Exam Handout for Physics/Astronomy Level II - Optics, Thermodynamics and Heat Engines, Exams of Thermodynamics

Download Exam Handout for Physics/Astronomy Level II - Optics, Thermodynamics and Heat Engines and more Exams Thermodynamics in PDF only on Docsity! The Handbook of Mathematics, Physics and Astronomy Data is provided KEELE UNIVERSITY EXAMINATIONS, 2011/12 Level II Wednesday 11th January 2012, 09:30 - 11:30 PHYSICS/ASTROPHYSICS PHY-20027 Optics and Thermodynamics Candidates should attempt ALL of PART A and ONE question from each of PARTS B and C. PART A yields 40% of the marks, PART B yields 30%, PART C yields 30% NOT TO BE REMOVED FROM THE EXAMINATION HALL PHY-20027 Page 1 of 7 PART A Answer all TEN questions A1 Sketch a Michelson Interferometer, identifying all relevant compo- nents. [4] A2 A Michelson Interferometer is used to produce an interference pat- tern using a monochromatic light source. Describe the arrangement of the mirrors that produces: (a) circular fringes; [2] (b) straight fringes. [2] A3 The near point of an eye is located 1m away. Determine the focal length of the lens required to allow clear vision of an object placed 30 cm in front of the eye. [4] A4 Light from a monochromatic light source is incident on a slit with width a = 0.75mm. A diffraction pattern is observed on a screen located at a distance L = 2m. The first minimum is measured at a distance y = 1.35mm from the central maximum. What is the wavelength of the light? [4] A5 State Fourier’s theorem. For a generic function f(x) write down its representation as a Fourier series. [4] A6 Sketch the Carnot cycle in a P − V diagram, stating each of the thermodynamical processes involved. [4] PHY-20027 Page 2 of 7 B2 (a) Consider a rectangular glass block (with refractive index ng = 1.5) in vacuum. A beam of natural light strikes the first surface at an angle of incidence equal to the polarisation angle. Show that the transmitted beam also strikes the second surface at the polarisation angle. [12] (b) What is relative orientation of the polarisation axes of two stacked polaroid filters, if the intensity of the incident unpo- larised light is reduced by a factor 4? [8] (c) i. Explain briefly what is meant by the terms linear polarisa- tion and circular polarisation. [5] ii. Derive the state of polarisation of the wave described by: E = E0 cos ( π 2 − (kz − ωt) ) î + E0 sin(kz − ωt) ĵ [5] /Cont’d PHY-20027 Page 5 of 7 PART C Answer ONE out of TWO questions C1 Consider an engine, running on 1 mole of an ideal gas, based on the following cycle: 1. Isothermal compression at temperature T1, from volume Va to volume Vb; 2. Ignition, releasing heat Q2, causing an increase in pressure whilst not being allowed to expand; 3. Isothermal expansion at temperature T2; 4. Cooling at constant volume to temperature T1. (a) Draw this cycle in P − V and S − T diagrams (where P is pressure and S is entropy), and label the diagrams to indicate the four steps. [10] (b) The efficiency of this engine is given by: η = R(T2 − T1) ln r CV(T2 − T1) + RT2 ln r , where r = Va/Vb is the compression ratio and CV is the heat capacity for the exploding gas. i. By considering the heat absorbed by the gas and the work done by it, derive the above equation for the efficiency of the engine. [15] ii. Discuss how the efficiency of this engine can be optimised, and compare it with the efficiency of the Carnot cycle. [5] /Cont’d PHY-20027 Page 6 of 7 C2 The Clausius–Clapeyron equation for the phase transition between liquid and vapour can be used to find that the boiling point at pressure P has a temperature Tboil = ( nR L ln P0 P + 1 T0 )−1 , where T0 is the boiling temperature at pressure P0. (a) Consider 1 kg of water (≡ 55.4 mole) at 0◦C, on a mountain summit where the pressure is half that at sea level. For water, L = 2.27× 106 J kg−1 and CV,liquid = 4.187× 103 J kg−1 K−1. i. Show that Tboil = 354.4 K. [3] ii. Calculate the total energy required to turn the water into vapour. [5] (b) The water vapour, at its mountain summit boiling temperature, is then taken in a thermally and pressure isolated container to sea level. There it is maintained in thermal isolation but allowed to come into pressure equilibrium with its surroundings. For water vapour, the ratio of specific heats γ = 1.32. i. Show that the temperature increases to T = 419.2 K. [6] ii. Calculate the amount of work done on the vapour. [6] (c) Finally, the vapour is allowed to cool to a temperature of 0◦C. For water vapour, CP = 1.996× 103 J kg−1 K−1. i. Calculate the total energy released by the water. [5] ii. Compare your answers to parts (bii) and (di), and give a physical explanation for the difference. [5] PHY-20027 Page 7 of 7