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IIT JAM 2009

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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-2009(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. Q.1 Isothermal compressibility T of a substance is defined as T 1 V . Its value V P T for n moles of an ideal gas will be (a) 1 P (b) n P (c) 1 P (d) n P Q.2 A space crew has a life support system that can last only for 1000 hours. What minimum speed would be required for safe travel of the crew between two space stations separated by a fixed distance of 1.08 x 1012 km? (a) c (b) 3 c (c) 2 c 2 (d) c 5 Q.3 An oscillating voltage V(t) = V0 cos t is applied across a parallel plate capacitor having a plate separation d. The displacement current density through the capacitor is V t V0 cos t d (a) 0 V0 cos t d (c) 0 0 V0 cos t d (b) 0 0 V0 cos t d (d) 0 V0 sin t d 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 Q.4 Two spherical nuclei have mass numbers 216 and 64 with their radii R1 and R2 , respectively. The ratio (a) 1.0 R1 is R2 (b) 1.5 (c) 2.0 (d) 2.5 Q.5 In the Fourier series of the periodic function (shown in the figure) f x sin x n cos nx n sin nx n 0 2 Which of the following coefficients are nonzero? (a) n for odd n (b) n for even n (c) n for odd n (d) n for even n Q.6 A particle is moving in space with O as the origin. Some possible expression for its position, velocity and acceleration in cylindrical coordinates ( , , z) are given below. Which one of these is correct? d d dz z zz and velocity v (a) Position vector r dt dt dt d 2 d d d d 2 z d dz z and acceleration a 2 z (b) Velocity v dt dt dt 2 dt dt dt dt d d dz z zz and velocity v (c) Position vector r dt dt dt d d dz z zz and velocity v (d) Position vector r dt dt dt Q.7 Which one of the following is an INCORRECT Boolean expression? (a) P Q PQ Q (b) P Q P Q P (c) P P Q Q (d) P Q R P Q R PQ R PQ R Q 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 Q.8 Monochromatic X-rays of wavelength 1 are incident on a simple cubic crystal. The first order Bragg reflection from (311) plane occurs at angle of 30 0 from the plane. The lattice parameter of the crystal in is (a) 1 (b) 11 2 (c) 3 (d) 11 exists in space. What will be the total sin cos Q.9 An electric field E r r charge enclosed in a sphere of unit radius centered at the origin? (a) 4 0 (b) 4 0 ( + ) (c) 4 0 ( - ) (d) 4 0 Q.10 A thin massless rod of length 2l has equal point masses m attached at its ends (see figure). The rod is rotating about an axis passing through its centre and making angle dL with it. The magnitude of the rate of change of its angular momentum is dt m l (a) 2ml2 2 sin cos (b) 2ml2 2 sin (c) 2ml2 2 sin2 (d) 2ml2 2 cos2 Q.11 A battery with a constant emf and internal resistance ri provides power to an external circuit with a load resistance made up by combining resistance RL and 2RL in parallel. For what value of RL will the power delivered to the load be maximum? (a) RL ri 4 (b) RL ri 2 (c) RL 2 ri 3 (d) RL 3 ri 2 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 Q.12 Among the following displacement versus time plots, which ones may represent an over-damped oscillator? (P) (Q) x t x t t t (R) (S) x t x t t t (a) only (P) and (Q) (b) only (P) and (R) (c) only (P) and (S) (d) only (P), (R) and (S) Q.13 A wave packet in a certain medium is constructed by superposing waves of frequency around 0 = 100 and the corresponding wave-number k with k0 = 10 as given in the table below k 81.00 9.0 90.25 9.5 100.00 10.0 110.25 10.5 121.00 11.0 Find the ratio v /vp of the group velocity v and the phase velocity vp. (a) 1 2 (b) 1 (c) 3 2 (d) 2 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 Q.14 A box containing 2 moles of a diatomic ideal gas at temperature T0 is connected to another identical box containing 2 moles of a monatomic ideal gas at temperature 5T0. There are no thermal losses and the heat capacity of the boxes is negligible. Find the final temperature of the mixture of gases (ignore the vibrational degrees of freedom for the diatomic molecules). (a) T0 (b) 1.5 T0 (c) 2.5 T0 (d) 3T0 Q.15 Moment of inertia of a solid cylinder of mass m, height h and radius r about an axis (shown in the figure by dashed line) passing through its centre of mass and perpendicular to its symmetry axis is (a) 1 2 1 mr mh 2 4 12 (b) 1 2 1 mr mh 2 2 8 (c) 1 2 1 mr mh 2 2 12 (d) 1 2 1 mr mh 2 2 4 r h 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 A parallel beam of light of diameter 1.8 cm contains two wavelengths 4999.75 and 5000.25 . The light is incident perpendicularly on a large diffraction grating with 5000 lines per centimeter. (a) Using Rayleigh criterion, find the least order at which the two wavelengths are resolved. (12) (b) What will be the angular separation (in radians) of the two wavelengths at order n =2? (9) Q.17 A block of mass M is free to slide on a frictionless horizontal floor. The block has a cylindrical cavity of radius R in the middle of it. The centre of mass (CM) of the block lies on the dashed line passing through the centre of the cavity (see figure). Initially the CM of the block is at a horizontal distance X1 from the origin. Now a point particle of mass m is released from point A into the cavity. There is negligible friction between the particle and the cavity surface. When the particle reaches point B, the CM of the blocks is at a distance X2 from the origin. Find (X2 X1). (21) R X1 Q.18 The electric field of an electromagnetic wave propagating through vacuum is given cos 100 3 x 100 y t . by E r , t E0 z (a) What is the wave vector k ? Hence find the value of . (9) at the origin. What is (b) At the time t = 0 there is a point charge q with velocity v v0 x the instantaneous Lorentz force acting on the particle? (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 6 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.19 Consider two infinitely long wires parallel to Y the z-axis carrying the same current I. One wire passes through the point L with coordinates (-1, 1) and the other through M with coordinates (-1, 1) in L 1,1 x the XY planes shown in the figure. The direction of 1, 1 direction. (a) Find the value of X the current in both the wires is in the positive z- B dl along the semicircular closed path of radius 2 units shown in the figure. (9) (b) A third long wire carrying current I and also perpendicular to the XY plane is placed at a point N with coordinates (x, 0) so that the magnetic field at the origin is doubled. Find x and the direction of the current in the third wire. (12) Q.20 A particle of mass m is confined in a one-dimensional box of unit length. At time t =0 the wavefunction of the particle is (x, 0) = Asin2 xcos x, where A is the normalization constant. (a) Write the wavefunction (x, t) at a later time t. (12) (b) Find the expectation values of momentum and energy at t = 0. (9) 20V Q.21 For the transistor circuit shown in the figure dc = 100 and VBE = 0.7V. Determine the base current IB, the collector-emitter 2k voltage VCE, the emitter voltage VE, the base 450k Vo voltage VB and the saturation current ICsat. 10 F Vi 10 F 1 k (21) 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 a Q.22 The equation of state of one mole of a van der Waals gas is P 2 V b RT V and its internal energy U(T, V) is given by U T , V U 0 CV T a , where U0 and C V V can be taken as constants. (a) Prove that in a reversible adiabatic process the temperature and volume satisfy the equation T V b R / CV = constant. (12) (b) Calculate the change in entropy of the gas when it undergoes a reversible isothermal expansion from volume V0 to 2V0 . (9) Q.23 (a) Find the normalized eigenvector of the matrix M 0 1 1 , corresponding 0 to its positive eigenvalue. (9) (b) The normalized eigenvectors of the matrix N 0 i i are and with the 1 2 0 eigenvalues 1 and 2, respectively and 1 > 2. If the eigenvector obtained in part (a) is expressed as = P 1 + Q 2, find the constants P and Q. (12) Q.24 A particle of mass m is thrown vertically up from the ground with initial speed v0 . As it moves it experiences a drag force Fdrag kv 2 , where v is the speed of the particle and k is a constant. Up to what height does the particle go and what is its speed when it reaches the ground again? (21) Q.25 Consider the radioactive transformation A B C with decay constants A and B for elements A and B; C is a stable element. Assume that at t = 0, NA = N0, NB = 0 and NC = 0, where NA, NB, and NC are the number of atoms of A, B and C, respectively. (a) Show that at any later time t the number of atoms NB of element B will be N B t t N 0 A tAt e e B t . B A (12) (b) Sketch qualitatively the variation of NA, NB and NC with time on three separate plots. (9) 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

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