THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2912 PHYSICS 2B (ADVANCED) ALL QUESTIONS HAVE THE VALUE SHOWN

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1 CC0936 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 91 PHYSICS B (ADVANCED) SEMESTER, 014 TIME ALLOWED: 3 HOURS ALL QUESTIONS HAVE THE VALUE SHOWN INSTRUCTIONS: This paper consists of 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 reasons 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. Nonprogrammable calculators are permitted.

2 CC0936 SEMESTER, 014 Page of 9 Table of constants Avogadro s number N A = mol 1 speed of light c = m.s 1 electronic charge e = C electron rest mass m e = kg electron rest energy E = 511 kev electron volt 1 ev = J proton rest mass m p = kg neutron rest mass m n = kg Planck s constant h = J.s Planck s constant (reduced) = J.s Boltzmann s constant k B = J.K 1 Stefan s constant σ = W.m.K 4 Coulomb constant (4πɛ 0 ) 1 = N.m.C permittivity of free space ɛ 0 = C.N 1.m permeability of free space µ 0 = 4π 10 7 kg.m.c gravitational constant G = N.m.kg atomic mass constant u = kg

3 CC0936 SEMESTER, 014 Page 3 of 9 SECTION A: ELECTROMAGNETIC PROPERTIES OF MATTER FORMULAS B V B V A = E dl A E x = V x E y = V y F E = qe E = 1 q 4πɛ 0 r r Φ E = E = V E z = V z E da = q enclosed ɛ 0 E = 1 3(p r)r r p V = 1 p r D da = q 4πɛ 0 r 5 4πɛ 0 r f W = 1 D Edr E = σ/ɛ 0 p = qd τ = p E U = p E p = ρ(r)rdv V = 1 4πɛ 0 q r Q = CV C = ɛ 0ɛ r A d U = 1 CV U = V Q P = N p σ b = P n ρ b = P D = ɛ 0 ɛ r E = ɛe D = σ f D = ɛ 0 E + P P = χ e ɛ 0 E [ P = N p coth(pe/kt ) p = αe α = 4πε 0 a 3 α = 3ɛ 0 N ] 1 pe/kt V H = Bi ɛ r 1 ɛ r + i = dq dt J = I/A i = nqv drift A R H = 1 net nq = E H JB J = nqv drift R = V/I ρ = E/J v drift = µe σ = 1/ρ R = ρl/a ρ = m J = ρ e nτ t. ω p = N e ε 0 m Φ B = B da F B = qv B µ = NiA n ω c = qb/m r L = mv B = µ 0 (H + M) B = µ 0 3(m r)r r m qb 4π r 5 τ B = µ B df B = idl B B = µh W = µ B M = χh W = V HdB B = µ r µ 0 H S = 1 E B db = µ 0 ids r µ 0 4π r 3

4 CC0936 SEMESTER, 014 Page 4 of 9 B da = 0 B(z) = E ds = dφ B dt B z = E y /c µ 0 ia E y (z + a ) 3/ x B(r) = µ 0i πr B ds = µ 0 i + µ 0 ɛ 0 dφ E dt µ 0ɛ 0 E y t = 0 E y = E y0 cos(ωt kx) B = µ 0 ni D da = q f U/V = B /µ 0 θ loss = sin 1 (1/ R m ) [ H ds = i f + dφ ( ) ] D T B c = B c (0) 1 dt T c

5 CC0936 SEMESTER, 014 Page 5 of 9 Please use a separate book for this section. Answer ALL QUESTIONS in this section. Each part of each question has an equal number of marks. 1. (a) Draw the typical shape of B vs H for ferromagnetic materials, noting on the diagram the remanence and coercivity. (b) The relative permeablity of copper is µ r = Is copper attracted to stronger or weaker magnetic fields? Justify your answer using the expression of the relevant potential energy. (c) A cylindrical rod of copper or radius a, length l is placed in a vertical solenoid, making an angle θ with its axis ẑ. Inside the solenoid, the magnetic intensity is assumed constant and uniform taking value H = niẑ where n is the umber of turns per unit length and I the current. Calculate the magnetization M inside the copper rod, and the total magnetic dipole of the copper rod. What is the torque on the rod due to the magnetic field? What is the net force on the rod? (d) A second identical copper rod is introduced into the solenoid. Both rods are now aligned with ẑ, on the central axis of the solenoid, at a distance d l > a from each other so that the fields they generate can be approximated as that of dipoles. Without deriving the full expression of the force, justify the existence and direction of a net magnetic force between the rods. (15 marks). A charge Q > 0 is placed at the centre of a water-filled sphere with radius a (the dielectric constant of water is ε r = 81). (a) Draw a diagram of the system, showing the polarization density vectors and indicating the induced volume and surface polarization charges with + and signs. (b) Find the electric field both inside and outside the sphere. (c) Find the polarization density P inside the sphere, and use P to determine the polarization surface charge density and the total surface polarization charge Q pol on the sphere. (d) Show that Q/ε r = Q Q pol, and, relating this to the electric field outside the sphere, explain the physical significance of this relationship. (15 marks)

6 CC0936 SEMESTER, 014 Page 6 of 9 3. We consider a neutral atom with polarizability α at a point B, and a point charge Q located at a fixed point A. (a) Briefly define the polarizability, and explain what its physical origins are. (b) Express the induced dipole p caused by the electric field of the point charge in terms of Q and the vector AB. (c) Express the electric field in A due to the neutral atom s induced dipole moment. (d) What is the net force of the charge on the dipole? Is it attractive or repulsive? The mass of the neutral atom is m = kg and its polarizability is α/(4πε 0 ) = m 3 ; calculate the acceleration due to the force between charge and atom at a distance AB = m for Q = C. (15 marks) 4. We consider a slab of doped silicon of thickness d = 10µm, with current I along the slab, immersed in a uniform magnetic field B 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 that shows the various forces and charge accumulations that are relevant, explain the Hall effect and its origin. (b) For I = 0.1A, and B = 0.1T, the voltage is measured to be V B V A = +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 temperature is increased by 0 C? Justify your answer. (d) A Hall sensor as as depicted in the figure, with N = m 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)

7 CC0936 SEMESTER, 014 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. (1 marks) 6. Consider a standard Stern-Gerlach experiment, with a beam of spin-1/ particles prepared in the state ψ = + e iπ/3. (a) Normalise this state vector. (b) What are the possible results of a measurement of the spin component S z, and with what probabilities do they occur? (c) What is the expectation value S z for this state? (d) What is the uncertainty S z for this state? (e) What are the possible results of a measurement of the spin component S y, and with what probabilities do they occur? Hint: the eigenstates for S y, expressed in terms of the eigenstates of S z, are: + y = 1 ( + + i ), y = 1 ( + i ), (f) Suppose that the S y measurement was performed, with the result S y = /. Subsequent to that result, a second measurement is performed to measure the spin component S z. What are the possible results of that measurement, and with what probabilities do they occur? (1 marks)

8 CC0936 SEMESTER, 014 Page 8 of 9 7. Consider spin-1/ particles prepared in the initial state ψ(t = 0) = + n with the direction ˆn = (ˆx + ŷ)/. These particles precess in a magnetic field B aligned in the z-direction according to the Hamiltonian where ω 0 = eb 0 /m e. H = µ B = ω 0 S z, (a) Express ψ(t = 0) in terms of its components in the z-basis + and. (b) Calculate ψ(t), the state at some later time t. Express your answer both in the standard basis ( + and ) and also in terms of its direction ˆn(t). (c) Calculate the expectation value S x as a function of time. Draw a plot of your answer. Hint: the S x operator in matrix notation is S x = ( ). (d) Determine the state ψ that describes a spin-1/ particle aligned in the opposite direction to one described by ψ(0). (e) Describe, in 50 words or less and with supporting equations if necessary, how you would could evolve ψ(0) to the state ψ. (1 marks) 8. Consider the entangled quantum state Φ 1 of two spin-1/ particles, given by Φ 1 = (a) What are the possible results of a measurement of the spin component S z of just the first particle, and with what probabilities do they occur? (b) Describe the possible results from measurements of the spin component S z of both particles. (That is, a measurement of S z of particle 1 and a measurement of S z of particle.) (c) In addition to performing experiments of the above type, in which measurements of the spin component S z 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/ particles that would result in anticorrelated outcomes when measurements of the spin component S x of both particles are performed. Justify your answer. (1 marks)

9 CC0936 SEMESTER, 014 Page 9 of 9 9. A particle with mass m is confined to a one-dimensional quantum well with boundaries at x = 0 and x = L. At t = 0, the particle is known to be in the left half of the well with a probability density that is uniform on the left half of the well. (a) What is the initial wavefunction ψ(x) of the particle? (b) Show that the components φ n ψ of this wavefunction in the energy eigenbasis ψ n (x) = /L sin(nπx/l), n = 1,, 3,... are given by: n even nπ φ n ψ = 0 n = 1, 5, 9,... 4 n = 3, 7, 11,... nπ Hint: you will find the following definite integral helpful. 1 π/ n even n sin(nx)dx = 0 n = 1, 5, 9,... 0 n = 3, 7, 11,... n (c) Find the probabilities that the particle is measured to have energy E 1, E, or E 3. (d) Show that this state has an infinite expectation value of the energy. (1 marks) THERE ARE NO MORE QUESTIONS.