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IIT JAM 2008

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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-2008(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 The product PQ of any two real, symmetric matrices P and Q is (a) symmetric for all P and Q (b) never symmetric (c) symmetric if PQ = QP (d) antisymmtric for all P and Q Q.2 The work done by a force in moving a particle of mass m from any point (x, y) to a neighboring point (x + dx, y + dy) is given by dW = 2xy dx + x2 dy. The work done for a complete cycle around a unit circle is (a) 0 (b) 1 (c) 3 (d) 2 Q.3 EFGH is a thin square plate of uniform density and y side 4a. Four point masses, each of mass m, are placed on the plate as shown in the figure. In the moment of intertia matrix I of the composite system, F m ( a, a,0) m (a, a,0) x (a) only Ixy is zero ( a, a,0) m ( a, a,0) m (b) only Ixz and Iyz are zero (c) all the product of intertia terms are zero G H (d) none of the product of intertia terms is zero Q.4 The chemical potential of an ideal Bose gas at any temperature is (a) necessarily negative (b) either zero or negative (c) neccessarily positive (d) either zero or positive Q.5 If the electrostatic potential at a point (x, y) is given by V = (2x + 4y) volts, the electrostatic energy density at that point (in J/m3) is (a) 5 0 (b) 10 0 (c) 20 0 (d) 1 0 2 x 4 y 2 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 1 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.6 In an intertial frame S, a stationary rod makes an angle with the x-axis. Another interial frame S moves with a velocity v with respect to S along the common x-x axis. As observed from S the angle made by the rod with the x -axis is . Which of the following statement is correct? (a) < (b) > (c) < if v is negative and > if v is positive (d) > if v is negative and < if v is positive Q.7 Consider a doped semiconductor having the electron and the hole mobilities n and p, respectively. Its intrinsic carrier density is ni. The hole concentration p for which the conductivity is minimum at a given temperature is (a) ni n p (b) ni p n (c) ni p n (d) ni n p Q.8 Two coherent plane waves of light of equal amplitude, and each of wavelengths 20 10 8 m , propagating at an angle of radian with respect to each other, fall 1080 almost normally on a screen. The fringe-width (in mm) on the screen is (a) 0.108 (b) 0.216 (c) 1.080 (d) 2.160 Q.9 A circular disc (in the horizontal xy-plane) is spinning about a vertical axis through its center O with a constant angular velocity . As viewed from the reference frame of the disc, a particle is observed to execute uniform circular motion, in the anticlockwise sense, centered at P. when the particle is at the point Q, which of the following figures correctly represents the directions of the Coriolis force Fcor and the centrifugal force Fcfg ? 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 F r , F cfg a b P Q F r P F cfg Q O c F cfg O F r Q F cfg F r d P P Q O O Q.10 The instantaneous position x(t) of a small block performing one-dimensional damped oscillations x(t) = Ae-rt cos( t + a). Here is the angular frequency, the damping coefficient, x t 0 0 and A the initial amplitude and the initial phase. If dx v, the values of A and (with n = 0, 1, 2, .) are dt t 0 (a) A v 2n 1 , 2 2 (b) A v , n (c) A v 2n 1 , 2 (d) A 2v 2n 1 , 2 Q.11 A photon of wavelength is incident on a free electron at rest and is scattered in the backward direction. The functional shift in its wavelength in terms of the Compton wavelength c of the electron is (a) C 2 (b) 2 C 3 (c) 3 C 2 (d) 2 C 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 The logic expression for the output Y of the following circuit is P Y Q R S (a) P Q QR S (b) P Q QR S (c) P Q QR S (d) P Q QR S Q.13 The activity of a radioactive sample is decreased to 75% of the initial value after 30 days. The half-life (in days) of the sample is approximately [You may use ln 3 1.1, ln 4 1.4] (a) 38 (b) 45 (c) 59 (d) 69 Q.14 The ratio of the second-neighbour distance to the nearest-neighbour distance in an fcc lattice is (a) 2 2 (b) 2 (c) 3 (d) 2 Q.15 A thermodynamic system is maintained at constant temperature and pressure. In thermodynamic equilibrium, its (a) Gibbs free energy is minimum (b) enthalpy is maximum (c) Helmholtz free energy is minimum (d) internal energy is zero 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 SUBJECTIVE QUESTIONS Q.16 A thin hollow cylinder of radius and length both P equal to L is closed at the bottom. A disc of radius L/2 is removed from the bottom as shown in the figure. This L object carries a uniform surface-charge density . L Calculate the electrostatic potential at the point P on the axis of the cylinder as shown in the figure. [you may use dx 2 x a 2 L L/2 ln x x 2 a 2 . ] (21) Q.17 A particle of mass 1kg is moving in a central force field given by 3 1 . F1 r 2 r r r (a) Assuming that the particle is moving in a circular orbit with angular momentum 2 J-s, find out the radius of the orbit. (12) (b) At t = 0, an additional force F2 v , where v is the instantaneous velocity of the particle, is switched on. Show that the magnetic of its angular momentum after a time 2 second is J s . e 1 (9) Q.18 An incompressible fluid is enclosed between two horizontal surface located at z = 0 and z = d. The fluid y z motion is two dimensional, and the velocity field w x, z , t z V x, z, t is given by V x , z , t u x , z , t x d where u x, z, t and w x, z, t are periodic functions of 2 the horizontal coordinate x with wavenumber k. L x z (a) If the vertical velocity w x, z , t A t cos kx sin , find the horizontal velocity d u x, z , t using the equation of continuity. What is the vorticity field V ? (15) 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 (b) Find the net fluid flux from a parallelepiped of size L where d as shown in the figure, 2 2 . k (6) Q.19 The wave function n(x) of a particle confined to a one-dimensional box of length L with rigid walls is given by n x 2 n x sin , n = 1, 2, 3, .. L L (a) Determine the energy eigenvalues. Also, determine the eigenvalues and the eigen functions of the momentum operator. (15) (b) Show that the energy eigenfunctions are not the eigenfunctions of the momentum operator. (6) Q.20. A mass and spring system consists of two block of mass m (<M). These blocks are connected with two identical springs with spring constant k as shown in the figure. The system is constrained to move along a straight line on a frictionless horizontal surface. The spring follows Hooke s law. Find the angular frequencies of the independent oscillations (normal modes). k k M M m Now, the masses M and m are interchanged and the new arrangement is shown in the following figure: k k m M m The ration of the new arrangement to that of the old arrangement, when the middle block remains stationary, is 2 . Find the ratio of the frequencies in the two arrangements when the middle block oscillates. (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 6 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.21 A half-wave plate and a quarter-wave plate are placed between a polarizer P1 and an analyzer P2 . All of these are parallel to each other and perpendicular to the direction of propagation of unpolarized incident light (see the figure). The optic-axis of the half-wave plate makes an angle of 30 0 with respect to the pass-axis of P1 and that of the quarterwave plate is parallel to the pass-axis of P1 . x z y P1 P2 Half-wave Quarter-wave plate plate (a) Determine the state of polarization for the light after passing through (i) the half-wave plate and (ii) the quarter-wave plate. (15) (b) What should be the orientation of the pass-axis of P2 with respect to that of P1 such that the intensity of the light emerging from P2 is maximum? (6) Q.22 Consider a system of N non-interacting distinguishable spin- particles, each of magnetic moment . The system is at an equilibrium temperature T in a magnetic field B such that n particles have their magnetic moments aligned parallel to B . (a) Find the energy E and the entropy S of the system. (b) Using the relationship between E and S, find T. Hence determine the ratio (9) n in terms N of , B and T. [Use ln N! = N ln N N] (12) Q.23 (a) An ideal gas, kept in contact with a heat reservoir, undergoes a quasistatic process in which its pressure gets doubled. Obtain the Maxwell relation from the differential form dF = - S dT P dV and evaluate the expression for the change in entropy of n moles of the gas. (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 (b) Using S = S(T, V), derive a general expression for ( U/ V)T, where U(S,V) is the internal energy. Evaluate it for the ideal gas as considered in part (a). Justify that the outcome is consistent with the expression for the average energy known from the kinetic theory. (12) Q.24 Consider an ideal Fermi gas consisting of N non-relativistic spin- 1 particles 2 confined to a length L in one dimension at 0 K. (a) Find an expression for the density of states and hence calculate the Fermi energy of the gas. (15) (b) Find the mean energy per particle in terms of the Fermi energy. (6) Q.25 A square loop of side L and mass M is made of a wire of cross-sectional area A and in the horizontal xy-plane, resistance R. The loop, moving with a constant velocity vo i . enters a region 0 x 2L having constant magnetic field Bk x b a z y vo i d L c x 0 B x 2L (a) Find an expression for the x-component of the force F acting on the loop in terms of its velocity v t , B, L and R. (12) (b) Find the speed of the loop as its side ad exits the field region at x = 2L and sketch its variation with x. (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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