EXAMEN GÉNÉRAL DE SYNTHÈSE – PARTIE ÉCRITE Programme

EXAMEN GÉNÉRAL DE SYNTHÈSE – PARTIE ÉCRITE
Programme de doctorat en génie physique
Jeudi 14 juin 2012
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 20 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
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
CONSTANTES PHYSIQUES :
=
e 1.602 ×10−19 C
=
me 9.109 ×10−31 kg
 =1.055 ×10−34 J ⋅ s
k B =8.617 ×10−5 eV ⋅ K −1
ÉQUATIONS PHYSIQUES :
ÉQUATIONS MATHÉMATIQUES
∇ ⋅ D = ρf
Intégrale
∇ ⋅ B =0
∫
∞
0
∞
π
2
2
1 π
e − a× d × =
2 a
× e −× d × =
∂B
∇×E= −
∂t
∫
∂D
∇ × H= J f +
∂t
Loi des cosinus
c 2 = a 2 + b 2 − 2ab cos γ
B=
µ0
4π
∫
f FD ( E ) =
f BE ( E ) =
J × rˆ
r
2
dV
Approximation de Stirling
n! ≈ n n e − n 2πn
1
e
( E −µ )
k BT
+1
1
e
( E −µ )
k BT
0
−1
Identité
∇ 2 A = ∇ ( ∇A ) − ∇×∇× A
Identités trigonométriques
sin ( α ± β=
) sin α cos β ± cos α sin β
c os ( α ± β ) = cos α cos β  sin α cos β
sin=
2θ 2 sin θ cos θ
cos =
2θ cos 2 θ − sin 2 θ
1
( cos ( α + β ) + cos ( α −β ) )
2
1
sin α=
sin β
( cos ( α − β ) − cos ( α + β ) )
2
1
sin α =
cos β
( sin ( α + β ) + sin ( α − β ) )
2
cos α=
cos β
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 2 de 11
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)
2.
Considering a plane wave moving in the z direction in such a medium for which


ˆ exp( j(t  kz)) and H  ˆjH exp( j (t  kz )), determine
E  iE
0
0
a)
b)
A relationship between H 0 and E0
The wave number k as a function of  , e and m .
(4 points)
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 20 gigawatt laser beam of 2mm diameter is moving in a glass plate for which m  m0 and
e  2.56´e0 , where m0 is the permeability of vacuum, and e0  8.8542´1012 Farad/m is the
1
where c  3´10 8 m/s is the speed of light in vacuum,
vacuum permittivity. Knowing that m0 
2
e0 c
calculate the values of E0 and of H 0 .
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 3 de 11
QUESTION 2 :
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
𝐻0 =
𝑝2 𝑚𝜔2 2
+
𝑥 .
2𝑚
2
One also knows that the normalized ground state wave function for such a particle takes the form
2
𝜓0 (𝑥) = 𝐾𝑒 −𝛽𝑥 .
(5 points)
a. Find the constants 𝐾 and 𝛽 as well as the energy 𝐸0 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 𝐻𝑝 = 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 𝜓0 (𝑥), will be found in state 𝜓′0 (𝑥).
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 4 de 11
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 N2 molecules (atomic weight 28).
(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 n0 in the tire. Write down a similar expression for the rate that
air passes from outside through the hole, replacing the inside density n0 with the outside
density n.
(4 points)
𝑑𝑛
b)
Using your results from (a), find the rate 𝑑𝑡0 by which the density inside the tire is changing.
Your answer should depend on the terms listed in part (a) as well as the volume of the tire V.
(3 points)
c)
Integrate your result from part (b) to find the elapsed time t as a function of m, A, V, T, n, ni
and nf, where ni and nf are the initial and final densities inside the tire.
(3 points)
d)
Let T = 300 K, V = 0.5 liters (500 cm3), 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:
3�
2
𝑚𝛽
𝑝(𝜐)𝑑𝜐 = � �
2𝜋
𝑒𝑥𝑝 �−
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
𝑚𝜐 2 𝛽�
2� 𝑑𝜐
Département de génie physique
Page 5 de 11
QUESTION 4 :
MÉCANIQUE SUPÉRIEURE
Coronal mass ejection
Consider a star with radius R and mass M.
(4 points)
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.
(2 points)
b)
Find the period of the orbit.
(4 points)
c)
Find the minimum escape velocity allowing the coronal matter to leave one star and end up on
the other.
(4 points)
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,
(2 points)
e)
Find the ratios of the velocities calculated in a) and c) and in a) and d).
(4 points)
f)
Discuss the overall transfer of mass between two stars if 1) their masses are equal and 2) if
their masses are different.
Note : Consider a sharp transition between the star and its environment.
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 6 de 11
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 nl=2.8, while refractive index of air is na=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 (2) 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.
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 7 de 11
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 (2) 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.
(2 points)
2)
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?
(4 points)
𝑝
3)
Find expressions for the radii of the Newton rings corresponding to the p’s maxima 𝑟𝑚𝑎𝑥
and
𝑝
p’s minima 𝑟𝑚𝑖𝑛 of the reflected intensity. Give numerical value for the radius of the first
𝑝
minima. In your derivations assume that 𝑟𝑚𝑖𝑛,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=
𝑚𝑎𝑥
𝑚𝑖𝑛
𝑅𝑡𝑜𝑡
− 𝑅𝑡𝑜𝑡
𝑚𝑎𝑥
𝑚𝑖𝑛
𝑅𝑡𝑜𝑡
+ 𝑅𝑡𝑜𝑡
(3 points)
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).
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 8 de 11
QUESTION 6 :
OPTIQUE 2
Optics: polarisation
A collimated beam of unpolarized light, of wavelength 632 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.
(12 points)
1.
Describe what is observed on the screen at the output of the system.
(8 points)
2.
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
• extra-ordinary index (along the optical axis)
no = 1,650
ne = 1,486
Note : Refraction in the prisms can be neglected. The beams can be assumed to propagate along the
optical axis of the system.
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 9 de 11
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, RH, defined as a Ey/JxBz ratio, is given by:
RH = 1/nq,
where Ey is the Hall electric field, Jx is the current density, Bz 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.
(4 points)
b)
Determine the Hall voltage VH measured upon application of a total current of 100 mA at the
presence of a magnetic field of 1 T considering that RH = -0.82 x 10-10 Ωm/T. Why does RH
exhibit a negative value?
(3 points)
c)
Calculate a ratio between the charge carrier density and the atomic density of silver. Compare
your obtained value with that expected theoretically.
(3 points)
d)
Calculate the lattice parameter of a silver film while considering the charge density calculated
in (c).
(3 points)
e)
Is the mean free path of the charge carriers smaller or larger than the lattice parameter
obtained in (d)? Justify your response.
Figure 1: Hall effect measurement geometry.
Useful parameters
Mass density of silver: 10.5 g/cm3
Atomic structure of silver: [Kr] d10 5s1
Atomic mass of silver: 107.87 g/mol
Resistivity of silver: 1.59×10−8 Ω•m at 20°C
Crystalline structure of silver: c.f.c.
�⃗
𝑑𝑣
�⃗
𝑣
��⃗ is the force acting on the charge
Charge carrier transport equation: 𝑚 � 𝑑𝑡 + 𝜏 � = 𝐹⃗ , where 𝐹
carriers, �𝑣⃗ is the mean velocity, and τ is the time to transit the mean free path.
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 10 de 11
QUESTION 8 :
PHYSIQUE DU SOLIDE 2
(3points)
a)
Identify the different mechanisms of polarization in a dielectric material.
(6 points)
b)
Here is the Clausius-Mossotti relationship:
ϵ − 1 Nα
=
ϵ + 2 3ϵ0
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 78.96 g / mol. The
value of the dielectric constant of a-Se is 6.7. Calculate the polarizability of a Se atom in
a-Se.
(2 points)
d)
Calculate the polarizability of a single Se atom, α', for which the atomic radius,
r, is 0.12 nm.
Note: The electronic polarizability of an individual atom is given by the following equation:
α’ = 4πε0r3.
(4 points)
e)
Compare the result obtained in d) with the value of the polarizability of a Se atom in
a-Se and explain this difference.
Examen général de synthèse - Partie écrite
Jeudi 14 juin 2012
Département de génie physique
Page 11 de 11