## CSIR NET PHYSICS (DEC - 2016)

Previous Year Question Paper with Solution.

1. Consider two radioactive atoms, each of which has a decay rate of 1 per year. The probability that at least one of them decays in the first two years is

(a)

(b)

(c) 1 – e^{–4}

(d) (1 – e^{–2})^{2}

Ans. (c)

Sol. According to Passion distribution.

The probability of not observing a decay in first two years.

1 is the mean decay of 1 year. 2 is the mean decay of 2 years.

Then, Not operating a decay in both atoms e^{–2}.e^{–2} and observing atleat one decay 1 – e^{–4}.

2. The Fourier transform of the function is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

3. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to gravity, the ball experiences a damping force of the form –v, where v is its instantaneous velocity and is a constant. Given the values m = 10 kg, = 10 kg/s, and , the distance travelled (in metres) in time t in seconds, is

(a) 10(t + 1 – e^{–t})

(b) 10(t – 1 + e^{–t})

(c) 5t^{2} – (1 – e^{t})

(d) 5t^{2}

Ans. (b)

Sol.

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

General solution of above equation is

x(t) = c_{1} + c_{2}t + 10e^{–t }... (3)

x(0) = 0, x*'* (0) = 0 (given) ... (4)

Eq. (4) put in Eq. (3), then we get

c_{1} = – 10 and c_{2} = 10 ... (5)

Eq. (5) put in Eq. (3)

x(t) = 10(t – 1 + e^{–t})

4. The matrix satisfies the equation

(a) M^{3} – M^{2} – 10M + 12I = 0

(b) M^{3} + M^{2} – 10M + 10I = 0

(c) M^{3} – M^{2} – 10M + 10I = 0

(d) M^{3} + M^{2} + 10M + 10I = 0

Ans. (c)

Sol. The characteristic equation is

Thus the matrix M satisfies the equation

M^{3} – M^{2} 10M + 10I = 0 then the correct option is (c)

5. The Laplace transform of

is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We can write

Hence the transform of f(t) is

6. A relativistic particle moves with a constant velocity v with respect to the laboratory frame. In time , measured in the rest frame of the particle, the distance that it travels in the laboratory frame is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. From Particle x*'*_{1} = 0 x*'*_{2} = 0 t_{initial} = t*'*_{1 }t_{finial} = t*'*_{2}

_{ }

_{ }

_{ }

7. A particle in two dimensions is in a potential V(x, y) = x + 2y. Which of the following (apart from the total energy of the particle) is also a constant of motion?

(a) p_{y} – 2p_{x}

(b) p_{x} – 2p_{y}

(c) p_{x} + 2p_{y}

(d) p_{y} + 2p_{x}

Ans. (a)

Sol. V(x, y) = x + 2y ... (1)

H = T + V ... (2)

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

= [py – 2p_{x}, H] = [py – 2p_{x}, x + 2y] = [p_{y}, 2y] – [2p_{x}, x] – [2p_{x}, x] = – 2 + 2 = 0

8. The dynamics of a particle governed by the Lagrangian describes

(a) an undamped simple harmonic oscillator

(b) a damped harmonic oscillator with a time varying damping factor

(c) an undamped harmonic oscillator with a time dependent frequency

(d) a free particle

Ans. (d)

Sol.

Lagrangian equation of motion

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

So motion is equivalent to free particle

9. The parabolic coordinates are related to the Cartesian coordinates (x, y) by x = and . The Lagrangian of a two-dimensional simple harmonic oscillator of mass m and angular frequency is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For two dimensional Harmonic oscillation

L = T – V ... (2)

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

Eq. (4) put in Eq. (3)

10. A conducting circular disc of radius r and resistivity rotates with an angular velocity in a magnetic field B perpendicular to it. A voltmeter is connected as shown in the figure below.

Assuming its internal resistance to be infinite, the reading on the voltmeter

(a) depends on , B, r and

(b) depends on , B and r, but not on

(c) is zero because the flux through the loop is not changing

(d) is zero because a current flows in the direction of B

Ans. (b)

Sol.

11. The charge per unit length of a circular wire of radius a in the xy-plane, with its centre at the origin, is , where is a constant and the angle is measured from the positive x-axis. The electric field at the centre of the circle is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. At centre O, direction of field is .

12. A screen has two slits, each of width w, with their centres at a distance 2w apart. It is illuminated by a monochromatic plane wave travelling along the x-axis.

The intensity of the interference pattern, measured on a distant screen, at an angle to the x-axis is

(a) zero for n = 1,2,3...

(b) maximum for n = 1,2,3....

(c) maximum for

(d) zero for n = 0 only

Ans. (a)

Sol. Maximum for n = 3 and zero for n = 1, 2, 3...

13. The electric field of an electromagnetic wave is , where and k are positive constants. This represents

(a) a linearly polarised wave travelling in the positive z-direction

(b) a circularly polarised wave travelling in the negative z-direction

(c) an elliptically polarised wave travelling in the negative z-direction

(d) an unpolarised wave travelling in the positive z-direction

Ans. (c)

Sol. E_{0} is amplitude along x-axis

2E_{0} amplitude along y-axis

So, we can say that wave is elliptically polarised.

14. Consider the two lowest normalized energy eigenfunctions and of a one dimensional system. They satisfy and , where is a real constant. The expectation value of the momentum operator in the state is

(a)

(b) 0

(c)

(d)

Ans. (b)

Sol.

Integrate by parts

15. Consider the operator acting on smooth functions of x. The commutator [a, cos x] is

(a) – sin x

(b) cos x

(c) – cos x

(d) 0

Ans. (a)

Sol.

[a cos x] = – sin x

16. Let be the lowering and raising operators of a simple harmonic oscillator in units where the mass, angular frequency and have been set to unity. If is the ground state of the oscillator and is a complex constant, the expectation value of in the state , is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

17. Consider the operator , where is the momentum operator, is the vector potential and q denotes the electric charge. If denotes the magnetic field, the z-component of the vector operator is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

So, z components is given by

18. Consider a gas of N classical particles in a two-dimensional square box of side L. If the total energy of the gas is E, the entropy (apart from an additive constant) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

19. Consider a continuous time random walk. If a step has taken place at time t = 0, the probability that the next step takes place between t and t + dt is given by bt dt, where b is a constant. What is the average time between successive steps?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Correct option is (d)

20. The partition function of a two-level system governed by the Hamiltonian is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

21. A silica particle of radius 0.1 µm is put in a container of water at T = 300 K. The densities of silica and water are 2000 kg/m^{3} and 1000 kg/m^{3}, respectively. Due to thermal fluctuations, the particle is not always at the bottom of the container. The average height of the particle above the base of the container is approximately

(a) 10^{–3} m

(b) 3 × 10^{–4} m

(c) 10^{–4} m

(d) 5 × 10^{–5} m

Ans. (c)

Sol. Average thermal kinetic energy thermal equilibrium of transport in 3-dimensional motion = 3/2 kT.

Average thermal kinetic energy = Average potential energy at height 'h'

= 1.5 × 10^{–4} m = 10^{–4} m.

22. Which of the following circuits implements the Boolean function F(A, B, C) = ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

23. A pair of parallel glass plates separated by a distance d is illuminated by white light as shown in the figure below. Also shown is the graph of the intensity of the reflected light I as a function of the wavelength recorded by a spectrometer.

Assuming that the interference takes place only between light reflected by the bottom surface of the top plate, the distance d is closest to

(a) 12 µm

(b) 24 µm

(c) 60 µm

(d) 120 µm

Ans. (d)

Sol. Condition for constructive interference is

Equation (2) put in equation (1)

24. The I-V characteristics of a device is , where T is the temperature and a and I_{S} are constants independent of T and V. Which one of the following plots is correct for a fixed applied voltage V?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

So graph between ln I and will be a straight line for large , so the graph would be

Correct option is (d)

25. The active medium in a blue LED (light emitting diode) is a Ga_{x}In_{1–x}N alloy. The band gaps of GaN and InN are 3.5 eV and 1.5 eV respectively. If the band gap of Ga_{x}In_{1–x}N varies approximately linearly with x, the value of x required for the emission of blue light of wavelength 400 nm is (take )

(a) 0.95

(b) 0.75

(c) 0.50

(d) 0.33

Ans. (b)

Sol. E_{gGaN} = 3.5 eV and E_{gInN }= 1.5 eV

Band Gap energy of Ga_{x} In_{1–x} N is E x.

For blue light of wavelength 400 nm, the band gap

Thus equating slopes we get;

26. A stable asymptotic solution of the equation is x = 2. If we take and , where and are both small, the ratio is approximately

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

on putting these values

can be neglected in the denominator

27. The 2 × 2 identity matrix I and the Pauli matrices do not form a group under matrix multiplication. The minimum number of 2 × 2 matrices, which includes these four matrices, and form a group (under matrix multiplication) is

(a) 20

(b) 8

(c) 12

(d) 16

Ans. (d)

Sol.

28. Given the values sin 45º = 0.7071, sin 50º = 0.7660, sin 55º = 0.8192 and sin 60º = 0.8660, the approximate value of sin 52º, computed by Newton's forward difference method, is

(a) 0.804

(b) 0.776

(c) 0.788

(d) 0.798

Ans. (c)

Sol. Given

h = 5, 52° = x_{0} + uh = 45° + uh

From table we have

= 0.7071 + 0.0825 – 0.0016 + 0.0000 = 0.7880

29. Let f(x, t) be a solution of the heat equation in one dimension. The initial condition at t = 0 is for . Then for all t > 0, f(x, t) is given by

[Useful integral: .]

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

The solution f(x, t) is given as

30. After a perfectly elastic collision of two identical balls, one of which was initially at rest, the velocities of both the balls are non-zero. The angle θ between the final velocities (in the lab frame) is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Angle between two particle

Conservation of momentum

conservation of kinetic energy

From (i) and (ii)

31. Consider circular orbits in a central force potential , where k > 0 and 0 < n < 2. If the time period of a circular orbit of radius R is T_{1} and that of radius 2R is T_{2}, then T_{2}/T_{1} is

(a)

(b)

(c)

(d) 2^{n}

Ans. (c)

Sol.

32. Consider a radioactive nucleus that is travelling at a speed c/2 with respect to the lab frame. It emits -rays of frequency v_{0} in its rest frame. There is a stationary detector (which is not on the path of the nucleus) in the lab. If a -ray photon is emitted when the nucleus is closest to the detector, its observed frequency at the detector is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

33. Suppose that free charges are present in a material of dielectric constant and resistivity . Using Ohm's law and the equation of continuity for charge, the time required for the charge density inside the material to decay 1/e is closest to

(a) 10^{–6} s

(b) 10^{6} s

(c) 10^{12} s

(d) 10 s

Ans. (d)

Sol.

34. A particle with charge –q moves with a uniform angular velocity in a circular orbit of radius a in the xy-plane, around a fixed charge +q, which is at the centre of the orbit at (0, 0, 0). Let the intensity of radiation at the point (0, 0, R) be I_{1} and at (2R, 0, 0) be I_{2}. The ratio I_{2}/I_{1}, for R >> a, is

(a) 4

(b)

(c)

(d) 8

Ans. (c)

Sol.

35. A parallel plate capacitor is formed by two circular conducting plates of radius a separated by a distance d, where d << a. It is being slowly charged by a current that is nearly constant. At an instant when the current is I, the magnetic induction between the plates at a distance a/2 from the centre of the plate, is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

36. Two uniformly charged insulating solid spheres A and B, both of radius a, carry total charges +Q and –Q, respectively. The spheres are placed touching each other as shown in the figure.

If the potential at the centre of the sphere A is V_{A} and that at the centre of B is V_{B}, then the difference V_{A} – V_{B} is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

37. A particle is scattered by a central potential , where V_{0} and µ are positive constants. If the momentum transfer is such that , the scattering cross-section in the Born approximation, as , depends on q as [You may use ]

(a) q^{–8}

(b) q^{–2}

(c) q^{2}

(d) q^{6}

Ans. (a)

Sol. The form factor is given for high energy as q

38. A particle in one dimension is in a potential . Its wavefunction is continuous everywhere. The discontinuity in at x = a is

(a)

(b)

(c)

(d) 0

Ans. (a)

Sol.

Integerates both side within limit

39. The dynamics of a free relativistic particle of mass m is governed by the

Dirac Hamiltonian , where is the momentum operator and and are four 4 × 4 Dirac matrices. The acceleration operator can be expressed as

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

If v_{x} velocity of x direction

From the Ehrenfest theorem

Similarly, acceleration is given by

40. A particle of charge q in one dimension is in a simple harmonic potential with angular frequency . It is subjected to a time-dependent electric field , where A and are positive constants and . If in the distant past the particle was in its ground state, the probability that it will be in the first excited state as is proportional to

(a)

(b)

(c) 0

(d)

Ans. (a)

Sol. Transition probability is proportional to where

41. Consider a random walk on an infinite two-dimensional triangular lattice, a part of which is shown in the figure below.

If the probabilities of moving to any of the nearest neighbour sites are equal, what is the probability that the walker returns to the starting at the end of exactly three steps?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. For walker to return to starting position it must move along an equivalent triangle in three steps.

For steps on any movement can result in equilateral triangle.

For step two, two out of six options will form equilateral triangle.

For step three, only one out of six options will form equilateral triangle.

Total probability

42. An atom has a non-degenerate ground-state and a doubly-degenerate excited state. The energy difference between the two states is . The specific heat at very low temperatures is given by

(a)

(b)

(c)

(d) k_{B}

Ans. (c)

Sol. Assume energy at ground state is 0 and energy at first excited state is . The partition function is Z = 1 +

43. The electrons in graphene can be thought of as a two-dimensional gas with a linear energy-momentum relation , where and v is a constant. If is the number of electrons per unit area, the energy per unit area is proportional to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The number of k state in range of k to k + dk:

Since, dispersion relation is E = |P|v =

The number of electron at T = 0° K is

The average energy at T = 0K is

44. In the circuit below, the input voltage V_{i} is 2V, V_{cc} = 16 V, and .

The value of R_{1} required to deliver 10 mW of power across R_{L} is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

45. Two sinusoidal signals are sent to an analog multiplier of scale factor 1 V^{–1} followed by a low pass filter (LPF).

If the roll-off frequency of the LPF is f_{c} = 5 Hz, the output voltage V_{out} is

(a) 5 V

(b) 25 V

(c) 100 V

(d) 50 V

Ans. (b)

Sol. After multiplying

46. The resistance of a sample is measured as a function of temperature, and the data are shown below.

The slope of R vs T graph, using a linear least-squares fit to the data, will be

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

47. Consider a one-dimensional chain of atoms with lattice constant a. The energy of an electron with wave-vector k is , where µ and are constants. If an electric field E is applied in the positive x-direction, the time dependent velocity of an electron is

(In the following B is the constant)

(a)

(b) proportional to E

(c) independent of E

(d)

Ans. (d)

Sol. In the presence of electric field E, we can write

Time dependent velocity of electron is

48. A thin rectangular conducting plate of length a and width b is placed in the xy-plane in two different orientations, as shown in the figures below. In both cases a magnetic field B is applied in the z-direction and a current flows in the x direction due to the applied voltage V.

If the Hall voltage across the y-direction in the two cases satisfy V_{2} = 2V_{1}, the ratio a:b must be

(a) 1:2

(b)

(c) 2:1

(d)

Ans. (d)

Sol. Since, Hall voltage is given by , where w is width of conducting plate.

Since, in case (I), V = I_{1}R_{1} and

49. Consider a hexagonal lattice with basis vectors as shown in the figure below.

If the lattice spacing is a = 1, the reciprocal lattice vectors are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. From the figure, we can write

Reciprocal lattice vectors are

50. In the L-S coupling scheme, the terms arising from two non-equivalent p-electrons are

(a) ^{3}S, ^{1}P, ^{3}P, ^{1}D, ^{3}D

(b) ^{1}S, ^{3}S, ^{1}P, ^{1}D

(c) ^{1}S, ^{3}S, ^{3}P, ^{3}D

(d) ^{1}S, ^{3}S, ^{1}P, ^{3}P, ^{1}D, ^{3}D

Ans. (d)

Sol. For pp configuration

Thus

Thus spectroscopic terms are^{}1S^{3}, S^{1}, P^{3}, P^{1}D, ^{3}D

51. The total spin of a hydrogen atom is due to the contribution of the spins of the electron and the proton. In the high temperature limit, the ratio of the number of atoms in the spin-1 state to the number in the spin-0 state is

(a) 2

(b) 3

(c) 1/2

(d) 1/3

Ans. (b)

Sol.

The dgenercy of quantum level is 2F + 1.

52. A two level system in a thermal (black body) environment can decay from the excited state by both spontaneous and thermally stimulated emission. At room temperature (300 K), the frequency below which thermal emission dominates over spontaneous emission is nearest to

(a) 10^{13} Hz

(b) 10^{8} Hz

(c) 10^{5} Hz

(d) 10^{11} Hz

Ans. (d)

Sol. At thermal equilibrium, the ratio of the number of spontaneous to stimulated emission is given by

The ratio is less than one, thus stimulated thermal emission dominates over sponaneous emission below all frequency of 10^{11} Hz

53. What should be the minimum energy of a photon for it to split an -particle at rest into a tritium and a proton?

(The masses of and are 4.0026 amu, 3.0161 amu and 1.0073 amu, respectively, and 1 amu 938 MeV.)

(a) 32.2 MeV

(b) 3 MeV

(c) 19.3 MeV

(d) 931.5 MeV

Ans. (c)

Sol. From conservation of energy

54. Which of the following reaction(s) is/are allowed by the conservation laws?

(a) both (i) and (ii)

(b) only (i)

(c) only (ii)

(d) neither (i) nor (ii)

Ans. (a)

Sol.

Reaction is allowed

Reaction is allowed

55. A particle, which is a composite state of three quarks u, d and s, has electric charge, spin and strangeness respectively, equal to

(a)

(b) 0, 0, –1

(c)

(d)

Ans. (c)

Sol. Charge, spin and strangers of Quarks u, d and s are given as

If a particle x is a composite of u, d and s, then net charge, spin and strangness on x is net charge = 0

net spin and net strangness = –1