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

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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.

1 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 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 and 11 pages. ENGLISH VERSION 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 1J4

2 CONSTANTES PHYSIQUES : e= m k e kg 34 = B = C J s = ev K 5 1 ÉQUATIONS PHYSIQUES : ÉQUATIONS MATHÉMATIQUES D =ρ f B = 0 B E = t D H = J f + t µ ˆ dv 4 r 1 E = µ e E = µ e 1 0 B = π J r f f ( ) ( ) B FD E k T ( ) ( ) B BE E k T Intégrale 0 e d = π a 1 π 0 e d = a 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 ( α±β ) = sin αcos β± cos αsin β c os( ) cos cos sin cos α ± β = α β α β sin θ= sin θcos θ cos θ= cos θ sin θ 1 cos α cos β = ( cos ( α +β ) + cos( α β )) 1 sin αsin β= cos ( α β) cos ( α+β ) 1 sin αcos β= sin ( α+β ) + sin ( α β ) ( ) ( ) Page de 11

3 QUESTION 1 : ÉLECTRICITÉ ET MAGNÉTISME Maxwell s equations predict the existence of electromagnetic waves. Using these equations, for a stationary, non conducting, homogeneous and linearly isotropic medium, without free charges and without free current density, for which e is the permittivity and m is the permeability, you are asked to (5 points) 1. Derive the wave equations governing the electric field and the magnetic field. (6 points). Considering a plane wave moving in the z direction in such a medium for which E îe exp( j( t kz)) and ˆ 0 H jh0 exp( j( t kz)), determine a) A relationship between H 0 and E 0 b) The wave number k as a function of, e and m. 3. Determine the relations between the amplitude of the Poynting vector and the electric and magnetic energy densities of the wave. (5 points) Numerical Application: A 0 gigawatt laser beam of mm diameter is moving in a glass plate for which m m 0 and e.56 e 0, where m 0 is the permeability of vacuum, and e Farad/m is the vacuum permittivity. Knowing that m 0 1 e 0 c where c m/s is the speed of light in vacuum, calculate the values of E 0 and of H 0. Page 3 de 11

4 QUESTION : MÉCANIQUE QUANTIQUE Harmonic oscillator and electric field A particle of charge q and mass m moves in one dimension in the presence of a harmonic potential of angular frequency ω. The Hamiltonian for this system is given by H 0 = p m + mω x. One also knows that the normalized ground state wave function for such a particle takes the form ψ 0 (x) = Ke βx. (5 points) a. Find the constants K and β as well as the energy E 0 of this ground state. (7 points) b. Determine the ground state wave function ψ 0 (x) and energy E 0 of the particle if an additional contribution H p = q E x is added to the Hamiltionian above due to the presence of an electric field E. Note: Here an expression for the exact solution is required. (8 points) c. Assuming that the electric field was turned on instantaneously, evaluate the probability that the system, initially in state ψ 0 (x), will be found in state ψ 0 (x). Page 4 de 11

5 QUESTION 3 : PHYSIQUE STATISTIQUE A cyclist likes to keep the tires on his bicycle inflated to 5 atm. He left the bicycle in a garage and returned 48 hours later to get the bicycle, only to find that the rear tire has deflated to 1.5 atm. Solve (a)-(c) to compute the diameter of the hole in the tire. Assume that the temperature T is constant and that the volume V of the tire does not decrease as it deflates. You may also assume that the air is made up entirely of N molecules (atomic weight 8). (10 points) a) Find the rate at which air molecules escape from the tire through the hole. The answer should depend on the area A of the hole, the temperature T, the mass m of the nitrogen molecules and the number density of nitrogen n 0 in the tire. Write down a similar expression for the rate that air passes from outside through the hole, replacing the inside density n 0 with the outside density n. b) Using your results from (a), find the rate dn 0 by which the density inside the tire is changing. dt Your answer should depend on the terms listed in part (a) as well as the volume of the tire V. c) Integrate your result from part (b) to find the elapsed time t as a function of m, A, V, T, n, n i and n f, where n i and n f are the initial and final densities inside the tire. d) Let T = 300 K, V = 0.5 liters (500 cm 3 ), t = 48 hours. Based on your result from (c) and the ideal gas law, find the diameter of the hole. Note : The velocity of the molecules of the air follows the Maxwell-Boltzmann distribution: p(υ)dυ = mβ 3 π exp mυ β dυ Page 5 de 11

6 QUESTION 4 : MÉCANIQUE SUPÉRIEURE Coronal mass ejection Consider a star with radius R and mass M. a) Find the minimum escape velocity allowing coronal matter to escape the gravitational field of the star. How does the escape velocity vary as a function of the ejection angle (defined with respect to the normal to the surface)? Consider now two identical stars of mass M orbiting around each other. ( points) b) Find the period of the orbit. c) Find the minimum escape velocity allowing the coronal matter to leave one star and end up on the other. d) Find the minimum escape velocity allowing the coronal matter to escape the gravitational field of both stars. If the distance between both stars is D=50R, ( points) e) Find the ratios of the velocities calculated in a) and c) and in a) and d). f) Discuss the overall transfer of mass between two stars if 1) their masses are equal and ) if their masses are different. Note : Consider a sharp transition between the star and its environment. Page 6 de 11

7 QUESTION 5 : OPTIQUE 1 Rings of Newton Figure 1. Schematic of a setup for observation of Newton rings. Consider a concave lens of radius R=1 cm in air placed on top of the planar perfect reflector (made of ideal conductor). Refractive index of the lens is n l =.8, while refractive index of air is n a =1.0. At the center of the lens r=0, z=0 the lens thickness is infinitely small. The lens is illuminated from the top by the monochromatic beam of wavelength λ=600nm. The incident beam is considered having propagation wavevector parallel to the Z axes. When looking under the microscope on the curved surface of the lens (air/lens interface) one observes dark and bright rings called Newton rings. These rings appear because of the interference between multiple waves. Particularly, the first wave (labeled as (1) in Figure 1) is due to reflection of the incident wave at the air/lens interface. The second wave (labeled as () in Figure 1) is due to transmission of the incident wave into the lens, its consequent reflection in the perfect reflector and the following transmission through the lens/air interface. In what follows we ignore contribution of the higher order waves which are due to multiple reflections in the lens between the air and perfect reflector interfaces. We also ignore refraction at the air/lens interface. Page 7 de 11

8 QUESTION 5 : OPTIQUE 1 (SUITE) (8 points) 1) Assuming normal incidence onto the air/lens and lens/reflector interfaces, using expressions for the Fresnel reflection and transmission coefficients for the electric field, and taking into account only the first two reflected waves (1) and () as shown in Figure 1, find electric field intensity of the total reflected wave at distance r from the lens center. Also, find the corresponding power reflection coefficient. ( points) ) At the center of the lens would you expect a maximum or a minimum of the light intensity (dark spot or a bright spot) as seen under the microscope? p 3) Find expressions for the radii of the Newton rings corresponding to the p s maxima r max and p p s minima r min of the reflected intensity. Give numerical value for the radius of the first p minima. In your derivations assume that r min,max R. (3 point) 4) Assuming that only two waves contribute to the interference pattern as explained in the introduction, find the relative intensity contrast between the minima and maxima and present its numerical value: I = R tot max min R tot R max min tot + R tot 5) Assuming that the incident planewave is not monochromatic but has a spectrum that spans an interval of wavelengths [λ, λ + Δλ], where Δλ = 6nm λ = 600nm, find the maximal ring order that would still be distinguishable under the microscope. (Hint: if the position of intensity maximum for one wavelength coincides with the position of the intensity minimum for another wavelength, then the contrast between the rings disappears). Page 8 de 11

9 QUESTION 6 : OPTIQUE Optics: polarisation A collimated beam of unpolarized light, of wavelength 63 nm, is launched along the optical axis of the optical system shown in the figure below. The optical system consists of two polarizers rotated by 45 placed on either side of two adjacent prisms made of calcite, a transparent uniaxial birefringent material. Each calcite prism has a square section, each side measuring 1 cm, and linearly variable thickness from 0 to 1 mm. The optical axis of the first prism is vertical, the optical axis of the second is horizontal. The transmitted beam is visualized on a white screen. (1 points) 1. Describe what is observed on the screen at the output of the system. (8 points). Describe a procedure allowing the determination of the wavelength of the incident light based on the observation on the screen. Figure : Optical system comprising: a polarizer at 45, a calcite prism with vertical optical axis (dashed line), a calcite prism with horizontal optical axis (dashed line), a polarizer at 45. A white screen is placed at the output of the system. The following data are given for calcite: ordinary index n o = 1,650 extra-ordinary index (along the optical axis) n e = 1,486 Note : Refraction in the prisms can be neglected. The beams can be assumed to propagate along the optical axis of the system. Page 9 de 11

10 QUESTION 7 : PHYSIQUE DU SOLIDE I You are going to perform a Hall effect measurement on a thin film of silver with a thickness d of 100 nm. (6 points) a) Show that the Hall coefficient, R H, defined as a E y /J x B z ratio, is given by: R H = 1/nq, where E y is the Hall electric field, J x is the current density, B z is the applied magnetic field, n is the charge carrier density, and q is the elementary charge. The geometry of the set-up is illustrated in Figure 1. b) Determine the Hall voltage V H measured upon application of a total current of 100 ma at the presence of a magnetic field of 1 T considering that R H = -0.8 x Ωm/T. Why does R H exhibit a negative value? c) Calculate a ratio between the charge carrier density and the atomic density of silver. Compare your obtained value with that expected theoretically. d) Calculate the lattice parameter of a silver film while considering the charge density calculated in (c). e) Is the mean free path of the charge carriers smaller or larger than the lattice parameter obtained in (d)? Justify your response. Useful parameters Figure 1: Hall effect measurement geometry. Atomic mass of silver: g/mol Mass density of silver: 10.5 g/cm 3 Resistivity of silver: Ω m at 0 C Atomic structure of silver: [Kr] d 10 5s 1 Crystalline structure of silver: c.f.c. Charge carrier transport equation: m dv dt + v τ = F, where F is the force acting on the charge carriers, v is the mean velocity, and τ is the time to transit the mean free path. Page 10 de 11

11 QUESTION 8 : PHYSIQUE DU SOLIDE (3points) a) Identify the different mechanisms of polarization in a dielectric material. (6 points) b) Here is the Clausius-Mossotti relationship: ϵ 1 ϵ + = Nα 3ϵ 0 Explain in a few lines the usefulness of the relationship and discuss the significance and nature of the physical variables in the relationship. (5 points) c) Amorphous selenium (a-se) has a density of 4.3 g cm -3 and a mass of g / mol. The value of the dielectric constant of a-se is 6.7. Calculate the polarizability of a Se atom in a-se. ( points) d) Calculate the polarizability of a single Se atom, α', for which the atomic radius, r, is 0.1 nm. Note: The electronic polarizability of an individual atom is given by the following equation: α = 4πε 0 r 3. e) Compare the result obtained in d) with the value of the polarizability of a Se atom in a-se and explain this difference. Page 11 de 11

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 18 juin Salle B-508.

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Salle B-58 de 9h3 à 3h3 NOTES : No documentation allowed. A non-programmable calculator is allowed. The candidate can answer