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1, Answer any four questions from the following :
(Each answer must not be more than 150 words long)
4xl0=40
(a) Using Euler's cquatioru; of motion for a force-free
rigid body, show that the kinetic energy remains
constant all along the motion of the rigid body.
(b) . The vibration of a string fixed at both ends is
represented by the equation
" y = 2 sin 3 cos 501It metre.
If the above stationary wave is produced due to
the superposition of two waves of same frequency,
velocity and amplitude travelling in opposite
directions
y1 =A sin
n (x- vt) and y2 =A sin ~n (x + vt)
/, 1'.
(i) find the equation of the component waves
"nd
(ii) find the distance between two consecutive
nudes of t.he ~tationary wave.
(c) Light rays of wavelength 5890 A and 5896 A fall
on a grating having 5000 lines/em. If a lens of
focal length 100 em is used to form spectra on a
screen, find the distance between the lines in the
third order.
(d) For a damped harmonic oscillator •with equation
mX+~:i +kx=O,
show tl1at the work done against the damping
force in an infinitesimal time 'dt' is equal to the
loss of energy of the mass 'm' during the same
time inWrval 'dt'.
(e) Explain the mechanisms of pulse dispersion in a
step index fibre.
2. (a) A moving particle of charge Ze hits a fixed charge
Z'e. Show that the Rutherford scattering
cross-section o{fl) for this phenomenon can be
briven by
!T{fl) "' ..!:_ ( ZZ'e
where 8 is the scattering angle and E lS the
energy of the incident particle.
(b) Discuss the significance of the impact parameter
and the scattering angle in the analysis of
30
Rutherford scattering phenomena. 10
3. (a) Show how the momentum components of a
moving particle transform under Lorentz
~
transformation. A particle of momentum P1 , rest
mass m1 is incident upon a stationary particle of
rest mass m2. Show that the velocity of the
centre of mass system is given by
->
v =
(b) When an external sinusoidal force is applied to a
vibrating sy~tem, we have a situation !-ike forced
vibration. Show that in the steady state, the
frequency of the forced vibration is the same as
25
thut of the external force. 15
4. (a) Describe the construction of a Michelson's
interferometer. If the movable mirror is moved
through a small distance d and the number of
fringes that cross the field of view is n, then show
that the wavelength of light is given by A: 2d/n. J.'J
(b) Show that the dispersion D and the resolving
power of a grating are given respectively by
(i) D:=dO, m
dA dcm; e '"'
(ii) R = }. = N.m,
M
where d is the grating element, m is the order
number and N is the Wtal number of rulings in
the grating. 10
(!
Draw the figures depicting the variation of the
electric field vector with time for linearly
polarized, circularly polarized and elliptically
polarized light beams respectively. Describe the
experimental method for detecting the state of
polarization of the light beam. 15
SECTION B
5. Answer any four questions from the following:
(Each answer must not be more than 150 words long)
4x10=40
{a) In an RC circuit a fully charged capacitor of 1 ~-tF
discharges through a resistor of 100 k.Q, After
how much time will the energy stored in the
capacitor become 1~ th of it.s initial value?
(b) Suppose a cavity of volume V contains black-body
radiation in equilibrium with the walls of the
cavity at a temperature T. For a reversible
adiabatic change of volume show that
VT'1 = constant.
If the initial Lemperature is 2000 K and the
volume is increased from 10 cm3 t.u 1250 cm3,
reversibly .and .adiabatically, what would be the
Gnal temperature of the radiation?
(c) Consider a system of two particles and three
quantum states. Discuss how the particles will
distribute themselves among the three states if
ihey obey
(i) Maxwell- BoHzmann statistics
(ii) Bose- Einstein statistics
(iii) Fermi- Dirac statistics.
Ignore the spin degree of freedom.
(d) Write a short note on Saba's ionization forn1ula
and discuss its applications.
(e) Calculate the radius of an oxygen molecule, if its
coefficient of viscosity at 15° C is 19•6 x 10-6 PI
(Pl "' Poiseuille) and the mean speed of the
oxygen molecules is 436 ms-1.
6. (a) What is the Method of Images ? What are the
necessary conditions which must be satisfied in
order to apply the image method ? Show that a
point ~barge placed at a distance above a perfect
conducting plane of infinite extent can be
replaced by itself, its image and an equipotential
surface in place of the conducting plane.
(b) State Poynting's theorem for a combined system
of charges and electromagnetic fields in the form
of continuity equation•. Explain the physical
20
sigiJificance of each term in the equation. 10
(c) Determine the mutual inductance of two coplanar
concentric circular loops of radii 1 m and 2 m.
(Usual assumptions may be used, where
required, to solve the problem)_ 10
7. {a) Derive Planck's law of black-body radiation.
Obtain its limiting forms for (i) very low
frequencies (ii) very high frequencies. 20
{b) Consider a system of N non-interacting bosons,
occupying a volume V, at a temperature T. Derive
an expression for n(E}, the average number of
bosons oceupying the energy state E. 16
(c) What do you understand by Bose- Einstein
condensation ? 1
8. (a) Derive an expression for the specific beat of a
solid based on Einstein's theory. Obtain the
limiting form of the specific heat at very low
temperatures.
(b) Far diamond, the
OE "' 1450 K Calculate
diamond at T = 290 K.
Einstein tempemture,
the specific heat of
(c) For a system of N non-interacting fermions,
endose1d m a volume V, at T "' 0, find an
expression for (i) the internal energy U, (iil the
16
pressure P. 15
(d) Metallic silver has a density of 10•5 x 103 kg m-3
and its atomic weight is 107. Taking one free
electron per silver atom, calculate the pressure of
the electron gas in silver at T = 0.
PHYSICS
Paper-II
INSTRUCTIONS
Candidates should attempt Question Nos. 1 and 5
which are c
mpulsory, and THREE of the
remaining question11, selecting at least
ONE question ftom each Section.
The number of marks ~:-arried by each question is
indicated at the end of the question.
ADswers must be written in ENGLISH only.
Assume suitable data, if considered necessary and
indicate the ~:>ame clearly.
List of Useful Constants
Mass of proton =1•673xlo-27 kg
Mass of neutron =1•675xlo-27 kg
Mass of electron = 9. 11 x 10-31 kg
Planck constant = 6 • 626 x lQ-34 J-s
Boltzmann constant = 1• 380 x lQ-23 J-K-t
Bohr magneton = 9 • 273 x lo-24 A-m 2
Electronic charge =1•602xl0-19 C
Atomic mass unit (u) =1•660x1Q-27 kg
=931 MeV
Velocity of light in
vacuum, c <=3xl08 m-s-1
m(fH) = 1•007825 u
m(fH) =2-014102 u
m({HJ =3•016049 u
m{12q =12•00000 u
m(fgNe) =19-992439 u
m{~He) =4•002603 u
m(~n} =1-008665 u
m(e) =0-000549 u
m(~t2 sm) = 151-919756 u
m(~~2 Eu) =151-91749 u
m(~2 Gd) = 151•919794 u
m(35 Cl] =35-450u
Unles:;o otherwise Indicated, symbols and
notations ha.ve usual meanings.
Section-A
1. Answer any four of the following :
(a) The wavelength and frequency of a
guided wave are related by
A=c/~v 2 -vg
Express the wave's group velocity !.!
in
terms of c and its phase velocity 1.1 P. l 0
(b) What is the minimum angle that the
angular momentum vector may make
with the z-axis, in the case .of 1 = 3? How
many levels are possible? Show the
angle in a vector diagram.
(c) An H 35CI molecule is found to have a
fundamental vibration at 2885. 9 em -l.
Its internuclear distance changes by
0 •OS nm. What will be the change tn
10
potential energy? 10
(d) Calculate the ~eman energy splitting of
the 2P112 and 2S112 states in sodium for
a magnetic field of 2 T. 10
(e) If 6' x• 6' y and 6' z are Pauli spin matrices,
prove the following relationship : 10
sin {d ,41) == d x sin !jl
2. (a) The wave function of a quantum particle
is W(X) == Aeii(U:- a21l/2m). Find {p"' ).
(b) For theE
expression for the probability that the
particles will be transmitted resonantly.
10
U(x) = (
0 XL
U0 O
(c) -If an electron is incident on a barrier
potential 0-100 eV and width 15 nm in
the problem (b) above, calculate the
transmission probability, if its energy is
0•060 eV. 10
3. (a) Pr-ove that i 2 ""i~ +i~ +i~ commutes
with all the three components of
angular momentum operator f. 20
(b) Discuss briefly the Stern-Gerlach
experiment with the help of a diagram.
Analyse the outcome of this experiment. 20
4, (a) Discuss the number of fundamental
vibrations of water molecule.
(b) Frequency separation between adjacent
lines in the rotational spectrum of
35ct 19F is .measured as 11• 2 GHz.
15
Calculate the interatomic spacing. 15
(c) The NMR spectrum of CH31 taken in a
60 MHz spectrometer gave a strong
signal at 130 Hz and chemical shift
2 • 16 ppm. Obtain the absorption
frequency, if the spectrum were to be
made in a 90 MHz spectrometer. What
would be the chemical shift in the new
spectrum?
Section-B
5. Answer any four of the following :
(a} {i) What is the Q-value of the reaction
10
1s2Eu (n., Plts28m?
5
(ii) What type of weak interaction can
occur for 152Eu? 5
(b) Lead shows superconductivity at 7 •19 K
for zero applied magnetic field. When
magnetic field of 0 • 08 T is applied,
superconductivity will not take place
at any temperature. Calculate the
applied magnetic field that will stop
superconductivity at 2 • 0 K. 10
(c) What is intemai conversion? Show that
the internal conversion coefficient
depends upon the atomic number of the
nucleus.
(d) Indicate with reasons, why each of the
following high-energy reactions/particle
!0
decays is either allowed or forbidden 10
(i} n- +p~no +n
(ii) no -ty+y+y
(iii) no -ty+y
(iv) n• -t!l+ +v,._
(v} n• -til+ +V,._
(e) Write down the quark composition of
the following : !0
(i} Neutron {n)
(ii) Proton (p)
{iii) Sigma-plus p::+)
(iv) Kaon-zero (K 0)
(u} Pion-zero (n°)
6. (a) Derive an expressia•n for the recoil
energy loss
ray. What
Explain.
in the emission of gamma
IS natural line width?
(b) What is Mlissbaucr effect? Define the
MOss bauer
quadrupole
splitting.
parameters-isomer shift,
splitting and magnetic
(c) For atomic transitions, resonance
fluorescence and absorption are
normal, but in nuclear ganuna decay
this does not occur. Why? Explain with
10
20
a typical illustrative example. 10
7. (a) Discuss Grand Unified The01y schemes
for Strong, Weak and Electromagnetic
interactions. What are experimentally
verifiable predictions of such schemes? 20
(b) Explain Kronig-Penney model for the
motion of an electron in a periodic
one-dimensional square well potential.
Show from the energy band diagram
that the materials can be classified
as conductor, semiconductor and
insulator. 20
8. (a) Calculate the resistivity of silicon Mving
atomic mass 28, and having density of
2•42xl03 kg-m-3 and a valence of 2.
Assume that the Fermi energy for
silicon is 2 eV. (Assume electron mean
free path to be 100 times the
interatomic spacing.) 20
(b) Discuss the construction and operation
of Enhancement MOSFET and
Depletion MOSFET. Give their
characteristics. How do they differ from
JFET? 20
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