final exam – written part - École Polytechnique de Montréal

ÉCOLE POLYTECHNIQUE 

Département de génie physique 

 

EXAMEN GÉNÉRAL DE SYNTHÈSE – PARTIE ÉCRITE 

Programme de doctorat en génie physique 

Vendredi 16 juin 2006 

au local B-413 

de 10h00 à 14h00 

NOTES : 

Aucune documentation permise. 

Calculatrice électronique permise. 

Le candidat répond à 6 questions de son choix sur les 8. 

Les questions ont toutes le même poids. 

Utiliser un cahier différent pour chaque question. 

Le questionnaire contient 11 pages. 

CONSTANTES UTILES : 

 

m e 

q 

= 1.054 x 10 -34 J.s 

= 9.1 x 10 -31 kg 

= 1.6 x 10 -19 C 

ENGLISH VERSION

QUESTION 1 : 

Skin effect 

For the following questions, you may consider a monochromatic plane wave of the form : exp ikz i t . 

a) Show that a good conductor behave as a dielectric with an effective permeability : 

eq 

i . 

b) For a material which is a good conductor, such as

QUESTION 2 : 

In a unidimensional box of length L where V(x) = 0 for 0 ≤ x ≤ L, and V(x) = ∞ elsewhere. We consider that the 

potential changes with time in the interval 0 ≤ x ≤ L as 

V 1 

(x) 

x 

L 

2 sin t 

(a) 

(b) 

Determine the probability that a ground state particle (n = 1) makes a transition to the first excited state 

(n = 2) (you can consider that t to answer this question) ? 

What is the probability that the ground state particle makes the transition to the second excited 

state (n = 3) ? 

(c) What happens to the results when 0 ? 

Examen général de synthèse - Partie écrite 


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Département de génie physique

CONSTANTES, INTÉGRALES ET RELATIONS UTILES 

constantes 

m e = 9.11x10 -31 kg, 1 eV = 1.6022x10 -19 J, c = 3x10 8 m/s, e = 1.602x10 -19 C 

h = 6.6262x10 -34 J s, a 0 = 0.529177x10 -10 m, 0 = 8.854x10 -12 C 2 /Nm 2 

intégrales 

0 

x n e qx dx 

n! 

q n 1 , n 1, q 0 

sin 

cos 

x 2n e ax2 dx 

0 

0 

e i x x2 dx e 2 4 

x 2n 1 e ax2 dx 

xcos(ax)dx 

xsin(ax)dx 

1 3 5(2n 1) 

2 n 1 a n a 

n! 

, 

n 1 

2a 

(a 0) 

x 

1 

a cos(ax) 

2 1 

a sin(ax) 

2 

identités trigonométriques 

a sin(ax) 

x 

a cos(ax) 

1 

sin sin 

2 cos( ) 1 

2 cos( ) cos cos 1 

2 cos( ) 1 

2 cos( ) 

1 

2 sin( ) 1 

2 sin( ) cos sin 1 

2 sin( ) 1 

2 sin( ) 



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QUESTION 3 : 

Consider a one-dimensional lattice and assume that the energy of the conduction electrons is given by : 

k 

c 

 

2 

k 

2 

* 

2m 

e 

. 

where 

c 

is the conduction band minimum and 

a) Show that the density of state g c ( ) of the electrons is 

* 

m 

e 

is the effective mass of the electrons. 

b) 

1 

2 

g 

c 

2 

 

2m 

e 

C 

c) Show that the electronic concentration n in the non-degenerate case 

(where one can use the Maxwell-Boltzmann statistics) can be written as: 

n 

N exp F c 

c 

k T 

; 

B 

Determine N c . ( k B 

is the Botzmann constant and 

F 

the Fermi level). 

d) By inspection, determine the concentration p of the holes. 

e) Calculate the intrinsic Fermi level as a function of the effectives masses. 

Usefull relation : 

0 

exp 

x 

1 

dx 

x 



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QUESTION 4 : 

A rigid pendulum consists of a mass m fixed at the end A of a rigid rod. This rod is of length L and of 

negligeable mass. The gravitational field is vertically oriented downward and constant in time. The end B of the 

rod (the pivot) is following a forced vertical movement, given by a function h(t), around the fixed point O. This 

configuration is illusrated on figure 1. 

Determine the Lagrangian and the Hamiltonian of this system, using the horizontal line passing through the 

fixed point O as the reference for the potential. 

Figure 1 



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QUESTION 5 : 

Propagation in a graded index lens 

A graded index lens with parallel end-faces has a refractive index with radial dependence 

n 2 (r) n 0 

2 

ar 2 

where r is the distance from the central axis of the lens in a cylindrical coordinate system and n 0 is the index on 

the central axis. 

Using the ray equation of geometrical optics 

d 

ds n d r 

ds 

 

n 

answer the following questions : 

1. Calculate a function r(z) describing the path of a ray incident at a distance r 0 from the center of the input 

face and perpendicular to this face. Use the paraxial approximation and eliminate negligible terms in 

your calculation if needed. 

2. What is the minimal thickness of the lens for an incident plane wave on the input face to be focused to a 

point on the output face? 

3. What is the transfer matrix of this lens? 

4. What is the transfer matrix of four of these lenses placed in sequence and in contact? Comment. 



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QUESTION 6 : 

The intensity of a source, i.e. the power emitted by the whole source per unit solid angle 

(W/steradian) in a given direction is given by: 

I( ) = I 0 cos n ( ) 

a) Calculate the power contained in a cone with an angular spread of ± . 

b) Calculate the total power emitted by the source . 



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QUESTION 7 : 

Fluctuations of magnetization in a paramagnet 

a) Show that for an electron with J 

M Ng J B 

0 

J 

b g ; 

g 

0H 

kT 

1 

2 

and g 2 that the general form of the magnetization 

Jb g b g c b gh b g 

1 

Where B J 1 2 coth J 1 2 1 2coth 

2 

J 

becomes M N 0 

tanhb 2g 

cosh 1 

cosh 1 

note : tanhb 2g 

b g 

; coth 2 

sinh b g b g b g 

sinh 

b g 

b) The partition function for one atom “i” is 

c b 

b 2g 

m J sinh J 

ZibT, Hg expbm 

g 

sinh 

m J 

b g b g 

1 2 

gh 

Zi T, H 2cosh 

2 pour J 1 2 et g 2 

and the probability that a spin has energy Ei g Hm 

0 

is 

P i , m 

exp 

e 

b 

Ei 

Zi T, 

H 

j 

g 

where m l 1 2, 

1 2q . 

expb mg 

Zi T, 

H 

b 

g 

Find the fluctuation (or variance) of the magnetization (m) of one atom 

m 

2 2 

m m 

note : 

tanhb g 

sech 2 b g 



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c) Find the fluctuation (or variance) of the total magnetization M using 

N 

2 

g 0 

m et M M 

2 2 

i 1 

d) Show that 

M 

2 

kT 

where 

M 

H 

is the magnetic susceptibility. 

e) The result in d) has the same form as that for energy fluctuations where 

E 

2 

kT 

2 

C V 

and C V 

are both response functions of a system. 

What is the response described by ? 

What is the response described by C V 

? 

Describe in one sentence the relationship between a response function and a fluctuation? 



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QUESTION 8 : 

CONSTANTES UTILES : 

= 1.054 x 10 -34 J.s 

m e = 9.1 x 10 -31 kg 

q = 1.6 x 10 -19 C 

N 

23 

6.02 10 at 

1 

mol 

A 

-1 

Gold (Au) (atomic mass196.97 g mol ) is a monovalent metal which solidifies in a face centered cubic 

structure, with a density of 

-3 

19.3 g cm . 

(1) Estimate the average separation R between an electron and a metal ion in the solid, and 

(2) the mean distance d between electrons. 

(3) Determine the mean free path ( ) of electrons at 273K if the resistivity (Au) = 22.8 n m à 0 C . Give 

the value of in nm. 

(4) Estimate de value of at 273 C . 

In all cases, clearly show your work and your assumptions. 



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final exam – written part - École Polytechnique de Montréal

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