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IIT JAM 2011

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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-2011(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. B Q.1 The line integral F dl , where F A path as shown in the figure below is x y x x2 y2 x2 y2 , along the semi-circular y y x 1,0 (a) -2 (b) 0 1,0 (c) 2 (d) 4 Q.2 Six simple harmonic oscillations each of same frequency and equal amplitude are superposed. The phase difference between any two consecutive oscillations i.e., n n 1 is constant, where n is the phase of the n th oscillation. If the resultant amplitude of the superposition is zero, what is the phase difference ? (a) 6 (b) 3 (c) 2 Q.3 A particle of mass m is moving in a potential V x (d) 2 1 a 2 2 m 0 x 2 2mx 2 where 0 and a are positive constants. The angular frequency of small oscillations for the simple harmonic motion of the particle about a stable minimum of the potential V(x) is (a) 2 0 (b) 2 0 (c) 4 0 (d) 4 2 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 1 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.4 Intensity of three different light beams after passing through an analyzer is found to vary as shown in the following graphs. Identify the option giving the correct states of polarization of the incident beams from graphs. 1 1 Graph 2 Graph 1 0.5 0.5 0 0 0 1 2 3 4 5 0 6 1 2 3 4 5 6 Analyzer Orientation (Radians) Analyzer Orientation (Radians) 1 Graph 3 0.5 0 0 1 2 3 4 5 6 Analyzer Orientation (Radians) (a) Graph1: Linear Polarization Graph2: Circular Polarization, Graph3: Elliptic Polarization (b) Graph 1: Circular Polarization, Graph 2: Linear Polarization, Graph 3: Elliptic Polarization (c) Graph 1: Unpolarized Graph 2: Circular Polarization, Graph 3: Linear Polarization (d) Graph 1: Unpolarized Graph 2: Elliptic Polarization, Graph 3: Circular Polarization Q.5 Which of the following circuit does not satisfy the Boolean expression AB A B F (a) F 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 (b) F (c) F (d) F Q6. Light described by the equation E = (90 V/m) [sin (6.28 x 1015 s-1) t + sin (12.56 x 10 15 s-1) t] is incident on a metal surface. The work function of the metal is 2.0 eV. Maximum kinetic energy of the photoelectrons will be (a) 2.14 eV (b) 4.28 eV (c) 6.28 eV (d) 12.56 eV Q.7 A gas of molecular mass m is at temperature T. If the gas obeys Maxwell-Boltzmann velocity distribution, the average speed of molecules is given by (a) k BT m (b) 2k B T m (c) 2k B T m (d) 8k B T m Q.8 Consider free expansion of one mole of an ideal gas in an adiabatic container from volume V1 to V2. The entropy change of the gas, calculated by considering a reversible process between the original state (V1, T) to the final state (V2, T) where T is the temperature of the system is denoted by S 1. The corresponding change in the entropy of the surrounding is S 2. Which of the following combinations is correct? 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 (a) S 1 = R ln (V1/V2), S2 = -R ln(V1/V2) (b) S 1 = - R ln (V1/V2), S 2 = R ln(V1/V2) (c) S 1 = R ln (V2/V1), S2 = 0 (d) S1 = - R ln (V2/V1), S2 = 0 Q.9 Equipotential surface corresponding to a particular charge distribution are given by 4x2 + (y -2)2 + z2 = Vi, where the values of Vi are constants. The electric field E at the origin is (a) E 0 (b) E 2 x (c) E 4 y (d) E 4 y Q.10 The wave function of a quantum mechanical particle is given by 3 4 x 1 x 2 x 5 5 where 1(x) and 2(x) are eigenfunctions with corresponding energy eigenvalues -1 eV and -2 eV, respectively. The energy of the particle in the state is (a) 41 eV 25 (b) 11 eV 5 (c) 36 eV 25 (d) 7 eV 5 Q.11 A rain drop falling vertically under gravity gathers moisture from the atmosphere at a rate given by dm kt 2 , where m is the instantaneous mass, t is time and k is a constant. dt The equation of motion of the rain drop is m dv dm v mg dt dt If the drop starts falling at t = 0, with zero initial velocity and initial mass m0 (given m0 = 2gm, k = 12 gm/s3 and g = 1000 cm/s2), the velocity (v) of the drop after one second is (a) 250 cm/s (b) 500 cm/s (c) 750 cm/s (d) 1000 cm/s such that P is skew (anti) and Q is Hermitian and Q Q.12 Given two (nxn) matrices P is necessarily a and Q Hermitian. Which one of the following combinations of P Hermitian matrix? Q (a) P Q (b) iP iQ (c) P Q (d) P 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.13 An X-ray diffraction (XRD) experiment is carried out on a crystalline solid having FCC structure at room temperature. The solid undergoes a phase transformation on cooling to -20oC and shows orthorhombic structure with small decreas in its unit cell lengths as compared to the FCC unit cell lengths. As a result the (311) line of the XRD pattern corresponding to the FCC system (a) will split into a doublet (b) will split into a triplet (c) will remain unchanged (d) will split into two separate doublets Q.14 A closed Gaussian surface consisting of a hemisphere and a circular disc of radius R, is placed in a uniform electric field E , as shown in the figure. The circular disc makes an angle = 30 o with the vertical. The flux of the electric field vector coming out of the curved surface of the hemisphere is (a) (b) 1 2 R E 2 E 3 R 2 E 2 (c) R2E (d) 2 R2E Q.15 In an experiment the resistance of a rectangular slab of a semiconductor is measured as a function of temperature. The semiconductor shows a resistance of 300 at 200K and 2 at 250 K. Its energy band gap is [Given: ln(15) = 2.708, ln(10) = 2.303] (a) 0.138 eV (b) 0.431 eV (c) 0.690 eV (d) 0.862 eV 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 3y 2 y Q.16. Consider a vector A 4 yx 2 x y Q(1,1) P(0,1) O(0,0) (a) Calculate the line integral R(1,0) x A d l from point P O along the path P Q R O as shown in the figure. (9) (b) Using Stokes s theorem appropriately, calculate A d l for the same path P Q R O. (12) J Q.17 An infinitely long solid cylindrical conductor of radius r1 , carries a uniform volume current density J . It is uniformly surrounded by another coaxial cylinder of a linear magnetic medium with permeability , up to radius r1 r2 as shown in the figure. (a) Determine the magnetic field H in the region r r1 and Air r2 magnetic induction B in the regions, r1 r r2 and r r1 , where r is the radial distance from the axis of the cylinder. (12) (b) Sketch the variation of regions. H with r in all the three (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.18 (a) Consider heat conduction in a medium. Let T(x, y, z, t) denoted the temperature of the medium at time t and position (x, y, z). Consider a volume V enclosed by a surface S inside the medium. The decrease in heat energy per unit volume per unit time is a a T t and outward flux of heat per unit area of the surface per unit time is b T , where a and b are material dependent constants. If there is no generation or loss of heat, show that T satisfies the equation T b k 2 T , with k . t a (9) (b) Now consider a thin annular shaped material enclosed between two concentric circles of radii r0 and 2r0 as shown in the figure. The temperature is 2T0 at r r0 and T0 at r 2r0 . Assuming steady state condition, find T as a function of radial distance r from T0 the centre O, for r0 r 2r0 . 2r0 O r0 (12) 2T0 Q.19 An ideal gas reversible engine operates in a closed cycle. The P-V diagram is shown below. ( P2 ,V1 ) P Adiabatic A ( P1 , V1 ) C(P 1 ,V2 ) V (a) Find out the efficiency of the reversible engine assuming both specific heats, CP and CV as constants. (12) (b) Identify the thermodynamic processes and draw the corresponding T-S diagram schematically. (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.20 (a) A solid having a simple cubic structure at room temperature with lattice parameter a and one valence electron per atom, is assumed to show free electron behavior. Calculate the magnitude of the Fermi wave vector and the corresponding equivalent temperature. (12) (b) Find the ratio of the magnitude of the Fermi wave vector to the radius of the largest sphere that can be inscribed within the first Brillouin zone of the solid. (9) Q.21 For the given circuit using an operational amplifier, the input is a sinusoidal signal of amplitude Vin = 1mV (peak-to-peak), 10k Vin 10k 1 F 10k 15V Vo 10k 10k 100 F (a) What is the lower cut-off frequency at which the gain is down by 3 dB as compare to the gain at midband? If the bandwidth of the amplifier is 1 MHz for unity gain, what will be the bandwidth of the given circuit? (12) (b) What is the output voltage (V0) at 15 kHz? (9) Q.22 A particle of mass m and angular momentum l is moving under the action of a central force f(r) along a circular path of radius a as shown in the figure. The force centre y O lies on the orbit. (a) Given the orbit equation in a central field motion. d 2u m 1 u 2 2 f , where u . 2 r d l u O Determine the form of force in terms of l , m, a and r . (9) r C (a,0) x (b) Calculate the total energy of the particle assuming that the potential energy V ( r ) 0 as r . (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 8 fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES Q.23 A particle of mass m moves in a V(x ) potential given by V(x) = for x<0 Vo =0 for 0<x<L = V0 for x>L I V 0 x 0 II x L x (a) Write down the general solutions for wave functions in regions I and II, if the energy of the particle E<V0. Using appropriate boundary conditions, find the equation that relates E to V0, m and L. (12) (b) Now, set V0 = 0 and assume that a beam of particles is incident on the infinite step potential (from x>0) with energy E (>0). Using the general solution for the wave function, calculate the reflection coefficient. (9) Q.24. A diffraction grating having N slits, each of width b and period d, is illuminated normally by a monochromatic plane wave of wavelength . (a) Obtain an expression for the highest diffraction order that can be observed. What is the phase difference between waves from first and Nth slit in the highest diffraction order? (9) (b) If alternate slits are covered with a retarder that retards the wave by , obtain an expression for intensity distribution of the Fraunhofer diffraction pattern? (12) Q.25 Unpolarized light is incident on an air-dielectric interface. The interface is the x-y plane, and the plane of incidence is y-z plane. The electric field of the reflected light is ik 2 exp given by E E o x 3 y z i t , where k is the propagation constant in air and is the angular frequency of the light. Assume magnetic permeability = 0 . (a) Determine the dielectric constant of the second medium. (b) Determine the direction of the Poynting vector in the dielectric medium. (12) (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 9

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