EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 16 juin Salle A-401.

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EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Jeudi 6 juin 06 Salle A-40 de 9h30 à 3h30 NOTES : No documentation allowed. A non-programmable calculator is allowed.

1 EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Jeudi 6 juin 06 Salle A-40 de 9h30 à 3h30 NOTES : No documentation allowed. A non-programmable calculator is allowed. The candidate can answer up to 6 questions of his choice. Each question is worth 0 points. Use a notebook different for each question, making sure to include the question number on it. This questionnaire contains 8 questions, 5 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 900, boul. Édouard-Montpetit 500, chemin de Polytechnique Montréal (Québec) Canada H3T J4

2 CONSTANTES PHYSIQUES : e =.60 x 0-9 C me = 9,09 x 0-3 Kg ħ =.055 x 0 x 0-34 J s kb =.38 x 0-3 J/K εε 0 = x 0 - F/M μμ 0 = 4ππ xx 0 7 N/A ÉQUATIONS PHYSIQUES : D =ρ f B = 0 B E = t D H = J f + t µ ˆ dv 4 r 0 B = π J r f f ( E) = ( µ ) B FD E k T e + = µ e ( E) ( ) B BE E k T Clausius-Mossotti relation nnnn 3εε 0 = εε rr εε rr + ÉQUATIONS MATHÉMATIQUES Intégrale xx 0 ee aaaa 0 ee xx dddd = ππ dddd = ππ aa xx ee aaxx dddd = 0 ππ 4aa 3/ Loi des cosinus c = a + b ab cos γ Approximation de Stirling n n n! n e π n Identité A= A A ( ) Identités trigonométriques sin(αα ± ββ) = sin αα cos ββ ± cos αα sin ββ cos(αα ± ββ) = cos αα cos ββ sin αα sin ββ sin θ= sin θcos θ cos θ= cos θ sin θ cos α cos β = ( cos ( α +β ) + cos( α β )) sin αsin β= cos ( α β) cos ( α+β ) sin αcos β= sin ( α+β ) + sin ( α β ) ( ) ( ) June 6, 06 Page de 5

3 QUESTION : ELECTRICITY AND MAGNETISM Induction-based power charging The Wireless Power Consortium (WPC) publishes a standard for wireless charging of electrical devices, notably cell phones and tablets. The energy transfer between the charging base and the device is based on magnetic induction. The charging base contains near the surface a circular coil of diameter φφ ss = 30 mm connected to an alternative-current (AC) source ii ss (tt) = ii ss0 cos(ωωωω). The device being charged contains near its back-surface a thin circular coil of diameter φφ dd = 5 mm. The following data is given. Inductance of the source coil LL ss 5 µh Inductance of the device coil LL dd 35 µh Mutual inductance MM ssss 5 µh Frequency of operation ff rr 00 khz Based on The Qi Wireless Power Transfer System, WPC, version.., April 06. Answer the following questions. ) (8 pts) Estimate the number of turns in the source coil and the device coil, based on their inductance value. Indicate the approximations used for this estimation. For the following questions, the operating point of the charging system is such that, across the device coil, ii dd =,5A cos(ππff rr ) VV dd = V cos(ππff rr ππ/3) ) (6 pts) Calculate the electrical power delivered to the device. 3) (6 pts) Write a formula for the calculation of the required current on the source coil to attain the operating point. June 6, 06 Page 3 de 5

4 QUESTION : QUANTUM MECHANICS 4 38 We consider the emission of an alpha particle ( HHHH) by a 9UU nucleus by tunnel effect. The kinetic energy of the particle is 4. MeV. The profiles of the potentials are indicated in the figure below. Before emission, the alpha particle is contained within the nuclear radius (rn = m). a) (5 pts) Give the definition of tunnel effect (3-5 lines). b) (0 pts) Find the height of the barrier (Vc (r=rn)) and the distance that the alpha particle has to travel by tunnel effect, by using a square potential to calculate the transmission coefficient by tunnel effect. c) (5 pts) Is the result obtained reasonable? Explain (3-lines). Formulas VV CC = ZZ ZZ ee 4ππεε 0 rr NN ππ mm(vv EE) κκ = h TT = 6 EE EE ee κκκκ VV 0 VV 0 Useful information ev=, J Mass of an alpha particle 377 MeV/c June 6, 06 Page 4 de 5

5 QUESTION 3 : STATISTICAL PHYSICS Consider a system of N particles in a box of fixed volume. Each particle may be in only two quantum states with energies 0 or E. The total energy of the system is U. (0 pts) a) Determine the entropy of this system. (0 pts) b) Calculate the temperature T and deduce U(T). i) What is U at T=0 K? Discuss the physical meaning of this result. ii) What is U at high temperature? Discuss the physical meaning of this result. Sterling approximation : lnx! = x ln x - x June 6, 06 Page 5 de 5

6 QUESTION 4 : CLASSICAL MECHANICS A bead of mass m slides without friction along a curved wire with shape z = f(r), where we use cylindrical coordinates (r, θ, z). The wire rotates around the z-axis at a constant angular velocity ω, keeping its shape fixed. Gravity acts downward, with an acceleration g > 0.. (8 pts) Using Newton s second law (F = m a) in an inertial frame, derive an expression for the radius r0 of a fixed circular orbit (i.e. a solution with r = r0 = const.).. (8 pts) What is the normal force the wire applies to the bead to keep it in a circular orbit? 3. (4 pts) Write down the one-dimensional Lagrangian L(r, dr/dt, t) for this dynamic system. Using this Lagrangian, obtain the equation of motion for r(t) and verify your result for r0 obtained in (a). June 6, 06 Page 6 de 5

7 QUESTION 5 : OPTICS I Interference. Fabry-Perot Interferometer Commercial systems for generating and detecting terahertz continuous waves (electromagnetic waves of constant amplitude and frequency) are often found in the following configuration. A photoconductive antenna on the silicon wafer acts as a point source of THz waves. This antenna is located at the focal point of a lens (focal length F) that is used to collimate the light emitted by the point source. A second lens identical to the first lens is placed at a distance L from the first one. It is used to focus light onto a point detector the form of a photoconductive antenna. In the following, all the coefficients of reflection and transmission are given by amplitude. The detector absorbs a fraction t D of the incident light, while a fraction r D of the incident light is reflected. The reflected light, therefore, returns to the source of emission. When it arrives at the source, a fraction of the light is lost due to scattering a source (the antenna), while a fraction r S is reflected back to the detector. In this way, a wave emitted by the source travels infinitely times between a source and a detector, and the system is known as a Fabri-Perot resonator. Consider in more details the path of a wave emitted by the point source. By writing the wave emitted by the point source in the complex form as E exp i ( kx ωt ), where k = ω c, answer the following questions: 0 ( ). (4 pts) Give an expression for En( ω,) t that describes a part of the original wave detected by the point detector after the emitted wave travels n times back and forth in the cavity of the Fabry- Perot resonator. A hint: for example for n= (that is to say after emission, a journey to the detector, a partial reflection on the detector, a journey to the source, a partial reflection on the source and a journey to the detector), the detector will register the following portion of the initial wave E( ω, t) = Et rr exp i(3 k( F+ L) ωt) 0 D D S ( ). (0 pts) Find an expression for the total complex amplitude E( ω,) t of the wave recorded by the detector as the sum of all partial contributions found in question, + E( ω,) t = En( ω,) t n= 0. next page June 6, 06 Page 7 de 5

8 3. (3 pts) Draw schematically the behavior of the intensity I ( ω ) registered by the detector (spectrogram) as a function of the source frequency using ( ω) for the spectral positions of the minima I = E( ω,) t. Give expressions ω min and maxima ωmax of the intensity I ( ω ). 4. (3 pts) Find the values of the minimal intensity I ( ω min ) and the maximal intensity I ( ω max ) well as a spectrogram contrast defined as I( ω ) I( ω ) I ω + I ω. ( max min ) ( ( max ) ( min )), as June 6, 06 Page 8 de 5

9 QUESTION 6 : OPTICS Fig. is a schematic diagram showing a simplified spectrometer consisting of a transmission diffraction grating with a groove density of 000 lines per millimeter and of a lens having a focal length f and focussing each wavelength onto a detector array having 5 elements, each 0 micrometers wide. We hypothesize that the distance between each element is null. The beam incident on the grating has a diameter φφ, a central wavelength λλ 0 and a bandwidth ΔΔΔΔ. Figure Schematic diagram of a simple spectrometer showing a parallel beam incident at angle αα into a diffraction grating. The lens of focal length f focuses each wavelength onto an element of the CCD detector array.. (6 pts) Find the angle αα for which the diffracted central wavelength propagates parallel to the optical axis, perpendicular to the diffraction grating, the lens and the detector array?. (6pts) For this spectrometer, the resolution may be limited by the diffraction grating or by the lens. In order to obtain a diffraction-limited resolution (limited by the lens), the diameter of the Airy disk produced by the lens on the detector must be smaller than the width of an element of the detector array. Find the diameter of the incident beam required to obtain a diffraction-limited spectrometer. 3. (4pts) What is the maximum bandwidth, ΔΔΔΔ, allowed for this spectrometer. 4. (4 pts) Identify two methods that would increase the allowed bandwidth. June 6, 06 Page 9 de 5

10 QUESTION 7 : SOLID STATE PHYSICS I For a free electron gas, the expression of permittivity as a function of frequency is given by a) (6 pts) Demonstrate the expression (). b) ( pt) What does ωω pp represent? εε(ωω) = ωω pp () ωω c) (5 pts) What happens to the incident light when ωω pp < ω and when ωω pp > ω? d) (.5 pts) Draw schematically the variation of the ideal reflectance in Figure. e) (.5 pts) Calculate the cutoff wavelength for silver that has a lattice parameter of 4,079 Å. The following figure shows the reflectance of silver as a function of energy of the incident wave: Figure : Reflectance as a function of energy for a silver surface. f) Explain in few words the following : i. (.5 pt) Why is the reflectance indicated in Figure lower than the one predicted in point (d)? ii. (.5 pt) What happened at point A (see Figure )? Note : ωω pp = nnee εε 0 mm Reflection coefficient : R = [(n-) + k ] / [(n+) + k ] June 6, 06 Page 0 de 5

11 QUESTION 8 : SOLID STATE PHYSICS A solid element submitted to a heat source can store thermal energy within atomic vibrations. Einstein model assumes that each atom is submitted to an elastic force proportional to its displacement relative to its equilibrium position, and is thus equivalent to three harmonic oscillators (three translational degree of freedom) with a fixed vibrational frequency ωe. This model enables one to establish links between thermal and elastic properties of solids. The figure below shows the comparison between experimental data of the heat capacity and the behavior predicted by Einstein and Debye models. Statistical physics applied to Einstein model, allows us to establish the following expression for the heat capacity of one mole of copper ΘE T ΘE e C= 3Nk A B T ΘE T ( e ), where Θ E = ωe kb is the Einstein temperature (30 K for copper). Copper (atomic number 9) exhibits a face center cubic structure, with a density at 300 K of 8.96 g/cm 3 and an atomic mass of 63,55 g/mol. a) (4 pts) Show that at room temperature, the heat capacity is given in good approximation by Dulong and Petit law C= 3Nk A B. b) (6 pts) Based on the data on the figure, estimate the energy required to increase the temperature of a cm 3 cube of copper from 93 K to 303K. c) (4 pts) Estimate the vibration frequency of the atoms at room temperature. next page June 6, 06 Page de 5

12 d) (4 pts) Estimate the elastic constant (spring constant) of the Cu-Cu bond. e) ( pts) Discuss the possibility of estimating the speed of sound using the experimentally determined Einstein temperature (explain the procedure or explain why it would not work). Constantes u = kg NA = 6.0 x 0 3 mol - ev =.60 x 0-9 J /(4πε0) = 9.0 x 0 9 June 6, 06 Page de 5

13 QUESTION 9 : SUPPLEMENTARY QUESTION This doctoral comprehensive examination has questions about signal processing: continuous system and discrete system Question continuous system A time invariant and causal linear system has an impulse response ht () with a Laplace transform L { ht ()} given by 3.5( s ) H() s = s +.5s a) (5 points) Find and represent in the complex plane the poles and zeros of H() s. b) (5 points) Determine the regions of convergence of H() s and the impulse response of ht (). c) (5 points) As shown in figure below, a signal xt () is applied at the input of the system. The signal xt () has the following characteristics : ) xt () = 0, pour tout t < 0 and, ) xt () is such that its Laplace transform is X() s = s 0.75 for t > 0. ) Find the convergence regions of the Laplace transform Y() s, obtained at the output of the system. ) Say if the Fourier Transform Y( f ) exists or not. If yes, determine Y( f ). Otherwise, explain the raison for which Y( f ) doesn t exist. Linear and time invariant system d) (5 points) Repeat the question c) but for a signal xt () = 0, for all t > 0 and xt () is such that its Laplace transform is X() s = s 0.75 for t < 0 e) (5 points) Repeat the problem in c) by considering the input signal xt () = 0for all t < 0 and xt () is such that its Laplace transform : 5 X() s = s , for t > 0 June 6, 06 Page 3 de 5

Question - discrete system The signal flow graph shown in the following figure represents a realisation of a causal digital system having a transfer function H (z). 0.

Indicate the region of convergence. Discuss about the stability, causality, linearity and time invariance (or variance) of that system.

14 Question - discrete system The signal flow graph shown in the following figure represents a realisation of a causal digital system having a transfer function H (z). 0.3 a) Determine the transfer function H (z). What is the system order? b) Determine the poles and zeros of H (z) and give a representation in the complex plan of z. Indicate the region of convergence. Discuss about the stability, causality, linearity and time invariance (or variance) of that system. c) Give the difference equation for that system and find its impulse response. d) Show that the transfer function H (z) can be represented by the system flow graph shown below. Determine the values of coefficients Ki, i =,,3, z - z - June 6, 06 Page 4 de 5

15 QUESTION 0 : SUPPLEMENTARY QUESTION Comprehensive exam, digital logic part You have to design a digital circuit composed of: a clock input H a Boolean input vector X3..0 (4 bits) a Boolean output Y N-input gates: o OR(cost is N+), AND(cost is N+), XOR(cost is N) o NOR(cost is N+), NAND(cost is N+), XNOR(cost is N) D-flip-flops (cost is 0) The circuit samples the value of X3..0 at each rising edge of the clock H. Normally, at each clock cycle, one and only one bit of X3..0 is changed. Here is an example of a normal behaviour: You have to design a circuit that will detect any behaviour different from the normal behaviour (0,, 3 or 4 bits are modified). Here is an example of an abnormal behaviour: 0 0 (two bits have changed) At power-up, Y is false. As soon as an abnormal behaviour is detected, Y becomes true (on the same rising edge of the clock H as the one that has sampled the abnormal X3..0) and remains true forever. Propose your best circuit that minimizes the cost as much as possible. June 6, 06 Page 5 de 5

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 23 novembre Salle B-406.

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Jeudi 3 novembre 07 Salle B-406 de 9h30 à 3h30 NOTES : No documentation allowed. A non-programmable calculator is allowed.