9323 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2012 PHYSICS 2B SEMESTER 2, 2013 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. 9323 SEMESTER 2, 2013 Page 2 of 10 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 9323 SEMESTER 2, 2013 Page 3 of 10 SECTION A: ELECTROMAGNETIC PROPERTIES OF MATTER = r0 μ = μrμ0 c = 1√ 0μ0 v = c n n = √ rμr F = 1 4pi q1q2 r2 r E = 1 4pi q r2 r 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 · n 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 ] 9323 SEMESTER 2, 2013 Page 4 of 10 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 = NiAn 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 9323 SEMESTER 2, 2013 Page 5 of 10 Please use a separate book for this section. Answer ALL QUESTIONS in this section. 1. Give brief physical explanations for the following questions. (a) Why as the temperature increases does the electrical conductivity decrease in met- als but increase in electrolytes (b) Why do two circular current loops of the same size and with the same current di- rection attract each other when one is placed on top of the other but repel each other when they are placed side by side on the same plane What happens in these two situations if the directions of current are opposite in the two loops (c) What are the similarities and differences for the microscopic basis of paramag- netism versus the microscopic basis for orientational polarization in dielectric ma- terials What are the similarities and differences for the microscopic basis of dia- magnetism with the microscopic basis for electronic polarization in dielectric ma- terials (15 marks) 2. Consider the following questions given the density of water ρ = 103 kg.m 3, the molar mass of waterM = 18× 10 3 kg, and 1 atm = 1.013× 105 Pa. (a) The electric dipole moment of the water molecule has a value of 6.2× 10 30 C.m. Find the maximum electric polarization of water vapour at 100 C and atmospheric pressure. (b) Use the Langevin equation (when pE/kBT 1) and the fact that the dielectric constant of water at 20 C is r = 81 to determine the electric dipole moment, p, of the water molecule. (c) Comment on and explain any discrepancy between the two values of the electric dipole moment. (15 marks) 9323 SEMESTER 2, 2013 Page 6 of 10 3. A parallel plate capacitor has a square plate area of L2 = 0.25 m2 and a plate separation of d = 0.01 m. The potential difference between the plates is V = 3.0 kV. A dielectric slab with a dielectric constant of εr = 10 completely fills the space between the plates of the capacitor. The dielectric slab is partially removed while the capacitor is not connected to a battery. The slab is withdrawn in a direction parallel to the plates through a distance x (when the slab is fully inserted x = 0, when the space is half filled x = L/2, and when the slab is fully removed x = L). Assume the dielectric slab is removed very slowly so that the kinetic energy of the slab is zero and that edge effects at the ends of the capacitor can be ignored. Show and justify the following relationships as functions of x: (a) Capacitance C = ε0L d [x+ εr(L x)] , (b) Potential difference V = Qd ε0L[x+ εr(L x)] , (c) Energy stored by the capacitor Ucap = Q2d 2ε0L[x+ εr(L x)] , (d) Change in the energy stored by the capacitor Ucap = Ucap(x) Ucap(0) = Q 2d 2ε0L [ 1 x+ εr(L x) 1 εrL ] , (e) Work done by the external force Wme = Ucap , (f) External force in removing the dielectric slab Fme = Q2d 2ε0L εr 1 [x+ εr(L x)]2 . (g) Evaluate the external force for the given device parameters, and state its direction and physical significance. (18 marks) 9323 SEMESTER 2, 2013 Page 7 of 10 4. The Earth’s magnetic field arises from circulating currents in its core, and can be approxi- mated as that of a magnetic dipole at the Earth’s centre. The axis of rotation of the Earth’s dipole can be taken to coincide with the axis of rotation. On the equator (6400 km from the centre), the field strength is 4 × 10 5 T, with the B-field directed horizontally with respect to the surface of the Earth and pointing due North. Recall that at a large distance, the magnetic field of a magnetic dipole is given by B(r) = μ0 4pi 3(m · r)r r2m r5 . (a) Find Earth’s magnetic dipole moment. (b) The magnetic properties of the Earth can be modeled as arising from a simple current loop with a radius 0.1 times that of the Earth. Based on this, estimate the current. (c) Find the field strength at the geographical North Pole. Which way does the B field point there A sketch may help. (12 marks) 9323 SEMESTER 2, 2013 Page 8 of 10 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 spectrum of a quantum system. (b) The Born rule. (c) A Bell inequality. (12 marks) 6. Consider a standard Stern-Gerlach experiment, with a beam of spin-1/2 particles prepared in the state |ψ〉 = 2|+〉 e2ipi/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 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| 〉) , (d) Suppose that the Sz measurement was performed, with the result Sz = ~/2. Subsequent to that result, a second measurement is performed to measure the spin component Sy. What are the possible results of that measurement, and with what probabilities do they occur (e) Determine the direction ~n = (sin θ cosφ, sin θ sinφ, cos θ) for which an S~n mea- surement of |ψ〉 will give the result S~n = +~/2 with certainty (that is, with proba- bility equal to 1). (12 marks) 9323 SEMESTER 2, 2013 Page 9 of 10 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 to the z-direction, ~B = B0z , 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) If instead the magnetic field was parallel to the x-direction, ~B = B0x , calculate |ψ(t)〉. (d) Qualitatively describe the evolution of the spin-1/2 particle in part (c) in 50 words or less. (12 marks) 8. An electron is confined to a one-dimensional quantum well of width L = 0.100 nm. (a) If the height of the potential can be considered extremely large, calculate the ener- gies of the three lowest energy eigenstates. Hint: The formula for the energies of an infinite well can be obtained from the ki- netic energy expressionE = p2/(2m), where the momentum takes quantized values pn = 2pi~/λn for the allowed wavelengths λn = 2L/n. (b) Draw a diagram to show the probability density functions (the functions |ψ(x)|2) for the ground state and first excited state. (c) 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). (d) 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) 9323 SEMESTER 2, 2013 Page 10 of 10 9. Consider the entangled quantum state |Ψ〉12 of two spin-1/2 particles, given by |Ψ〉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 less than 100 words, describe Einstein’s interpretation of the measurements de- scribed in part (c). (12 marks) THERE ARE NOMORE QUESTIONS.
