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THE UNIVERSITY OF HULL ___________________ Department of Physical Sciences (Physics) Level 3 Examination January 2008 FUNDAMENTAL TOPICS IN PHYSICS (Module No. 04306) TUESDAY 15 JANUARY 2008 2 hours (13.30 – 15.30) Answer THREE questions, TWO from Section A and ONE from Section B. Do not open or turn over this exam paper, or start to write anything until told to do so by the Invigilator. Starting to write before permitted to do so may be seen as an attempt to use Unfair Means. Module 04306 CONTINUED Page 1 of 7 SECTION A: ELECTRODYNAMICS 1. An electromagnetic (EM) plane wave travelling in a vacuum in the z-direction has an E field of amplitude E0 polarised in the x-direction, i.e., E E0iˆ cost kz and an H field given by H H 0 cos(t kz) , where H0 is the vector amplitude of the H field. H (i) Using the Maxwell equation E 0 , show that the H field is polarised in the t y-direction with an amplitude of H 0 k / 0 E0 E0 / 0 c , where c k is the speed of light. [8 marks] (ii) The Poynting vector for an EM wave is given by N E H . a. Discuss briefly the physical significance of the Poynting vector. b. Using your result in (i), calculate the magnitude and direction of the Poynting vector for the EM wave. c. Show that the magnitude of the time averaged Poynting vector is E02 /( 2 0 c) . [6 marks] (iii) The Sun has a power output of 3.8×1026W. If the earth-sun distance is 1.5×1011m, use your result in (ii) to estimate the average amplitude of the E and H fields produced by the sun’s radiation at a distance equal to the earth’s orbit. [6 marks] [c = 3 ×108m s-1; 0 4 10 7 Hm -1 ; SI unit for E field is Vm-1; SI unit for H field is Am-1] Module 04306 CONTINUED Page 2 of 7 2. An electromagnetic wave is incident at a boundary between two dielectric media of refractive index n1 and n2 respectively. The amplitude reflection and transmission coefficients are E ' n cos i n2 cos t rN 0 1 E0 N n1 cos i n2 cos t and, (1) E '' 2n1 cos i t N 0 E0 N n1 cos i n2 cos t (2) when the electric field of the incident beam is polarised perpendicular to the plane of incidence. i and t are the angles of incidence and transmission respectively. (i) Use equation (1) to show that the reflected beam is in phase with the incident beam when n1 > n2, and that the two beams are out of phase by when n1 < n2. [4 marks] (ii) Use equation (1) to find an expression for RN , the coefficient of reflection for energy flow, at normal incidence. Evaluate RN for (a) a GaAs/air, and (b) a glass/air interface, taking the refractive index of GaAs and glass as 3.5 and 1.5 respectively. Hence, give a reason why the cleaved facets of GaAs are often used to form a laser cavity whereas the polished ends of a Nd3+:glass rod are unlikely to be used. [6 marks] (iii) The refractive index of a good conductor at wavelength 0 can be expressed as n 0 (1 i) (3) where is the skin depth. Use equations (1), (2) and (3) to show that for normal incidence at a boundary between a dielectric and good conductor (a) the reflected wave has approximately the same amplitude as the incident wave and is completely out of phase with it, and (b) the transmitted beam is extremely weak and out of phase by -/4 with the incident beam. State any approximations used. [6 marks] (iv) Copper has a skin depth of 6.6 10-5 m at a frequency of 1 MHz. What is the phase velocity of the transmitted wave in copper? Find the intensity transmission coefficient from air to copper at normal incidence. [4 marks] [Velocity of light in vacuo c = 3 108 ms-1] Module 04306 CONTINUED Page 3 of 7 3. When an external electric field Eex is applied to a medium, dipoles are induced in individual molecules and the induced molecular dipole moment p is linearly dependent on the local electric field Eloc given by E loc E ne ar E L E p E e x . (1) where EL and Ep are the Lorenz and macroscopic polarisation fields respectively. (i) Use the Lorenz model to explain the origin of the components of Eloc. You should use a diagram to clarify your explanation. [8 marks] (ii) Using Coulomb’s Law and integration with spherical coordinates, show that EL = P/30, where P is the macroscopic polarisation and 0 the permittivity of free space. [6 marks] (iii) Explain why local field effects are not usually considered when describing the polarisation of gases [2 marks] (iv) In a gas, the dielectric constant r is related to the molecular polarisability of a molecule by the formula r 1 N 0 where N is the number of molecules per unit volume. Find for CO2 at STP (P = 105 Nm-2, T = 273K) if r = 1.00099. [4 marks] [Boltzmann’s constant k = 1.381 10-23 JK-1; 0 = 8.854 10-12 Fm-1] Module 04306 CONTINUED Page 4 of 7 4. (i) Sketch the generalised equivalent circuit representation for an element of transmission line. Give a brief explanation of each component in this equivalent circuit. How is this simplified for an ideal transmission line? [4 marks] (ii) The voltage, V, and current, i, on an ideal transmission line satisfy V i L x t i V C x t and where C is the capacitance per unit length and L the inductance per unit length of line. (a) Use these equations to show that V and i obey wave equations where the phase velocity u 1 . LC (b) Assuming a solution to the wave equation of the form V Vo sin[ (t x / u)] show that V and i are in phase and that Z, the ratio V/i, is Z L . C (c) What is Z called? Give a brief explanation of its physical significance. [9 marks] (iii) An overhead power transmission line with C = 4.03 10-12 F.m-1 and L = 2.76 10-6H.m-1 connects a dynamo and step-up transformer to a sub-station located 150km away. If a lightning strike produces a sudden short-circuit at the dynamo explain qualitatively what happens on the line. How long will it take the wave produced to reach the sub-station and what fraction of the voltage wave is reflected at the sub-station if it forms a load of 900? [7 marks] Module 04306 CONTINUED Page 5 of 7 SECTION B: ACOUSTICS 5. (i) The wave equation for the transverse displacement y on a string of mass per unit length L under tension T is 2 y 1 2 y x 2 c 2 t 2 where the phase velocity is c2 = T/L. Assuming a general solution for y of the form y(x,t) = Aej(t + kx)+ Bej(t - kx) show that the normal modes (eigenfunctions) for a string of length L, rigidly fixed at both ends are yn(x,t) = Ansin(n x/L) expjn t where n = 1, 2, 3 …. is an integer. Hence derive the expression for n, the characteristic angular frequency of the string. Discuss how, in principle, yn(x,t) could be used to analyse the harmonic content of the note from a plucked guitar string and from a struck piano string and explain why these have different ‘acoustic’ qualities. [10 marks] (ii) A recent paper reports a fixed/fixed string formed by a single carbon nanotube. If L = 1.110-16kgm-1 and T = 810-10N calculate the wave velocity on the string. If L = 300nm what is its fundamental frequency and does this lie in the ‘acoustic’ range? [4 marks] (iii) Why do gases support only longitudinal acoustic waves? The pressure amplitude po in a plane acoustic wave varying as p = po sin(t –kx) in air is measured as po = 0.01Pa. Calculate the corresponding particle velocity amplitude and determine the particle displacement if = 200rad.s-1 (for air take = 1.25 kgm-3and c = 342m/s). [6 marks] Module 04306 CONTINUED Page 6 of 7 6. (i) A sound wave travels from a material of specific acoustic impedance Z1 into one of specific acoustic impedance Z2. From the conditions that the acoustic pressure and the particle velocity are continuous at the boundary show that the reflection coefficient rp for pressure is: rp pro Z 2 Z1 pio Z1 Z 2 [7 marks] (ii) A sound wave travels from air (Zair = 428 rayls) into concrete (Zcon= 8 106 rayls). Determine rp and express this in decibels. [3 marks] (iii) (a) What is understood in acoustics by the term reverberation? (b) Discuss briefly how the Sabine reverberation time T is defined. Calculate T for a cubic enclosure of dimensions 3m 3m 3 m if its entire inner surface is coated with acoustic plaster (sound absorption coefficient = 0.2). You may assume T = 55.2V/ac where V is the enclosure volume, a i Si where Si is the area of wall with sound absorption i coefficient I and c = 340ms-1 is the sound speed for air. (c) Comment on the acoustic quality of this enclosure. [6 marks] (iv) Give a short account of the operating principle and constructional features of either the moving coil (‘dynamic’) microphone or the electret microphone. [4 marks] Module 04306 END Page 7 of 7