PHYSICS
Paper-I
ITime Allowed: Three Hours I IMaximum Marks 300 I

INSTRUCTIONS
Each question is printed both in-Hindi and in English.
Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space .provided for the purpose. No "marks will be given for the answers written in a medium other than that specified in the Admission
Certificate.
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.
The number of marks carried by each question is indicated, at the end of the question.
Assume suitable data if considered necessary and indicate'the same dearly.
Symbols/notations carry their usual meanings, unless otherwise indicated.

Section-A

1. Answer all the six, parts below 10x6=60
(a)With an appropriate diagram, show that in the Rutherford
scattering, the orbit of the particle is a hyperbola.
Obtain an expression for impact parameter.

(b)Imagine that a rigid body is rotating about a fIXed ·point
with angular velocity . Assuming that the coordinate axes
coincide with the prIncipal axes, if T stands for kinetic energy
and G for external torque acting on the body, show that
<img src='./qimages/1047-1b.jpg'>

A spaceship measures 50m in length on ground and it measured
a length of 49.7m in space as observed from the ground.
Find out the speed of the spaceship.

Write down the one-dimensional harmonic oscillator
differential equation under damping and its solution for the
lightly damped condition,with the meanings of symbols.
Determine the dependent energy in the lightly damped condition.


The Fraunhofer single-slit diffraction intensity is given by
I=I0 sin2x/x2
where x =piedy/lamdal with l as the distance from slit to source,
d the slit Width, y the detector distance and lamda the wavelength.
What is the value of cumulative iniensity
<img src='./qimages/1047-1e.jpg'>

A-plane wave has the following
expression for its electrIc field
<img src='./qimages/1047-1f.jpg'>
If the phase difference is defined as delta=beta-alpha under what
conditions do we achieve elliptic polarization? What are the
conditions for circular polarization?

Determine the number of degrees of
freedom for a rigid body­
(i)moving freely in space of three dimensions;
(ii)having one point fixed;
(iii)having two points fixed. 30

(b)Prove that the time taken by the earth to travel over half
of its orbit separated by the minor axis remote from the sun is
two days more than half a year. Given,the period of the earth is 365 days
and eccentricity of the orbit 1/60. 10
With an appropriate diagram, deduce
the velocity profile for streamline flow of
a liquid through a capillary of circular
cross-section. Deduce also the fraction
of liquid which flows through the
section up to a distance a/2 from
the axis, where a is tne radius of the
capillary. 20

Explain the physical significance of
group velocity from the concept of phase
velocity with relevant expressions. 15
Prove that the group velocity Vg of electromagnetic waves in a
dispersive medium with refractive index. n(lamda0) at
wavelength lamda0 is given by
<img src='./qimages/1047-3b.jpg'>
where c is the free space velocity of light.
Find the time taken for the
electromagnetic pulse to travel a
distance D. 20
When a thin film of a transparent material is put behind one
of the slits in Young's double-slit interference
experiment, the zero-order fringe moves
to the position previously occupied by
the fourth-order bright fringe. The index
of refraction of the film is n =1.2 and
the wavelength of light, lamda=5000A.
Determine the thickness of the film. 10


In relation to a plane diffraction grating
having 5000 lines per cm and irradiated
by light of wavelength 6000A, answer
the following: 15
(i)What is the highest order spectrum
which may be observed?
If the width of opaque space is
exactly twice that of the
transparent space,which order of
spectra will be absent?

For calcite the refractive indices of
ordinary and extraordinary rays are 1·65836 and 1·48641 at
lamda0 5893 A respectively. A left circularly polarized
beam of this wavelength is incident
normally on such crystal of thickness
0.005141 mm having its optic axis cut
parallel to the surface. What will be the
state of polarization of the emergent
beam? 15
(b)Bring out the essential differences
between the physical principles of
spontaneous and stimulated emission
of radiation. why is it difficult to get
efficient lasing action in case of an ideal
two-level material system? Can you
propose a scheme to enhance
efficiency? Discuss. 15
show with proper mathematical analysis that the ratio of einstein's A and B coefficients
depends upon the energy seperation between the two energy levels participating in the optical
transitions.what is the physical significance of A coefficient?justify the statement,
"it is very difficult to develope an X-ray laser".

5. Answer all the six parts below: lOx'6=60
Find whether the discharge of a condenser through the inductive
circuit is oscillatory when C=0.1mewF,L=10mH and R=200omega
If it is oscillatory,calculate its frequency.

the four arms of a Wheatstone bridge have the following resistances:
AB=l00oms BC=l0oms, CD=5oms, DA=60oms
A galvanometer of I5oam resistance is connected across BD Calculate
the current through the galvanometer when a potential difference
of 10 volts is maintained across AC.

(c)Write down the expressions for Bose-Einstein and Fermi-Dirac
distribution functions, and show how Fermi-Dirac distribution
leads to the explanation of Pauli's exclusion principle.
Find out the total electric potential energy of a single
spherical object of unifonn charge density total charge Q and radius R.

long solenoid has 220 turns/cm; its diameter is 3·2cm.
Inside the solenoid at its centre,we place a 130-turn close­packed
coil of diameter 2.1cm along its axis. The current in the solenoid is
increased from zero to l·5 amperes at a steady rate over a period
of 0.16 second. What is the magnitude of the induced
e.m.f. that appears in the central coil when the current in the
solenoid is being changed?

(f)consider Maxwell's equation In differential form in media.
For assume epsilon =Epsilon0 ealphat and mew=mew0 ealphat
and show that the relevant wave equation for a plane wave propagating along x-direction is
<img src='./qimages/1047-5f.jpg'>

An electrical circuit consists of a
resistance inductance L and
capacitance C in series. if a charge is
put on the capacitor at some 'instant,
determine the condition that Vc, the
voltage across the capacitor is
subsequently oscillatory. Derive an
expression for the quality factor Q of the
circuit by cpnsidering the decay of the
oscillation, using the result that the
amplitude falls by a factor of e in period. 20

resistor 6·2 Momega) and a capacitor 2·4 mewF) are
connected in series and a 12 V battery of negligible internal resistance
is connected across their combination·
(i)What is the capacitive time constant of this circuit?
(ii)At what time, after the battery is connected, does the potential
difference across the capacitor become 5.6 5+5=10
(c)Determine the torque experienced by an electric dipole of moment
p if placed in an electric field E in nonaligned state.also show
that the interaction energy of two dipoles of moments. P1 and P2
separated by a displacement r is

<img src='./qimages/1047-6c.jpg'>

assuming the expression for field due to a dipole. 20
the electric potential of a grounded conducted sphere of radius a
in a uniform electric field is given as
<img src='./qimages/1047-6d.jpg'>
"Find the surface charge distribution. 10

7.(a)Consider a plane wave travelling along
the positive y-direction incident upon a
glass of refractive index n =1.6 Find
the transmission,coefficient. 25

Justify which of the four Maxwell's
equations imply that there are no
magnetic monopoles. How these
equations would have been written if
they were? 15
In deriVing the Rayleigh-Jeans law, we
count the number of modes dn
corresponding to a wave number k for a
photon gas in a cubical box. Consider
a cubical container of volume V
containing such gas in equilibrium.
Calculate the differential number of
allowable normal modes of frequency omega. 20


8.(a)What led van der Waals to modify the ideal gas equation?
using the concepts of critical temperature Tc, pressure Pc and volume Vc,
show that the critical constant for areal gas is 8/3. 5+15=20
Find out the expressions for van der Waals constants a and b. 20
Calculate the values of van der Waals constants a and b for oxygen with
Tc= 154.2K, Pc= 49.7 atmosphere and R 80cm3 atmosphere/K. 20