PHYSICS
Paper-I
ITime Allowed -Three Hours I IMaximum Marks 200I
INSTRUCTIONS
Candidates should attempt Question Nos. 1 and 5
which are compulsory, and any THREE of the
remaining questions, selecting at least
ONE question from each Section.
All questions carry equal marks.
The number of marks carried by each part of a question is indicated against each.
Answers must be written in ENGLISH only.
Assume suitable data, if necessary, and indicate the same clearly.
Unless otherwise indicated, symbolic notations carry usual meaning.
Useful Constants
Electron charge 1-602x10-19 C
Electron rest mass =9· 109 x 10-31 kg
Proton mass 1-672 x 10-27 kg

Vacuum permittivity 8· 854 x 10-12 farad/m Vacuum permeability 1· 257 x 10-6 henry/m Velocity of light in free space =3x108 mjs Boltzmann constant "" 1·38 x 10-23 J IK Electron volt 1-604 10-19 J Planck's constant 6 -62 x 10-34 Stefan's constant (al "" W m-2 K-4 Avogadro's number ." -6·02 x 1026 kmol-1 Gas constant S-31x103 Jkmol-1K-1
cxp 2 -7183

Section-A

1. Answer any four of the following
Establish the relation between the angular momentum and torque of a particle. Show that "this relation leads one to the principle of conservation of angular momentum. 10
How can one introduce the constraints of motion through, the concept of generalized coordinate systems? Write down the set of transformation equations for a system of N particles relating the generalized coordinates with the real coordinates. 10
Consider a system of two thin lenses of focal lengths 15 cm and -20 em separated by a distance of 25 em in air. Determine the system matrix. For an object of height 1 cm placed at a distance of 27'5 cm in front of the convex lens, find the size and position of the image. 10
A slit 0·25 mm wide is placed in front of a convex lens and illuminated by plane waves of wavelength 500 nm. The Fraunhofer diffraction pattern is formed in the focal plane of the lens. In the pattern, the distance from the third minimum on the left to the third minimum on the right is found to be 3 mm. Find the focal length of the lens. 10
Between multi mode and single-mode fibres, explain critically how and why single-mode fibre is chosen for communication. 5

Why are the two wavelengths 1·30 new m and 1·55 new m important in single-mode fiber optical communication system? 5
2. For a particle of mass m moving with velocities vs and v rrelative to the space and rotating axis, respectively, show that the equation of motion is obtained as F x vr) -mwx(wxr) mar .with w and a r being the angular velocity and,acceleration in rotating coordinates; 15
Which term in the above equation represents the Cariolis force? 5
Discuss the important role of Coriolis forces m the circulation pattern of winds. 5
Show that the magnitudes of the centripetal and Coriolis accelerations of the earth while rotating counter clockwise about the north pole are around 0·034 m sec-and (1· 46 x 10-4v) m sec-2 respectively. In obtaining these results, consider the ratio of sidereal days to solar days in a year as (366·5/365·5) and the radius of the earth along the equator as 6400 km. 15

3. An electron of rest mass m0 moves with a velocity v such that its total energy is double of its rest mass energy. What is the electron velocity? 10

At what velocity, the electron momentum will be m0c? 5
What is birefringence? Indicate how it can be used to obtain plane and circularly polarized light. 15
A left circularly polarized beam of light having lambdao 5893 A is incident on a calcite crystal with its optic axis cut parallel to the surface. The crystal has thickness d 0·005141 mm, no 1·65836 and n e 1·48641 at this lambda 0.What will be the state of polarization of the incident beam? 10 .

4. With a suitable diagram, deduce an expression for numerical aperture for an optical fiber having refractive indices of core and cladding n1 and n2, respectively and being placed in a medium of index no. 15
What do you mean by underfilled and
overfilled conditions with reference to
numerical aperture in exciting light in
fiber? 5+5
Consider a bare fiber having n1 1.48 and n 2 (air).Find out the NA. What is the maximum incident angle up to which light can be guided through the fiber? 5+5
Why is the NA of the single-mode optical fiber low as compared to multimode fiber? 5
Section-B

5. Answer any four of the following
Consider the circuit shown below:

Determine its resonance frequency
Determine its quality factor Q and width of resonance delta f. 6
Consider a long, line charge with charge density PI 10-6 coulomb/roo Find the force acting on a dust particle carrying coulomb, 1 metre away from the
line charge in free space. 10
te) The electric field for a uniform plane wave in free space is given by +Y)10e j10z . Determine the corre­sponding magnetic field vector. 10

Show that when the temperature T of a radiating object is not too different from the surrounding temperature To, the object obeys Newton's law of cooling. 10


The following inputs are given:


Surface temperature of the sun, To 5500 K
Radius of the sun, R 7 X 1010 em
Radius of the earth, r 6·4xl08 em Distance between the sun and the earth,
D=1·5x1013 em
B-JGT-K-QIA/24 5 P.T.G.

Assume that the earth and the sun both
absorb all electromagnetic radiations
incident on them, and that the earth is
at a constant temperature T over the
day-night cycle. Calculate T. 10
6. A spherical capacitor is made of
concentric conductors of radii a and b
a). The total charge on the inner
sphere of radius a is -Q.
Derive an expression· for the
capacitance C. 8
The earth may be modeled as
a spherical capacitor with
6-5 x 106 m and b 00.
Determine if the medium
surrounding the earth is free space. 2

For two isotropic media with III J.l2
and £1 i fmd an expression for
the Brewster angle eb for parallel
polarization. 10
The electromagnetic field inside a device
is given by
yE0 sin (kxx}e-Jkzz where k x T11t
a
Eo -kz sin zjkx cos(kxx)]e-jkzz
WJl CllJl
Obtain an expression for the z·
component of time-averaged Poynting

vector S. 10

B-JGT-K-QIA/24 6

A point charge is located near the corner of a horizontal and a vertical plate as shown below

Obtain an expression for the
electrostatic potential p using the
image method. 10
7. A long wire of radius a carries I amperes
ofcurrent. The magnetic field surround­
ing it is given by for a. 21tp
Obtain expressions for ti) the magnetic
energy stored per unit' length in the
region b P a and the equivalent
inductance L per unit length. 8+2
Consider a perfectly conducting half­
space as shown below:
x


B-JGT-K-QIA/24 7 p.T.a.

A uniform plane wave gIven by
.
E t
oe
Eo -jkz
A
110 120n ohm llo
is incident normally on the boundary. Write down the expressions for the reflected electric and magnetic fields. 10

N particles are distributed among three states having energies E E kT and E '2kT. If the total equilibrium energy of the system is 1000kT) what is the value of 15


Explain why the distribution. of speeds of molecules emerging through a small hole ill an effusive molecular beam source is not a Maxwellian distribution. 5


8. Let f be the Fermi-Dirac distribution function) then show that­
u isa. at the

-(TatEJ maxlmum
Fermi level; 20
J IS I ae symmetric about the
Fermi level. 5

A radiation gas of temperature T fills a cavity of volume V. The system expands adiabatically and reversibly to a volume equal to 8 V. By what factor does the
. ..­
temperature change?
15


B-JGT-K-QIA/24 8 BS-12*500