CC0936 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 91 PHYSICS B (ADVANCED SEMESTER, 015 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.
CC0936 SEMESTER, 015 Page of 13 Table of constants Avogadro s number N A = 6.0 10 3 mol 1 speed of light c =.998 10 8 m.s 1 electronic charge e = 1.60 10 19 C electron rest mass m e = 9.110 10 31 kg electron rest energy E = 511 kev electron volt 1 ev = 1.60 10 19 J proton rest mass m p = 1.673 10 7 kg neutron rest mass m n = 1.675 10 7 kg Planck s constant h = 6.66 10 34 J.s Planck s constant (reduced = 1.055 10 34 J.s Boltzmann s constant k B = 1.380 10 3 J.K 1 Stefan s constant σ = 5.670 10 8 W.m.K 4 Coulomb constant (4πɛ 0 1 = 8.988 10 9 N.m.C permittivity of free space ɛ 0 = 8.854 10 1 C.N 1.m permeability of free space µ 0 = 4π 10 7 kg.m.c gravitational constant G = 6.673 10 11 N.m.kg atomic mass constant u = 1.660 10 7 kg
CC0936 SEMESTER, 015 Page 3 of 13 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 rr 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(rrdv 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 D = ρf 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 rr 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
CC0936 SEMESTER, 015 Page 4 of 13 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
CC0936 SEMESTER, 015 Page 5 of 13 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. Briefly answer the following questions: (a How can the Hall effect help find the sign of the charge of the charge carriers in a semiconductor? (b How is it that P = NαE for a gas but not for a solid or a liquid? (c In class we discussed the result J s = M ˆn. Explain the symbols and briefly discuss the physical meaning of this result. (d What is meant by a constitutive relation? Give two examples. (15 marks
CC0936 SEMESTER, 015 Page 6 of 13. Magnetohydrodynamic generators exploit the flow of a charged fluid to generate electricity: A fluid containing positive and negative charges flows at a velocity v in a channel through a magnetic field B 0 orthogonal to v. The channel is lined with two electrodes parallel to v separated by a distance w (see figure. For sub-questions (a-(b the switch S is open, so that no current flows in the circuit. Electrode B I S v B 0 w V BA R y x Electrode A (a Draw a diagram of the channel, indicating the trajectory of positive and negative charges. Explain how a voltage difference develops between the electrodes. (b Derive and justify that the steady-state voltage difference between the electrodes is V ss = vb 0 w. Power can be generated using fast plasma jets between superconducting magnets. Calculate the voltage difference for a plasma at v = 300 m/s in a magnetic field of B 0 = 1 T with w = 0.1 m. (c When a resistor with resistance R is connected between the electrodes A and B by closing switch S, the voltage drops and V BA V ss. The current density between the plates is then given by j = σ(vb 0 E y ŷ (1 where σ is the conductivity of the fluid and E y is the y component of the electric field. The plasma has σ = 0.36 Ω 1 m 1, and the electrodes have a surface area A = 0.1 m. Show that the short circuit current I max (when R = 0 and V = 0 is and evaluate this expression numerically. I max = AσvB 0 (d Using Eq. (1, the relation between voltage V BA and E y, and Ohm s law for the resistor R show that the voltage difference between the plates for arbitrary R can be expressed as R V BA = V ss, R + R where R = w/(σa is the resistance of the plasma. (15 marks
CC0936 SEMESTER, 015 Page 7 of 13 3. All sub-questions can be answered independently, using answers provided within the previous sub-questions. We study an antenna of length h and circular cross-section (with radius r centered on the origin and parallel to ŷ (see figure. We consider the antenna to be static: free charges are immobilized in the antenna, with a free charge density distribution given by ρ(y = ρ 0 y/h. We consider r h and as a consequence assume all fields inside the rod are along ŷ and depend on y only: that is, E(x, y, z = E(yŷ. r y h x -h (a Explain why at points far away from the rod, the electric field and potential due to the rod are that of an electric dipole. (b Show that the dipole moment of the rod is p = 3 πr h ρ 0 ŷ. (c Evaluate the dipole moment using the equation in (b. What is the direction and magnitude of the electric field due to the rod at position (x, y, z = (5 m, 0, 0? You are given ρ 0 = 0.00 C.m 3, r = mm, h = 0 cm. (d For this sub-question we assume the rod is a dielectric with relative permittivity ε r. Using the differential form of Gauss law, show that the electric field inside the rod takes the form ( ρ0 y E(y = + E 0 ŷ, hε 0 ε r where E 0 is the field at the origin. (e Assuming everywhere outside the rod that the field is that of a point dipole placed at the origin but with same p as calculated in sub question (b, and remembering that D y is continuous at the rod s top and bottom interface, find an expression for E 0. Discuss the validity of the assumption in this question. (15 marks
CC0936 SEMESTER, 015 Page 8 of 13 4. Consider a long solenoid with its center at the origin and its axis aligned with the z axis. A current flows in the solenoid, producing a uniform magnetic field B 0 in the z direction. A solid platinum ball (µ r = 1.0003 of radius a is placed at the center of the solenoid. Magnetization of the ball creates a dipole field outside and a uniform field inside, given by B M = µ 0 M/3 where M is the magnetization density. The total magnetic field is obtained by superposing these magnetization fields with B 0. (a Find the magnetic dipole moment, m, of the ball in terms of M, and write down the total magnetic fields inside and outside the ball (within the solenoid. Draw a diagram that shows the magnetic field lines due to the magnetization fields (i.e., exclude B 0. (b Using the defining relationship, M = χ m H in, show that the total magnetic field strength inside the ball is related to B 0 by B in = 3µ rb 0 µ r +. Calculate the ratio B in /B 0, and comment on the effect of the magnetization on the magnetic field inside the ball. (c Show that the magnetization density can be expressed in terms of B 0 as M = 3(µ r 1 µ 0 (µ r + B 0. (d Find the potential energy of the ball inside the solenoid and determine the work that needs to be done to pull the ball outside the solenoid. Using the values, B 0 = 0.01 T, a = 0.1 m, calculate this potential energy, and compare with the gravitational potential energy required to lift the ball by its height, U = mga (use m = 90 kg, and g = 10 m /s. How much do you need to increase B 0 to make the magnetic potential energy competitive with the gravitational potential energy? (15 marks
CC0936 SEMESTER, 015 Page 9 of 13 SECTION B: QUANTUM PHYSICS Formula sheet Spin-1/ operators as matrices: S x = ( 0 1 1 0 S y = ( 0 i i 0 S z = ( 1 0 0 1 ( S n = cos θ sin θ e iφ sin θ e iφ cos θ + x = 1 ( 1 1 + y = 1 ( 1 i ( 1 + = 0 ( cos θ + n = sin θ eiφ x = 1 ( 1 1 y = 1 ( 1 i ( 0 = n = 1 ( sin θ cos θ eiφ Solution to the Schrödinger equation: ψ(0 = c 1 E 1 + c E + ψ(t = c 1 e ie 1t/ E 1 + c e ie t/ E + Quantum harmonic oscillator equations: (â ˆx = mω + â â = mω (ˆx + i ˆp mω (â mω ˆp = i â â = mω (ˆx i ˆp mω [â, â ] = 1 â n = n n 1 â n = n + 1 n + 1 Ĥ n = ω(n + 1/ n â â n = n n
CC0936 SEMESTER, 015 Page 10 of 13 Please use a separate book for this section. Answer ALL QUESTIONS in this section. 5. Explain briefly (write less than 50 words for each part the following. (a The significance of the normalisation condition for a state vector ψ describing a spin-1/ particle. (b The significance of the Hamiltonian H of a quantum system. (c The connection between the state vector and the wavefunction for a particle. (1 marks 6. Consider a standard Stern-Gerlach experiment, with a beam of spin-1/ particles prepared in the state ψ = + + e iπ/6. (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 are the possible results of a measurement of the spin component S x, and with what probabilities do they occur? (d Suppose that the S z measurement was performed, with the result S z = /. Subsequent to that result, a second measurement is performed to measure the spin component S x. What are the possible results of that measurement, and with what probabilities do they occur? (e Using matrix notation, construct a spin operator S n for a direction ˆn = (1/ x + (1/ŷ + (1/ ẑ. Give expressions for the eigenstates and the corresponding eigenvalues for this operator. (1 marks
CC0936 SEMESTER, 015 Page 11 of 13 7. An electron is at rest in a magnetic field B = B 0 ŷ, where B 0 is constant. The energy of the electron is given by µ B. Assume the electron magnetic moment is given by µ = es/m where S is the electron spin. (a Pretend for a moment that the electron is classical: the electron experiences a torque µ B. Assume that at time t = 0, the spin vector is S = Sẑ. Give a brief qualitative description of the time evolution of the spin. Include a diagram to illustrate your answer. The electron, of course, is a quantum particle. The rest of this question concerns the quantum mechanical description of the electron. (b Show that the Hamiltonian in the usual basis is represented by H = ω ( 0 0 i, i 0 and identify the value of ω 0. (c What are the eigenvalues for energy, and what are the corresponding eigenvectors? (d Assume the electron is in the following state at time t = 0: ψ(t = 0 = +. What are the possible results of measurement of S z at time t = 1 π/ω 0, and with what probabilities do they occur? (1 marks
CC0936 SEMESTER, 015 Page 1 of 13 8. Consider the entangled quantum state Ψ 1 of two spin-1/ particles, described by Ψ 1 = 1 + 1 1 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 The given state is written in the S z basis. Express the basis vectors + j and j (with j = 1, in terms of + x j and x j, and hence show that the given state may be rewritten in the S x basis as: Ψ 1 = 1 + x 1 x + 1 x 1 + x. (d Your answer for (b should include correlations between the outcomes of the spin measurements for the two electrons. Briefly explain: (i why Albert Einstein referred to the correlations as spukhafte Fernwirkungen (spooky action at a distance; (ii what John Bell established about the correlations. (1 marks
CC0936 SEMESTER, 015 Page 13 of 13 9. For a harmonic oscillator, consider the ground state 0 (also known as the vacuum state, which satisfies â 0 = 0. (a Show that the expectation values of ˆx and ˆp are given by: 0 ˆx 0 = 0, 0 ˆp 0 = 0. (b Calculate the expectation values of ˆx and ˆp in this state 0. (c Calculate the uncertainties x and p for this state 0. (d What is the product x p for the state 0? Explain why the vacuum state 0 is called a minimum-uncertainty state. (1 marks THERE ARE NO MORE QUESTIONS.