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IIT JAM 2012

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Brijmohan
Banaras Hindu University (BHU), Varanasi
PhD Condensed matter Physics

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fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES IIT-JAM-2012(PHYSICS) IMPORTAN NOTE FOR CANDIDTES Attempt ALL 25 questions. Questions 1-15(objective questions) carry six marks each and questions 1625(subjective questions) carry twenty one marks each. Q1. Given a function of both position x and time t, the value of f dx df x, t , x (where f ) is x dt dt (a) Q2. 2 f x 2 f x (c) f x (d) df dx If F is a constant vector and r is the position vector then F r would be (a) r F (b) F (c) F r (d) r F Q3. (b) Three masses m, 2m and 3m are moving in x-y plane with speeds 3u, 2u and u, respectively, as shown in the figure. The three masses collide at the same time at P and stick together. The velocity of the resulting mass would be y 2m 2u 60 m 3u P x 60 u 3m (a) u 3y x 12 (b) u 3y x 12 (c) u 3y x 12 (d) u 3y x 12 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 1 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q4. The figure shows a thin square sheet of metal of uniform density along with possible choices for a set of principal axes (indicated by dashed lines) of the moment of inertia, lying in the plane of the sheet. The correct choice(s) for the principal axes would be q p (a) p, q, and r Q5. (b) p and r r (c) p and q (d) p only A lightly damped harmonic oscillator loses energy at the rate of 1% per minute. The decrease in amplitude of the oscillator per minute will be closest to (a) 0.5% Q6. (b) 1% (c) 1.5% (d) 2% A parallel plate air-gap capacitor is made up of two plates of area 10 cm2 each kept at a distance of 0.88 mm. A sine wave of amplitude 10 V and frequency 50 Hz is applied across the capacitor as shown in the figure. The amplitude of the displacement current density (in mA/m2) between the plates will be closest to ~ (a) 0.03 Q7. (b) 0.30 (c) 3.00s (d) 30.00 A tiny dust particle of mass 1.4 10-11 kg is floating in air at 300K. Ignoring gravity, its rms speed (in m/s) due to random collisions with air molecules will be closest to (a) 0.3 (b) 3 (c) 30 (d) 300 fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 2 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q8. When the temperature of a blackbody is doubled, the maximum value of its spectral energy density, with respect to that at initial temperature, would become (a) Q9. 1 times 16 (b) 8 times (c) 16 times (d) 32 times Light takes 4 hours to cover the distance from Sun to Neptune. If you travel in a spaceship at a speed 0.99c (where c is the speed of light in vacuum), the time (in minutes) required to cover the same distance measured with a clock on the spaceship will be approximately (a) 34 Q10. 60 27 (b) 56 (c) 85 (d) 144 Co is a radioactive nucleus of half-life 2 ln 2 108 s . The activity of 10g of 60 27 Co in disintegrations per second is (a) Q11. 1 1010 5 (b) 5 1010 (c) 1 1014 5 (d) 5 1014 An X-ray beam of wavelength 1.54 is diffracted from the (110) planes of a solid with a cubic lattice of lattice constant 3.08 . The first-order Bragg diffraction occurs at 1 (a) sin 1 4 Q12. 1 (b) sin 1 2 2 1 (c) sin 1 2 1 (d) sin 1 2 The Boolean expression P P Q , where P and Q are the inputs to a circuit, represents the following logic gate (a) AND (b) NAND (c) NOT (d) OR fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 3 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q13. Group I contains x- and y- components of the electric field and Group II contains the type of polarization of light. Group I Group II E0 cos t kz 2 E y E 0 sin t kz 1. Linearly Polarized Q. E x E 0 sin t kz E y E 0 cos t kz 2. Circularly Polarized R. E x E1 sin t kz E y E2 sin t kz 3. Unpolarized S. E x E0 sin t kz E y E 0 sin t kz 4 4. Elliptically Polarized P. Ex The correct set of matches is (a) P 4; Q 2; R 4; S 1 (c) P 4; Q 2; R 1; S 4 Q14. (b) P 1; Q 3; R 1; S 4 (d) P 3; Q 1; R 3; S 2 For a liquid to vapour phase transition at Ttr, which of the following plots between specific Gibbs free energy g and temperature T is correct? g g (a) (b) Liquid Liquid Vapour T Ttr Vapour T Ttr g g (c) (d) Liquid Vapour Ttr Liquid T Vapour Ttr T fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 4 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q15. A segment of a circular wire of radius R, extending from 0 to / 2 , carries a constant linear charge density . The electric field at origin O is y y (a) x 4 0 R (b) 1 1 x y 4 0 R 2 2 (c) 1 1 y x 4 0 R 2 2 (d) 0 R O x fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 5 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES SUBJECTIVE QUESTIONS Q.16 The P-V diagram below represents an ideal monatomic gas cycle for 1 mole of a gas. In terms of the gas constant R, calculate the temperatures at the points J, K, L and M. Also calculate the heat rejected and heat absorbed during the cycle, and the efficiency of the cycle. P kPa K 160 80 M J 0.03 Q.17 L 0.10 V m3 2 kg of a liquid (specific heat = 2000 J K-1 kg-1, independent of temperature) is heated from 200 K to 400 K by either of the following two processes P1 and P2: P1: bringing it in contact with a reservoir at 400 K. P2: bringing it first in contact with a reservoir at 300 K till equilibrium is reached, and then bringing it in contact with another reservoir at 400 K. Calculate the change in the entropy of the liquid and that of the universe in processes P1 and P2. Neglect any change in volume of the liquid. fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 6 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.18 (a) Two concentric, conducting spherical shells of radii R1 and R2 (R1 < R2) are maintained at potentials V1 and V2, respectively. Find the potential and electric field in the region R1 < r < R2 . (b) z A polarized dielectric cube of side G l is kept on the x-y plane as F shown. If the polarization in the P positive constant, then find all the bound surface C D , where k is a cube is P kxx E O y charge densities and volume charge A B density. x Q.19 A water cannon starts shooting a jet of water horizontally, at t = 0, into a heavy trolley of mass M placed on a horizontal ground. The nozzle diameter of the water cannon is d, the density of water is , and the speed of water coming out of the nozzle is u. Find the speed of the trolley as a function of time. Assume that all the water from the jet is collected in the trolley. Neglect all frictional losses. WATER CANNON Q.20 A long straight solenoid of radius R and n turns per unit length carries a current I t , where is a constant. t is time and remains finite. The axis of the solenoid is along the z-axis. Find the magnetic field, electric field and the Poynting vector inside the solenoid. Show these vectors at some instant t1 at any point (i) on the axis of the solenoid, and (ii) at a distance r (< R) from the axis. fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 7 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.21 2 1 In the operational amplifier circuit shown below, input voltages V1 V and V2 V 3 2 R4 50 k are applied. V2 V1 R2 50 k 15V P Vo Q R1 50 k 15V R 3 50 k (a) Determine the current flowing through resistance R4 and the output voltage Vo. (b) In the above circuit, if V1 is grounded and square pulses of peak voltage 1V and frequency 100 Hz are applied at V2, determine the voltage and phase change of the output pulses. Q.22 z A particle of mass m is confined in a potential-box of sides Lx, Ly, and Lz, 0, 0, Lz as shown in the figure. By solving the Schr dinger equation of the particle, find its eigenfunctions and energy 0, L , 0 0, 0, 0 y eigenvalues. y Lx , 0, 0 x Q.23 A particle of mass m and charge q moves in the presence of a time-independent magnetic field B r . Set up Newton s equation of motion for the particle. Since for a magnetic field B 0 , one can write B A , where A is a function of dA d position. Calculate as seen by the moving particle. Show that p qA , where p dt dt is the momentum of the particle, can be written as q times the gradient of a function. fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 8 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.24 Consider a periodic function f (x), with periodicity 2 , 0 x c f x 0 x 2 where c is a constant. (a) Expand f (x) in a Fourier series. (b) From the result obtained in (a), show that 1 1 1 1 1 ...... 4 3 5 7 9 Q.25 Two orthogonally polarized beams (each of wavelength 0.5 m and with polarization marked in the figure) are incident on a two-prism assembly and emerge along x-direction, as shown. The prisms are of identical material and n o and n e are the refractive indices of the o-ray and e-ray, respectively. Use sin sin 3 1 , and n o . 3 4 y Optic axis x Beam 2 P P uy Beam 1 30 30 Optic axis (a) Find the value of and n e. (b) If the right hand side prism starts sliding down with the vertical component of the velocity uy = 1 m/s, what would be the minimum time after which the state of polarization of the emergent beam would repeat itself? fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498 Website: http://www.physicsbyfiziks.com Email: fiziks.physics@gmail.com 9

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