CSIR NET PHYSICS (JUNE 2016)
Previous Year Question Paper with Solution.

1.    The radius of convergence of the Taylor series expansion of the function around x = 0, is

(a)

(b)

(c)

(d) 1

Ans. (c)

Sol.

Assume the corresponding complex function

So, f(z) is analytic at z = 0 and we can expand the function into Taylor series about z = 0.

If we draw a circle centered at z = 0 and of largest possible radius units, i.e. then Talyor series expansion of f(z) about z = 0 will converge for all points of the region as f (z) is analytic in the region .

Therefore, radius of convergence will be .

2.    The value of the contour integral around the unit circle C traversed in the anti-clockwise direction, is

(a) 0

(b) 2

(c)

(d)

Ans. (c)

Sol.

Condition of singularity:

For an unit cirucle C centered at the origin, ln 3 lies within C.

Since, ln 3 is a simple pole, then

3.    The Gauss hypergeometric function F(a, b, c; z), defined by the Taylor series expansion around z = 0 as

satisfies the equation relation

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given Gauss hypergeometric function

4.    Let X and Y be two independent random variables, each of which follow a normal distribution with the same standard deviation , but with means +µ and –µ, respectively. Then the sum X + Y follows a

(a) distribution with two peaks at +µ and mean 0 and standard deviation

(b) normal distribution with mean 0 and standard deviation 2

(c) distribution with two peaks at +µ and mean 0 and standard deviation 2

(d) normal distribution with mean 0 and standard deviation

Ans. (d)

Sol. If X and Y are two independent random variables which follow normal (Gaussian) distribution with the following specification.

Mean of X = µX, Mean of Y = µY

Variance of Variance of

then X + Y will also follow normal distribution and

Mean of (X + Y) = µX + µY and

Variance of

Given :

So, µX+Y = µX + µY = 0

Standard deviation

5.    Using dimensional analysis, Planck defined a characteristic temperature TP from powers of the gravitational constant G, Planck's constant h, Boltzmann constant kB and the speed of light c in vacuum. The expression for TP is proportional to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Dimension of h = ML2T–1 ; Dimension of G = M–1L3T–2

Dimension of kB = ML2T–2K–1; Dimension of c = LT–1

So, dimension of

= = K = Dimension of temperature

6.    Let (x, t) and (x', t') be the coordinate systems used by the observers O and O', respectively. Observer O' moves with a velocity v = c along their common positive x-axis. If x+ = x + ct and x = x – ct are the linear combinations of the coordinates, the Lorentz transformation relating O and O' takes the form

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

7.    A ball of mass m, initially at rest, is dropped from a height of 5 meters. If the coefficient of restitution is 0.9, the speed of the ball just before it hits the floor the second time is approximately (take g = 9.8 m/s2)

(a) 9.80 m/s

(b) 9.10 m/s

(c) 8.91 m/s

(d) 7.02 m/s

Ans. (c)

Sol. We know that

Coefficient of restitution

8.    Four equal charges of +Q each are kept at the vertices of a square of side R. A particle of mass m and charge +Q is placed in the plane of the square at a short distance a(<< R) from the center. If the motion of the particle is confined to the plane, it will undergo small oscillations with an angular frequency

(a)

(b)

(c)

(d)

Ans. (c)

Sol. According laplace equation

According symmetry

9.    The Hamiltonian of a system with generalized coordinate and momentum (q, p) is H = p2 q2. A solution of the Hamiltonian equation of motion is (in the following A and B are constants)

(a)

(b)

(c)

(d)

Ans. (a)

Sol. H = q2p2

We know that

Divide (i) and (ii),

ln q + ln p = constant, qp = constant

10.    Two parallel plate capacitors, separated by distances x and 1.1x respectively, have a dielectric material of dielectric constant 3.0 inserted between the plates, and are connected to a battery of voltage V. The difference in charge on the second capacitor compared to the first is

(a) +66%

(b) +20%

(c) –3.3%

(d) –10%

Ans. (d)

Sol.

11.    The half space regions x > 0 and x < 0 are filled with dielectric media of dielectric constants and respectively. There is a uniform electric field in each part. In the right half, the electric field makes an angle to the interface. The corresponding angle in the left half satisfies

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

12.    The x- and z-components of a static magnetic field in a region are Bx = B0(x2 – y2) and Bz = 0, respectively. Which of the following solutions for its y-component is consistent with the Maxwell equations ?

(a) By = B0xy

(b) By = –2B0xy

(c) By = –B0 (x2 – y2)

(d)

Ans. (b)

Sol.

13.    A magnetic field B is in the region x > 0 and zero elsewhere. A rectangular loop, in the xy-plane, of sides l (along the x-direction) and h (along the y-direction) is inserted into the x > 0 region from the x < 0 region at a constant velocity . Which of the following values of l and h will generate the largest EMF?

(a) l = 8, h = 3

(b) l = 4, h = 6

(c) l = 6, h = 4

(d) l = 12, h = 2

Ans. (b)

Sol. e.m.f. = vBh

e.m.f. will be largest

if 'h' is largest

14.    The state of a particle of mass m in a one-dimensional rigid box in the interval 0 to L is given by the normalised wavefunction

.

If its energy is measured, the possible outcomes and the average value of energy are, respectively.

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

15.    If and are the components of the angular momentum operator in three dimensions, the commutator may be simplified to

(a)

(b)

(c)

(d) 0

Ans. (a)

Sol.

16.    Suppose that the Coulomb potential of the hydrogen atom is changed by adding an inverse-square term such that the total potential is , where g is a constant. The energy eigenvalues Enlm in the modified potential

(a) depend on n and l, but not on m

(b) depend on n but not on l and m

(c) depend on n and m, but not on l

(d) depend explicitly on all three quantum numbers n, l and m

Ans. (a)

Sol. Hamiltonain of hydrogen atom,

En depends on n and l but not m.

17.    The eigenstates corresponding to eigenvalues E1 and E2 of a time-independent Hamiltonian are and respectively. If at t = 0, the system is in a state the value of at time t will be

(a) 1

(b)

(c)

(d)

Ans. (a)

Sol.

18.    The specific heat per molecule of a gas of diatomic molecules at high temperatures is

(a) 8kB

(b) 3.5 kB

(c) 4.5 kB

(d) 3kB

Ans. (b, c)

Sol.

Correct option is (b) and (c)

19.    When an ideal monatomic gas is expanded adiabatically from an initial volume V0 to 3V0, its temperature changes from T0 to T. Then the ratio T/T0 is

(a)

(b)

(c)

(d) 3

Ans. (b)

Sol. PV5/3 = constant

20.    A box of volume V containing N molecules of an ideal gas, is divided by a wall with a hole into two compartments. If the volume of the smaller compartment is V/3, the variance of the number of particles in it, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The probability of smaller compartment of box

The probability of larger compartment of box

Variance of the number of particles (N)

21.    A gas of non-relativistic classical particles in one-dimension is subjected to a potential , where is a constant). The partition function is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

22.    The dependence of current I on the voltage V of a certain device is given by

where I0 and V0 are constants. In an experiment the current I is measured as the voltage V applied across the device is increased. The parameters V0 and can be graphically determined as

(a) the slope and the y-intercept of the I-V2 graph

(b) the negative of the ratio of the y-intercept and the slope, and the y-intercept of the I-V2 graph

(c) the slope and the y-intercept of the graph

(d) the negative of the ratio of the y-intercept and the slope, and the y-intercept of the graph

Ans. (d)

Sol.

on y – axis, intercept will be

on x – axis, intercept will be = V0

23.    In the schematic figure given below, assume that the propagation delay of each logic gate is tgate.

The propagation delay of the circuit will be maximum when the logic inputs A and B make the transition

(a)

(b)

(c)

(d)

Ans. (d)

Sol. To make maximum proportional delay = 4 Tpd, when transition will be (0, 0) to (0, 1) then P.D. will be = 4Tpd.

24.    Given the input voltage Vi, which of the following waveforms correctly represents the output voltage V0 in the circuit shown below?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

R2 = 10K and R1 = 5K

25.    The intensity distribution of a red LED on an absorbing layer of material is a Gaussian centered at the wavelength = 660 nm and width 20 nm. If the absorption coefficients varies with wavelength as , where and K are positive constants, the light emerging from the absorber will be

(a) blue shifted retaining the Gaussian intensity distribution

(b) blue shifted with an asymmetric intensity distribution

(c) red shifted retaining the Gaussian intensity distribution

(d) red shifted with an asymmetric intensity distribution

Ans. (d)

Sol. After passing through LED, the intensity of transmitted beam will be where Ii is incident intensity is absorption coefficient t is thickness of LED. Given

Hence,

Ii has Gaussian shape but after passing through LED, shape function will also include factor, hence shape would not be Gaussian. Further, transmitted intensity includes factor which means wavelengths higher than will have higher intensities. Therefore, peak intensity will shift to higher wavelength (red shift).

26.    What is the Fourier transform , where is the Dirac delta-function?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. According to given definition of Fourier transform

using the above mentioned property,

27.    The integral equation is equivalent to the differential equation

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given integral solution

Using the integral representation of Dirac – Delta function,

28.    A part of the group multiplication table for a six element group G = {e, a, b, c, d, f} is shown below. (In the following e is the identity element of G).

The entries x, y and z should be

(a) x = a, y = d and z = c

(b) x = c, y = a and z = d

(c) x = c, y = d and z = a

(d) x = a, y = c and z = d

Ans. (d)

Sol. G {e, a, b, c, d, f}

From the given tabel, a22 = a * a = b, a23 = a * b = e

a33 = x = b * b = (a * a) * b = a * (a * b) = a * e = a [as 'e' is the identity element]

Property of group multiplication table:

No element should be repeated in any rows or column of the multiplication table.

So, y should not be equal to 'd'.

z should be either 'd' or 'c'.

Only option (d) satisfies all the conditions.

29.    In finding the roots of the polynomial f(x) = 3x3 – 4x – 5 using the iterative Newton-Raphson method, the initial gues is taken to be x = 2. In the next iteration its value is nearest to

(a) 1.671

(b) 1.656

(c) 1.559

(d) 1.551

Ans. (b)

Sol. Given polynomial: f(x) = 3x3 – 4x – 5, f '(x) = 9x2 – 4

Initial guess value x0 = 2

Next guess value

30.    For a particle of energy E and momentum p (in a frame F), the rapidity y is defined as . In a frame F' moving with velocity v = (0, 0, c) with respect to F, the rapidity y' will be

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Using energy momentum transformation we get

31.    A canonical transformation (q, p) (Q, P) is made through the generating function F(q, P) = q2P on the Hamiltonian , where and are constants. The equations of motion for (Q, P) are

(a)

(b)

(c)

(d)

Ans. (b)

Sol. F(q, P) = q2P, differential relation

Eq. (i) put in Eq. (ii)

32.    The Lagrangian of a system moving in three dimensions is

The independent constant(s) of motion is/are

(a) energy alone

(b) only energy, one component of the linear momentum and one component of the angular momentum

(c) only energy and one component of the linear momentum

(d) only energy and one component of the angular momentum

Ans. (a)

Sol.

None of the coordinate is cyclic so linear momentum is not conserved, there is no or or term so no component of angular momentum is conserved.

L doesn't explicity depend on 't' so energy is conserved.

33.    Consider a sphere S1 of radius R which carries a uniform charge of density . A smaller sphere S2 of radius is cut out and removed from it. The centers of the two spheres are separated by the vector , as shown in the figure.

The electric field at a point P inside S2 is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Electric field inside cavity = field of big sphere + field of small sphere

34.    The values of the electric and magnetic fields in a particular reference frame (in Gausian units) are and , respectively. An inertial observer moving with respect to this frame measures the magnitude of the electric field to be . The magnitude of the magnetic field measured by him is

(a) 5

(b) 9

(c) 0

(d) 1

Ans. (c)

Sol. Let the observer is moving along x – direction

35.    A loop of radius a, carrying a current I, is placed in a uniform magnetic field B. If the normal to the loop is denoted by , the force F and the torque T on the loop are

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

36.    A wavelength has a square cross-section of side 2a. For the TM modes of wavevector k, the transverse electromagnetic modes are obtained in terms of a function which obeys the equation

with the boundary condition . The frequency of the lowest mode is given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

For lowest mode, m = n = 1

37.    Consider a particle of mass m in a potential . The change in the ground state energy, compared to the simple harmonic potential , to first order in g is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

38.    The energy levels for a particle of mass m in the potential , determined in the WKB approximation , where a, b are the turning points and n = 0, 1, 2, ....), are

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

At turning point total energy equals potential energy

By energy quantization condition

39.    A particle of mass m moves in one-dimension under the influence of the potential , where is a positive constant. The uncertainty in the product in its ground state is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

40.    The ground state energy of a particle of mass m in the potential , estimated using the normalized trail wavefunction is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

41.    Consider a gas of Cs atoms at a number density of 1012 atoms/cc. When the typical inter-particle distance is equal to the thermal de-Broglie wavelength of the particles, the temperature of the gas is nearest to (Take the mass of a Cs atom to be 22.7 × 10–26 kg).

(a) 1 × 10–9 K

(b) 5 × 10–5 K

(c) 1 × 10–3 K

(d) 2 × 10–8 K

Ans. (d)

Sol.

42.    The internal energy E(T) of a system at a fixed volume is found to depend on the temperature T as E(T) = aT2 + bT4. Then the entropy S(T), as a function of temperature, is

(a)

(b) 2aT2 + 4bT4

(c)

(d) 2at + 2bT3

Ans. (c)

Sol. E = aT2 + bT4

43.    A radioactive element X decays to Y, which in turn decays to a stable element Z. The decay constant from X to Y is , and that from Y to Z is . If, to begin with, there are only N0 atoms of X, at short times (t << 1/ as well as 1/) the number of atoms of Z will be

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Let Nx, Ny and Nz are the number of atoms of X, Y and Z elements respectively. At t =0, Nx = N0.

According to successive radioactive transformation;

From solving equation (i), (ii) and (iii), we get

(Here constant of integration is taken as zero).

44.    Two completely overlapping semi-circular parallel plates comprise a capacitive transducer. One of the plates is rotated by an angle of 10º relative to their common center. Ignoring edge effects, the ratio, In : Io of sensitivity of the transducer in the new configuration with respect to the original one, is

(a) 8 : 9

(b) 11 : 12

(c) 17 : 18

(d) 35 : 36

Ans. (c)

Sol. The sensitivity of the transducer would be maximum if two plates completely parallel to each other. The plates are semicircular hence, the plate area will be extended over 1800. Therefore, after the rotation through 100. The new overlapping area will be 1700. Hence, new sensitivity after rotation becomes.

45.    The state diagram that detects three or more consecutive 1's in a serial bit stream is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. It is satisfying option (d) the state diagram of question will be that only.

46.    The decay constants fp of the heavy pseudoscalar mesons, in the heavy quark limit, are related to their masses mp by the relation , where a is an empirical parameter to be determined. The values mp = 6400 + 160 MeV and fp = 180 + 15 MeV correspond to uncorrelated measurement of a meson. The error on the estimate of a is

(a) 175 (MeV)3/2

(b) 900 (MeV)3/2

(c) 1200 (MeV)3/2

(d) 2400 (MeV)3/2

Ans. (c)

Sol.

Taking log on both sides, ln

Taking partial differentiation on both sides, we get

Here, out of given options, (c) shows the approximate result (~1200).

47.    Consider electrons in graphene, which is a planar monatomic layer of carbon atoms. If the dispersion relation of the electrons is taken to be = ck (where c is constant) over the entire k-space, then the Fermi energy depends on the number density of electrons as

(a)

(b)

(c)

(d)

Ans. (a)

Sol. For graphene, dispersion relation is given as ...(i)

For 2 - dimensional structure, the density of state is given by

From equation (i),

Putting value of dk from equation (iii), into equation (ii), we get,

Now, the number of electrons at 0K can be calculated by integrating above equation in limit o to

Thus, total number of electrons

We can take, as density in two dimension. Therefore, by replacing, , equation (v)

48.    Suppose the frequency of phonons in a one-dimensional chain of atoms is proportional to the wavevector. If n is the number density of atoms and c is the speed of the phonons, then the Debye frequency is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Since, the frequency of phonons is a one – dimensional chain of atoms is proportional to wave vector (k)

Where, c is constant of proportionality and is speed of phonos. For one dimensional chain the density of state can be given as

Also, we know that number density, (Since, linear chain)

49.    The band energy of an electron in a crystal for a particular k-direction has the form , where A and B are positive constants and 0 < ka < . The electron has a hole-like behaviour over the following range of k:

(a)

(b)

(c)

(d)

Ans. (a, d)

Sol. Band energy is given as . The electron will behave as hole when its effective (m*) is negative. The effective mass (m*) remains negative for those k – values which lie in between point of inflexion and top of the band .

Therefore, ka should lie between

Correct options are (a) and (d)

50.    The ground state electronic configuration of 22Ti is [ar]3d2 4s2. Which state, in the standard spectroscopic notations, is not possible in this configuration?

(a) 1F3

(b) 1S0

(c) 1D2

(d) 3P0

Ans. (a)

Sol. Electronic configuration of 22Ti = [Ar]3d24s2

Empty orbit is 3d2.

There are two electrons in d - state.

Total angular momentum, J = (L + S) to (L – S)

Now, according to Hund's rule, for equivalent electrons in d-state, 1S0, 1D2, 1G4, 3P0,1,2, 3F2, 3, 4 are allowed spectrosopic terms. Hence, it is clear that is not possible term in the [Ar]3d24s2.

51.    In a normal Zeeman effect experiment using a magnetic field of strength 0.3 T, the splitting between the components of a 660 nm spectral line is

(a) 12 pm

(b) 10 pm

(c) 8 pm

(d) 6 pm

Ans. (d)

Sol. The value number separation between consecutive components is equal to the separation between consecutive zeeman levels, and is given by

52.    The separation between the energy levels of a two-level atom is 2 eV. Suppose that 4 × 1020 atoms are in the ground state and 7 × 1020 atoms are pumped into the excited state just before lasing starts. How much energy will be released in a single laser pulse ?

(a) 24.6 J

(b) 22.4 J

(c) 98 J

(d) 48 J

Ans. (d)

Sol. Let N1 and N2 be the populations of the two levelsin laser. The N1 is ground state population and N2 is excited state populations.

Therefore, according to questions, N1 = 4 × 1020 atoms, N2 = 7 × 1020 atoms

Now, energy (released in single laser pulse

53.    In the large hadron collider (LHC), two equal energy proton beams traverse in opposite directions along a circular path of length 27 km. If the total center of mass energy of a proton-proton pair is 14 TeV, which of the following is the best approximation for the proper time taken by a proton to traverse the entire path?

(a) 12 ns

(b) 1.2 µs

(c) 1.2 ns

(d) 0.12 µs

Ans. (a)

Sol. In the center of mass, both protons have same energy. The energy of protons = 14 TeV

Rest mass of the proton = 938 MeV = 938 × 106 eV

Let the proper time is then the time taken will be t = .

The speed of the proton will be approximately equal to speed of light.

54.    Let ES denote the contribution of the surface energy per nucleon in the liquid drop model. The ratio is

(a) 2 : 3

(b) 4 : 3

(c) 5 : 3

(d) 3 : 2

Ans. (b)

Sol. Surface energy per nucleon, Es = as A–1/3 ...(i)

From equations (ii) and (iii), we get

55.    According to the shell model, the nuclear magnetic moment of the nucleus is (Given that for a proton gl = 1, gs = 5.586, and for a neutron gl = 0, gs = – 3.826).

(a) –1.913 µN

(b) 14.414 µN

(c) 4.793 µN

(d) 0

Ans. (c)

Sol. For 13Al27, Z = 13 (odd), N = A – Z = 27 – 13 = 14 (even).

therefore, last proton will be in shell.

Also, for odd proton,

From equation, (i), we get