## CSIR NET PHYSICS (JUNE 2012)

Previous Year Question Paper with Solution.

1. A vector perpendicular to any vector that lies on the plane defined by x + y + z = 5, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Any vector perpendicular to any vector lies on x + y + z = 5 is

2. The eigenvalues of the matrix, are

(a) (1, 4, 9)

(b) (0, 7, 7)

(c) (0, 1, 13)

(d) (0, 0, 14)

Ans. (d)

Sol. Eigen values of a matrix have rows or columns that are scalar multiple of a particular row or column respectively are Tr(A), 0, 0,.......

Eigen values of the given matrix = 14, 0, 0

3. The first few terms in the Laurent series for is the region 1 < |z| < 2, and around z = 1 is

(a)

(b)

(c)

(d) 2(z – 1) + 5(z – 1)^{2} + 7(z – 1)^{3} + .....

Ans. (b)

Sol.

4. The radioactive decay of a certain material satisfies Poisson statistics with a mean rate of per second, what should be the minimum duration of counting (in seconds) so that the relative error is less than 1%?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Accuracy = 10^{–2}; Mean rate = s^{–1}; Minimum duration of counting

5. Let be the real part of an analytic function f(z) of the complex variable z = x + iy. The imaginary part of f(z) is

(a) y + xy

(b) xy

(c) y

(d) y^{2} – x^{2}

Ans. (a)

Sol.

Compare equations (1) and (2), taking h(y) = y; and

6. Let y(x) be a continuous real function in the range 0 and 2, satisfying the inhomogeneous differential equation:

. The value of dy/dx at the point x = /2

(a) is continuous

(b) has a discontinuity of 3

(c) has a discontinuity of 1/3

(d) has a discontinuity of 1

Ans. (d)

Sol.

7. A ball is picked at random from one of two boxes that contain 2 black and 3 white and 3 black and 4 white balls respectively. What is the probability that it is white?

(a) 34/70

(b) 41/70

(c) 36/70

(d) 29/70

Ans. (b)

Sol.

Probability of picking up white ball

8. The bob of a simple pendulum, which undergoes small oscillations, is immersed in water. Which of the following figures best represents the phase space diagram for the pendulum?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. It is a case of damped oscillator where the amplitude 'x' and momentum decreases towards zero. It is an inward spiral.

9. Two events, separated by a (spatial) distance 9 × 10^{9} m, are simultaneous in one inertial frame. The time interval between these two events in a frame moving with a constant speed 0.8c (where the speed of light c = 3 × 10^{8} m/s) is:

(a) 60 s

(b) 40 s

(c) 20 s

(d) 0 s

Ans. (b)

Sol.

10. If the Lagrangian of a particle moving in one dimension is given by , the Hamiltonian is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

We know that

Eq. (3), (1) put in Eq. (2)

11. A horizontal circular platform rotates with a constant angular velocity directed vertically upwards. A person seated at the centre shoots a bullet of mass 'm' horizontally with speed 'v'. The acceleration of the bullet, in the reference frame of the shooter, is

(a) 2v to his right

(b) 2v to his left

(c) v to his right

(d) v to his left

Ans. (a)

Sol.

12. The magnetic field corresponding to the vector potential, where is a constant vector, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

13. An electromagnetic wave is incident on a water-air interface. The phase of the perpendicular component of the electric field, , of the reflected wave into the water is found to remain the same for all angles of incidence. The phase of the magnetic field H.

(a) does not change

(b) changes by 3/2

(c) changes by /2

(d) changes by

Ans. (d)

Sol. Phase changes of magnetic field on reflection from a dielectric interface

n_{1} > n_{2} (water to air); . Phase changes by .

14. The magnetic field at a distance R from a long straight wire carrying a steady current I is proportional to

(a) IR

(b) I/R^{2}

(c) I^{2}/R^{2}

(d) I/R

Ans. (d)

Sol. Amphere's circuital law

15. The component along an arbitrary direction , with direction cosines (n_{x}, n_{y}, n_{z}), of the spin of a spin-½ particle is measured. The result is:

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol.

16. A particle of mass m is in a cubic box of size a. The potential inside the box is zero and infinte outside. If the particle is in an eigenstate of energy , its wave function is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The energy of the particle .

The energy of the particle in cubic box of size a is

[where, n_{x}, n_{y}, and n_{z} = 1, 2, 3, 4....]

For energy , the possible values of (n_{x}, n_{y}, n_{z}) = (1, 2, 3).

The wave function the particle

On substituting the values of

17. Let denote the eigenfunctions of a Hamiltonian for a spherically symmetric potential V(r). The wavefunction is an eigenfunction only of

(a) H, L^{2} and L_{z}

(b) H and L_{z}

(c) H and L^{2}

(d) L^{2} and L_{z}

Ans. (c)

Sol. The wavefunction

For spherically symmetric potential , where E_{n} depends only an n and independent of *l* and m.

So is an eigen function of H.

When we operate H, L^{2} and L_{z} on a given wavefunction, then in the case of H and L^{2}, there is no change in the form of wavefunction but in case of L_{z}, wavefunction changes, so the given wavefunction is the eigen function of H and L^{2} only.

18. The commutator [x^{2}, p^{2}] is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. [x^{2}, p^{2}] = x[x, p^{2}] + [x, p^{2}]x [By using distributive property]

19. Consider a system of non-interacting particles in d dimensions obeying the dispersion relation = Ak^{s}, where is the energy, k is the wavevector, 's' is an integer and A a constant. The density of states, N(), is proportional to

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The dispersion relation

The volume of sphere in d– dimension in k–space

**[See appendix C on page 504 in Statistical Mechanics-II edition by R.K. Pathria]**

The number of microstates

The number of microstates in the range to + d is proportional to

The density of states .

20. The number of ways in which N identical bosons can be distributed in two energy levels, is

(a) N + 1

(b) N(N–1)/2

(c) N(N + 1)/2

(d) N

Ans. (a)

Sol. The number of ways in which N identical bosons can be distributed in two energy levels

Number of ways

21. The free energy of a gas N particled in a volume V and at a temperature T is , where a_{0} is a constnat and k_{B} denotes the Boltzmann constant. The internal energy of gas is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The Free energy, F = Nk_{B}T ln [a_{0}V (k_{B}T)^{5/2}/N] = k_{B}T ln [a_{0}V (k_{B}T)^{5/2}/N]^{N}; F = –k_{B}T ln Q

[Extra negative sign is due to the wrong sign in the given question]

22. In the op-amp circuit shown in the figure below, the input voltage V_{i} is 1V. The value of the output V_{0} is

(a) –0.33 V

(b) –0.50 V

(c) –1.00 V

(d) –0.25 V

Ans. (c)

Sol. Apply Nodal Analysis:

23. An LED operates at 1.5 V and 5 mA in forward bias. Assuming an 80% external efficiency of the LED, how many photons are emitted per second?

(a) 5.0 × 10^{16}

(b) 1.5 × 10^{16}

(c) 0.8 × 10^{16}

(d) 2.5 × 10^{16}

Ans. (d)

Sol. Power = (VI) × efficiency = (1.5 V) (5 mA) (0.8 watt) = nhv.

[E_{g} of semiconductor is not given but answer will be correct. If we use GaAs semiconductor.]

Semiconductor LED, E_{g} = 1.43eV nhv = 6 watt

24. The transistor in the given circuit has and . If the load resistance R_{L} = 1000, the voltage and current gain are, respectively.

(a) –35, +35

(b) 35, –35

(c) 35, –0.97

(d) 0.98, –35

Ans. (a)

Sol. h_{fe} = 35; h_{ie} = 1000 = R_{i}; R_{L} = 1000

For common emitter voltage gain = h_{fe} = – 35 R_{i} = h_{ie} = 1000

25. The experimentally measured transmission spectra of metal, insulator and semiconductor thin films are shown in the figure. It can be inferred that I, II and III correspond respectively, to

(a) Insulator, semiconductor and metal

(b) Semiconductor, metal and insulator

(c) Metal, semiconductor and insulator

(d) Insulator, metal and semiconductor

Ans. (a)

Sol. Transmittance of metal does not depend on wavelength (). Hence III is metal.

26. The eigenvalues of the antisymmetric matrix, where n_{1}, n_{2} and n_{3} are the components of a unit vector, are

(a) 0, i, –i

(b) 0, 1, –1

(c) 0, 1 + i, –1 –i

(d) 0, 0, 0

Ans. (a)

Sol.

27. Which of the following limits exists?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

28. A bag contains many balls, each with a number painted on it. There are exactly n balls which have the number n (namely one ball with 1, two balls with 2, and so on until N balls with N on them). An experiment consists of choosing a ball at random, noting the number on it and returning it to the bag. If the experiment is repeated a large number of times, the average value of the number will tend to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Take N = 4.

In general,

Average value of the number

29. The value of the integral is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Since the two poles are on real axis, therefore, required integral = Real part (i × sum of the residues)

30. The Poisson bracket {|r|, |p|} has the value

(a) |r||p|

(b)

(c) 3

(d) 1

Ans. (b)

Sol.

31. Consider the motion of a classical particle in a one dimensional double-well potential . If the particle is displaced infinitesimally from the minimum the positive x-axis (and friction is neglected), then

(a) the particle will execute simple harmonic motion in the right well with an angular frequency

(b) the particle will execute simple harmonic motion in the right well with an angular frequency

(c) the particle will switch between the right and left wells

(d) the particle will approach the bottom of the right well and settle there

Ans. (b)

Sol.

For unit mass

32. What is the proper time interval between the occurence of two events if in one inertial frame the events are separated by 7.5 × 10^{5} m and occur 6.5 s apart?

(a) 6.50 s

(b) 6.00 s

(c) 5.75 s

(d) 5.00 s

Ans. (b)

Sol.

Proper time interval between two events

Eq. (1), (2) put in Eq. (3)

33. A free particle described by a plane wave and moving in the positive z-direction undergoes scattering by a potential . If V_{0} is changes to 2V_{0}, keeping R fixed, then the differential scattering cross-section, in the Born approximation,

(a) increases to four times the original value

(b) increases to twice the original value

(c) decreases to half the original value

(d) decreases to one fourth the original value

Ans. (a)

Sol. The scattering amplitude in the born approximation,

The differential cross-section

34. A variational calculation is done with the normalized trial wavefunction for the one dimensional potential well

The ground state energy is estimated to be

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

35. A particle in one-dimension is in the potential

If there is at least one bound state, the minimum depth of the potential is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Solve the above equation,

For wavefunction to be finite at

For the wavefunction to be zero at x = 0, B = 0.

36. Which of the following is a self-adjoint operator in the spherical polar coordinate system ?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. To calculate self adjoint we use condition

On integrating by parts

37. Which of the following quantities is Lorentz invariant?

(a) |E × B|^{2}

(b) |E|^{2} – |B|^{2}

(c) |E|^{2} + |B|^{2}

(d) |E|^{2} |B|^{2}

Ans. (b)

Sol. E^{2} – B^{2} is Lorentz invariant.

38. Charges Q, Q and –2Q are placed on the vertices of an equilateral ABC of sides of length a, as shown in the figure

The dipole moment of this configuration of charges, irrespective of the choice of origin, is

(a)

(b)

(c)

(d) 0

Ans. (c)

Sol.

So, we can say that dipole moment does not depend upon the choice of the origin.

39. The vector potential due to a magnetic moment 'm' at a point 'r' is given by . If is directed along the positive z-axis, the x-component of the magnetic field, at the point r, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

The x-component of this term = zero

40. A system has two normal modes of vibration, with frequencies and . What is the probability that at temperature T, the system has an energy less than ? [In the following and Z is the partition function]

(a) x^{3/2} (x + 2x^{2})/Z

(b) x^{3/2} (1 + x + x^{2})/Z

(c) x^{3/2} (1 + 2x^{2})/Z

(d) x^{3/2} (1 + x + 2x^{2})/Z

Ans. (d)

Sol. The possible energies of the system are given by

The possible energies with corresponding degeneracies (that are less than )

The probability that at temperature T, the system has energy less than

41. The magnetization M of a ferromagnet, as a function of the temperature T and the magnetic field H, is described by the equation . In these units, the zero-field magnetic susceptibility in terms of M(0) = M(H = 0) is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

42. Bose condensation occurs in liquid He^{4} kept at ambient pressure at 2.17 K. At which temperature will Bose condensation occur in He^{4} in gaseous state, the density of which is 1000 times smaller than that of liquid He^{4} ? (Assume that it is a perfect Bose gas)

(a) 2.17 mK

(b) 21.7 mK

(c) 21.7 µK

(d) 2.17 µK

Ans. (b)

Sol. Bose condensation Temperature , n = particle density

43. Consider black body radiation contained in a cavity whose walls are at temperature T. The radiation is in equilibrium with the walls of the cavity. If the temperature of the walls is increased to 2T and the radiation is allowed to come to equilibrium at the new temperature, entropy of the radiation increases by a factor of

(a) 2

(b) 4

(c) 8

(d) 16

Ans. (c)

Sol. In black-body radiation, entropy,

44. The output 0, of the given circuit in cases I and II, where

Case I : A, B = 1; C, D = 0; E, F = 1 and G = 0

Case II : A, B = 0; C, D = 0; E, F = 0 and G = 1 are respectively

(a) 1, 0

(b) 0, 1

(c) 0, 0

(d) 1, 1

Ans. (d)

Sol.

Output is (1)

45. A resistance strain gauge is fastened to a steel fixture and subjected to a stress of 1000 kg/m^{2}. If the gauge factor is 3 and the modulus of elasticity of steel is 2 × 10^{10} kg/m^{2}, then the fractional change in resistance of the strain gauge due to the applied stress is:

(Note: The gauge factor is defined as the ratio of the fractional change in resistance to the fractional change in length.)

(a) 1.5 × 10^{–7}

(b) 3.0 × 10^{–7}

(c) 0.16 × 10^{–10}

(d) 0.5 × 10^{–7}

Ans. (a)

Sol.

Fractional change in resistance.

46. Consider a sinusoidal waveform of amplitude 1V and frequency f_{0}. Starting from an arbitrary intial time, the waveform is sampled at intervals of 1/(2f_{0}). If the corresponding Fourier spectrum peaks at a frequency and an amplitude , then

(a)

(b)

(c)

(d)

Ans. (d)

Sol. is the nyquist frequency which is always of sampling rate,

value of sine wave

47. The first absorption spectrum of ^{12}C^{16}O is at 3.842 cm^{–1} while at of ^{13}C^{16}O is at 3.673 cm^{–1}. The ratio of their moments of inertia is

(a) 1.851

(b) 1.286

(c) 1.046

(d) 1.038

Ans. (c)

Sol. First absorption spectra corresponds to J = 0 J = 1 and corresponding frequency is 2Bc.

48. The spin-orbit interaction in an atom is given by , where L and S denote the orbital and spin angular moment, respectively of electron. The splitting between the levels ^{2}P_{3/2} and ^{2}P_{1/2} is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

49. The spectral line corresponding to an atomic transition from J = 1 to J = 0 states splits in a magnetic field of 1 KG into three components seperated by 1.6 × 10^{–3} Å. If the zero field spectral line corresponding to 1849Å, what is the g-factor corresponding to the J = 1 state? (You may use .)

(a) 2

(b) 3/2

(c) 1

(d) 1/2

Ans. (c)

Sol.

50. The energy required to create a lattice vacancy in a crystal is equal to 1 eV. The ratio of the number densities of vacancies n(1200 K)/n(300 K), when the crystal is at equilibrium at 1200 K and 300 K, respectively is approximately

(a) exp(–30)

(b) exp(–15)

(c) exp(15)

(d) exp(30)

Ans. (d)

Sol. Use Boltzman factor, (1 ev = 11605k)

51. The dispersion relation of phonons in a solid is given by

The velocity of the phonons at large wavelength is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

For larger , (k_{x}a, k_{y}a, k_{z}a) are small.

52. Consider an electron in a box of length L with periodic boundary condition . If the electron is in the with energy , what is the correction to its energy, to second order of perturbation theory, when it is subjected to weak periodic potential V(x) = V_{0} cos gx, where g is an integral multiple of the ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

53. The ground state of nucleus has spin-parity , while the first excited state has . The electromagnetic radiation emitted when the nucleus makes a transition from the first excited state to the ground state are

(a) E2 and E3

(b) M2 and E3

(c) E2 and M3

(d) M2 and M3

Ans. (c)

Sol. No parity change;

For E* _{l}* type, , (for no parity change )

For M* _{l}* type, , (for no parity change )

, No parity change E2; = 3, No parity change M3

54. The dominant interactions underlying the following processes

(a) A: strong, B: electromagnetic and C: weak

(b) A: strong, B: weak and C: weak

(c) A: weak, B: electromagnetic and C: strong

(d) A: weak, B: electromagnetic and C: weak

Ans. (a)

Sol.

(Not conserved), I_{3} is the Z component of isotopic spin Weak interaction

(B) This reaction involves µ^{+} and µ^{–} which are leptons which are charged

Electromagnatic interaction

(A) involves hadrons; (Conserved) Strong interaction.

55. If a Higgs boson of mass m_{H} with a speed decays into a pair photons, then the invariant mass of the photon pair is

[**Note:** The invariant mass of a system of two particles, with four-momenta p_{1} and p_{2} is (p_{1} + p_{2})^{2}]

(a)

(b) m_{H}

(c)

(d)

Ans. (b)

Sol. The mass of photon pair = mass of the initial particle = m_{H}