EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 14 juin Salle A-552.

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EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Jeudi 14 juin 2018 Salle A-552 de 9h30 à 13h30 NOTES : No documentation allowed.

1 EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Jeudi 14 juin 2018 Salle A-552 de 9h30 à 13h30 NOTES : No documentation allowed. A non-programmable calculator is allowed. The candidate answers to 6 questions of his choice among 8. Each question is worth 20 points. Use a different notebook for each question, making sure to include the question number on it. This questionnaire contains 8 questions, 11 pages. ENGLISH VERSION Département de génie physique Pavillon principal Téléphone : Télécopieur : Courriel : info@phys.polymtl.ca Adresse postale C.P. 6079, succ. Centre-ville Montréal (Québec) Canada H3C 3A7 Campus de l Université de Montréal 2900, boul. Édouard-Montpetit 2500, chemin de Polytechnique Montréal (Québec) Canada H3T 1J4

2 å x n = 1 n=0 FORMULAS AND CONVERSION 1- x, x <1 * * ò Y v Y v dx = a ò Y v Y v dy = a ò H 2 v (y )e - y 2 dy = ap 1/2 2 v v! v!=v(v-1)(v-2) 1 u = 931,494 MeV/c 2 PHYSICAL CONSTANTS e = x C me = 9,109 x Kg ħ = x 10 x Js kb = x J/K ε 0 = x F/M μ 0 = 4π x 10 7 N/A 2 PHYSICS EQUATIONS : D f B 0 B E t D H J f t 0 B J r f f 4 ˆ 2 dv r 1 E B FD E k T e 1 1 E B BE E k T e 1 Clausius-Mossotti relation nα = ε r 1 3ε 0 ε r + 2 MATHEMATICS EQUATIONS Integrals x 0 e ax2 0 e x dx = π 2 dx = 1 2 π a x 2 e ax2 dx = 0 π 4a 3/2 Law of cosines c a b 2ab cos Stirling s approximation n n n! n e 2 n Identity 2 A A A Trigonometric identities sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β sin 2 2 sin cos 2 2 cos 2 cos sin 1 cos cos cos 2 cos 1 sin sin cos 2 cos 1 sin cos sin 2 sin Page 2 de 11

3 QUESTION 1 : ELECTROMAGNETISM Consider a long, coaxial cable of radius b and length l, with a center conductor of radius a. The center conductor is made of material having resistivity and linear magnetic permeability. The outer shield is a perfect conductor and is shorted to the inner conductor at the left end. At t = 0 a voltage V0 is suddenly applied at the right end and remains constant thereafter. In this problem you can assume that the current is uniform along the length of the cable, that l >> b and that the capacitance C of the system is zero. (a) (4 pts) Draw the electrical circuit equivalent of the system. (b) (8 pts) Determine the resistance R and the inductance L of the system. (c) (8 pts) Determine the current I(t) as a function of time. Page 3 de 11

4 QUESTION 2 : QUANTUM MECHANICS We consider a spin ½ particle subjected to a magnetic field B given by: B = B cos(ωt) z (B is a constant) The state vector of this particle, ψ 0 (t), fulfills the following eigenvalue equation: where S x is the spin operator along x axis. S x ψ 0 (t = 0) = h 2 ψ 0(t = 0) (a) (10 pts) Knowing that ψ 0 (t = 0) can be expressed as a function of the eigenstates of S z + et, S z is the spin operator along z axis, find the state vector of the particle at t > 0 ψ 0 (t > 0). Note that B varies with time. (b) (10 pts) Find the expected value <S z> at t > 0. The Hamiltonian is given by : H = ω 0 S z where ω 0 = e B(t) (e, the elementary charge, m the electron mc masse, and c the speed of light). Note : t The solution of the differential equation f (t) = y(t). f(t) is : f(t) = f(0). exp ( y(t ). dt ) 0 Page 4 de 11

5 QUESTION 3 : STATISTICAL PHYSICS Magnetic susceptibility of an electron gas at a temperature T=0 K. An electron of mass m, momentum p and spin s = ±1 in a magnetic field H has a total energy given by ε s = p2 2m + s μ B H where μ B is Bohr magneton. Here, we will assume that the system is at a temperature T=0 K. Accordingly, the chemical potential is equal to the Fermi energy μ 0 and the electrons will fill all the states with an energy ε μ 0. a) (5 pts) What is the maximum momentum p ±,max, that the electrons can take when in the spin states s = ±1. b) (5 pts) Show that the number of electrons N ± in the spin states s = ±1 is given by N ± = 4πV 3h 3 (p ±,max) 3 c) (5 pts) Assuming that the magnetic field H is such that μ 0 μ B H, compute the total number of electrons N = N + + N in a gas at T=0 K that occupies a volume V. What is the magnetization M = μ B (N + N ) of this electron gas. d) (5 pts) Show that the magnetic susceptibility of this electron gas is χ = M H = 3N(μ B) 3 2μ The following relation could be useful: (1 + x) p 1 + px when x 1. Page 5 de 11

QUESTION 4 : CLASSICAL MECHANICS Uniform hemispherical lawn sprinkler A lawn water sprinkler consists of a hemisphere of radius R containing hundreds of small holes from which tiny water jets are

The spatial distribution of holes is non uniform and positioned such that the lawn receives the same amount of water per unit surface everywhere within the range of the sprinkler (uniform irrigation,

6 QUESTION 4 : CLASSICAL MECHANICS Uniform hemispherical lawn sprinkler A lawn water sprinkler consists of a hemisphere of radius R containing hundreds of small holes from which tiny water jets are emitted at constant output velocity and flow rate. The spatial distribution of holes is non uniform and positioned such that the lawn receives the same amount of water per unit surface everywhere within the range of the sprinkler (uniform irrigation, Fig. A). A) (5 pts) Find the expression of the range of a given water jet as a function of its angular position at the surface of the sprinkler. Define any necessary variables and take θ = 0 (horizontal) as the angular reference) B) (5 pts) As illustrated in Fig. B-C, each point on the ground will be irrigated by two water jets of respective angles θ1 et θ2 with the horizontal. Give the relation between any two paired angles θ1 and θ2 which achieve the same range. C) (10 pts) What is the spatial distribution n(θ) of holes at the surface which would enable a uniform irrigation?) Hint : Suppose that the water jets have ballistic trajectories, i.e. you could replace the fluids with marbles ejected at constant interval and constant initial velocity from the holes and the result would be the same. Useful identities: surface element in spherical coordinates; surface element in spherical coordinates; Page 6 de 11

7 QUESTION 5 : OPTICS I Optics : Interferometric velocimetry Interferometry allows the measurement of the translation speed of partially reflecting objects such as blood cells. We study here a simplified model where the moving object is a perfectly reflecting mirror. laser i 1 v Figure : Interferometric system for speed measurement of the mirror on the right. A monochromatic laser with 1 µm wavelength is launched in the interferometric system represented on the Figure comprising two 50/50 beam-splitters and three perfect mirrors. The object having its speed measured is the mirror represented on the right of the Figure. 1) (4 pts) On a drawing similar to the Figure, trace the two optical paths which participate in the interference. One is a reference path and the other depends on the moving mirror s position. 2) (8 pts) Give an expression for the time dependence of the difference of photo-currents i = i 2 i 1 in the two detectors as a function of the speed v of the moving mirror. 3) (4 pts) What is the oscillation frequency of the photo-current difference when the mirror s speed is v = 1 cm/s? 4) (4 pts) Suppose now that the coherence length of the laser source is 10 cm, what precaution must be taken to ensure the success of the speed measurement? In case you wish to lift an ambiguity on the exact phase of each optical path, it is specified that the beam-splitters are based on dielectric thin films applying a phase of either 0 or π radians depending on the incidence side. i 2 φ t = 0 φ t = 0 φ r = π φ r = 0 Page 7 de 11

$QUESTION 6 : OPTICS 2 We are designing an optical system consisting of an optical fibre, a lens and a diffraction grating. The optical system is shown in Figure 1.$

8 QUESTION 6 : OPTICS 2 We are designing an optical system consisting of an optical fibre, a lens and a diffraction grating. The optical system is shown in Figure 1. Figure 1 Final optical system Each question assesses your understanding of these individual components. Part 1 : (5 pts) Snell s Laws applied to Fibre Optics a) Name the optical principle allowing for propagation of light through optical fibres almost losslessly. b) The numerical aperture (NA) of an optical system is defined by: NA = n sin θ where θ is the half-angle of the cone of acceptance of light of the optical system. The NA may also be used in the context of optical fibres. Let us look at the simplest step-index optical fibre. With the fibre seen as a bidimensional structure, show that its NA can be written as: NA = n 1 2 n 2 2 where n 1 is the index of the core of the fibre (the part transmitting light) and n 2 is the index of the cladding (the part that does not transmit light). Use variables shown in Figure 2. (next page) Page 8 de 11

QUESTION 6 : OPTICS 2 (CONT D) Figure 2 Variables used to describe the fibre, lens and light beam Note : Here, only consider light transmitted by the fundamental mode, within the fibre optics core.

9 QUESTION 6 : OPTICS 2 (CONT D) Figure 2 Variables used to describe the fibre, lens and light beam Note : Here, only consider light transmitted by the fundamental mode, within the fibre optics core. To help you, you may use the identity: sin(90 α) = cos (α). Part 2: (5 pts) Thin lens After the optical fibre, you place a thin lens to create a parallel beam of diameter D, as shown in Figure 2. Using the paraxial equation, what should the focal length f be? Write an equation relating the focal length, the diameter and the numerical aperture of the fibre optics. Part 3 : (5 pts) Dispersion Light from the optical fibre is polychromatic. The center of the spectrum is in the red at a wavelength of 632nm. The half-width at half maximum of the spectrum is 30nm. The spectrum hence ranges from 617nm to 647nm. You fear that using this simple lens causes chromatic dispersion. a) In your words, define the phenomenon of chromatic dispersion. b) Propose a solution to mitigate the effect of chromatic dispersion. Briefly explain the implementation of your solution. Part 4 : (5 pts) Diffraction grating You place a diffraction grating 25mm from your lens, in a Littrow configuration: the incidence angle equals the exit angle for the first order of diffraction, at the central wavelength, as shown in Figure 1. The diffraction grating has 1000 lines per millimetre and is used in transmission. What is the incidence angle for this grating in Littrow configuration? Page 9 de 11

10 QUESTION 7 : SOLID STATE PHYSICS I Free electron gaz model applied to Na Let us consider a thin film of sodium (Na). The film, resting on an isolating substrate, is shaped as a rectangular conductor for a resistance measurement as illustrated on the figure below. The dimensions of the conducting film are : length l = 10 cm, width w = 100 µm and thickness t = 1.0 µm. Applying a voltage, VA VB, of 100 mv between the extremities of the sample lead to a current I of 2.0 ma. The measurement is carried at T = 300K. V B l V A I t w Based on the free electron model, one can use this measurement to estimate the mean free path of the conducting electrons. a) (5 pts) From the experimental data, considering Ohm s law along with the link between the electrical resistance and resistivity, determine the numerical value of the electrical resistivity of Na. b) (5 pts) Admitting that Na has a body center cubic structure, with a lattice parameter of a = nm, and that each Na atom provides a single conduction electron: determine the numerical value of the free electron density in the material. c) (5 pts) Exploit the free electron model to estimate the value of the Fermi energy (in ev) of Na. d) (5 pts) Based on the results found in a), b) and c), and considering Drude formula, find the expression and numerical value of the free electrons mean free path. Reminder : the Drude relation for electrical conductivity is given by : 2 ne. m Page 10 de 11

11 QUESTION 8 : SOLID STATE PHYSICS 2 Consider a one-dimensional crystal for which the interatomic distance is a. In the thigh binding approximation, the energy of an electron as a function of the wave vector k of the electron, ε(k), is given for wave functions of type s by: (k)=s - 2s cos(ka) where s is the energy level of the state s, and s is the overlap integral of the wave functions s. a) (3 pts) Draw (k) in the 1st Brillouin zone as well as for a representation in repeated zone. What is the width of this band? b) (4 pts) Determine the effective mass of the electrons me * at k=0. c) (5 pts) Determine the velocity v(k) of the electrons and plot v(k) in the repeated zone representation. Why is the speed v = 0 at Bragg's plans? d) (8 pts) A constant field E is applied in space and in time. Assume that, at t = 0, the wave vector of the electron is k = 0. Also assume that the electron does not undergo collisions in the presence of the field. Show that the position x(t) of the electron in real space corresponds to an oscillatory motion. Discuss the fact that an oscillatory motion is expected with the application of a constant field. Is this possible? Page 11 de 11

EXAMEN GÉNÉRAL DE SYNTHÈSE PARTIE ÉCRITE Programme de doctorat en génie physique. Jeudi 14 juin Salle A-503. de 9h30 à 13h30

EXAMEN GÉNÉRAL DE SYNTHÈSE PARTIE ÉCRITE Programme de doctorat en génie physique Salle A-503 de 9h30 à 13h30 NOTES : No documentation allowed. A non-programmable calculator is allowed. The candidate can