. I ..
I I
PHYSICS
Paper I
I Time Allowed Three Hours I I Maximum Marks: 2ool
INSTRUCTIONS
Candidates should attempt Questions No. I and
5 which are compulsory, and any THREE of the
remaining questions, selecting at least ONE
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 1·602 x 10-i9. C
Electron rest mass 9·109 x 10-31 kg
Proton mass 1·67Z x w-27 kg
Vacuum permittivity 8·854 x 10-12 farad/m
Vacuum permeability (J.Lo) 1·257 X 10-6 henry/m
D-VSF-L-CUA 1 [Contd.)
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Velocity of light in free space 3 x 108 rnls
Boltzmann constant 1·38 x 10-23 J/K
Electron volt 1·602 x 10-19 J
Planck's constant 6·62 x 10-34 J-s
Stefan's constant 5·67 x W m
Avogadro's number 6·02 x 1026 kmor1
Gas constant 8·31 x 103 J kmoi-1 K-1
exp 2·7183
SECTION A
1. Answer any four of the following 4x10=40
particle of mass moves according to the
equations x a cos wt, y a sin wt, z where
w are constants. Obtain the instantaneous.
velocity and linear momentum vectors in terms
of the Cartesian components and hence the
angular momentum. Find the force F and the
torque N acting on the particle and verify that
the angular momentum L and the torque N
satisfy the relation dL/dt=N.

particle of rest mass moving at a velocity
collides with a stationary particle of rest
mass the particles stick together, show
that the speed of the composite ball is equal to
<img src='./qimages/1483-1b.jpg'>
plano-convex lens of crown glass is connected
to a concavo-convex lens of flint glass so that
the convex surface of the first and the concave
surface of the second fit exactly. If the
combination is achromatic and the combined
focal length is -50cm, determine the radii of
curvature of the faces of the flint glass lens. The
dispersive power of the crown glass is one-half
of that of the flint glass and the refractive
indices for yellow light are respectively 1.500
and 1·625 respectively.
diffraction grating has 5000 lines per cm. For
illumination at normal incidence, determine the
dispersive power of the grating in the second
order spectrum in the range of wavelengths
around 500 nm.

(e)Consider a Gaussian pulse propagating in pure
silica, at lamda0 0·85mewm, along z-direction. As the
pulse propagates, it gets broadened according to
the formula
<img src='./qimages/1483-1e.jpg'>
where is the pulse width after propagation
through a distance z and t0 is the pulse
width at z 0. Given that alpha 3 x 10-26 S2/m at
lamda0 0·85mewm. Calculate the distance at which
root2 T0.
2.(a)A rigid body rotates with the angular velocity
about an axis through the origin O and having
direction cosines n. Show that the moment
of inertia of the rigid body about the axis is
<img src='./qimages/1483-2a.jpg'>
where the symbols have their usual meanings. 20
(b)The principal moments of inertia of a body at a
point are given as 200, 300 and 450 gm.cm2
Write down the equation of the ellipsoid of
inertia at that point.
light spring of relaxed length is suspended
from a point. It carries a mass at its lower
free end, which stretches it through a distance l.
Show that the vertical oscillation of the system
is simple harmonic in nature and has time period
T 2pie root where g is the acceleration due to
gravity.

3.(a)In the steady state forced vibration of the
damped harmonic oscillator, show that the
amplitude of the driven system is maximum
when root w2 -2b2 and further the value of
the maximum amplitude is f/2b root w2-b2, where
the symbols have their usual significance.
(Here you start from the steady state solution) 5

a double slit experiment the slits are 2 mm
apart and are illuminated with a mixture of two
wavelengths, A 750nm and lamda= 900nm. At
what minimum distance from the common
central bright fringe on a screen 2m distant
from the slits will a bright fringe from one
interference pattern coincide with a bright fringe
from the other 10
(c)What is Fermat's principle Derive the laws of
reflection and refraction with the help of this
principle, when the light is incident on a curved
surface.
4.(a)Explain the diffraction at straight edge with the
help of Cornu's spiral.
(b)Distinguish between the intensity patterns due
to diffraction from a narrow slit and a straight edge. 15

SECTION B
5. Answer any four questions from the following 4xJ0;40
(a)The electric field in a medium is given by
<img src='./qimages/1483-5a.jpg'>
where E0 is a constant vector with dimensions of
the electric field. Prove that E cannot have a
component along the unit vector zcap parallel to
the z-axis. Here alpha is a positive constant.
(b)Show that the elemental quantity of heat deltaQ is
not a total differential.
(c)Prove that the Maxwell's equations in a medium
contain the conservation of charge in differential
form.
inductor of inductance 5 H is suddenly
connected to a 10V d.c. power supply through a
resistor of 10ohm. After what time will the current
in the circuit be 1/l0th of its steady. state
value?
(e)Consider the Earth as a black body. Radiations
from the Sun arrive at the surface of the Earth
with an average intensity of S watts/m2. If the
reflection coefficient of the Earth's surface is alpha,
determine the temperature of the Earth under
equilibrium conditions.

6.(a)Write down the expression for the
Bose Einstein distribution function and explain
the meaning of the symbols used. 10
(b)Consider a gas of photons in equilibrium and
contained in a volume V at a temperature T.
Using the Bose-Einstein statistics, calculate the
total energy of the photon gas. 20

the magnetic field B at a point with position
vector r is uniform, show that the corresponding
vector potential is given by
<img src='./qimages/1483-6c.jpg'>

7.(a)Consider an infinite current sheet with a uniform
current density K Show that the
magnetic field H at a point away from the sheet
is
<img src='./qimages/1483-7a.jpg'>
where n is a unit normal vector directed from
the current sheet to the point.
(b)The plane y 5 carries a current of density
10z (Amp/m).Calculate the value of the
magnetic field H at the point 15

8.(a)Consider an infinite line charge with charge
density p coulomb/meter located at a distance
d meters from a grounded conducting plane z=0.
Determine:
(i)the magnitude of the potential V for z 0
and
(ii)the surface charge density induced on the
conducting plane.
(b)The current density in spherical co-ordinates is
given by
<img src='./qimages/1483-8b.jpg'>

where r and theta are unit vectors. Calculate the
amount of current passing through a hemisphere
of radius 20cm. 15