TIFR GS 2017 Question Paper Physics Y

PHY – Y
GS-2017-Y (Physics)
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Wrtten Test in PHYSICS – December 11, 2016
Duration : 3 (Three) Hours

NAME:_____________________________________REF. CODE:________
Please Read These Instructions Carefully Before Attempting the Questions
1. Please fill in details about name, reference code etc. on the answer sheet and
question paper. Use only blue/black ballpoint pen. The Answer Sheet is machine
–readable and will not read other colours.
2. This test consists of three parts: Section A, Section B and Section C. You must
answer questions according to the programme you are applying for.
Candidates applying for Must answer Should not attempt
Integrated M.Sc.-Ph.D. Section A + Section B Section C
Ph.D. Section A + Section C Section B
Section A has 25 questions : 1 – 15 are multiple-choice; 16 – 25 are numerical.
Section B has 15 questions : 26 – 35 are multiple-choice; 36 – 40 are symbolic.
Section C has 15 questions : 41 – 50 are multiple-choice; 51 – 55 are symbolic.
3. Indicate your ANSWER ON THE ANSWER SHEET as follows.
Multiple choice questions have four options (a), (b), (c) and (d), of which only
one option is correct. Indicate the answers by filling up the bubble on the
Answer Sheet corresponding to the correct option. If more than one bubble is
filled in, it will be treated as not answered.
Numerical questions have answers which are 3 (three) digit integers. Indicate
the answers by filling in the corresponding bubbles on the Answer Sheet.
Unless all three bubbles for a given question are filled, it will be treated as not
answered. (See inside for details.)
Symbolic questions have answers which are a number, a short formula or a
word. Indicate the answers by writing in the boxes on the Answer Sheet next
to the appropriate question numbers. (See inside for details.)
4. The marking for these questions shall be as follows.
If the answer is Multiple-choice Numerical Symbolic
Correct ൅͵ ൅ͷ ൅ͷ
Incorrect െͳ Ͳ Ͳ
Not attempted Ͳ Ͳ Ͳ
Multiple options marked Ͳ  
Note that only multiple-choice type questions have negative marking.

continued on next page…

5. Candidates are advised to mark the Answer Sheet only when they are sure of
the answer. Till then, they may mark the answers on the question paper.
6. Rough work may be done on blank pages of the question paper. If needed, you
may ask for extra sheets from the invigilators.
7. Use of scientific, non-programmable calculators is permitted. Calculators
which plot graphs are NOT allowed. Multiple-use devices, such as cell phones,
smartphones, etc. CANNOT be used as calculators.
8. Do NOT ask the invigilators for clarifications regarding the questions. They
have been instructed not to respond to any such queries. In case a
correction/clarification is deemed necessary, it will be announced in the
examination hall.
9. A list of useful physical constants is given on the next page. Make sure to use
only these values in answering the questions, especially those of numeric type.

USEFUL CONSTANTS
Symbol Name/Definition Value
ܿ speed of light in vacuum ͵ ൈ ͳͲ଼ m s-1
԰ reduced Planck constant (ൌ ݄Ȁʹߨ) ͳǤͲͶ ൈ ͳͲିଷସ Js
‫ܩ‬ே gravitational constant ͸Ǥ͸͹ ൈ ͳͲିଵଵ m3 kg-1 s-2
‫ٖܯ‬ solar mass ͳǤͻͺͻ ൈ ͳͲଷ଴ kg
ߝ଴ permittivity of free space ͺǤͺͷ ൈ ͳͲିଵଶ F m-1
ߤ଴ permeability of free space Ͷߨ ൈ ͳͲି଻ N A-2
݁ electron charge (magnitude) ͳǤ͸ ൈ ͳͲିଵଽ C
݉௘ electron mass ͻǤͳ ൈ ͳͲିଷଵ kg
ൌ ͲǤͷ MeV/c 2
ܽ଴ Bohr radius 0.51 Å
ionisation potential of H atom 13.6 eV
ܰ஺ Avogadro number ͸ǤͲʹ͵ ൈ ͳͲଶଷ mol-1
݇஻ Boltzmann constant ͳǤ͵ͺ ൈ ͳͲିଶଷ J K-1
ൌ ͺǤ͸ͳ͹͵ ൈ ͳͲିହ eV K-1
ܴ ൌ ܰ஺ ݇஻ gas constant ͺǤ͵ͳ J mol-1 K-1
ߛ ൌ ‫ܥ‬௣ Ȁ‫ܥ‬௏ ratio of specific heats: monatomic gas 1.67
diatomic gas 1.40
ߪ Stefan-Boltzmann constant ͷǤ͸͹ ൈ ͳͲି଼ W m-2 K-4
ߙ fine structure constant (ൌ ݁ ଶ ȀͶߨߝ଴ ԰ܿ) 1/137
݃ acceleration due to gravity 9.8 m s-2
ܴா radius of the Earth ͸ǤͶ ൈ ͳͲଷ Km
ܴௌ radius of the Sun ͹ ൈ ͳͲହ Km
݉௣ proton mass (ൎ ʹͲͲͲ݉௘ ) ͳǤ͹ ൈ ͳͲିଶ଻ kg
ൌ ͻ͵ͺǤʹ MeV/c 2
݉௡ neutron mass (ൎ ʹͲͲͲ݉௘ ) ͳǤ͹ ൈ ͳͲିଶ଻ kg
ൌ ͻ͵ͻǤ͸ MeV/c 2
UNIT CONVERSIONS
Symbol Name/Definition Value
1 A.U. mean distance of Earth from Sun ͳǤͷ ൈ ͳͲଽ km
1 a.m.u. atomic mass unit ͳǤ͸ ൈ ͳͲିଶ଻ kg
ൌ ͻ͵ͳǤͷ MeV/c 2
1 eV electron Volt ͳǤ͸ ൈ ͳͲିଵଽ J
1T Tesla ͳͲସ gauss
1 bar mean atmospheric pressure at 00 C ͳǤͲͳ ൈ ͳͲହ Pa (= N m-2)
1Å Ångstrom unit 10-8 cm
԰ܿ conversion constant ͵Ǥͳ͸ ൈ ͳͲିଶ଺ J m-1
ൌ ͲǤͳͻ͹͵ GeV fm

GS-2017-Y (Physics)
Section A
Q. 1 – 25 : to be attempted by ALL candidates

PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

The following questions (1 – 15) are all of multiple-choice type. For
every question, four options (a), (b), (c) and (d) are given, of which only
one is correct. Indicate the correct option on the OMR by filling the
bubble.

A correct answer will be awarded ൅૜ marks and an incorrect answer
will be awarded െ૚ mark. If the question is not attempted, no marks
will be awarded.

1. Denote the commutator of two matrices ‫ ܣ‬and ‫ ܤ‬by ሾ‫ܣ‬ǡ ‫ܤ‬ሿ ൌ ‫ ܤܣ‬െ ‫ ܣܤ‬and the anti-
commutator by ሼ‫ܣ‬ǡ ‫ܤ‬ሽ ൌ ‫ ܤܣ‬൅ ‫ܣܤ‬Ǥ

If ሼ‫ܣ‬ǡ ‫ܤ‬ሽ ൌ Ͳ, we can write ሾ‫ܣ‬ǡ ‫ܥܤ‬ሿ ൌ

(a) െ‫ܤ‬ሾ‫ܣ‬ǡ ‫ܥ‬ሿ (b) ‫ܤ‬ሼ‫ܣ‬ǡ ‫ܥ‬ሽ (c) ሾ‫ܣ‬ǡ ‫ܥ‬ሿ‫ܤ‬ (d) െ‫ܤ‬ሼ‫ܣ‬ǡ ‫ܥ‬ሽ

2. Consider the waveform ‫ݔ‬ሺ‫ݐ‬ሻ shown in the diagram below.
‫ݔ‬
ͳ
െʹܶ െܶ
‫ݐ‬
Ͳ ܶ ʹܶ ͵ܶ

െͳ
The Fourier series for ‫ݔ‬ሺ‫ݐ‬ሻwhich gives the closest approximation to this waveform is

ʹ ߨ‫ͳ ݐ‬ Ͷߨ‫ͳ ݐ‬ ͵ߨ‫ݐ‬
(a) ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ൤ ‘• െ ‘• ൅ ‘• ൅  ǥ൨
ߨ ܶ ʹ ܶ ͵ ܶ
ʹ ʹߨ‫ͳ ݐ‬ Ͷߨ‫ͳ ݐ‬ ͸ߨ‫ݐ‬
(b) ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ൤െ ‘• ൅ ‘• െ ‘• ൅  ǥ൨
ߨ ܶ ʹ ܶ ͵ ܶ
ʹ ߨ‫ͳ ݐ‬ ʹߨ‫ͳ ݐ‬ ͵ߨ‫ݐ‬
(c) ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ൤•‹ െ •‹ ൅ •‹ ൅  ǥ൨
ߨ ܶ ʹ ܶ ͵ ܶ
ʹ ߨ‫ͳ ݐ‬ ʹߨ‫ͳ ݐ‬ ͵ߨ‫ݐ‬
(d) ‫ݔ‬ሺ‫ݐ‬ሻ ൌ ൤െ•‹ ൅ •‹ െ •‹ ൅  ǥ൨
ߨ ܶ ʹ ܶ ͵ ܶ

3. A solid tetrahedron (solid with four plane sides) has the following projections
(drawn to scale) when seen from three different sides:

Left Top Right
When viewed from the front, its projection will be

(a) (b)

(c) (d)

4. A small elastic ball of mass ݉ is placed at the apex of a Ͷͷ௢ inclined plane as
shown in the figure below.

The ball is allowed to slip without friction down the plane (along the dotted line),
hit the ground (as shown) and bounce along it. If the height of the inclined plane
is ݄ and the coefficient of restitution between the ball and the ground is ͲǤͷ, then
the distance AB, as marked on the figure, will be

(a) ൫ͳ ൅ ξʹ൯݄ (b) ͵ξʹ݄ (c) ͵݄ (d) ʹ݄

5. Two masses ͵݉ and ʹ݉ are suspended vertically by
identical massless springs, each of stiffness constant ݇. ݇
The mass ʹ݉is suspended from the mass ͵݉ and the
͵݉
mass ͵݉is suspended from a rigid support, as shown in
the figure. If only vertical motion is permitted, the ݇
frequencies of small oscillations of this system are ʹ݉

݇ ͵݇ (b ݇ ݇ ݇ ݇ ʹ݇ ͵݇
(a) ඨ ǡඨ ඨ ǡඨ (c) ඨ ǡඨ (d) ඨ ǡඨ
݉ ʹ݉ ) ʹ݉ ͵݉ ݉ ͸݉ ͵݉ ʹ݉

6. Two long hollow conducting cylinders, each of height ݄, are placed concentrically
on the ground, as shown in the figure (top view). The outer cylinder is grounded,
while the inner cylinder is insulated. A positive charge (the black dot in the figure)
is placed between the cylinders at a height ݄Ȁʹ from the ground.

Which of the following figures gives the most accurate representation (top view) of
the lines of force ?

(a) (b)

(c) (d)

7. A common model for the distribution of charge in a hydrogen atom has a point-like
proton of charge ൅‫ݍ‬଴ at the centre and an electron with a static charge density
distribution
‫ݍ‬଴ ିଶ௥Ȁ௔
ߩሺ‫ݎ‬ሻ ൌ  െ ݁
ߨܽଷ
ሬԦ at ‫ ݎ‬ൌ ܽ due to this system of charges will
where ܽ is a constant. The electric field ‫ܧ‬
be

ͷ‫ݍ‬଴ ͷ‫ݍ‬଴
(a) െ Ͷߨ߳ ݁ ଶ ܽଶ ‫ݎ‬Ƹ (b) െ ‫ݎ‬Ƹ
଴ Ͷߨ߳଴ ݁ܽଶ

͵‫ݍ‬଴ ͷ‫ݍ‬଴
(c) Ͷߨ߳ ݁ ଶ ܽଶ ‫ݎ‬Ƹ  (d)  ‫ݎ‬Ƹ
଴ Ͷߨ߳଴ ݁ ଶ ܽଶ

8. A rectangular metallic loop with sides ‫ܮ‬ଵ and ‫ܮ‬ଶ is
placed in the vertical plane, making an angle ߮ with
respect to the ‫ݔ‬-axis, as shown in the figure, and a
spatially uniform magnetic field ‫ܤ‬ሬԦ  ൌ ‫ݕܤ‬ො is applied.
The loop is free to rotate about the‫ݖ‬Ƹ axis (shown in
the figure with a double line). The magnetic field
changes with time at a constant rate
݀‫ܤ‬
ൌߢ
݀‫ݐ‬
If the resistance of the loop is R, the torque ߬
required to prevent the loop from rotating will be

ሺ‫ܮ‬ଵ ‫ܮ‬ଶ ሻଶ ሺ‫ܮ‬ଵ ‫ܮ‬ଶ ሻଶ
(a) ߢ‫ܤ‬ •‹ ߮ ‫ݖ‬Ƹ (b) െߢ‫ܤ‬ •‹ ʹ߮ ‫ݖ‬Ƹ
ʹܴ ʹܴ
ሺ‫ܮ‬ଵ ‫ܮ‬ଶ ሻଶ ሺ‫ܮ‬ଵ ‫ܮ‬ଶ ሻଶ
(c) ߢ‫ܤ‬ •‹ ߮ ‘• ߮ ‫ݖ‬Ƹ  (d) െߢ‫ܤ‬ •‹ ߮ ‫ݖ‬Ƹ
ܴ ܴ

9. Consider the 1-D asymmetric double-well potential ܸሺ‫ݔ‬ሻas sketched below.
ܸሺ‫ݔ‬ሻ

‫ݔ‬

The probability distribution ‫݌‬ሺ‫ݔ‬ሻ of a particle in the ground state of this potential is
best represented by

(a) ‫݌‬ሺ‫ݔ‬ሻ (b) ‫݌‬ሺ‫ݔ‬ሻ

‫ݔ‬ ‫ݔ‬
(c) ‫݌‬ሺ‫ݔ‬ሻ (d) ‫݌‬ሺ‫ݔ‬ሻ

‫ݔ‬ ‫ݔ‬

10. The normalized wave function of a particle can be written as
ஶ
ͳ ௡
Ȳሺ‫ݔ‬ሻ ൌ ܰ ෍ ൬ ൰ ߮௡ ሺ‫ݔ‬ሻ
௡ୀ଴
ξ͹
where ߮௡ ሺ‫ݔ‬ሻ are the normalized energy eigenfunctions of a given Hamiltonian. The
value of ܰ is

(a) ට൫͸ െ ʹξ͹൯Τ͹ (b) ඥͳΤ͹

(c) ඥ͵Τ͹ (d) ඥ͸Τ͹

11. Consider a system of non-interacting particles with integer angular momentum ‫ ܬ‬at
a temperature ܶ. This system is placed in a magnetic field ‫ ܤ‬in the z direction. The
energy of a state with ‫ܬ‬௭ ൌ ݉԰ is
‫ܧ‬௠ ൌ ݉ߤ஻ ‫ܤ‬
with ߤ஻ ൐ ͲǤ The fractional magnetization of the particles as a function of ߤ஻ ‫ܤ‬Ȁ݇஻ ܶ
can be represented as

(a) (b)

(c) (d)

12. The separation between neighbouring absorption lines in a pure rotational spectrum
of the hydrogen bromide (HBr) molecule is ʹǤʹ͵ meV. If this molecule is considered
as a rigid rotor and the atomic mass number of Br is 80, the corresponding
absorption line separation in deuterium bromide (DBr) molecule, in units of meV,
would be

(a) ʹǤʹ͵Ͷ (b) ͳǤͳͳͷ (c) ͶǤͶ͸ͳ (d) ͳǤͳʹͺ

13. Consider a 2-D square lattice. The ratio of the kinetic energy of a free electron at a
corner of the first Brillouin zone (‫ܧ‬௖ ) to that of an electron at the midpoint of a side
face of the same zone (‫ܧ‬௠ ) is ‫ܧ‬௖ Τ‫ܧ‬௠ ൌ

(a) ʹ (b) ξʹ (c) 1 (d) ͳȀʹ

14. A current source produces a square wave ‫ܫ‬ሺ‫ݐ‬ሻ of ͳǤͲ V peak-to-peak voltage and is
used to drive the RC circuit shown below.

Which of the following represents the correct voltage across the capacitor C ?

(a) (b)

(c) (d)

15. The output (Y) of the following circuit will be

(a) ‫ܥ‬ҧ (b) ‫ܤ‬ത (c) ‫ܣ‬ҧ (d) ‫ܣ‬ҧ ൅ ‫ ܤ‬൅ ‫ܥ‬ҧ

Section A continues (to be answered by ALL candidates)
PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

The answers to the following questions (16 – 25) are all integers of 3 (three) digits
each. You may round off decimal parts, e.g. ૚૛૛Ǥ ૞ ൑ ࢞ ൏ ૚૛૜Ǥ ૞ǡas ࢞ ൌ ૚૛૜ and e.g.
૚૛૜Ǥ ૞ ൑ ࢞ ൏ ૚૛૝Ǥ ૞as ࢞ ൌ ૚૛૝ and so on.

Use only values of constants given in the table ‘USEFUL CONSTANTS’.

Answer these questions on the OMR by filling in bubbles as you did for your reference
code. Note that if the answer is, e.g. 25, you must fill in 025 and if it is, e.g. 5, you
must fill in 005. If it is 0, you must fill in 000. If the zeros are not filled in (where
required), the answer will not be credited.

A correct answer will be awarded ൅૞ marks.
There are NO NEGATIVE MARKS for these questions.

16. A space telescope in orbit around the Earth discovers a new planet, which is observed
to move around the Sun by an angle of ͶǤ͹ʹ milliradians in a year. Assuming a
circular orbit, estimate the distance, in A.U., of the planet from the Sun.
Ans = 121
17. The matrix
ͳͲͲξʹ ‫ݔ‬ Ͳ
ቌ െ‫ݔ‬ Ͳ െ‫ ݔ‬ቍ
Ͳ ‫ݔ‬ ͳͲͲξʹ
where ࢞ ൐ Ͳ, is known to have two equal eigenvalues. Find the value of ‫ݔ‬.

Ans = 050

18. A beam of plane microwaves of wavelength 12 cm strikes the surface of a dielectric
at Ͷͷ୭ . If the refractive index of the dielectric is 4/3, what will be the wavelength, in
units of mm, of the microwaves inside the dielectric?
Ans = 090

19. A system of particles occupying single-particle levels and obeying Maxwell-
Boltzmann statistics is in thermal equilibrium with a heat reservoir at temperatureܶ.
If the population distribution in the non-degenerate energy levels is as shown in the
table below, what would be the temperature of the system in degree Kelvin?

Energy (eV) PopulationΨ
͵ͲǤ͵Ͳ ͵Ǥͳ͸
Given +5 marks to
ʹͳǤ͸Ͳ ͺǤ͸ͻ
all students.
ͳ͵ǤͲͳ ʹ͵ǤͷͶ
ͶǤ͵ͳ ͸ͶǤ͸ͳ

20. A thermally isolated container stores ܰଶ gas at ʹ͹ǤʹͶԨ at one atmospheric pressure.
Suddenly the pressure of the gas is increased to two atmospheric pressures.
Assuming ܰଶ to behave as an ideal gas, estimate the change in temperature of the
gas, in Celsius degrees (Ԩ).
Ans = 066

21. A signal is to be sent from a coaxial cable with impedance 40 Ω into a second coaxial
cable with impedance 60 Ω. We can prevent reflection at the joint between the cables,
by adding an impedance in parallel to the second cable. What should be the value,
in units of Ohms (Ω), of that impedance?
Ans = 120

22. An AC voltage source has an internal resistance of 50 Ω and is specified to deliver an
rms voltage of 50 V to a matched load. If you connect this AC source to a cathode-
ray oscilloscope with 1 MΩ input setting, what will be the peak-to-peak voltage you
observe?
Ans = 283

23. The energy of an electron in the ground state of the ‡ atom is െ͹ͻ‡. Considering
the Bohr model of the atom, what would be 10 times the first ionization potential for
a ‡ା ion, in units of eV?

Ans = 246

24. A quantum mechanical system consists of a one-dimensional infinite box, as
indicated in the figures below.

‫ܧ‬଴

3 (three) identical non-interacting spin-½ particles, are first placed in the box, and
the ground state energy of the system is found to be ‫ܧ‬଴ ൌ 18 eV. If 7 (seven) such
identical particles are placed in the box, what will be the ground state energy, in
units of eV?
Ans = 132

25. Cosmic ray muons, which decay spontaneously with proper lifetime ʹǤʹߤ s, are
produced in the atmosphere, at a height of 5 km above sea level. These move straight
downwards at 98% of the speed of light.
Find the percent ratio ͳͲͲ ൈ ሺܰ஺ Ȁܰ஻ ሻ of the number of muons measured at the top of
two mountains A and B, which are at heights 4,848 m and 2,682 m respectively
above mean sea level.

Ans = 052

GS-2017-Y (Physics)
Section B
Q. 26 – 40 : to be attempted only by all candidates for
Integrated M.Sc.-Ph.D. programme.
(Candidates for Ph.D. programme will get NO credit for attempting this section.)

PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

The following questions (26 – 35) are all of multiple-choice type. For
every question, four options (a), (b), (c) and (d) are given, of which only
one is correct. Indicate the correct option on the OMR by filling in the
bubble next to the correct label.

A correct answer will be awarded ൅૜ marks and an incorrect answer
will be awarded െ૚ mark. If the question is not attempted, no marks
will be awarded.

26. A unitary matrix ܷ is expanded in terms of a Hermitian matrix ‫ܪ‬, such that

ܷ ൌ ݁ ௜గுȀଶ
If we know that

ͳൗ Ͳ ξ͵ൗ
ʹ ʹ
‫ܪ‬ൌ‫Ͳ ۇ‬ ͳ Ͳ ‫ۊ‬
ξ͵ Ͳ െ ͳൗʹ‫ی‬
‫ ۉ‬ൗʹ

then ܷ must be

݅ Τʹ Ͳ ݅ξ͵Τʹ ݅ ͳΤʹ ξ͵Τʹ
(a) ቌ Ͳ ݅ Ͳ ቍ (b) ቌ ͳΤʹ ݅ ͳ Τ ʹቍ
݅ξ͵Τʹ Ͳ െ ݅ Τʹ ξ͵Τʹ ͳΤʹ ݅

ͳ Ͳ ξ͵ ʹ݅ ͳ ξ͵Τʹ
(c) ቌ Ͳ ʹ Ͳቍ (d) ቌ ͳ ʹ݅ Ͳ ቍ
ξ͵ Ͳ െͳ ξ͵Τʹ Ͳ ʹ݅

27. A liquid is flowing through a capillary tube of inner radius r under the influence of
an external pressure P. The uncertainties in the measurements of P and r are found
to be 2% and 1%, respectively. The uncertainty in the flow of liquid per second is

(a) ͵Ǥ͸ͳΨ (b) 2.23% (c) 2.83% (d) 4.47%

28. A uniform solid sphere ଵ of radius ‫ ݎ‬and mass ݉ is rolling S1
without slipping on top of another sphere ଶ of radius ܴ, as N ‫ݎ‬
shown in the figure. Initially, ଵ was at rest directly on top ܴߠ P
of ଶ , and then it started rolling down under the influence
of gravity. The point of contact P subtends an instantaneous
angle ߠ from the topmost point N of the lower sphere at the S2
centre of the lower sphere.
At what minimum value of ߠ will the spheres lose contact?

ͷ ͷ ʹ ͳʹ
(a) ‘•ିଵ (b) ‘•ିଵ (c) ‘•ିଵ (d) ‘•ିଵ
ͳ͵ ͳʹ ͵ ͳ͵

29. An electromagnetic wave in free space is described by
ͳ
‫ܧ‬ሬԦ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ǡ ‫ݐ‬ሻ ൌ  ‫ݖ‬Ƹ ‫ܧ‬଴  ‘• ൫݇‫ ݔ‬െ ξ͵݇‫ ݕ‬െ ʹ߱‫ݐ‬൯
ʹ
The Poynting vector associated with this wave is along the direction

(a) ‫ݔ‬ො ൅ ξ͵‫ݕ‬ො (b) ξ͵‫ݔ‬ො ൅  ‫ݕ‬ො

(c) ‫ݔ‬ො െ ξ͵‫ݕ‬ො (d) െξ͵‫ݔ‬ො ൅  ‫ݕ‬ො

30. Electrons in a given system of hydrogen atoms are described by the wave function
߰ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ͲǤͺΨଵ଴଴ ൅ ͲǤ͸݁ ௜గΤଷ Ψଷଵଵ
where the Ψ௡ℓ௠ denote normalized energy eigenstates. If ൫‫ܮ‬෠௫ ǡ ‫ܮ‬෠௬ ǡ ‫ܮ‬෠௭ ൯ are the
components of the orbital angular momentum operator, the expectation value of ‫ܮ‬෠ଶ௫
in this system is

(a) ͳǤͷ԰ଶ (b) ͲǤͳͺ԰ଶ (c) ͲǤ͵͸԰ଶ (d) Zero

31. In two dimensions, two metals A and B, have the number density of free electrons
in the ratio
݊஺ ‫݊ ׷‬஻  ൌ ͳ ‫ʹ ׷‬

The ratio of their Fermi energies is

(a) ʹǣ ͵ (b) ͳǣ ͺ (c) 1:2 (d) ͳǣ Ͷ

32. One mole of monoatomic ideal gas is
initially at pressure ܲ଴ and volume ܸ଴ .
The gas then undergoes a three-stage
cycle consisting of the following
processes:
i) An isothermal expansion till it reaches
volume ʹܸ଴, and heat Q flows into the
gas
ii) An isobaric compression back to the
original volume ܸ଴
The corresponding P-V diagram is
iii) An isochoric increase in pressure till shown above.
the original pressure ܲ଴ is regained.

The efficiency of this cycle can be expressed as

Ͷܳ െ ʹܴܶ଴ Ͷܳ ൅ ʹܴܶ଴
(a) ߳ ൌ  (b) ߳ ൌ
Ͷܳ ൅ ͵ܴܶ଴ Ͷܳ െ ͵ܴܶ଴
Ͷܳ െ ʹܴܶ଴ Ͷܳ ൅ ʹܴܶ଴
(c) ߳ ൌ  (d) ߳ ൌ
Ͷܳ ൅ ܴܶ଴ Ͷܳ ൅ ܴܶ଴

33. A deuteron of mass ‫ ܯ‬and binding energy ‫ ܤ‬is struck by a gamma ray photon of
energy‫ܧ‬ఊ , and is observed to disintegrate into a neutron and a proton. If ‫ ܿܯ ا ܤ‬ଶ ,
the minimum value of ‫ܧ‬ఊ must be

‫ܤ‬ଶ ͳ ‫ܤ‬ଶ
(a) ʹ‫ ܤ‬൅ (b) ቆ͵‫ ܤ‬൅ ቇ
ʹ‫ ܿܯ‬ଶ ʹ ‫ ܿܯ‬ଶ

‫ܤ‬ଶ ͳ ‫ܤ‬ଶ
(c) ‫ ܤ‬൅ (d) ቆʹ‫ ܤ‬൅ ቇ
‫ ܿܯ‬ଶ ʹ ‫ ܿܯ‬ଶ

34. Light passes through a narrow slit and gives the
Fraunhofer diffraction pattern shown in the adjacent
figure.

Which of the following could be the shape of the slit?

(a) (c)

S L
(b) (d)

Z +
35. For exact calculation and minimum complexity, two four-digit binary numbers can
be added with
(a) 3 full adders and 1 half-adder (b) 2 full adders and 2 half-adders
(c) 1 full adder and 3 half-adders (d) 4 full adders

Part B continues on next page
(to be attempted by all candidates for Int. M.Sc.-Ph.D. programme)
PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

Each of the following questions (36 – 40) must be answered by a word or
a number or a simple mathematical expression, which must be written
down clearly and legibly using only black/blue ballpoint pen. If there are
any cancellations, overwriting or erasures, the question will be
considered un-answered.

Use only values of constants given in the table ‘USEFUL CONSTANTS’.

A correct answer will be awarded ൅૞ marks.
There are NO NEGATIVE MARKS for these questions.

36. Which digital logic gate is mimicked by the following silicon diode and silicon
transistor circuit?

Ans = AND or (A.B)

37. Consider the following situations.

A B

݀
‫ݍ‬
݀ ‫ݍ‬

݀

In situation A, two semi-infinite earthed conducting planes meet at right-angles. A
point charge ‫ݍ‬, is placed at a distance ݀ from each plane, as shown in the figure A.
The magnitude of the force exerted on the charge ‫ ݍ‬is denoted ‫ܨ‬஺ .
In situation B, the same charge ‫ ݍ‬is kept at the same distance ݀ from an infinite
earthed conducting plane, as shown in the figure B. The magnitude of the force
exerted on the charge ‫ ݍ‬is denoted ‫ܨ‬஻ .
Find the numerical ratio ‫ܨ‬஺ Τ‫ܨ‬஻

Ans = 0.9142

38. Two identical bosons may occupy any of two energy levels Ͳ, ߝ, where ߝ ൐ Ͳ. The
lowest energy state is doubly-degenerate and the excited state is non-degenerate.
Assume that the two-particle system is in thermal equilibrium at a temperature T.
Calculate the average energy ‫ ۄܧۃ‬. What will be the leading term of

‫ۄܧۃ‬
ߝ
‡š’ ቀെ ቁ
݇஻ ܶ
at low temperature?

Ans = 2*epsilon/3

39. Evaluate the expression
஺ ௫೙షభ ௫೙షమ ௫య ௫మ ௫భ

݊Ǩ න ݀‫ݔ‬௡ିଵ න ݀‫ݔ‬௡ିଶ න ݀‫ݔ‬௡ିଷ ǥන ݀‫ݔ‬ଶ න ݀‫ݔ‬ଵ න ݀‫ݔ‬଴ 
଴ ଴ ଴ ଴ ଴ ଴
Ans = A^n
40. In outer space, where the effects of gravity can be neglected, a
drop of liquid assumes a spherical shape. However, when
disturbed it undergoes shape oscillations (see figure). The
frequency ߥ of oscillation of a drop depends on its equilibrium
radius, its density and the surface tension.

What would be the numerical value of the ratio ߥୌ୥ Ȁߥୌమ ଴ of the
frequencies of oscillation between a drop of mercury (Hg) and a
drop of water (H2O) of the same equilibrium radius ?

You may use the following data:
Liquid Density in gm cm-3 Surface tension in N m-1
water 1.0 0.073
mercury 13.6 0.487

Ans = 0.7

GS-2017-Y (Physics)
Section C
Q. 41 – 55 : to be attempted only by candidates for Ph.D. programme.
(Candidates for Integrated M.Sc.-Ph.D. programme will get NO credit
for attempting this section.)

PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

The following questions (41 – 50) are all of multiple-choice type. For
every question, four options (a), (b), (c) and (d) are given, of which only
one is correct. Indicate the correct option on the OMR by filling in the
bubble next to the correct label.

A correct answer will be awarded ൅૜ marks and an incorrect answer
will be awarded െ૚ mark. If the question is not attempted, no marks
will be awarded.

41. The value of the integral
∞
݀‫ݔ‬
න
଴ ‫ݔ‬ସ ൅ Ͷ
Is
ߨ ߨ ߨ
(a) (b) (c)  (d) ߨ
ͺ Ͷ ʹ

42. The Lagrangian of a system described by a single generalised coordinate ‫ݍ‬is
ͳ
‫ ܮ‬ൌ ‫ݍ‬ሶ •‹ଶ ‫ݍ‬
ʹ
Its Hamiltonian is
ͳ
(a) Zero (b) ‫ݍ‬ሶ ൬‫ ݌‬െ •‹ଶ ‫ݍ‬൰
ʹ
(c) െ‫ݍ‬ሶ •‹ଶ ‫ݍ‬ (d) ‘–†‡ˆ‹‡†

43. A one-dimensional quantum harmonic oscillator of natural frequency ߱ is in
thermal equilibrium with a heat bath at temperatureܶ. The mean value ‫ ۄܧۃ‬of the
energy of the oscillator can be written as

԰߱ ԰‫ݓ‬ ԰߱ ԰߱
(a) ‘–Š൬ ൰ (b) • Š൬ ൰
ʹ ʹ݇஻ ܶ ʹ ʹ݇஻ ܶ
԰߱ ԰߱ ԰߱ ԰߱
(c) •‡ Š൬ ൰ (d) –ƒŠ ൬ ൰
ʹ ʹ݇஻ ܶ ʹ ʹ݇஻ ܶ

44. Consider a spherical shell with radius ܴ such that
the potential on the surface of the shell in spherical
coordinates is given by,
ܸሺ‫ ݎ‬ൌ ܴǡ ߠǡ ߮ሻ ൌ ܸ଴  ‘• ଶ ߠ P
ܴ ߠ
where the angle ߠ is shown in the figure. There are
no charges except for those on the shell. The C
potential outside the shell at the point P a distance
ʹܴ away from its center C (see figure) is

ܸ଴ ܸ଴
(a) ܸ ൌ ሺͳ ൅ ʹ ‘•ଶ ߠሻ (c) ܸ ൌ ሺͳ ൅ ‘•ଶ ߠሻ
ͺ ͺ

ܸ ܸ଴
(b) ܸ ൌ  ଴ ሺͳ െ ‘•ଶ ߠሻ (d) ܸ ൌ ሺെʹ ‘• ߠ ൅ ‘• ଷ ߠሻ
Ͷ ʹ

45. A quantum mechanical system which has stationary states ȁͳ‫ ۄ‬, ȁʹ‫ ۄ‬and ȁ͵‫ ۄ‬,
corresponding to energy levels 0 eV, 1 eV and 2 eV respectively, is perturbed by
a potential of the form
ܸ෠ ൌ ߝȁͳ‫͵ۃۄ‬ȁ ൅ ߝȁ͵‫ͳۃۄ‬ȁ
where, in eV, Ͳ ൏ ߝ ‫ͳ ا‬.
The new ground state, correct to order ߝ, is approximately.

ߝ ߝ ߝ
(a) ቀͳ െ ቁ ȁͳ‫ ۄ‬൅ ȁ͵‫ۄ‬ (b) ȁͳ‫ ۄ‬൅ ȁʹ‫ ۄ‬െ ߝȁ͵‫ۄ‬
ʹ ʹ ʹ

ߝ ߝ
(c) ȁͳ‫ ۄ‬െ ȁ͵‫ۄ‬ (d) ȁͳ‫ ۄ‬൅ ȁ͵‫ۄ‬
ʹ ʹ

46. Hydrogen atoms in the atmosphere of a star are in thermal equilibrium, with an
average kinetic energy of 1 eV. The ratio of the number of hydrogen atoms in the
2nd excited state (݊ ൌ ͵) to the number in the ground state (݊ ൌ ͳ) is

(a) ͷǤ͸ʹ ൈ ͳͲି଺ (b) ͵Ǥͳ͸ ൈ ͳͲିଵଵ

(c) ͵Ǥͳ͸ ൈ ͳͲି଼ (d) ͳǤ͵͵ ൈ ͳͲି଼

47. A photomultiplier tube is used to detect identical light pulses each of which
consists of a fixed number of photons. The photoelectric efficiency is 10%, i.e. a
photon has 10% probability of causing the emission of a detectable photo-electron.
The photomultiplier gain is ͳͲ଺ .

‫ܫ‬ሺ‫ݐ‬ሻ

‫ܫ‬୫ୟ୶
‫ݐ‬
ʹͲ•
The typical output current, as a function of time, is shown by the figure below for
a few pulses, where ‫ܫ‬୫ୟ୶ is 80 μA. It follows that the number of photons in each
pulse is

(a) ͷ ൈ ͳͲ଺ (b) ͷͲ
(c) ͺͲͲ (d) ͷ

48. A subatomic particle ߰ and its excited state ߰ ‫ כ‬have rest masses 3.1 GeVȀܿ ଶ and
3.7 GeVȀܿ ଶ respectively. A table of its assigned quantum numbers is given below.

Angular Momentum Parity C-Parity Isospin Electric charge
‫ ܬ‬ൌ ͳ ܲ ൌ  െͳ ‫ ܥ‬ൌ  െͳ ‫ ܫ‬ൌ Ͳ ܳ ൌ Ͳ

If ߨ ଴‫ כ‬is an excited state of ߨ ଴ with a mass of about 1.3 GeVȀܿ ଶ , which of the
following reactions is possible when the above quantum numbers are conserved?

(a) ߰ ‫ כ‬՜ ߛߛ (b) ߰ ‫ כ‬՜  ߨ ଴ ߨ ଴
(c) ߰ ‫ כ‬՜ ߰ߨ ା ߨ ି (d) ߰ ‫ כ‬՜ ߰ߨ ଴‫כ‬

49. The cosmic microwave background radiation in the Universe has a blackbody
distribution corresponding to a temperature 2.735 K. In a certain cosmological
model, it was assumed that the universe consists purely of radiation and is
undergoing adiabatic expansion. In this model it was predicted that the volume of
the Universe will be tripled in the next ͳͲଵ଴ yrs. The corresponding blackbody
radiation temperature would be

(a) 0.9116 K (b) 2.078 K
(c) 1.526 K (d) 1.896 K

50. The following circuit is fed with an input sine wave of frequency ͷͲ Hz.

Which of the following graphs (solid line is input and dashed line is output) best
represents the correct situation?

(a) (b)

(c) (d)

Part C continues on next page
(to be attempted by all candidates for Ph.D. programme)
PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER

Each of the following questions (51 – 55) must be answered by a word or
a number or a simple mathematical expression, which must be written
down clearly and legibly using only black/blue ballpoint pen. If there are
any cancellations, overwriting or erasures, the question will be
considered un-answered.

Use only values of constants given in the table ‘USEFUL CONSTANTS’.

A correct answer will be awarded ൅૞ marks.
There are NO NEGATIVE MARKS for these questions.

51. Electrons in a metal are scattered by both impurities and phonons. The impurity
scattering time is ͺ ൈ ͳͲିଵଶ s and the phonon scattering time is ʹ ൈ ͳͲିଵଶ s. Taking
the density of electrons to be ͵ ൈ ͳͲଵସ m-3, find the conductivity of the metal in units
of A V-1 m-1. [Assume that the effective mass of the electrons is the same as that of
a free electron.]

Ans = 13.5 x 10-6

52. A particle of mass ݉, confined to one dimension ‫ݔ‬, is in the ground state of a
harmonic oscillator potential with a normalized wave function
ଵ
ʹܽ ସ మ
Ȳ଴ ሺ‫ݔ‬ሻ ൌ ൬ ൰ ݁ ି௔௫
ߨ

where ܽ ൌ ݉߱Τʹ԰. Find the expectation value of ‫ ଼ ݔ‬in terms of the parameter ܽ.

Ans = 105/(256*a^4)

53. Write down ‫ݔ‬ሺ‫ݐ‬ሻ, where ‫ݔ‬ሺ‫ݐ‬ሻ is the solution of the following differential equation

݀ ݀
൬ ൅ ʹ൰ ൬ ൅ ͳ൰ ‫ ݔ‬ൌ ͳǡ
݀‫ݐ‬ ݀‫ݐ‬

with the boundary conditions
݀‫ݔ‬ ͳ
ฬ ൌ Ͳǡ‫ݔ‬ሺ‫ݐ‬ሻȁ௧ୀ଴ ൌ െ
݀‫ ݐ‬௧ୀ଴ ʹ

Ans = exp(-2t) -2exp(-t) + 1/2

54. Assume that the crystal structure of metallic copper (Cu) results in a density of
atoms ߩେ୳ ൌ ͺǤͶ͸ ൈ  ͳͲଶ଼m-3. Each Cu atom in the crystal donates one electron to
the conduction band, which leads, for the 3-D Fermi gas, to a density of states

ͳ ʹ݉‫ כ‬ଷȀଶ ଵȀଶ
݃ሺߝሻ ൌ ൬ ൰ ߝ
ʹߨ ଶ ԰ଶ

where ݉‫ כ‬is the effective mass of the conduction electrons. In the low temperature
limit (i.e. ܶ ൌ Ͳ‫)ܭ‬, find the Fermi energy ‫ܧ‬ி , in units of eV. You may assume ݉‫ כ‬to
be equal to the free electron mass ݉௘ .

Ans = 6.87

55. In a theoretical model of the nucleus, the binding energy per nucleon was predicted
as shown in the figure below.

If a nucleus of mass number A ൌ ʹͶͲ undergoes a symmetric fission to two
daughter nuclei each of mass number A ൌ ͳʹͲ, write down the amount of energy
released in this process, in units of MeV, using this theoretical model.

Ans = 240