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IIT JAM 2010

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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-2010(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 A matrix is given by M 1 i 2 1 1 . The eigenvalues of the M are i (a) real and positive (b) purely imaginary with modulus 1 (c) complex with modulus 1 (d) real and negative Q.2 The value of at which the first-order peak in X-ray ( =1.53 ) diffraction corresponding to (111) plane of a simple cubic structure with the lattice constant, = 2.65 , is approximately (a) 15o (b) 30o (c) 45o (d) 60o Q.3 The magnetic field associated with the electric field vector E E0 sin kz t j is given by E (a) B 0 sin kz t i c E (b) B 0 sin kz t i c E (c) B 0 sin kz t j c E (d) B 0 sin kz t k c Q.4 Consider the following truth table A The logic expression for F is B C F 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 (a) AB + BC + CA (b) A B AC B C (c) CAB AB (d) C A B AB 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.5 The following are the plots of the temperature dependence of the magnetic susceptibility for three different samples. a (T ) (T ) 0 T c b (T ) 0 0 T T The plots a, b and c correspond to (a) ferromagnet, paramagnet and diamagnet, respectively (b) paramagnet, diamagnet and ferromagnet, respectively (c) ferromagnet, diamagnet and paramagnet, respectively (d) diamagnet, paramagnet and ferromagnet, respectively Q.6 The equation of a surface of revolution is z 3 2 3 2 x y 2 2 2 is The unit normal to the surface at the point A , 0 , 1 3 (a) 3 2 i k 5 10 (b) 3 2 i k 5 10 (c) 3 2 i k 5 5 (d) 3 2 i k 10 10 Q.7 A gas of molecules each having mass m is in thermal equilibrium at a temperature T. Let vx, vy, vz be the Cartesian components of velocity, v , of a molecules. The mean value of v x v y v z is 2 (a) 1 2 2 (c) 2 2 kmT k BT m B (b) 1 2 2 (d) 2 2 kmT B kBT m 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 circular platform is rotating with a uniform angular speed counterclockwise about an axis passing through its centre and perpendicular to its plane as shown in the figure. A person of mass m walks radially inward with a uniform speed v on the platform. The magnitude and the direction of the Coriolis force (with respect to the direction along which the person walks) is (a) 2m v towards his left (b) 2m v towards his front (c) 2m v towards his right (d) 2m v towards his back Q.9 A quarter-wave plate is placed in between a polarizer and a photo-director. When the optic axis of the quarter-wave plate is kept initially parallel to the pass axis of the polarizer and perpendicular to the direction of light propagation. The intensity of light passing through the quarter-wave plate is measured to be I0 (see figure). If the quarter wave plate is now rotated by 45 0 about an axis parallel to the light propagation, what would be the intensity of the emergent light measured by the photo-director? Direction of rotation of quarter wave plate Polaroid (a) Io 2 Quarter wave plate (b) I0 2 Photo - detector I (c) 0 2 2 (d) I 0 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.10 A particle of mass m, moving with a velocity v v0 i j , collides elastically with another particle of mass 2m which is at rest initially. Here, v0 is a constant. Which of the following statements is correct? i j (a) The direction along which the centre of mass moves before collision is 2 (b) The speed of the particle of mass m before collision in the center of mass frame is 2v 0 . (c) After collision the speed of the particle with mass 2m in the centre of mass frame is 2 v0 . 3 (d) The speed of the particle of mass 2m before collision in the center of mass frame is 2v 0 . Q.11 A trapped air bubble of volume V0 is released from a depth h measured from the water surface in a large water tank. The volume of the bubble grows to 2 V0 as it reaches just below the surface. The temperature of the water and the pressure above the surface of water (105 N/m2) remain constant throughout the process. If the density of water is 1000 kg/ m3 and the acceleration due to gravity is 10m/ s2, then the depth h is (a) 1 m (b) 10 m (c) 50 m (d) 100 m Q.12 A particle of mass m is confined in a two-dimensional infinite square well potential of side a. The eigen-energy of the particle in a given state is E (a) 4-fold degenerate 25 2 2 . The state is ma 2 (b) 3-fold degenerate (c) 2-fold degenerate (d) Non-degenerate Q.13 For a wave in a medium the angular frequency and the wave vector k are related 2 by, 2 0 c 2 k 2 where 0 and c are constants. The product of group and phase velocities, i.e v g .v p is (a) 0.25 c2 (b) 0.4 c2 (c) 0.5 c2 (d) c2 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 Three identical non-interacting particles, each of spin and mass m, are moving in a one-dimensional infinite potential well given by, for 0 x a for x 0 and x a 0 V x The energy of the lowest energy state of the system is 2 2 (a) ma 2 2 2 2 (b) ma 2 3 2 2 (c) ma 2 5 2 2 (d) 2ma 2 Q.15. Assume that z 0 plane is the interface between two linear and homogenous dielectrics (see figure). The relative permittivities are r 5 for z > 0 and r 4 for k V m . If there are no z < 0. The electric field in the region z > 0 is E 1 3i 5 j 4k free charges on the interface, the electric field in the region z < 0 is given by z r 5 z 0 r 4 3 5 (a) E 2 i j k k V m 4 4 kV m 5 (c) E 2 3i j 5k kV m 5 (b) E 2 3i j k kV m 5 (d) E 2 3i j 5k 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 particle of mass 2/3 kg is subjected to a potential energy function V(x) = (3x2 2x3) J, where x 0 and expressed in meters. (a) Sketch V(x) with respect to x in the range 0 to 2 m. Mark the positions of all the maxima and minima. What is the maximum value of the potential energy? (12) (b) Supposing the particle is released at x = 4/3 m, find its velocity when it reaches x = 3/2 m. (9) y Q.17 A vector field is given by, a x j yi F r 2 x j yi ar0 x2 y2 for x 2 y 2 r02 region - I 2r0 for x 2 y 2 r02 region - II r0 x Here a and r0 are two constants. (a) Find the curl of this field in both the regions. (b) Find the line integral F dl (9) along the closed semicircular path of radius 2r0 as shown in the figure below. (12) Q.18 A solid cylinder C of mass 10 kg rolls without P slipping in an inclined plane which offers friction. The angle of the inclined plane with the horizontal is 30o as shown in the figure below. A massless inextensible string is wrapped around the cylinder and it passes over a frictionless pulley, P. The other end of the string holds a C block B of mass 2 kg as shown in the figure. Take g = 10 ms2. B 30 o (a) Draw the free body diagram of the cylinder and the block. If the block B moves down by 0.01 m, how much distance does C move along the incline? (9) (b) If the block B moves with an acceleration of 0.05 g upward, find the magnitude of the frictional force. (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 (a) A photon of initial momentum p 0 collides with an electron of rest mass m0 moving with relativistic momentum P and energy E. The change in wavelength of the photon after scattering by an angle is given by 2c 0 p0 P 2 sin , where c is the speed of E cP 2 light and 0 is the wavelength of the incident photon before scattering. What will be the value of when the electron is moving in a direction opposite to that of the incident photon with momentum P and energy E? Show that the value of becomes independent of the wavelength of the incident photon when the electron is at rest before collision. (12) (b) In a Compton experiment, the ultraviolet light of wavelength 2000 is scattered from an electron at rest. What should be the minimum resolving power of an optical instrument to measure the Compton shift, if the observation is made at 90 o with respect to the direction of the incident light? (9) Q.20 A conducting spherical shell of radius R1 carries a total charge Q. A spherical layer of a linear, homogeneous and isotropic dielectric of dielectric constant K and outer radius R2 (>R1) covers the shell as shown in figure. R2 R1 (a) Find the electric field and the polarization vector P inside the dielectric. From this P , calculate the surface bound charge density, b, on the outer surface of the the dielectric layer and the volume bound charge density b , inside the dielectric. (12) (b) Calculate the electrostatic energy stored in the region R1 r R2. (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 7 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.21 Two spaceships A and B of proper length 50 m each are moving with relativistic speeds 0.8c and 0.6c, respectively, in the same direction with respect to a stationary observer S. Here c is the speed of light. At t = 0, the spaceship A is just behind the spaceship B as shown below. A B S (a) Find the time taken by the spaceship A to completely overtake the spaceship B (that is the back of A is in line with the front of B) as seen by the observer S. (9) (b) Find the time taken by the spaceship A to completely overtake the spaceship B as seen by an observer in the spaceship B. (12) Q.22 A free particles of mass m with energy V0/2 is incident from left on a step potential of height V0 as shown in the figure below. V (x) V0 region - I 0 x region - II Writing down the time independent Schr dinger equation in both the regions, obtain the corresponding general solutions. Apply the boundary conditions to find the wave functions in both the regions. Determine the reflection coefficient R. What is the transmission coefficient T? (21) Q.23 For the transistor shown in the figure, the dc current 5V gain dc = 50 and VBE = 0.7 V. The switch S is initially 500 open. 10k (a) Calculate the voltage at point A. If the switch S is now closed, what would be the voltage at point A? (12) S A (b) Draw the dc load line and find the Q-point of the circuit 10 k B E 260 with the switch S remaining closed. (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 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.24 Consider a capacitor placed in free space, consisting of two concentric circular parallel plates of radii r. The separation z between the plates oscillates with a constant frequency , i.e. z t z 0 z1 cos t . Here z 0 and z1 ( z 0 ) are constants. The separation z t ( r ) is varied in such a way that the voltage V0 across the capacitor remains constant. (a) Calculate the displacement current density and the displacement current between the plates through a concentric circle of radius r . 2 (12) (b) Calculate the magnetic field vector ( H ) between the plates at a distance r 2 axis of the capacitor. from the (9) Q.25 One mole of an ideal monatomic gas in an initial state a with pressure, pi and volume Vi is to be taken to a final state d with p f Vi B through the path a b c d as shown in the figure below for a particular value of B (> 2). Here a b and c d are adiabatic paths while b c is an isotherm with temperature T0. States b and c correspond to p1 ,V1 and ( p 2 ,V2 , respectively. Find the ratio V2 and the total work done by the gas in terms of p i , Vi , T0 and B. V1 (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 9

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