CC0933
THE UNIVERSITY OF SYDNEY
FACULTY OF SCIENCE
INTERMEDIATE PHYSICS
PHYS 2012 PHYSICS 2B
SEMESTER 2, 2014 TIME ALLOWED: 3 HOURS
ALL QUESTIONS HAVE THE VALUE SHOWN
INSTRUCTIONS:
This paper consists of 2 sections.
Section A Electromagnetic Properties of Matter 60 marks
Section B Quantum Physics 60 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. Non-
programmable calculators are permitted.
CC0933 SEMESTER 2, 2014 Page 2 of 9
Table of constants
Avogadro’s number NA = 6.022× 1023 mol−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 E = 511 keV
electron volt 1 eV = 1.602× 10−19 J
proton 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) ~ = 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 (4pi0)
−1 = 8.988× 109 N.m2.C−2
permittivity of free space 0 = 8.854× 10−12 C2.N−1.m−2
permeability of free space µ0 = 4pi × 10−7 kg.m.C−2
gravitational constant G = 6.673× 10−11 N.m2.kg−2
atomic mass constant u = 1.660× 10−27 kg
gas constant R = 8.314 J.mol−1.K
CC0933 SEMESTER 2, 2014 Page 3 of 9
SECTION A: ELECTROMAGNETIC PROPERTIES OF MATTER
= r0 µ = µrµ0
c =
1√
0µ0
v =
c
n
n =
√
rµr
F =
1
4pi
q1q2
r2
r̂ E =
1
4pi
q
r2
r̂ V =
1
4pi
q
r
E = −∇V Vb − Va = −
∫ b
a
E · dl
∮
E · dl = 0
FE = qE∮
E · dA = qenclosed
0
=
qf − qb
0
=
qf
r0∮
D · dA = qf D = E = σf
∮
D · dA =
∫
ρfdV
σb = P · n̂ ρb = −∇ ·P E = 1
0
(D−P) P = χe0E r = 1 + χe
C =
Q
V
C =
qf
V
C =
A
d
Cseries =
1
C−11 + C2
−1 + · · · Cparallel = C1 + C2 + · · ·
U =
1
2
Q2
C
=
1
2
CV 2 =
1
2
QV W =
1
2

∫
E2dV u =
1
2
E2
p = qd τ = p× E U = −p · E
P = np Jb = −∂P
∂t
p = αE
α = 4pi0a
3 α =
30
n
r − 1
r + 2
P = np
[
coth(pE/kT )− 1
pE/kT
]
CC0933 SEMESTER 2, 2014 Page 4 of 9
F = qv ×B dF = idl×B dB = µ0
4pi
idl× r
r3
Fc =
mν2
r
ΦB =
∫
B · dA
∮
B · dA = 0
emf =
∮
E · dl = −dΦB
dt∮
B · dl = µ0itotal
∮
H · dl = ifree
pm = NiAn̂ τ = pm ×B U = −pm ·B
M = χmH B = µ0(H+M) B = µH = µrµ0H µr = 1 + χm
W = V
∫
HdB
i =
dq
dt
i = nqνdriftA i =
∫
J · dA
R =
V
I
R = ρ
L
A
σ =
1
ρ
J = σE ρ =
m
e2nτ
CC0933 SEMESTER 2, 2014 Page 5 of 9
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
1. A chargeQ > 0 is placed at the centre of a water-filled sphere with radius a (the dielectric
constant of water is εr = 81).
(a) Draw a diagram of the system, showing the polarization density vectors and indi-
cating the induced polarization charges with + and − signs.
(b) Find the electric field both inside and outside the sphere.
(c) Find the polarization densityP inside the sphere, and useP to determine the polar-
ization surface charge density and the total polarization charge Qpol on the sphere.
(d) Show that Q/εr = Q−Qpol, and explain the physical significance of this relation-
ship.
(15 marks)
2. A long solenoid with lengthL and radius a has n turns per unit length and carries a current
I . Assume that there is a uniform magnetic field inside the solenoid parallel to its axis,
given by B0 = µ0nI .
(a) A cylindrical iron bar of length L and radius a/2 is fully inserted into the solenoid
so that its axis coincides with that of the solenoid. Draw a diagram for a cross
section of the solenoid, showing the direction of the magnetization density and the
direction of currents in the solenoid and the iron bar. Find the total magnetic field
everywhere in the solenoid, and sketch it as a function of the radial distance from
the axis in units of B0 (use µr = 1000 for the iron bar).
(b) Repeat the question in (a) for the case where the iron bar is replaced with an iden-
tical platinum bar (µr = 1.0003).
(c) Next consider the case where the iron bar in (a) is replaced with a cylindrical shell
made of iron, which has length L, outer radius a, and inner radius a/2 (i.e., it has a
hole of radius a/2 in the centre). Repeat the question in (a) for this configuration.
(15 marks)
CC0933 SEMESTER 2, 2014 Page 6 of 9
3. Give brief physical answers to the following questions.
(a) Why does filling a capacitor with a dielectric material which has a large εr increase
its capacity for charge storage and hence storage of potential energy?
(b) State whether the expression σ = Nq2τ/m for the conductivity of a material over-
estimates or underestimates the conductivity of (i) electrons in metals and (ii) ions
in electrolytes, and then explain why. Here N , q, τ and m are the number density,
charge, mean collison time, and mass of the charge carriers, respectively.
(c) Discuss the temperature dependence of magnetization in paramagnetic, diamag-
netic and ferromagnetic materials and give a physical explanation for each be-
haviour.
(15 marks)
4. We consider a slab of doped silicon of thickness d = 10µm, with current I along the
slab, immersed in a uniform mangetic fieldB parallel to the slab and perpendicular to the
direction of the current (see figure below).
Figure 1: Geometry for Question 4(a-c).
(a) Using a schematic of the slab showing the various forces and charge accumulations
that are relevant, explain the Hall effect and its origin.
(b) For I = 0.1A, and B = 0.12T, the voltage is measured to be VB − VA = +500mV.
Are the predominant charge carriers positive or negative? Calculate the density of
carriers N .
(c) Maintaining the same current, would the voltage increase or decrease if the temper-
ature is increased by 20◦C? Justify your answer.
(d) A Hall sensor as as depicted in the figure, with N = 1.5 × 1022m−3 is used as
an electronic compass in a mobile phone. The Earth’s magnetic field in Sydney
is 55µT. Assuming the smallest voltage change that can be reliably detected is
∆V = 100µV, determine the smallest error ∆θ with which the magnetic north
can be determined with this sensor, for I = 0.1A. How could this be improved?
(15 marks)
CC0933 SEMESTER 2, 2014 Page 7 of 9
SECTION B: QUANTUM PHYSICS
There is no formula sheet for the Quantum Physics Section
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
5. Explain briefly (less than 50 words each) what is meant by each of the following.
(a) The Stern-Gerlach experiment.
(b) The Hamiltonian operator.
(c) A quantum dot.
(12 marks)
6. Consider a standard Stern-Gerlach experiment, with a beam of spin-1/2 particles prepared
in the state
|ψ〉 = |+〉 − 2eipi/3|−〉 .
(a) Normalise this state vector.
(b) What are the possible results of a measurement of the spin component Sz, and with
what probabilities do they occur?
(c) What is the expectation value 〈Sz〉 for this state?
(d) What is the uncertainty ∆Sz for this state?
(e) What are the possible results of a measurement of the spin component Sy, and with
what probabilities do they occur? Hint: the eigenstates for Sy, expressed in terms
of the eigenstates of Sz, are:
|+〉y = 1√
2
(|+〉+ i|−〉) , |−〉y = 1√
2
(|+〉 − i|−〉) ,
(f) Suppose that the Sy measurement was performed, with the result Sy = −~/2.
Subsequent to that result, a second measurement is performed to measure the spin
component Sz. What are the possible results of that measurement, and with what
probabilities do they occur?
(12 marks)
CC0933 SEMESTER 2, 2014 Page 8 of 9
7. Consider a spin-1/2 particle with a magnetic moment µ. At time t = 0, the state of the
particle is |ψ(t = 0)〉 = |−〉x = 1√2(|+〉 − |−〉). The Pauli spin matrices in the |+〉, |−〉
basis are given by
Sx =
~
2
(
0 1
1 0
)
, Sy =
~
2
(
0 −i
i 0
)
, Sz =
~
2
(
1 0
0 −1
)
.
(a) If the particle evolves in a uniform magnetic field parallel in the z-direction, ~B =
B0zˆ, calculate |ψ(t)〉, the state of the particle (in the |+〉, |−〉 basis) at some later
time t. Hint: The Hamiltonian operator is given by H = ω0Sz, where ω0 = µB0.
(b) At time t, the observable Sx is measured. What is the probability that the value
+~/2 will occur? Draw a plot of this probability as a function of time t.
(c) For what direction(s) of the magnetic field ~B would the state |−〉x be a stationary
state?
(12 marks)
8. Consider the entangled quantum state |Φ〉12 of two spin-1/2 particles, given by
|Φ〉12 = 1√
2
|+〉1|+〉2 − 1√
2
|−〉1|−〉2 .
(a) What are the possible results of a measurement of the spin component Sz of just
the first particle, and with what probabilities do they occur?
(b) Describe the possible results from measurements of the spin component Sz of both
particles. (That is, a measurement of Sz of particle 1 and a measurement of Sz of
particle 2.)
(c) In addition to performing experiments of the above type, in which measurements of
the spin component Sz of both particles are performed, what else must be measured
in order to demonstrate a test of Bell nonlocality?
(d) Write down an entangled state of two spin-1/2 particles that would result in anticor-
related outcomes when measurements of the spin component Sx of both particles
are performed. Justify your answer.
(12 marks)
CC0933 SEMESTER 2, 2014 Page 9 of 9
9. An electron is confined to a one-dimensional quantum well of width L = 0.250 nm.
(a) If the height of the potential can be considered extremely large, show that the al-
lowed de Broglie wavelengths of the electron in this well are given by λn = 2L/n,
for n = 1, 2, 3, . . .
(b) Using the facts that the momentum of a quantum particle with de Broglie wave-
length λn is given by pn = 2pi~/λn and that the kinetic energy is En = p2n/(2m),
calculate the energies of the three lowest energy eigenstates.
(c) If you were to perform spectroscopy on this system, what would be the minimum
energy needed to induce a transition from the ground state?
(d) Draw a diagram to show the probability density functions (the functions |ψ(x)|2)
for the ground state and first excited state.
(e) Draw these same functions for the case where the height of the potential is not very
large (but still higher than the energy of the second excited state), clearly marking
the difference in the form of the functions compared with part (c).
(f) Will the energies of the finite well be higher or lower than the corresponding ener-
gies of the infinite well? Justify your answer.
(12 marks)
THERE ARE NOMORE QUESTIONS.