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IIT JAM 2006

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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-2006(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 In a crystalline solid, the energy band structure (E-k relation) for an electron of mass m is given by E 2 k 2k 3 . The effective mass of the electron in the crystal is 2m (a) m (b) 2 m 3 (c) m 2 (d) 2m Q.2 Two electric dipoles P1 and P2 are placed at (0, 0, 0) and (1, 0, 0) respectively, with both of them pointing in the +z direction. Without changing the orientations of the dipoles P2 is moved to (0, 2, 0). The ratio of the electrostatic potential energy of the dipoles after moving to that before moving is (a) 1 16 (b) 1 2 (c) 1 4 (d) 1 8 Q.3 The truth table for the given circuit is J Q K (a) J 0 0 1 1 K 0 1 0 1 Q 1 0 1 0 (b) J 0 0 1 1 K 0 1 0 1 Q 1 0 0 1 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 (c) J 0 0 1 1 K 0 1 0 1 (d) Q 0 1 0 1 J 0 0 1 1 K 0 1 0 1 Q 0 1 1 0 Q.4 At a given point in space the total light wave is composed of three phasors P1 = a, P2 a i a e and P3 e i . The intensity of light at this point is 2 2 (a) 4 a 2 cos 2 2 (b) 4 a 2 cos 4 2 (c) a 2 cos 2 (d) 4 a 2 cos 2 2 Q.5 A small magnetic dipole is kept at the origin in the x-y plane. One wire L1 is located at z = -a in the x-z plane with a current I flowing in the positive x direction. Another wire L2 is at z =+a in y-z plane with the same current I as in L1, flowing in the positive y direction. The angle made by the magnetic dipole with respect to the positive x-axis is (a) 225o (b) 120o (c) 45o (d) 270o Q.6 The ratio of the inner radii of two glass tubes of same length is 2 . A fluid of viscosity 8.0 cP flows through the first tube, and another fluid of viscosity 0.8 cP flows through the second one when equal pressure difference is applied across both of them. The ratio of the flow rate in the first tube to that in the second tube is (b) 10 2 (a) 1.6 2 (c) (d) 0.4 Q.7 The relation between angular frequency and wave number k for given type of waves is 2 = k + k3. The wave number k 0 for which the phase velocity equals the group velocity is (a) 3 1 (b) 3 (c) 1 (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 2 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.8 A neutron of mass mn = 10 -27 kg is moving inside a nucleus to be a cubical box of size10 -14 m with impenetrable walls. Take 10 -14 Js and 1 MeV 10-13 J. An estimate of the energy in MeV of the neutron is (a) 80 MeV (b) 1 MeV 8 (c) 8 MeV (d) 1 MeV 80 Q.9 A spring-mass system has undamped natural angular frequency 0 = 100 rad s-1. The solution x(t) at critical damping is given by x t x0 1 0 t exp 0 t , where x0 is a constant. The system experiences the maximum damping force at time (a) 0.01 s (b) 0.1 s (c) 0.01 s (d) 0.1 s Q.10 In an intrinsic semiconductor, the free carrier concentration n (in cm-3) varies with temperature T (in Kelvin) as shown in the figure below. The band gap of the semiconductor is (use Boltzmann constant k B = 8.625 x 10-3 eVK-1 ) 36.5 35.5 ln n 34.5 33.5 (a) 1.44 eV 32.5 (b) 0.72 eV (c) 1.38 eV 3 1000 / 2 (d) 0.69 eV 4 Q.11 E x, y, z , t A 3i j exp i t kz represents an electromagnetic wave. Possible directions of the fast axis of a quarter wave plate which convert this wave into a circularly polarized wave are (a) (b) (c) (d) 1 2 1 2 1 2 1 2 7i j and 3i 4 j and 3i 4 j and 7i j and 1 2 i 7 j 1 2 1 2 4i 3 j 4i 3 j 1 i 7 j 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 A particle of rest mass m0 is moving uniformly in a straight line with relativistic velocity c, where c is the velocity of light in vacuum and 0< <1. The phase velocity of the de Broglie wave associated with the particle is (a) c (b) c (c) c (d) c 2 Q.13 Electrons of energy E coming in from x = - impinge upon a potential barrier of width 2a and height V0 centered at the origin with V0 E , as shown in the figure below. Let k 2 m V0 E . In the region a x a, the electrons is a linear combination of V V0 a (a) e kx and e kx (b) e ikx and e kx a x (c) e ikx and e ikx (d) e ikx and e kx Q.14 A solid melts into a liquid via first order phase transition. The relationship between the pressure P and the temperature T of the phase transition is P = -2T + P0 , where P0 is a constant. The entropy change associated with the phase transition is 1.0 J mole-1 K -1. The dP Calusius-Clapeyron equation for the latent heat is L T v . Here v vliquid v solid dT is the change in molar volume at the phase transition. The correct statement relating the values of the volumes is (a) vliquid vsolid (c) vliquid v solid (b) vliquid v solid 1 1 2 (d) v quid v solid 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 a Q.15 The symmertric part of P a 2 b is b a a 2 (b) b a a 1 (c) b a 1 b a 1 b2 b b2 a a 2 (d) b a 1 2 1 (a) a 2 ba 2 ba 1 b 2 b a 1 b2 SUBJECTIVE QUESTIONS Q.16 Consider a Body Centered Cubic (BCC) crystal with lattice constant a . Determine (a) the Miler indices for the (1, 0, 0) plane, (9) (b) the number of atoms per unit area in the (1, 1, 1) plane. (12) a2 Q.17 The equation of state of a gas is P V V nRT , while during an adiabatic process the gas obeys PV K , where a and K are positive constants. All other symbols P have their (P, V) to (P1, 2V) and then adiabatically from (P1, 2V) to 1 , V1 , 2 where P1 P . (21) Q.18 A conducting sphere of radius RA has a charge Q. It is surrounded by a dielectric spherical shell of inner radius RA and outer radius RB (as shown in the figure below) having electrical permittivity r 0 r . (a) Find the surface bound charge density at r = RA. (b) Find the total electrostatic energy stored in the dielectric (region B) (12) (9) RA A B RB 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 Q.19 For the transistor circuit shown below, evaluate VE , R B and RC , given I C 1mA , VCE = 3.8 V, VBE = 0.7 V and VCC = 10V use the approximation IC IE. VCC 10V RC RB VC (21) VE 24k 1.3 k , yz 2 Q.20 For the vector field V xz 2 i j z x2 y2 k (a) Calculate the volume integral of the divergence of V out of the region defined by a x a, -b y b and 0 z c. (12) (b) Calculate the flux of V out of the region through the surface at z = c. Hence deduce the net flux through the rest of the boundary of the region. (9) Q.21 The spherical surface of a plano-convex lens of radius of curvature R = 1m is gently placed on a flat plate. The space between them is filled with a transparent liquid of refractive index 1.55. The refractive indices of the lens and the flat plate are 1.5 and 1.6 respectively. The radius of the sixteenth dark Newton s ring in the reflected light of wavelength is found to be 5 mm. (a) Determine the wavelength (in microns) of the light. (12) (b) Now the transparent liquid is completely removed from the space between the lens and the flat plate. Find the radius (in mm) of the twentieth dark ring in the reflected light after this change. (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 6 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.22 A resistor of 1 k and an inductor of 5 mH are connected in series with a battery of emf 4 V through a switch. The switch is closed at time t = 0. In the following, you may use e 3 20. (a) Find the current flowing in the circuit at t = 15 micro-second. (9) (b) Find the heat dissipated through the resistor during the first 15 micro-second. (12) Q.23 A photon of energy Eph collides with an electron at rest and gets scattered at an angle 60o with respect to the direction of the incident photon. The ratio of the relativistic kinetic energy T of the recoiled electron and the incident photon energy Eph is 0.05. (a) Determine the wavelength of the incident photon in terms of the Compton h wavelength c m c , where h, me , c are Planck s constant, electron rest mass and e velocity of light respectively. (b) What is the total energy Ee of the recoiled electron in units of its rest mass? (12) (9) such that v 3 v . The v Q.24. A particle moves in a plane with velocity v vr r r 4 time dependence of the magnitude of the velocity v 5t . It is given that r = 1, = 0 and vr > 0 at t = 0. (In the following, you may use e 3 20 .) (a) Determine the trajectory r( ) of the particle. (b) At what time will become 4 radian? Q.25. A body of mass 1kg moves under the influence of a central force, with a potential energy function V r exp 3r / 2 joule, where r is in meters. It is found to move in a 5r 2 circular orbit of radius r = 2m. (In the following, you may use e3 20). (a) Find its angular momentum L and total energy E. (b) A piece of mass m1 = 0.5 kg breaks off suddenly from the body and begins to fall radially inwards with velocity v = 10 cm s-1. What are the values of angular momentum L2 and total energy E2 of the remaining piece, assuming that the potential energy function remains the same? 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

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