CSIR NET PHYSICS (DEC 2014)
Previous Year Question Paper with Solution.
1. A particle of mass m is moving in the potential where a, b are positive constants. The frequency of small oscillations about a point of stable equilibrium is
(a)
(b)
(c)
(d)
Ans. (b)
Sol.
For stable equilibrium
2. The radius of Earth is approximately 6400 km. The height h at which the acceleration due to Earth's gravity differs from g at the Earth's surface by approximately 1% is
(a) 64 km
(b) 48 km
(c) 32 km
(d) 16 km
Ans. (c)
Sol. Acceleration due to gravity at heigh H
3. According to the special theory of relativity, the speed v of a free particle of mass m and total energy E is:
(a)
(b)
(c)
(d)
Ans. (c)
Sol.
4. Let denote the position vector of any point in three-dimensional space, and
. Then
(a)
(b)
(c)
(d)
Ans. (d)
Sol. Given: Position vector of any point in three dimensional space is
Therefore,
5. The column vector is a simultaneous eigenvector of
and
if
(a) b = 0 or a = 0
(b) b = a or b = –2a
(c) b = 2a or b = –a
(d) b = a/2 or b = –a/2
Ans. (b)
Sol. Given: The eigenvector is a simultaneous eigenvector of
and
Therefore,
and
For the non-trivial solution of a and b from the above equations, we should have
6. The principal value of the integral is
(a)
(b)
(c)
(d)
Ans. (a)
Sol. Given integral:
Assume, the complex function to be
Singular points of f(z) : z = 0 is pole of order 3
Since, the pole lie on the real axis, therefore,
So, the required integral Imaginary part of
7. Two independent random variables m and n, which can take the integer values 0, 1, 2, ...., , follow the Poisson distribution, with distinct mean values µ and v respectively. Then
(a) the probability distribution of the random variable l = m + n is a binomial distribution
(b) the probability distribution of the random variable r = m – n is also a Poisson distribution
(c) the variance of the random variable l = m + n is equal to µ + v
(d) the mean value of the random variable r = m – n is equal to 0.
Ans. (c)
Sol. Given: m and n are two independent random variables, following Poisson distribution and the corresponding mean values are given to be .
Now, consider the random variable l = m + n
Probability of 'L' no. of successes for the random variable l i.e
(Here, we have assumed that out of L number of successes of random variable z, x number of successes correspond to random variable m and (L-x) number of variables correspond to random variable n)
Therefore,
Using the binomial expansion, , we get
So, the random variable l = m + n also folow the poisson distribution having mean value and for a poisson distribution mean and variance of the distribution is same.
8. The Laurent series expansion of the function f(z) = ez + e1/z about z = 0 is given by
(a)
(b)
(c)
(d)
Ans. (c)
Sol. Given the complex function: f(z) = ez + e1/z
Since, ez is not analytic at z = and e1/z is not analytic at z = 0,
Therefore, the laurrent series expansion will be convergent for 0 < |z| < .
9. The equation of motion of a system described by the time-dependent Lagrangian is
(a)
(b)
(c)
(d)
Ans. (a)
Sol.
Lagrangian equation of motion.
10. A solid sphere of radius R has a charge density, given by , where r is the raidal coordinate and
, a and R are positive constants. If the magnitude of the electric field at r = R/2 is 1.25 times that at r = R, then the value of a is
(a) 2
(b) 1
(c)
(d)
Ans. (b)
Sol. According to Gauss's law
a = 1
11. The electrostatic lines of force due to a system of four point charges is sketched below.
At a large distance r, the leading asymptotic behaviour of the electrostatic potential is proportional to
(a) r
(b) r–1
(b) r–2
(c) r–3
Ans. (d)
Sol. Qmonopole moment
Dipole moment
So, we can say that quadropale moment is exist
12. A charged particle moves in a helical path under the influence of a constant magnetic field. The initial velocity is such that the component along the magnetic field is twice the component in the plane normal to the magnetic field. The ratio l/R of the pitch l to the radius R of the helical path is
(a)
(b)
(c)
(d)
Ans. (b)
Sol.
l is the pitch
From equation (i) and (ii) we get
13. A parallel beam of light of wavelength is incident normally on a thin polymer film with air on both sides. If the film has a refractive index n > 1, then second-order bright fringes can be observed in reflection when the thickness of the film is
(a) /4n
(b) /2n
(c) 3/4n
(d) /n
Ans. (c)
Sol. For normal incidence the condition for mth order bright fringes
where 'n' is the refractive index and is the wave length of the incidence wave.
Now, for second order bright fringes
14. Consider the normalized wavefunction where
is a simultaneous normalized eigenfunction of the angular momentum operators L2 and Lz, with eigenvalues
and
respectively. If
is an eigenfunction of the operator Lx with eigenvalue
, then
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Given the normalized wavefunction:
Now,
Using the operations of the ladder operators i.e.
Since, Φ is an eigenfunction of with eigenvalue
, therefore
15. Let x and p denote, respectively, the coordinate and momentum operators satisfying the canonical commutation relation [x, p] = i in natural units . Then the commutator [x, pe–p] is
(a) i(1 – p)e–p
(b) i(1 – p2)e–p
(c) i(1 – e–p)
(d) ipe–p
Ans. (a)
Sol. First Method:
(Using the property [A,BC] = [A,B]C + B[a,C])
= ie-p – ipe–p = i (1 – p)e–p
Second Method:
16. Suppose the Hamiltonian of a conservative system in classical mechanics is H = xp, where
is a constant and x and p are the position and momentum respectively. The corresponding Hamiltonian in quantum mechanics, in the coordinate representation, is
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Given: Hamiltonian of a conservative system in classical mechanics is H = xp
In Quantum mechnics, the position and momentum operator does not commute i.e.
Therefore, the corresponding Hamiltonian in quantum mechanics can be written as
17. Let and
denote, the normalized eigenstates of a particle with energy eigenvalues E1 and E2 respectively, with E2 > E1. At time t = 0 the particle is prepared in a state
. The shortest time T at which
will be orthogonal to
is
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Given: At t = 0, the wavefunction of the particle is
The wave function of the particle at t = T is
Since, ψ(t = T) and
ψ(t = 0) are orthogonal, then
For shortest time taking n = 0,
18. The I-V characteristics of the diode in the circuit below is given by
where V is measured in volts and I is measured in amperes.
The current I in the circuit is
(a) 10.0 mA
(b) 9.3 mA
(c) 6.2 mA
(d) 6.7 mA
Ans. (c)
Sol.
19. A junction is made between a metal of work function WM, and a doped semiconductor of work function WS with WM > WS. If the electric field at the interface has to be increased by a factor of 3, then the dopant concentration in the semiconductor would have to be
(a) increased by a factor of 9
(b) decreased by a factor of 3
(c) increased by a factor of 3
(d) decreased by a factor of
Ans. (a)
Sol. Electric field at metal-semiconductor interface proportional , where Nd is dopant concentration.
Hence, if Nd is increased by 3 times
Then, E increases 9 times.
20. In a measurement of the viscous drag force experienced by spherical particles in a liquid, the force is found to be proportional to V1/3 where V is the measured volume of each particle. If V is measured to be 30 mm3, with an uncertainty of 2.7 mm3, the resulting relative percentage uncertainty in the measured force is
(a) 2.08
(b) 0.09
(c) 6
(d) 3
Ans. (d)
Sol. Viscous drag force, (given) ... (1)
Where, C is some constant.
Take logarithm both sides, we get
Differentiating equation (2), we get
Percentage error in force is
Given, = 2.7 mm3, V = 30 mm3
Hence percentage error in force is 3.0%.
21. The pressure P of a fluid is related to its number density by the equation of state
where a and b are constants. If the initial volume of the fluid is V0, the work done on the system when it is compressed so as to increase the number density from an initial value of
to
is
(a)
(b)
(c)
(d)
Ans. (d)
Sol. The presence of fluid P is given by
The density,
The work done on the system,
Negative sign shows that work done on the system.
22. The Hamiltonian of a classical particle moving in one dimension is where
is a positive constant and p and q are its momentum and position respectively. Given that its total energy E < E0 the available volume of phase space depends on E0 as
(a)
(b) E0
(c)
(d) is independent of E0
Ans. (a)
Sol. The hamiltonian of a classical particle
The resultant phase diagram is ellipse in p and q2.
Available volume of phase space will have dimension which is product of p and q.
Since, and
The available volume of phase space
23. An ideal Bose gas is confined inside a container that is connected to a particle reservoir. Each particle can occupy a discrete set of single-particle quantum states. If the probability that a particular quantum state is unoccupied is 0.1, then the average number of bosons in that state is
(a) 8
(b) 9
(c) 10
(d) 11
Ans. (b)
Sol. The most probable distribution function for Boson
Average number of particles in a state of energy.
= probability of occupancy
The probability of occupancy + probability of unoccupancy = 1
The probability of occupancy = 1 – 0.1 = 0.9
Let total number of particles N = 10
Then average number of particles = 0.9 × 10 = 9
24. In low density oxygen gas at loe temperature, only the translational and rotational modes of the molecules are excited. The specific heat per molecule of the gas is
(a)
(b) kB
(c)
(d)
Ans. (d)
Sol. Oxygen molecule is diatomic
Total degree of freedom of a oxygen molecule = 3 × 2 = 6
Translational DOF = 3
Rotationa DOF = 2
Vibrational DOF = 1
According to equipartition theorem the average thermal kinetic energy associated with each DOF is
Then average energy of a oxygen molecule at low temperature
The specific heat of a molecule
25. Consider the amplifier circuit comprising of the two op-amps A1 sand A2 as shown in the figure If the input ac signal source has an impedance of 50k, which of the following statements is true?
(a) A1 is required in the circuit because the source impedance is much greater than r
(b) A1 is required in the circuit because the source impedance is much less than R
(c) A1 can be eliminated from the circuit without affecting the overall gain
(d) A1 is required in the circuit if the output has to follow the phase of the input signal
Ans. (a)
Sol. A1 is required in the circuit to reduce to source impedance otherwise if source is directly applied to A2 the gain will decrease by large amount.
26. A plane electromagnetic wave incident normally on the surface of a material is partially reflected. Measurements on the standing wave in the region in front of the interface show that the ratio of the electric field amplitude at the maxima and the minima is 5. The ratio of the reflected intensity to the incident intensity is
(a) 4/9
(b) 2/3
(c) 2/5
(d) 1/5
Ans. (a)
Sol. Let a and b are the amplitude of the incidence wave.
Therefore, the intensity ratio of incidence wave is
27. The scalar and vector potentials and
are determined up to a gauge transformation
and
where
is an arbitrary continuous and differentiable function of
and t. If we further impose the Lorenz gauge condition
then a possible choice for the gauge function
is (where
are nonzero constants with
)
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
28. A non-relativistic particle of mass m and charge e, moving with a velocity and acceleration
, emits radiation of intensity I. What is the intensity of the radiation emitted by a particle of mass m/2, charge 2e, velocity
and acceleration
?
(a) 16 I
(b) 8 I
(c) 4 I
(d) 2 I
Ans. (a)
Sol. An accelerating particle is emitting radiation and the intensity of the radiation is
29. Let and
be complex numbers. Which of the following sets of matrices forms a group under matrix multiplication?
(a)
(b)
(c)
(d)
Ans. (d)
Sol. SU(2) is a group of special unitary matrices (having determinant 1) of order 2, under matrix multiplication as the law of composition. The elements of this group can be represented as
where the determinant of the matrix
30. The expression (where
is the Levi-Civita symbol,
are the position, momentum and angular momentum respectively, and {A, B} represents the Poisson bracket of (A and B) simplifies to
(a) 0
(b) 6
(c)
(d)
Ans. (b)
Sol.
31. A mechanical system is described by the Hamiltonian . As a result of the canonical transformation generated by
, the Hamiltonian in the new coordinate Q and momentum P becomes
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
Differential relation for F(q, Q)
32. Let where
are the Pauli matrices. If
and
are two arbitrary constant vectors in three dimensions, the commutator
is equal to (in the following I is the identity matrix)
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
Therefore,
33. Consider the function of a complex variable
. The singularities of f(z) are as follows:
(a) branch points at z = 1 and z = ; and a pole at z = 0 only for
(b) branch points at z = 1 and z = ; and a pole at z = 0 for all θ other than 0 < θ < 2π
(c) branch points at z = 1 and z = ; and a pole at z = 0 for all θ
(d) branch points at z = 0, z = 1 and z =
Ans. (b)
Sol. Given Complex function:
Therefore, f(z) is not differentiable at z = 0,1, ∞ .
The points z = 1, comes from ln (1 – z) and since ln (1 – z) is a infinite valued function of z, then z = 1,
are two branch points of f(z).
On the other hand, z = 0 comes from i.e z = 0 is a simple pole of f(z). So, the residue of the function f(z) at z = 0 is
For the principal branch (k = 0) between if we take
= 0, then the residue of the function f(z) at z = 0 will be zero. But for the other branches of the function f(z), the residue will be non-zero. So, z = 0 will be a pole of f(z) for all
other than
34. The function satisfies the differential equation
(a)
(b)
(c)
(d)
Ans. (c)
Sol. The solution of the Bessel differential equation
So, for the given function f(x), k will be 1. Therefore, it satisfies the differential equation
35. The probe Mangalyaan was sent recently to explore the planet Mars. The inter-planetary parts of the trajectory is approximately a half-ellipse with the Earth (at the time of launch), Sun and Mars (at the time the probe reaches the destination) forming the major axis. Assuming that the orbits of Earth and Mars are approximately circular with radii RE and RM, respectively, the velocity (with respect to the Sun) of the probe during its voyage when it is at a distance r(RE << r << RM) from the Sun, neglecting the effect of Earth and Mars, is
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Major axis of ellipse = 2a = RE + RM
There is no effect of earth and mars so.
Total energy of the Mangalyaan is
If v be its speed when it is at a distance r from sun then,
36. A large MOS transistor consists of N individual transistors connected in parallel. If the only form of noise in each transistor is 1/f noise, then the equivalent voltage noise spectral density for the MOS transistor is
(a) 1/N times that of a single transistor
(b) 1/N2 times that of a single transistor
(c) N times that of a single transistor
(d) N2 times that of a single transistor
Ans. (a)
Sol.
N–Mos transistors connected is parallel at transistor are having same I/P.
N-transistors having each FT. f(k) combined FT is Nf(k).
We need ....... spectral density is given by |f(x)|
For N-transistors N|f(x)| for is given by
.
Spectral density becomes 1/N times.
37. Consider a particle of mass m in the potential V(x) = a|x|, a > 0. The energy eigen-values En(n = 0, 1, 2,....), in the WKB approximation, are
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Given: A particle of mass m is moving under the following potential: V(x) = a |x|
At classical turning points, E = V(x)
Using the quantization condition of WKB approximation,
According to the question, and
(for non-rigid walls)
Therefore,
So, Energy eigenvalues of the particle
38. When laser light of wavelength falls on a metal scale with 1mm engravings at a grazing angle of incidence, it is diffracted to form a vertical chain of diffraction spots on a screen kept perpendicular to the scale. If the wavelength of the laser is increased by 200nm, the angle of the first-order diffraction spot changes from 5º to
(a) 6.60º
(b) 5.14º
(c) 5.018º
(d) 5.21º
Ans. (b)
Sol. The meter scale with engravings will act as a reflection grating and the condition for the nth order diffraction maxima is
where d is separation between two consecutive rulings on the scale and is angle of diffraction.
Now,
where is change in the laser wavelength and
is corresponding change in angle of diffraction.
Therefore, the angle of diffraction will be 5.130.
39. Let (where c0 and c1 are constants with
) be a linear combination of the wavefunctions of the ground and first excited states of the one-dimensional harmonic oscillator. For what value of c0 is the expectation value
a maximum?
(a)
(b)
(c)
(d)
Ans. (c)
Sol. Given: with
Expectation value of position =
Using the relations, , we get
The expectation value of position is maximum when c0c1 is maximum. The maximum value of c0c1 occurs for c0 = c1. Since,
, then
Therefore, the maximum of expectation value of position is .
40. Consider a Low Pass (LP) and a High Pass (HP) filter with cut-off frequencies fLP and fHP, respectively, connected in series or in parallel configurations as shown in the Figures A and B below.
Which of the following statements is correct?
(a) For fHP < fLP, A acts as a Band Pass filter and B acts as a Band Reject filter
(b) For fHP > fLP, A stops the signal from passing through and B passes the signal without filtering
(c) For fHP < fLP, A acts as a Band Pass filter and B passes the signal without filtering
(d) For fHP > fLP, A passes the signal without filtering and B acts as a Band Reject filter
Ans. (c)
Sol.
fHP < fLP
A act as band pass
B passes without filltering.
For figure B parallel means additing.
Means all signal passes without filltering.
41. A collection N of non-interacting spins Si, i = 1, 2, ....., N, is kept in an external magnetic field B at a temperature T. The Hamiltonian of the system is
. What should be the minimum value of
for which the mean value
?
(a)
(b) 2 ln 2
(c)
(d) N ln 2
Ans. (c)
Sol. Each spin can take value ±1. The energy of each spin in external magnetic field B with Si = +1 and Si = –1 are and
respectively.
The partition function of each spin
The partition function of the system
Helmholtz free energy A = –kT ln Z
Magnetization,
Average value of spin
Let x =
42. When a gas expands adiabatically from volume V1 to V2 by a quasi-static reversible process, it cools from temperature T1 to T2. If now the same process is carried out adiabatically and irreversibly, and is the temperature of the gas when it has equilibrated, then
(a)
(b)
(c)
(d)
Ans. (b)
Sol. For reversible adiabatic process the change in entropy = 0 and for irreversible process
> 0
Sf – Si> 0
As entropy in final state for irreversible process will be larger in the comparsion of reversible process so work done will be smaller and kinetic energy will be larger for irreversible processes i.e. T'2 > T2.
43. A random walker takes a step of unit length in the positive direction with probability 2/3 and a step of unit length in the negative direction with probability 1/3. The mean displacement of the walker after n steps is
(a) n/3
(b) n/8
(c) 2n/3
(d) 0
Ans. (a)
Sol. The average or mean displacement ,
After one step,
After n steps, the mean displacement
44. The Hamiltonian H0 for a three-state quantum system is given by the matrix . When perturbed by
where
, the resulting shift in the energy eigenvalue E0 = 2 is
(a)
(b)
(c)
(d)
Ans. (c)
Sol. Given: Unperturbed Hamiltonian and Perturbed Hamiltonian
The unperturbed hamiltonian has unperturbed energy eigenvalues E0 = 1, 2, 2, out of which E0 = 2 is a degenerate eigenvalue having degenaracy 2. The nomralized eigenvectors corresponding to these eigenvalues are respectively
If is the energy correction due to the presence of the perturbation H' in the unperturbed eigen value E0 = 2, then
45. The ground state energy of the attractive delta function potential , where b > 0, calculated with the variational trial function
is
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Given: Potential and variational trial function
46. If the binding energy B of a nucleus (mass number A and charge Z) is given by where aV = 16 MeV, aS = 16 MeV, asym = 24 MeV and ac = 0.75 MeV, then the Z for the most stable isobar for a nucleus with A = 216 is
(a) 68
(b) 72
(c) 84
(d) 92
Ans. (c)
Sol.
47. Consider the crystal structure of sodium chloride which is modeled as a set of touching spheres. Each sodium atom has a radius r1 and each chlorine atom has a radius r2. The centres of the spheres form a simple cubic lattice. The packing fraction of this system is
(a)
(b)
(c)
(d)
Ans. (b)
Sol. Radius of Na atom = r1
Radius of cl atom = r2
48. Consider two crystalline solids, one of which has a simple cubic structure, and the other has a tetragonal structure. The effective spring constant between atoms in the c-direction is half the effective spring constant between atoms in the a and b directions. At low temperatures, the behaviour of the lattice contribution to the specific heat will depend as a function of temperature T as
(a) T2 for the tetragonal solid, but as T3 for the simple cubic solid
(b) T for the tetragonal solid and as T3 for the simple cubic solid
(c) T for both solids
(d) T3 for both solids
Ans. (d)
Sol. Lattice heat capacity of a 3-dimensional crystal is independent of crystal system. Hence, lattice heat capacity for both crystals would be proportional to T3 (Debye law).
49. An atomic transition in a magnetic field 1 Tesla shows Zeeman splitting. Given that the Bohr magneton µB = 9.27 × 10–24 J/T, and the wavelength corresponding to the transition is 250 nm, the separation in the Zeeman spectral lines is approximately
(a) 0.01 nm
(b) 0.1 nm
(c) 1.0 nm
(d) 10 nm
Ans. (a)
Sol. Given: The transition under magnetic field will show normal zeeman splitting.
This normal zeeman pattern will consist of three equidistant spectral lines having energy separation and the corresponding wavelength separation can be found out as follows:
50. If the leading anharmonic correction to the energy of the n-th vibrational level of a diatomic molecule is with xe = 0.001, the total number of energy levels possible is approximately
(a) 500
(b) 1000
(c) 250
(d) 750
Ans. (a)
Sol. The energy of the nth vibrational level of a diatomic molecule is
For the highest possible energy level, energy will be maximum. Therefore,
The total number of energy levels possible is
51. A superconducting ring carries a steady current in the presence of a magnetic field normal to the plane of the ring. Identify the incorrect statement.
(a) The flux passing through the superconductor is quantized in units of hc/e
(b) The current and the magnetic field in the superconductor are time independent
(c) The current density and
are related by the equation
, where
is a constant
(d) The superconductor shows an energy gap which is proportional to the transition temperature of the superconductor
Ans. (a)
Sol. Option (a) is incorrect statement about super conductors as flux is quantized in the units of .
52. The effective spin-spin interaction between the electron spin and the proton spin
in the ground state of the Hydrogen atom is given by
. As a result of this interaction, the energy levels split by an amount
(a)
(b)
(c)
(d)
Ans. (c)
Sol. Effective spin-spin interaction between electron spin and proton spin in the ground state of hydrogen atom is
The corresponding shift in energy is
The difference in energy levels corresponding to triplet state s = 1 and singlet state s = 0 is
53. In deep inelastic scattering, electrons are scattered off protons to determine if a proton has any internal structure. The energy of the electron for this must be at least
(a) 1.25 × 109 eV
(b) 1.25 × 1012 eV
(c) 1.25 × 106 eV
(d) 1.25 × 108 eV
Ans. (a)
Sol. The De-Broglie wavelength of electron should be of the order of the dimensions of proton (~ 1 fm).
Alternatively, we know hc = 1242 eV.nm = 1.242 × 109 eV.fm
54. The power density of sunlight incident on a solar cell is 100 mW/cm2. Its short circuit current density is 30 mA/cm2 and the open circuit voltage is 0.7 V. If the fill factor of the solar cell decreases from 0.8 to 0.5 then the percentage efficiency will decrease from
(a) 42.0 to 26.2
(b) 24.0 to 16.8
(c) 21.0 to 10.5
(d) 16.8 to 10.5
Ans. (d)
Sol. Fill factor of solar cell is given by
where Im and Vm are currents and voltages corresponding to maximum power.
ISC and VOC are short circuit current and open circuit voltages, respectively.
Given that, ISC = 30 mA/cm2
VOC = 0.7 volt
Therefore, if = 0.8 then maximum power density drawn from solar cell is
Percentage efficiency for e, is given by
55. Consider the four processes
(i)
(ii)
(iii)
(iv)
Which of the above is/are forbidden for free particles?
(a) only (ii)
(b) (ii) and (iv)
(c) (i) and (iv)
(d) (i) and (ii)
Ans. (d)
Sol. Proton is stable particle. So, free proton will not decay.
for free proton. In
, the charge is not conserved.