Find the total kinetic energy of the ball after it begins to roll without slipping if the mass of the ball is M = kg.. Two masses, and m, are connected to a wall and to each other by two springs with m spring constants k and k, as shown in the figure. The masses can move without friction only along k m k m the horizontal x-direction, as indicated in the figure. (a) Set up the equations of motion for the two coupled oscillators in terms of the deviations x x and x about the equilibrium positions of the two masses (see figure). (b) For k = k, k = k, m = 3m and m = m find the two characteristic frequencies of the coupled oscillators. i t Hint: Solve the equations of motion using the ansatz x() t Be ω, =,. (c) For the characteristic frequencies determined in (b), find the ratio of the oscillation amplitudes B, of the two masses. Describe what the coupled oscillations look like. x x 3. A long coaxial cable carries a current I. The current flows down the surface of the conducting inner cylinder of radius a, and back along the conducting outer cylinder of radius b. (a) Use Ampere s Law to determine the magnetic field:. inside the inner cylinder,. in the region between the conducting cylinders, 3. outside the coaxial cable. (b) Find the magnetic energy stored in a section of length l of the coaxial cable. 3 Hint: the magnetic energy is given by: W mag = B d r. μ 0 Page of 4

3 4. (a) Use the equation of continuity (for charge density ρ and current density J), Ohm s σρ ρ law, and Gauss law to derive the equation + = 0 for a linear, homogeneous, ε t conducting medium. Here σ is the conductivity and ε is the electric permittivity of the medium. Using the equation derived in part (a) (b) Show that any existing charge density ρ () t in the conductor will be damped exponentially in time. (c) Find the time required for ρ () t to be reduced to / e of its initial value. 5. Two mutually coherent beams having parallel electric fields are described by π E E = 3cos ks ωt + 5 π E = 4cos E P ks ωt + 6 with amplitudes in kv/m. The beams interfere at a point P as shown in the figure. The phase difference due to their path at P is π / 3, the first beam having a longer path. At the point of superposition, calculate (a) the irradiances I and I of the individual beams; (b) the irradiance I due to their interference; (c) the fringe visibility. To obtain maximum visibility what should be the relation between the amplitudes of the two fields? [Hint: the irradiance of a beam is given by I = ε ce 0 0, where ε 0 = C / Nm is 8 the permittivity of vacuum and c = 3 0 m/s is the speed of light; and the visibility is given by ( I max Imin )/( Imax + Imin ).] 6. Right circularly polarized light passes through two identical quarter-wave plates whose optic axes make an angle θ with each other. The refractive index for the extraordinary rays is less than that for the ordinary rays. What is the outgoing polarization state if (a) θ = 0 o? (b) θ = 45 o? (c) θ = 90 o? (d) Are two quarter-wave plates identical in their effect to a single half-wave plate? If any of the above answers is circular polarization, state whether it is right or left circular polarization. If any answers involve plane polarization, state the plane of polarization. Page 3 of 4

4 [Note: In right circular polarization, the electric field vector rotates clockwise as seen by an observed looking toward the source.] 7. Each of two identical bodies has an internal energy E = CT, where C is a constant and T is the temperature. The initial temperatures of the bodies are and T ( < T), respectively, T and they are to be used as source of work by connecting them to an ideal Carnot heat engine and bring them to a common final temperature. You may ignore the volume change for each body in the process. (a) What is the final temperature T f of the system? Hint: The total entropy of the system does not change, i.e., Δ S+Δ S = 0. (b) What is the work delivered? 8. A protein can be treated as a chain of N units connected by covalent bonds. A simplified zipper model assumes that () each unit exists in one of two possible states: helix [h] with a negative energy ε and coil [c] with energy 0, and () all of the helical units occur contiguously in a single region. Initiation of a helical sequence can occur at any point along the chain, and the helical region grows from there. Here are some examples of conformations for a -unit chain: ccchhhhhccc, allowed, helix length = 5, energy = 5ε hhhcccccccc, allowed, helix length = 3, energy = 3ε cchhcchhccc, forbidden (because the helix units are not contiguous) Consider a chain having N units. (a) Without explicit calculation, make a qualitative plot of the mean length of the helix n as a function of temperature. What is n in the limit of very low temperature? (b) Make a qualitative plot for the heat capacity C v (at constant volume) as a function of the temperature T. (c) Write an expression for the number of possible conformations Ω n of a chain with an n - unit long helical region. Note that initiation of a helical sequence can occur at any point along the chain. Hint: you should obtain different expressions for n = 0 and n 0. (d) Calculate the partition function over all possible helix lengths n. (e) What is the probability of finding an n -unit helix in the chain? Hint: You may find the following formulas useful: N n ( N + x = x x) /( x ) n= N n N+ N nx = [ Nx ( N + ) x + ] x/ ( x ) n= Page 4 of 4

5 UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART II Qualifying Examination August 6, 006, 9:00 a.m. to :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, 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.

6 . (a) Show that sin( a) exp( ia σ) = cos( a) σ 0 + i a σ, a where a = ( ax, ay, az) is a 3D vector, σ = ( σ x, σ y, σz) with the usual Pauli matrices 0 0 i 0 σ =, σ, σ x = 0 y = i 0 z 0, 0 and σ 0 = is the unit matrix. By definition, the dot product a σ = axσ x + ayσ y + azσ z. 0 Hint: Any matrix can be written as a linear combination of the unit matrix and the three Pauli matrices. Tr exp( i a σ )exp( i b σ ), where a and b are two arbitrary 3D vectors. (b) Calculate the trace [ ]. The stationary current density J of overdamped Brownian particles moving along the x-axis (e.g., molecules or ions in a channel protein) is constant and can be written as dp( x) J = D + f( x) p( x), 0 x L, dx where px ( ) is the corresponding probability density [i.e., pxdx ( ) is the probability of finding a particle in the region ( x, x+ dx) ], D is the diffusion constant and f ( x) = dv( x)/ dx is a position dependent force [derived from the potential energy V( x) ] acting on the particles. Assuming the boundary conditions p(0) = p0 and p( L) = pl, show that p0exp [ V0 / D] plexp [ VL / D] J = D, where V0 = V(0) and VL = V( L). Hint: Find first the general solution constant. L 0 [ V x ] exp ( ) / D dx px ( ) for J = 0 and then apply the method of variation of 3. A spaceship travels at V = 0.75c (c is speed of light in vacuum) relative to Earth. (a) If the spaceship fires a small rocket in the forward direction with what velocity v (relative to the ship) must it be fired for it to travel at v = 0.95c relative to Earth? The rocket is aimed towards a target situated at a distance d = 00 km(with respect to the spaceship) in front of the spaceship. (b) Find the distance d between the spaceship and target, at the moment when the rocket was launched, as measured by an observer on Earth. (c) For an observer on Earth, how long it takes the rocket to hit its target? 4. Students of quantum mechanics often believe, incorrectly, that the fundamental equation of quantum mechanics is Ĥ ψ = E ψ, Page of 5

7 φ where Ĥ is the hamiltonian operator, ψ is any quantum state, and E is its energy. (a) Why is this wrong? (b) What is the correct general form of the Schrödinger equation (for which he received the Nobel prize)? (c) State the Heisenberg uncertainty principle. 5. A one dimensional harmonic oscillator of mass m and angular frequency ω is initially in the state ψ (0) = +, 5 5 where n denotes the eigenstate of the hamiltonian, with eigenvalue En = ( n+ ) ω. (a) What is the state ψ () t at positive times? (b) What is the expectation value for the energy? (c) What is the expectation value of the position as a function of time? ( n+ ) (Note: you can use the result n+ xˆ n = ω for the matrix element of the position operator.) m 6. A particle of mass m moves in a three-dimensional central potential V( r) which vanishes as r. We are given that the wave function of a bound state (an exact eigenstate of the energy) is 3 ψ () r = Cr e αr cosθ, where C and α are constants ( α > 0 ) and θ is the angle between r and the z -azis. (a) What is the magnitude of the angular momentum of this state? (b) What is the value of the projection of the angular momentum along the z -axis? (c) What is the energy? Hint: for this wave function, in the limit of large r you can approximate ψ() r = α ψ() r. (d) Can you determine V( r)? (A clear description of the procedure, without actual calculations will do). Hint: You do not really need to use the following information to answer question (a) and (b), but here it is: (i) The square of the angular momentum operator is Lˆ = sinθ sinθ θ, and θ sin θ φ (ii) the z -component of the angular momentum operator is Lz ˆ = i. 7. Solve any three of the following problems. P: Franck-Hertz Experiment The figure below (left) shows the wiring diagram of a typical Franck-Hertz experiment, where the tube is filled with mercury. Grid is operated in the neighborhood of.5 V, and the retarding potential is of the same order. The anode currents are on the order of nano Ampere. Two sets of data obtained by a student for the excitation potential are shown below (right); both curves are obtained at 95 C. The filament voltage was.5 V for curve C and.85 V for curve D. Page 3 of 5

8 (a) A significant decrease in the electron current is seen every time the potential on grid is increased by ~5 V. What is the significance of this? Briefly explain. If the tube is changed to a neon-filled one, would you expect any changes in the electron current versus accelerating voltage graph? (b) The Frank Hertz experimental setup can be used for measuring the ionization potential, which is the energy required to remove an electron from the atom. In this case, instead of observing the bombarding electron beam, one detects the ions that are formed. The figure on the right shows the ion current versus accelerating voltage. Estimate the ionization potential of Hg from the graph. Hint: first obtain the contact potential from part (a)-electron current vs. accelerating voltage graph. P: Density of Solids The densities of most solids lie in the range between 0.5 and 0 times that of water. When the densities of various elements are plotted against their atomic numbers one observes periodic decreases and increases in their values. A general trend towards increasing density arises because atoms that make up solids get heavier as their atomic number increases. (a) What would the density of the elements be if they all had a simple cubic crystal structure with nearest neighbor separation a =0.3 nm. [note that the mass of an atom is Au, where A is the relative atomic mass and u = kg]. Sketch your result on the given figure. (b) For elements in the h group (A = 57 to 7) there is a striking linear increase in their densities. Why? Give a brief explanation. P3: High-pass filter Suppose that you wish to detect a rapidly varying voltage signal. However, the signal is superimposed on a large DC voltage level that would damage your voltmeter if it were in contact with it. You would like to build a simple passive circuit that allows only the high frequency signal to pass through. Page 4 of 5

9 (a) Sketch a circuit using only a resistor R and a capacitor C that would do the job for you. Indicate the points at which you measure the input and output voltages. (b) Derive a simple relation between the magnitude of the output and input voltage (in terms of R, C and ω). (c) Suppose that R = kω and the signal is MHz. Suggest a value for the capacitor C. P4. Polycyclic aromatic hydrocarbon (PAH) molecules are seen almost everywhere in the space between stars ( interstellar space ). The specific heat of a PAH molecule with N atoms can be approximated by 3( N ) k( T/ 300 K), T < 300 K CT ( ) = 3 (N ) k, T > 300 K 3 where T is the temperature of the PAH, and k = JK is the Boltzmann constant. In the interstellar space, on an average, an energetic photon has energy hν = 6eV. (a) For a PAH molecule with N = 00 atoms, what temperature T f would it reach after it absorbs such a photon? Hint: assume that the PAH stays in the vibrational ground state before the photon absorption event occurs so that its initial temperature T i = 0. 8 (b) If we assume a graphite grain with 500 million carbon atoms ( N = 5 0 ; corresponding to a spherical radius of 00 Å ) at an initial temperature of T = 0K, what temperature would it reach 9 after absorbing a photon of energy 6 ev ( ev = J ). f Hint: assume that all the photon energy goes into kinetic energy of the PAH so that hν = C( T) dt. T i T i P5. In interstellar space, PAH molecules have three emission features associated with vibrations of the C- H bond: the C-H stretch at λ 33. μm, in-plane C-H bending at λ 86. μm, and out-of-plane C-H bending at λ3. 3μm. These stretching and bending modes can be approximated as harmonic oscillators. If the hydrogen (H) atom is replaced by a deuterium (D) atom, show that: (a) The reduced mass of the C-H oscillator will increase by a factor 3/7, and the frequency will be reduced by a factor ~ 7 / 3. (The atomic mass of H, D and C are mh = u, m = u and mc = u, D respectively.) (b) The zero-point energy of the C-H bond will be lowered by ΔE(CH) ΔE(CD) eV [ h = Js is the Planck constant, c = 3 0 cms is the speed of light and 9 ev =.6 0 J ]. 9 0 P6. A cosmic ray with energy 0 ev traveling at speed of light c = 3 0 cms has a cross section 9 for ionizing collision of H of σ 0 cm. (a) On the average, how far does the cosmic ray travel between ionizing events (i.e., what is the ionization 4 3 mean-free-path ) in a typical dense interstellar cloud with a density of n = 0 cm? (b) What is the corresponding mean ionizing collision time? Page 5 of 5

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

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