examen gÉnÉral de synthÈse Épreuve Écrite programme …

Département de génie physique Pavillon principal

Téléphone : 514-340-4787

Télécopieur : 514-340-3218 Courriel : [email protected]

Adresse postale C.P. 6079, succ. Centre-ville

Montréal (Québec) Canada H3C 3A7

www.polymtl.ca

Campus de l’Université de Montréal

2900, boul. Édouard-Montpetit

2500, chemin de Polytechnique

Montréal (Québec) Canada H3T 1J4

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

Predoc exam – written part Department of Engineering Physics

June 14, 2018

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FORMULAS AND CONVERSION

xn

n=0

¥

å =1

1- x, x <1

v !=v(v-1)(v-2)…

1 u = 931,494 MeV/c2

PHYSICAL CONSTANTS MATHEMATICS EQUATIONS e = 1.602 x 10-19 C Integrals

me = 9,109 x 10-31 Kg

ħ = 1.055 x 10 x 10-34Js

kB = 1.381 x 10-23 J/K

𝜀0 = 8.854 x 10-12 F/M

𝜇0 = 4𝜋 𝑥 10−7 N/A2

PHYSICS EQUATIONS :

f D

0 B

t

BE

ft

DH J

0

2

ˆdV

4

J r

Br

BFD E k T

1f E

e 1

BBE E k T

1f E

e 1

Clausius-Mossotti relation 𝑛𝛼

3𝜀0=

𝜀𝑟 − 1

𝜀𝑟 + 2

∫ √𝑥

∞

0

𝑒−𝑥 𝑑𝑥 = √𝜋

2

∫ 𝑒−𝑎𝑥2

∞

0

𝑑𝑥 =1

2 √

𝜋

𝑎

∫ 𝑥2 𝑒−𝑎𝑥2𝑑𝑥 =

∞

0

√𝜋

4𝑎3/2

Law of cosines 2 2 2

c a b 2ab cos

Stirling’s approximation n n

n! n e 2 n

Identity

2A 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 2cos 2 cos sin

1

cos cos cos cos2

1

sin sin cos cos2

1

sin cos sin sin2

Yv

*

-¥

¥

ò Yvdx = a Y

v

*

-¥

¥

ò Yvdy = a H

v

2(y )e -y 2

-¥

¥

ò dy = ap1/22vv !


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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.


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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, |𝜓0(𝑡)⟩, fulfills the following eigenvalue equation:

�̂�𝑥|𝜓0(𝑡 = 0)⟩ = ℎ̅

2|𝜓0(𝑡 = 0)⟩

where �̂�𝑥 is the spin operator along x axis.

(a) (10 pts) Knowing that |𝜓0(𝑡 = 0)⟩ can be expressed as a function of the eigenstates of

�̂�𝑧 |+⟩ et |−⟩, �̂�𝑧 is the spin operator along z axis, find the state vector of the particle at 𝑡 > 0

|𝜓0(𝑡 > 0)⟩. Note that B varies with time.

(b) (10 pts) Find the expected value <�̂�𝑧> at t > 0.

The Hamiltonian is given by : �̂� = 𝜔0�̂�𝑧 where 𝜔0 =|𝑒|𝐵(𝑡)

𝑚𝑐 (e, the elementary charge, m the electron

masse, and c the speed of light).

Note :

The solution of the differential equation 𝑓̇(𝑡) = 𝑦(𝑡). 𝑓(𝑡) is : 𝑓(𝑡) = 𝑓(0). 𝑒𝑥𝑝 (∫ 𝑦(𝑡′)𝑡

0. 𝑑𝑡′)


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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

𝜀𝑠 =𝑝2

2𝑚+ 𝑠 𝜇𝐵 𝐻

where 𝜇𝐵 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 𝜇0 and the electrons will fill all the states

with an energy 𝜀 ≤ 𝜇0.

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

3ℎ3(𝑝±,max)

3

c) (5 pts) Assuming that the magnetic field 𝐻 is such that 𝜇0 ≫ 𝜇𝐵𝐻, 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 = 𝜇𝐵(𝑁+ − 𝑁−) of this electron gas.

d) (5 pts) Show that the magnetic susceptibility of this electron gas is

𝜒 =𝜕𝑀

𝜕𝐻=

3𝑁(𝜇𝐵)3

2𝜇

The following relation could be useful:

(1 + 𝑥)𝑝 ≈ 1 + 𝑝𝑥 when 𝑥 ≪ 1.


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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;


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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.

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 = 𝑖2 − 𝑖1 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.

laser

i 1

i 2

v

φr = π

φt = 0φt = 0

φr = 0


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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 = √𝑛12 − 𝑛2

2

where 𝑛1 is the index of the core of the fibre (the part transmitting light) and 𝑛2 is the index of the

cladding (the part that does not transmit light). Use variables shown in Figure 2.

(next page)


June 14, 2018

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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?


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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.

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 = 0.423

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 :

2ne

m

.

l

w

t

I VA

VB


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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?

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