NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF PHYSICS ADVANCED PLACEMENT TEST (SAMPLE) INTRODUCTION TO THERMODYNAMICS AND OPTICS

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1 NAIONAL UNIVERSIY OF SINGAPORE DEPARMEN OF PHYSICS ADVANCED PLACEMEN ES (SAMPLE) PC4 INRODUCION O HERMODYNAMICS AND OPICS MMM-YYYY----ime Allowed: Hours INSRUCIONS O CANDIDAES his paper contains five short questions in Part I and three long questions in Part II It comprises 5 printed pages including this one he total marks for Part I is 40 and that for Part II is 60 3 his is a CLOSED BOOK test 4 Non-Programmable calculator is allowed to be used 5 Answer ALL questions 6 Answers to the questions are to be written in the answer booklets 7 A list of constants and formulae are given on Page and 3

2 hermal physics and kinetic theory: (i) Kinetic mean free path is given by l = π n v d, where nv is the number density and d is the molecular diameter (ii) Maxwell Boltzmann distribution in 3D: P( v ) = 4π v π m k 3 / (a) Root-mean-square speed: v (b) Average speed: v av m v exp kb 3k B = m B rms 8k B = πm k (c) Most-probable speed: B v mp = m (iii) Ideal gas equation: p V = n R (iv) Van der Waals equation: a n ( p + ) ( V n b) = n R V hermodynamics: (i) First law of thermodynamics: ΔU = qin + Win Gas expansion work done by the gas: W out = p dv (ii) Carnot heat-engine efficiency e C C = H (v) One form of adiabat: p V (vi) Stefan Boltzmann equation: (vii) Planck radiation equation: P rad γ = cons tant π h c ( λ) = 5 h c λ exp λ kb (viii) Wien displacement law: λ peak 898x0 = 3 m K (ix) Convection equation: (x) Conduction equation: P conv P ccond 4 P rad = σ A e = h A Δ Δ = k A L (xi) Linear thermal expansion: Δ L = α L Δ dq (iii) Entropy rev ds = (iv) Enthalpy H = E + PV (v) Helmholtz free energy F = E S (vi) Gibbs free energy G = H S Geometric optics: (i) p and q are object and image distances measured respectively on opposite sides of the lens or of the refracting surface: (a) Object image relation (thin lens): + =, p q f where = ( n ) f R R n n n (b) Refracting-surface equation: n + = p q R (ii) Gullstrand equation: (a) Effective power: d P + P P P n P e = (b) Front-vertex refracting power: P f P = P + P d n (c) Back-vertex refracting power: P b P = P + P d n

3 (iii) Spherical-mirror equation: + =, where p q f = f R F f (iv) F-number: = # D (v) Numerical aperture: NA = n sinθ Wave optics: (i) Circular aperture (Airy s disc): first diffraction λ minimum is at sin θ = a λ (ii) Slit: first diffraction minimum is at sin θ = a (iii) N-slit intensity pattern: where φ = General: π d sinθ λ I = I o sin ( N φ / ), sin ( φ / ) (i) For a circle, the arc length subtended by conical angle α is given by s = r α θ r A (vi) Snell s law: n sinθ = n sinθ (a) Critical angle: sin θc = n / n (b) Brewster angle: tan θp = n /n (vii) Wave relation: v = f λ n (viii) Abbe number: D v = n n (iv) Single-slit diffraction pattern: sin ( δ / ) π I = I o, where δ = a sinθ λ ( δ / ) F (iii) loga (b c) = logab + logac (iv) loga b = logd b / logd a n+ n x (v) x dx = + c except for n = for n + which x dx = ln x + c C (ii) For a sphere, the surface area subtended by conical angle θ is given by A = π r ( cos θ ) Universal constants: Gas constant R = 834 J K mol Boltzmann constant kb = 38x0 3 J K Stefan Boltzmann constant σ = 5670x0 8 W m K 4 Avogadro s number NA = 60x0 3 mol Permittivity of free space εo = 8854x0 Fm Planck constant h = 666x0 34 J s Speed of light in vacuum c = 998x0 8 m s 3

4 PC4 Physics II Part Answer all FIVE questions All questions carry 8 marks each he dark nebula in interstellar space is thought to contain 50 hydrogen (H) atoms per cubic centimetre having a root-mean-square speed of 700 m s [Hint: H has an atomic weight of 0 g mol and a diameter of 0 Å] (i) Compute the mean free path of these H atoms (ii) Compute their mean temperature he Joule-homson coefficient μ J is the rate of change of temperature of a gas with pressure, μ J d = dp H, when it is expanded adiabatically against a constant pressure (ie, at constant enthalpy H) According to Euler s chain rule, this in turn is given by d dp H dh dp = dh d P Hence or otherwise, deduce the value of μ J for an ideal gas 3 At sufficiently low temperatures, the molar heat capacity C of non-metals varies according 3 to the Debye 3 law as C = k, where k = 940 J mol K, and θ is a materials θ property called the Debye temperature For NaCl, θ = 8 K Compute the heat per mol required to raise the temperature of NaCl from K to 30 K 4 he Arecibo radio telescope in Puerto Rico has a 305-m-diameter collector dish and uses radio waves of wavelength 75 cm, instead of visible light, to form images Compute the diffraction-limited feature size that that can be resolved according to the Rayleigh criteria 4

5 on the moon at a distance of 363,000 km Deduce whether it would be able to resolve a family-size car roaming on the surface of the moon 5 Unpolarised light of intensity I o is incident on a stack of three polarisers he transmission axis of the third polariser is always perpendicular to the first polariser, while that of the second polariser is oriented at an angle α with respect to this third polariser (see diagram below) Derive an expression for the intensity of light I 3 transmitted through the third polariser in terms of α and I o I 0 α I 3 transmission axis 5

6 Part Answer all HREE questions All questions carry 0 marks each 6 he Brayton cycle is often used to describe the operation of gas turbine engines It comprises the following steps, as shown below: a b: adiabatic compression of air, initially at atmospheric pressure p, by a rotary compressor to pressure p ; b c: isobaric heating at pressure p due to combustion of fuel injected into the hot air; c d: adiabatic expansion of the hot gases through the rotating turbine blades to atmospheric pressure p ; d a: isobaric cooling at pressure p due to heat rejection into atmosphere pressure p b c p a d volume (i) Sketch the temperature vs entropy state diagram Indicate on that diagram the locations of a, b, c and d, and the direction of the cycle (ii) Derive an expression for the thermal efficiency of this cycle in terms of the pressure ratio p /p and the heat capacity ratio of the working gas γ 6

7 7 Light from a tungsten filament lamp is focussed by a condenser assembly onto a slit which is the entrance to an optical instrument he height of the filament h is 30 mm (see diagram below) he condenser assembly comprises two fused-silica plano-convex lenses of focal lengths 50 mm and 00 mm, and identical diameters of 50 mm, oriented as shown below and aligned to the optical axis defined by the centres of the filament and of the slit he centre of the filament is at the focal point (ie, 50 mm to the left) of lens he refractive index of fused silica is 45 (i) Compute the F-number and the radius of curvature of lens (ii) Compute the location of the image and the minimum slit width d required to collect the entire image (iii) Explain briefly why it is more desirable to use fused silica lenses than soda lime glass lenses lamp condenser assembly slit h = 30 mm d lens f = 50 mm lens f = 00 mm 7

8 8 he information in a compact disc (CD) is encoded as 08-μm-wide bumps on a 6-μmwide spiral track on a reflective metallised surface he spiral puts subsequent segments of the track parallel to each other (schematic shown below) A semiconductor laser with a vacuum wavelength of 079-μm is focussed to a spot (also shown) within this track By spinning the CD, the laser spot is made to scan along the track As it passes over a bump, roughly half of the spot is reflected from the track surface, and the remainder from the top of the bump he height of these bumps is chosen to give destructive interference between these two reflections so that the reflected intensity is minimised he bump and track surface are covered by a millimetre-thick polycarbonate layer with a refractive index n of 8, through which the laser passes before reflection laser spot 6 μm bump track surface CD track (i) Compute the wavelength and frequency of the laser radiation inside the polycarbonate layer Write down whether this laser is visible to the human eye (ii) Deduce the minimum height of the bump required for the first destructive interference of the reflected laser light PH (iii) Compute the first-order diffraction angle of green light in air due to the periodicity perpendicular to the track direction when the CD surface is illuminated by white light at normal [Hint: he vacuum wavelength of green light is 550 nm] - END OF PAPER - 8

NATIONAL UNIVERSITY OF SINGAPORE

NATIONAL UNIVERSITY OF SINGAPORE PC1142 Physics ll (Semester l: AY 2009-10, 28 November 2009) Time Allowed:2 Hours INSTRUCTIONS TO CANDIDATES 1. This examination paper comprises EIGHT (8) printed pages