NATIONAL UNIVERSITY OF SINGAPORE

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1 NATIONAL UNIVERSITY OF SINGAPORE PC4 Physics II (Semester I: AY , 6 November) Time Allowed: Hours INSTRUCTIONS TO CANDIDATES This examiatio paper comprises EIGHT (8) prited pages with FIVE (5) short questios i Part ad THREE (3) log questios i Part Aswer ALL questios 3 The total marks for Part is 40 ad for Part is 60 4 Aswers to the questios are to be writte i the aswer booklet provided 5 This is a CLOSED BOOK examiatio Calculators are allowed 6 A list of physical costats ad formulae is give o pages ad 3

2 Thermal physics ad kietic theory: (i) Kietic mea free path is give by v d, where v is the umber desity ad d is the molecular diameter (ii) Maxwell Boltzma distributio i 3D: P( v ) 4 v m k T 3 / (a) Root-mea-square speed: v (b) Average speed: v av m v exp kb T 3k BT m B rms 8k BT m k T (c) Most-probable speed: B v mp m (iii) Ideal gas equatio: p V R T (iv) Va der Waals equatio: a ( p ) ( V b) R T V Thermodyamics: (i) First law of thermodyamics: U = qi + Wi Gas expasio work doe by the gas: W out p dv (ii) Carot heat-egie efficiecy e T C C TH (v) Oe form of adiabat: p V (vi) Stefa Boltzma equatio: (vii) Plack radiatio equatio: P rad cos tat h c ( ) 5 h c exp kb T (viii) Wie displacemet law: peak 898x0 T 3 m K (ix) Covectio equatio: (x) Coductio equatio: P cov P ccod 4 P rad A e T h A T T k A L (xi) Liear thermal expasio: L L T dq (iii) Etropy rev ds T (iv) Ethalpy H = E + PV (v) Helmholtz free eergy F = E TS (vi) Gibbs free eergy G = H TS Geometric optics: (i) p ad q are object ad image distaces measured respectively o opposite sides of the les or of the refractig surface: (a) Object image relatio (thi les):, p q f where ( ) f R R (b) Refractig-surface equatio: p q R (ii) Gullstrad equatio: (a) Effective power: d P P P P P e (b) Frot-vertex refractig power: P f P P P d (c) Back-vertex refractig power: P b P P P d

3 (iii) Spherical-mirror equatio: p q f f R F f (iv) F-umber: # D (v) Numerical aperture: NA si, where (vi) Sell s law: si si (a) Critical agle: si c = / (b) Brewster agle: ta p = / (vii) Wave relatio: v f (viii) Abbe umber: D v F C Wave optics: (i) Circular aperture (Airy s disc): first diffractio miimum is at si a (ii) Slit: first diffractio miimum is at si a (iii) N-slit itesity patter: where Geeral: d si I I o si ( N / ) si / (i) For a circle, the arc legth subteded by coical agle is give by s = r r A, (iv) Sigle-slit diffractio patter: si ( / ) I I o, where a si / (iii) loga (b c) = logab + logac (iv) loga b = logd b / logd a x (v) x dx c except for = for which x dx l x c (ii) For a sphere, the surface area subteded by coical agle is give by A = r ( cos ) Uiversal costats: Gas costat R = 834 J K mol Boltzma costat kb = 38x0 3 J K Stefa Boltzma costat = 5670x0 8 W m K 4 Avogadro s umber NA = 60x0 3 mol Permittivity of free space o = 8854x0 Fm Plack costat h = 666x0 34 J s Speed of light i vacuum c = 998x0 8 m s 3

4 PC4 Physics II Part Aswer all FIVE questios All questios carry 8 marks each Oe way to separate two isotopes of uraium, uraium-35 ad uraium-38, to make uclear fuel is by gaseous diffusio of their hexafluorides UF 6 through a porous membrae The 35 UF 6 molecules (molecular mass, 0349 kg mol ) diffuse slightly faster tha the 38 UF 6 molecules (035 kg mol ) due to their slightly higher average molecular speed Compute the ratio of the average speed of 35 UF 6 to that of 38 UF 6 at the same temperature For a spherical greybody with surface area A, emissivity e, ad whose heat capacity is give by the Dulog Petit law ( C 3 N k B, where N is the umber of atoms i the body ad k B is the Boltzma costat), show that the time t for pure radiative coolig of this N k B body from a iitial temperature T o to temperature T f is t 3 3, e A Tf To assumig that the temperature of the surroudigs is sufficietly low 3 At sufficietly low temperatures, the molar heat capacity c of o-metals varies accordig 3 to the Debye T 3 T law as c A, where A is a materials-idepedet costat ad is a materials-specific property called the Debye temperature Derive a expressio for the molar etropy of these materials as a fuctio of temperature, assumig that the etropy at 0 K is zero 4

5 4 A les of diameter 50 mm ad focal legth 0 mm is used to form a image of a star of egligible agular width Compute the diameter of the Airy s disc of this star at the focal plae for a wavelegth of 550 m [ m is 0 9 m] 5 Upolarised light is icidet ormally o a Wollasto prism, which cosists of two rightagle-triagle calcite prisms with perpedicular optic axes as show below The light separates ito two polarised beams The extraordiary refractive idex e of calcite is 486 (which applies to light with electric field parallel to the optic axis), while the ordiary refractive idex o is 658 (which applies to light with electric field perpedicular to the optic axis) Compute ad sketch o a diagram the directio of travel of these beams ad idicate the directio of their polarisatio y x 45º optic axis parallel to z directio 45º optic axis parallel to y directio 5

6 Part Aswer all THREE questios All questios carry 0 marks each 6 The Leoir cycle ca be used to model iteral-combustio pulsed jet egies It comprises the followig steps, as show below: a b: isochoric heatig of the workig gas due to combustio; b c: adiabatic expasio of the hot gas; c a: isobaric coolig due to heat rejectio ito atmosphere ad retur to begiig of cycle pressure p b p a c volume (i) Sketch the temperature vs etropy state diagram for this process Idicate o that diagram the locatios of a, b ad c, ad the directio of the cycle [8 marks] (ii) Derive a expressio for the thermal efficiecy of this cycle i terms of the pressure ratio p /p ad the heat capacity ratio of the workig gas [ marks] 6

7 7 Tiy glass spheres are ofte used as leses to focus light ito ad out of optical fibres A sphere of diameter 00 mm ad refractive idex 50 is located at a distace f b i frot of the optical fibre such that paraxial rays are focussed ito the fibre, as show below f b sphere fibre (i) Briefly explai the meaig of paraxial rays [4 marks] (ii) Compute the required value of f b [Hit: The sphere caot be cosidered as a thi les] [6 marks] 7

8 8 Cosider a ideal plaar mirror waveguide as show below, i which a thi film of dielectric material of thickess z ad refractive idex is bouded by two parallel perfect mirrors Cosider further that a s-polarised light wave of wavelegth o, where o is the vacuum wavelegth, is repeatedly reflected by these two mirrors at a bouce agle B with respect to the surface, ad with a reflectio phase shift of perfect mirror B B perfect mirror z (i) Derive a expressio for the time take for this light to travel a distace L alog the waveguide, i terms of the vacuum speed of light c ad other ecessary parameters [8 marks] side view showig wavefrots: b B d mirror z a B c mirror (ii) I order for the light wave to survive ad travel alog the waveguide, costructive iterferece of the twice-reflected wave eeds to be satisfied (ie, the wave trai at cd must be i phase with that at ab, see side view above) Hece or otherwise derive a equatio for the allowed values of B [ marks] - END OF PAPER - PH 8