GATE question papers: Physics 2009 (PH)

Some Useful Symbols

Speed of light in free space: c

Boltzmann constant: k_B

Electron charge: e

Planck's constant: h

Rest mass of electron: m_e

Rest mass of proton: m_p

Rest mass of neutron: m_n

Permeability of free space: m_o

Permittivity of free space: e_o

Q. 1 - Q. 20 carry one mark each

1. The value of the contour integral,, for a circle C of radius r with center at the origin is

(A) 2pr (B) (C) pr² (D) r

2. An electrostatic field exists in a given region R. Choose the WRONG statement.

(A) Circulation of is zero

(B) can always be expressed as the gradient of a scalar field

(D) The work done in a closed path lying entirely in r is zero

3. The LaGrange of a free particle in spherical polar co-ordinates is given by L =. The quantity that is conserved is

(A) (B) (C) (D)

4. A conducting loop L of surface area S is moving with a velocity in a magnetic field , B₀ is a positive constant of suitable dimensions. The emf induced, V_emf, in the loop is given by

(A) (B)

5. The eigen values of the matrix A = are

(A) Real and distinct (B) complex and distinct

6. s_i(i = 1, 2, 3) represent the Pauia spin matrices. Which one of the following is NOT true?

(A) s_is_j+s_js_i= 2d_ij (B) Tr(s_i)= 0

7. Which one of the functions given below represents the bound state eigenfunction of the operator - in the region, 0 £ x< ¥, with the eigenvalue-4?

(A) A₀e^2x (B) A₀cosh 2x (C) A₀e^-^2x (D) A₀sinh 2x

8. Pick the WRONG statement.

(A) The nuclear force is independent of electric charge

(B) The Yukawa potential is proportional to r^-¹ exp, where r is the separation between two nucleons

(D) The nucleons interact among each other by the exchange of mesons

9. If p and q are position and momentum variables, which one of the following is NOT a canonical transformation?

(A) Q = aq and P =, for a ¹ 0

(B) Q = aq + bp and P = bq +ap for a,b real and a² - b² = 1

(D) Q = P and P = - q

10. The common Mode Rejection Ratio (CMRR) of a differential amplifier using an operational amplifier 100 dB. The output voltage for a differential input of 200 mV is 2v. The common mode gain is

(A) 10 (B) 0.1 (C) 30 dB (D) 10 dB

11. In an insulating solid which one of the following physical phenomena is a consequence of Pauli's exclusion principle?

(A) Ionic conductivity (B) Ferromagnetism

12. Which one of the following curves gives the solution of the differential equation , where k₁,k₂ and k₃ are positive constants with initial conditions x = 0 at t = 0 ?

13. Identify which one is a first order phase transition?

(A) A liquid to gas transition at its critical temperature.

(B) A liquid to gas transition close to its triple point.

(D) A metal to superconductor transition in the absence of a magnetic field.

14. Group I lists some physical phenomena while Group II gives some physical parameters. Match the phenomena with the corresponding parameter.

Group I Group II

P. Doppler Broadening 1. Moment of inertia

Q. Natural Broadening 2. Refractive index

R. Rotational spectrum 3. Lifetime of the energy level

S. Total internal reflection 4. Pressure

(A) P - 4, Q - 3, R - 1, S - 2 (B) P - 3, Q - 2, R - 1, S - 4

15. The separation between the first Stokes and corresponding anti-Stokes lines of the rotational Raman spectrum in terms of the rotational constant, B is

(A) 2B (B) 4B (C) 6B (D) 12B

16. A superconducting ring is cooled in the presence of a magnetic field below its critical temperature (T_C). The total magnetic flux that passes through the ring is

(A) zero (B) (C) (D)

17. In a cubic crystal, atoms of mass M₁ lie on one set of planes and atoms of mass M₂ lie on planes interleaved between those of the first et. If C the force constant between nearest neighbour planes, the frequency of lattice vibrations for the optical phonon branch with wavevector k = 0 is

(A) (B)

18. In the quark model which one of the following represents a proton?

(A) udd (B) uud (C) (D)

19.

The circuit shown above

(A) is a common-emitter amplifier

(B) uses a pnp transistor

(D) has a voltage gain less than one

20. Consider a nucleus with N neutrons and Z protons. If m_p, m_n and BE represent the mass of the proton, the mass of the neutron and the binding energy of the nucleus respectively and c is the velocity of light in free space, the mass of the nucleus is given by

(A) Nm_n + Zm_p (B) Nm_p + Zm_n

Q.21 to Q.60 carry two marks each.

21. The magnetic field (in A m^-1) inside a long solid cylindrical conductor of radius a = 0.1 m is, . What is the total current (in A) in the conductor?

(A) (B) (C) (D)

22. Which one of the following current densities, , can generate the magnetic vector potential

(A) (B) (C) (D)

23. The value of the integral , where the contour C is the circle |z| = 3/2 is

(A) 2pie (B) pie (C) -2pie (D) -pie

24. In a non-conducting medium characterized by e = e₀, m = m₀ and conductivity s = 0, the electric field (in V m^-1) is given by . The magnetic field, (in A^-1), is given by

(A) (B)

25. A cylindrical rod of length L and radius r, made of an inhomogeneous dielectric, is placed with its axis along the z direction with one end at the origin as shown below.

If the rod carries a polarization, , the volume bound charge inside the dielectric is

(A) Zero (B) 10pr²L (C) -5pr²L (D) -5pr²L²

26. Let T_ij = , where e_ijk is the Levi-Civita density, defined to be zero if two of the indices coincide and +1 and -1 depending on whether ijk is even or odd permutaion of 1, 2, 3. Then b₃ is equal to

(A) 2a₃ (B) -2a₃ (C) a₃ (D) -a₃

27. The dependence of the magnetic susceptibility (c) of a material with temperature (T) can be represented by cµ , where q is the Curie-Weiss temperature. The plot of magnetic susceptibility versus temperature is sketched in the figure, as curves P, Q and R with curve Q having q = 0. Which one of the following statements is correct?

(A) Curve R represents a paramagnet and Q a ferromagnet

(B) Curve Q represents a ferromagnet and P an antiferromagnet

(D) Curve R represents an antiferromagnet and Q a ferromagnet

28. The dielectric constant of a material at optical frequencies is mainly due to

(A) ionic polarizability (B) electronic polarizability

29. An electron of wavevector , velocity and effective mass m_e is removed from a filled energy band. The resulting hole has wavevector velocity , and effective mass m_h. Which one of the following staatement sis correct?

(A) m_h = -m_e (B) m_h = m_e

30. In a diatomic molecule, the internuclear separation of the ground and first excited electronic state are the same as shown in the figure. If the molecule is initially in the lowest vibrational state of the ground state, then the absorption spectrum will appear as

31. Five energy levels of a system including the ground state are shown below. Their lifetimes and the allowed electric dipole transitions are also marked.

Which one of the following transitions is the most suitable for a continuous wave (CW) laser?

(A) 1 ® 0 (B) 2 ® 0 (C) 4 ® 2 (D) 4 ® 3

32. Assuming the mean life time of a muon (in its ret frame) to be 2 ´ 10^-⁶, its life time in the laboratory frame, when it is moving with a velocity 0.95c is

(A) 6.4´10^-6s (B) 0.62´10^-6s (C) 2.16´10^-6s (D) 0.19´10^-6s

33. Cesium has a nuclear spin of 7/2. The hyperfine spectrum of the D lines of the cesium atom will consist of

(A) 10 lines (B) 4 lines (C) 6 lines (D) 14 lines

34. The probability that an energy level e at a temperature T is unoccupied by a fermion of chemical potential m is given by

(A) (B) (C) (D)

35. Consider the following expression for the mass of a nucleus with Z protons and A nucleons:

M(A,Z) =(f(A) + yZ + zZ²). Here F(A) is a function of A,

y = -4a_A,

z = a_cA^-1/3 +4a_AA^-1,

a_A and a_care constants of suitable dimensions. For a fixed A, the expression of Z for the most stable nucleus is

(A) Z = (B) Z =

36. The de Broglie wavelength of mass m with average momentum p at a temperature T in three dimensions is given by

(A) l = (B) l = c (C) l = (D) l =

37.

Assuming an ideal voltage source, Thevenin's resistance and Thevenin's voltage respectively for the above circuit are

(A) 15 m and 7.5 V (B) 20 m and 5 V (C) 10 W and 10 V (D) 30 W and 15 V

38. Let |nñ and |pñ denote the isospin states with I = 1/2, I₃ = 1/2 and I 1/2, I₃ = - 1/2 of a nucleon respectively. Which one of the following two-nucleon states has I = 0, I₃ = 0?

(A) (B)

39. An amplifier of gain 1000 is made into a feedback amplifier by feeding 9.9% of its output voltage in series with the input opposing. If f₁ = 20 Hz and f_H = 200 kHz for the amplifier without feedback, then due to the feedback

(A) the gain decreases by 10 times

(B) the output resistance increases by 10 times

(D) the input resistance decreases by 100 times

40.

Pick the correct statement based on the above circuit.

(A) The maximum Zener current, I_Z(max), when R_L = 10 kW is 15 mA

(B) The minimum Zener current, I_Z(min), when R_L = 10 kW is 5 mA

(D) The power dissipated across the Zener when R_L = 10 kW and V_in = 20 V is 100 mW

41. The disintegration energy is defined to be the difference in the rest energy between the initial and final states. Consider the following process:

The emitted a particle has a kinetic energy 5.17 MeV. The value of the disintegration energy is

(A) 5.26 MeV (B) 5.17 MeV (C) 5.08 MeV (D) 2.59 MeV

42. A classical particle is moving in an external potential field V(x, y, z) which is invariant under the following infinitesimal transformations

x ® x¢ = x + dx,

y ® y¢ = y dy,

where R_z is the matrix corresponding to rotation about the z axis. The conserved quantities are (the symbols have their usual meaning)

(A) P_x, P_z, L_z (B) P_x, P_y, L_z, E (C) P_y, L_z, E (D) P_y, P_z, L_x, E

43. The spin function of a free particle, in the basis in which S_z is diagonal, can be written as with eigenvalues and -, respectively. In the given basis, the normalized eigenfunction of S_y with eigenvalue -

(A) (B) (C) (D)

44. and represent two physical characteristics of a quantum system. If is Hermitian, then for the produt to be Hermitian, it is sufficient that

(A) is Hermitian (B) is anti-Hermitian

45. Consider the set of vectors in three-dimensional ral vector space

R³, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statements is true?

(A) S is not a linearly independent set. (B) S is a basis for R³

46. For a Fermi gas of n particles in three dimensions at t = 0 K, the Fermi energy, E_F is proportional to

(A) N^2/3 (B) N^3/2(C) N³(D) N²

47. The Lagrangian of a diatomic molecule is given by L = , where m is the mass of each of the atoms and x₁ and x₂ are the displacements of atoms measured from the equilibrium position and k > 0. The normal frequencies are

(A) (B) (C) (D)

(A) (B)

49. The Lagrangian of a particle of mass m moving in one dimension is L = exp (at) , where a and k are positive constants. The equation of motion of the particle is

(A) (B) (C) (D)

50. The monochromatic waves having frequencies w and w + Dw (Dw << w) and corresponding wavelengths l and l - Dl (Dl << l) of same polarization, traveling along x-axis are superimposed on each other. The phase velocity and group velocity of the resultant wave are respectively given by

(A) (B) (C) (D)

Common Data Questions

Common Data for Questions 51 and 52:

Consider a two level quantum system with energies e₁ = 0 and e₂ = e

51. The Helmholtz free energy of the system is given by

(A) -k_BT ln (B) k_BT ln

52. The specific heat of the system is given by

(A) (B)

Common Data for Questions 53 and 54:

A free particle of mass m moves along the x direction. At t = 0, the normalized wave function of the particle is given by y (x, 0) = , where a is a real constant