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THE UNIVERSITY OF SYDNEY
FACULTY OF SCIENCE
INTERMEDIATE PHYSICS
PHYS 2012 PHYSICS 2B
NOVEMBER 2007 TIME ALLOWED: 3 HOURS
ALL QUESTIONS HAVE THE VALUE SHOWN
INSTRUCTIONS:
This paper consists of 2 sections.
Section A Quantum Physics 75 marks Section B Electromagnetic Properties of Matter 75 marks
Candidates should attempt all questions.
USE A SEPARATE ANSWER BOOK FOR EACH SECTION.
In answering the questions in this paper, it is particularly important to give rea- sons for your answer. Only partial marks will be awarded for correct answers with inadequate reasons.
No written material of any kind may be taken into the examination room. Calcu- lators are permitted.
Table of constants
Avogadro’s number NA = 6. 022 × 1023 mole−^1
speed of light c = 2. 998 × 108 m.s−^1
electronic charge e = 1. 602 × 10 −^19 C
electron rest mass me = 9. 110 × 10 −^31 kg
electron rest energy energy = 511 keV
electron volt 1 eV = 1. 602 × 10 −^19 J
protron rest mass mp = 1. 673 × 10 −^27 kg
neutron rest mass mn = 1. 675 × 10 −^27 kg
Planck’s constant h = 6. 626 × 10 −^34 J.s
Planck’s constant (reduced) ¯h = 1. 055 × 10 −^34 J.s
Boltzmann’s constant kB = 1. 380 × 10 −^23 J.K−^1
Stefan’s constant σ = 5. 670 × 10 −^8 W.m−^2 .K−^4
Coulomb constant (^4) π^10 = 8. 988 × 109 N.m^2 .C−^2
permittivity of free space 0 = 8. 854 × 10 −^12 C^2 .N−^1 .m−^2
permeability of free space μ 0 = 4 π × 10 −^7 kg.m.C−^2
gravitational constant G = 6. 673 × 10 −^11 N.m^2 .kg−^2
atomic mass constant u = 1. 660 × 10 −^27 kg
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
- (a) (i) State one piece of experimental evidence which demonstrates that the electron behaves as a classical (Newtonian) particle, and state one (different) piece of experimental evidence which demonstrates that the electron behaves as a wave.
(ii) These two results used to be known as the “wave-particle para- dox”. Explain briefly what is paradoxical about these results.
(b) (i) Describe briefly how using a wave packet as a model by which to describe an electron, “solves” the wave-particle paradox.
(ii) Describe briefly how the wave packet model leads, as a conse- quence, to the Uncertainty Principle.
(c) The “seeing ability”, or resolution, of any optical instrument is roughly the same as the wavelength being used. If the size of an atom is of the order of 0.1 nm, how fast must an electron travel to have a wavelength small enough to “see” the atom?
(15 marks)
- A very simple photoconductor consists of a piece of semiconducting ma- terial. When light shines on it, its conductivity changes noticeably.
(a) Explain, with the aid of an appropriate energy band diagram, why its conductivity changes. (Be careful to state whether its conductivity in- creases or decreases when it is illuminated.)
(b) When light shines on a photoconductive material, a electrical current flows. Do holes, as well as electrons, contribute to this current? Ex- plain your answer.
(c) Explain why such a simple photoconductor will not conduct if it is illuminated by electromagnetic radiation of frequency much less than visible light.
(d) A potassium chloride crystal has an energy band gap of 7.6 eV above the topmost occupied band, which is full. Would you expect this crystal to be opaque or transparent to light of wavelength 140 nm? Explain your reasoning.
(e) In the kind of photovoltaic device needed to convert solar radiation to useable energy, it is necessary to slow down the rate of recombination of the electron-hole pairs created by incident photons. Why is this necessary?
(15 marks)
- An electron, with total energy 9 U 0 , moving from left to right, encounters a rectangular potential barrier of height 10 U 0 and thickness L.
potential energy
!
distance (x)!
0
E! =! 9 U (^) 0
10 !U (^) 0
L
The values of the variables U 0 and L are chosen so that the dimensionless quantity ( 2 me U 0 L^2 /¯h^2 ) is small, of order ∼ 0. 01
(a) Use a model for this system in which the incident quantum particle is represented by a reasonably narrow wave packet. With the aid of simple diagrams, describe what the probability function would look like, before, during and after the electron encounters the barrier.
(b) Now use a model for this system in which a steady beam of wave- particles is incident upon the left-hand side of the barrier. Draw a care- fully labelled diagram to show what the probability function for the wave-particle beam should look like.
(c) If the wave-particle in question was a proton rather than an electron, redraw the diagram of part (b) showing very clearly how the two dia- grams differ from one another.
(d) Under what conditions is a discontinuous potential energy function a reasonable model of a real physics system?
(15 marks)
- (a) In terms of properties which can be observed in an experimental lab- oratory, state how a semiconductor is defined. Include especially the variation of its properties with temperature.
(b) Draw a careful energy band diagram for a simple intrinsic semicon- ductor. On this diagram show which energy levels are full at absolute zero temperature, and which are empty. Also, show clearly on that diagram what quantum mechanical feature distinguishes the semiconductor from a conductor.
(c) (i) Explain in words, with reference to your diagram, why this semi- conductor will conduct electricity at room temperature.
(ii) State in words how the conductivity of this material changes as the temperature increases further, and explain why this behaviour occurs.
(d) On a copy of the diagram you drew in part (b), indicate approximately where the Fermi energy should be, and explain in words why you drew it where you did.
(15 marks)
SECTION B
ELECTROMAGNETIC PROPERTIES OF MATTER
FORMULAS
Ex = −
∂V
∂x
Ey = −
∂V
∂y
Ez = −
∂V
∂z
FE = qE E =
4 π 0
q r^2
ˆr ΦE =
E · dA =
qenclosed 0
V =
4 π 0
p · ˆr r^2
D · dA = qf W =
D · Edr
E = σ/ 0 p = qd τ = p × E U = −p · E
V =
4 π 0
q r
Q = CV C =
0 rA d
U =
CV 2 P = np σb = P · ˆn
D = 0 rE = E D = σf D = 0 E + P
P = χe 0 E p = αE α =
n
r − 1 r + 2
P = np (coth(pE/kT ) − 1 /(pE/kT )) i =
dq dt
J = I/A
i = nqvdrif tA VH =
Bi net
RH =
nq
EH
JB
J = nqvdrif t R = V /I ρ = E/J
σ = 1/ρ R = ρL/A ρ =
m e^2 nτ
ΦB =
B · dA FB = qv × B μ = N iAnˆ
ωc = qB/m rL =
mv⊥ qB
B = μ 0 (H + M)
τ (^) B = μ × B dFB = idL × B B = μH W = −μ · B
M = χH W = V
HdB B = μrμ 0 H
S =
μ 0
E × B dB =
μ 0 4 π
ids × r r^3
prad = I/c or prad = 2I/c
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
- Briefly answer the following questions.
(a) We found in class that the electric field of a dipole decreases as 1 /r^3 at large distances. Explain why the field decays more rapidly for a dipole than for a bare charge.
(b) Why are dipoles important when considering the properties of dielec- tric media?
(c) In class we derived the result σb = P · nˆ. Explain the symbols and briefly discuss the physical interpretation of this result.
(d) What is meant by “drift velocity” in the context of conduction.
(e) Explain the difference between paramagnetic, ferromagnetic, and dia- magnetic media.
(f) Briefly explain why a permanent magnet can stick motionlessly to the steel front of a refrigerator?
(30 marks)
- In class we considered parallel plate capacitors with area A, plate separation d, and dielectric with εr between the plates.
(a) Show that the capacitance of such a capacitor is
C =
ε 0 εrA d
(b) In practice, capacitors rarely consist of only two plates, but look more like the figure below with multiple plates. Here the odd plates are connected to one pole of the battery and the even plates to the other pole. Derive the capacity of such a capacitor assuming it has M plates. Each plate has an area A and all plates are spaced by a distance d. You may ignore edge effects.
(15 marks)
- In class we discussed the motion of a particle in a magnetic field.
(a) Briefly discuss this motion. You may assume that there is no electric field.
(b) Briefly discuss what is meant by a magnetic mirror.
(c) Shown is a schematic of a tokamak fusion reactor. Explain how this design manages to confine the charged particles in order to prevent them from touching the reactor walls.
(15 marks)
