QUALIFICATION EXAM – WRITTEN PART Ph. D. Program in

Department Engineering Physics Main Building Téléphone : 514-340-4787 Télécopieur : 514-340-3218 Courriel : [email protected]

Adresse postale

P.O. Box 6079, Station Centre-ville Montréal (Québec) Canada H3C 3A7

www.polymtl.ca

2900, boul. Édouard-Montpetit Campus of the’Université of Montréal 2500, chemin de Polytechnique Montréal (Québec) Canada H3T 1J4

QUALIFICATION EXAM – WRITTEN PART Ph. D. Program in Engineering Physics

Thursday June 17, 2021

Room C-539.6 (Main Building)

from 9h30 to 13h30

NOTES : • No documentation is allowed.

• A non-programmable calculator is allowed.

• A page of mathematical and physics equation is provided at page 2.

• This examination contains 8 questions, 10 pages in total.

• Each question is worth 20 points.

• Provide solutions to no more than 6 questions of your choice. • Use a different notebook for each question, clearly making each notebook with the corresponding question number.

ENGLISH VERSION

Predoc exam – written part Department of Engineering Physics June 17th, 2021

Page 2 of 10

FORMULAS AND USEFUL RELATIONS

Constants

1 Å = 10�10m 1 eV = 1.602⇥ 10�19 J h = 6.626⇥ 10�34 J sc = 2.998⇥ 108 m/s kB = 1.381⇥ 10�23 J/K ✏0 = 8.854⇥ 10�12 C2/N m2

me = 9.109⇥ 10�31 kg mp = 1.672⇥ 10�27 kg |e| = 1.602⇥ 10�19 CNA = 6.023⇥ 1023 mol�1 a0 = 0.5291⇥ 10�10 m µ0 = 4⇡ ⇥ 10�7 N2/A2

Physics equations

r·D = ⇢f ; r·B = 0 ; r⇥E = �@B

@t; r⇥H = Jf+

@D

@t; B =

µ0

4⇡

ZJ⇥ r̂

|r|2 dV

fFD(E) =1

e(E�µ)/kBT + 1(distrib. Fermi-Dirac); fBE(E) =

1

e(E�µ)/kBT � 1(distrib. Bose-Einsten)

N↵

2"0=

"r � 1

"r + 2(Clausius-Mossotti relation); En =

n2⇡2~22mL2

(energy of an infinite 1D well)

IntegralsZ

dx

ex + 1= x� ln(ex + 1) ;

Zdx

ex � 1= ln(ex � 1)� x

Z 1

0xne�qxdx =

n!

qn+1, n > �1, q > 0 ;

Z 1

0e�↵2x2

dx =1

2↵

p⇡

Z 1

0x2ne�↵x2

dx =1 · 3 · 5 · · · (2n� 1)

2n+1↵n

r⇡

↵;

Z 1

0x2n+1e�↵x2

dx =n!

2↵n+1, (↵ > 0)

Trigonometric identities

cos(2a) = cos2 a� sin2 a = 2 cos2 a� 1 = 1� 2 sin2 a ; sin(2a) = 2 sin a cos a

sin a sin b =1

2cos(a� b)� 1

2cos(a+ b) ; cos a cos b =

1

2cos(a� b) +

1

2cos(a+ b)

sin a cos b =1

2sin(a+ b) +

1

2sin(a� b) ; cos a sin b =

1

2sin(a+ b)� 1

2sin(a� b)

cos a+ i sin a = eia ; cos a� i sin a = e�ia

Others

ex = 1 + x+x2

2!+

x3

3!+

x4

4!+ . . . ; (1 + x)n = 1 + nx+

n(n� 1)

2!x2 + . . .

c2 = a2 + b2 � 2ab cos � (cosine law) ; n! ⇡ nne�np2⇡n (Stirling approximation)

r2A = r(rA)�r⇥r⇥A (Identity) ;

1X

n=0

xn ⇡ 1

1� x, |x| < 1 (Serie)

p1± x2 ⇡ 1± x2

2, if x ⌧ 1


Page 3 of 10

QUESTION 1 : ELECTROMAGNETISM

Magnetic micrometer In many areas of technology, it is important to measure the thickness of dielectric and insulating components. One of the fast and straightforward techniques is the magnetic micrometer shown in the figure below.

Here, the micrometer consists of a flexible ring of high magnetic permeability µ (µ >> µ0), of medium radius r, of circular cross-section of radius a (a << r) which is cut to form a "C" shape ". The component (sheet, plate) to be measured is inserted into the air gap; by applying pressure, both sides of the ring come in contact with the component. A current I = I0 sinwt circulates in a coil made of N1 turns of wire, and an electromotive force e is measured at the terminals of a second coil comprising N2 turns of wire.

a) (7 pts) Determine the magnetic flux F circulating in the magnetic circuit and the magnetic field Bg in the gap filled with the component.

b) (5 pts) Calculate the mutual inductance M12 between the two coils; c) (4 pts) Determine the electromotive force e; d) (4 pts) Knowing that the voltmeter with which we measure e has a resolution DV, what

will be the resolution Dx for measuring the thickness x?


Page 4 of 10

QUESTION 2 : QUANTUM MECHANICS


Page 5 of 10

QUESTION 3 : STATISTICAL PHYSICS

A 1D random walk is occurring in discrete steps of length 𝐿. The total number of steps (𝑁) is the sum of the number of steps to the left (𝑁←) and to the right (𝑁→). Using,

𝑁 = 𝑁→ + 𝑁← ∆= 𝑁→ − 𝑁←

the probability of finding the particle at the position ∆𝐿 is

𝑃#(𝑁→, 𝑁←) = 𝑁!

𝑁→! 𝑁←!𝑝#→𝑞#← =

𝑁!

/12 (𝑁 + Δ)3 ! /12 (𝑁 − Δ)3 !

𝑝(#%&)/)𝑞(#*&)/) = 𝑃#(Δ)

a) (4 pts) Explain the origin of these terms: #!#→!#←!

, 𝑝#→, and 𝑞#←. b) (2 pts) For 𝑝 = 𝑞, what is the average position ⟨Δ𝐿⟩. Explain your reasoning.

Note: ⟨… ⟩ represents an average over all possible paths. c) (6 pts) The “root mean square” distance, 7⟨Δ)𝐿)⟩, is commonly used to estimate the

position with respect to the origin. Using again 𝑝 = 𝑞, calculate it for 𝑁 = 1, 2, 3 and 4. d) (2 pts) By inspection, what is the value of 7⟨Δ)𝐿)⟩ for any 𝑁.

Most of the fusion energy generated by the sun is occurring close to its center (𝑅 = 0). It is believed that the generated photons then undergo a symmetric random walk towards the surface, bouncing off matter after 𝐿 = 10*- m. This photon transport process dominates from the center to about 5/7 of the sun’s diameter (𝑅.), after which much faster processes help photons escape.

e) (6 pts) Assuming a 1D random walk, estimate the average time taken by a photon generated at the center to escape the sun. Assume that the refractive index of the sun is 1.

Data: 𝑅. = 5 × 10/ m; there are about 3 × 100 seconds in a year.


Page 6 of 10

QUESTION 4 : CLASSICAL MECHANICS

A frictionless rail supports a mass 𝑀. A smaller mass 𝑚 is attached to 𝑀 via a rigid massless string of length 𝑙.

(a) (3 pts) Write down the Lagrangian for this system.

(b) (6 pts) Obtain corresponding equations of motion. Approximate the final expressions for a case of small angle 𝜃 (to first order in 𝜃).

(c) (3 pts) Using results of part b, obtain 𝑥(𝑡) and use it to simplify the equation of motion for 𝜃.

(d) (4 pts) Find expressions for 𝜃(𝑡) and 𝑥(𝑡), assuming initial velocity 𝑣1.

(e) (3 pts) Write down special solutions from part (d) that correspond to normal modes of the system. Hint: consider either finite or null values of 𝑣1. How does the relative motion of the masses differ for these two modes?

(f) (1 pt) Consider motion types of part (e) in the limit of 𝑀 ≫ 𝑚. Is this an expected result?


Page 7 of 10

QUESTION 5 : GEOMETRICAL OPTICS

If we consider that the atmosphere is a homogeneous gas with temperature T = 300K and pressure P = 1 atmosphere, its refractive index as a function of the radius r can be expressed as:

. The variable R = 6400 x 103 m is the radius of the Earth and r is the distance from the center of the Earth. Hint: It is advisable to use Fermat's Principle in order to solve this problem. (a) (6 pts) Write the mathematical expression describing the optical path between two points located at the same altitude. (b) (10 pts) Calculate what must be r so that a collimated laser beam follows a circular path when emitted at sea level in a direction parallel to the surface (r = R). (c) (4 pts) How will the light paths be affected for the same value of r as that found in b) if the laser is not collimated, so if the laser has a certain divergence.


Page 8 of 10

QUESTION 6 : WAVES OPTICS

A Michelson interferometer is used to measure the refractive index of a liquid placed in one of the two arms of equal length L. A collimated tunable laser illuminates the interferometer at a selectable wavelength. A photodetector measures the output optical power. The instrument therefore produces a measurement of the output power 𝑃(𝜆) of the interference fringes exiting the interferometer as a function of the illumination wavelength.

Ideal conditions are assumed: perfectly transparent liquid, an exact 50/50 beamsplitter, perfect mirrors, etc.

a) (4pts) Write an expression for the accumulated phase difference Δ𝜙 between the vertical arm containing a liquid of index 𝑛(𝜆) and the horizontal arm of refractive index 𝑛 = 1.

b) (4 pts) Deduce then an expression for the measured power 𝑃(𝜆) at the photodetector.

c) (8 pts) Calculate the fringe spacing, i.e., the wavelength variation Δ𝜆 required to observer one full fringe at the photodetector.

d) (4 pts) Write this fringe spacing by involving the group index 𝑛2 = 𝑛 + 𝜔 3435

.

laser @�

P (�)


Page 9 of 10

QUESTION 7 : SOLID STATE PHYSICS 1

An ideal two-dimensional crystal consists of only one kind of atoms of mass m. Each atom has an equilibrium position located at a point of a square lattice 𝒓𝝉𝒔 = (𝜏𝑎, 𝑠𝑎), where τ,s=1,2,…N. We denote the displacements from the equilibrium position by (𝑥89, 𝑦89), i.e.,

𝒓𝝉𝒔 = (𝜏𝑎 + 𝑥89, 𝑠𝑎 + 𝑦89). In the case of harmonic approximation, we assume a crystal potential of the form 𝑉(𝑥89, 𝑦89) = ∑ Q𝑘:S(𝑥(8%:)9 − 𝑥89))+(𝑦8(9%:) − 𝑦89))T + 𝑘)S(𝑥8(9%:) − 𝑥89))+(𝑦(8%:)9 −8,9

𝑦89))TU.

(a) Determine the general phonon dispersion relation w4𝒒, where n is the band index and q the wavevector by:

- (5 pts) Writing down the equation of motion

- (10 pts) Finding a solution of the equation of motion in the form of a traveling wave in the crystal

(b) (5 pts) For the case 𝑘) = 0.1𝑘:, sketch w4𝒒 in the 𝑞=direction in the first Brillouin zone, as a function of the wavevector q of the form 𝒒 = W𝑞= , 𝑞>X =, 0 ≤ 𝜖 ≤ ?

@


Page 10 of 10

QUESTION 8 : SOLID STATE PHYSICS 2

Two-dimensional electron gas Consider a two-dimensional electron gas, which consists of non-interacting electrons

confined in a quantum well of width 𝐿. The gas is subjected to the following external

potential:

[𝑉 = 0|𝑧| < 𝐿

2_

𝑉 = 𝑉1|𝑧| > 𝐿2_

(a) (5 pts) Calculate and plot the density of states as a function of energy when 𝑉1 →

∞.

(b) (5 pts) Calculate and describe the behavior of the density of states in the case of

high energies.

(c) (5 pts) At 𝐿 = 10𝑛𝑚, up to which temperature the system remains a two-

dimensional electron gas.

(d) (5 pts) Without doing any calculations, describe the steps to follow in order to

identify the thickness range to achieve the two-dimensional electron gas when the

potential has a finite value of 100 meV and the temperature is 20 mK.

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QUALIFICATION EXAM – WRITTEN PART Ph. D. Program in