PHYSICS

Paper I
ITime Allowed: Three HoursI IMaximum Marks: 200I
INSTRUCTIONS
Candidates should attempt questions no. 1 and
5 which are compulsory, and any THREE ofthe

remaining questions, selecting at least ONE
question from each Section.
The number ofmarks carried by each question

is indicated at the end of the question.
Answers must be written in ENGLISH only.
Assume suitable'data, ifnecessary,

and indicate the same clearly.
Unless otherwise indicated, symbolic notations
carry usual meaning.

Useful Constants
Electron charge 1-602 x 10-19 C
Electron rest mass =9-109 x 10-31 kg
Proton mass =1·672 x 10-27 kg

p
Vacuum permittivity =8·854 x 10-12 farad/m
Vacuum permeability =1·257 x 10-6 henry/m
Velocity of light in free space x 108 m/s
Boltzmann constant 1-38 x 10-23 JIK

B-JGT-J-QIA 1 lContd.]
Electron volt =1·602 x 10-19 J
Planck's constant =6·62 x 10-34 Js
Stefan's constant 5·67 x10-8Wm-2K-4

Avogadro's constant 6·02 x 1026 kmol-1
Gas constant 8·31 x 103 J kmol-1 K-1
SECTION A
1. Answer any four questions from the following: (Each answer must not be more than 150 words long)
4xlO=40

Using Euler's equations of motion for a force-free rigid body, show that the kinetic energy remains constant all along the motion of the rigid body.


. The vibration of a string fixed at both ends is represented by the equation


7tX
2 sin 3 cos SOnt metre.
If the above stationary wave is produced due to the superposition of two waves of same frequency, velocity and amplitude in opposite directions
Yl A sin vt)

find the equation of the component waves and


find the distance between two consecutive nodes of the stationary wave.


B-JGT-J-QIA 2 rContd.]
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.
Cd) For a damped harmonic oscillator with equation mX kx
show that the work done against the damping force in an infinitesimal time is equal to the loss of energy of the mass during the same time interval
Explain the mechanisms of pulse dispersion in a step index fibre.
2. A moving particle of charge Ze hits a fixed charge Z'e. Show that the Rutherford scattering cross-section for this phenomenon can be given by
.!.
1 4 2E sin4
where 8 is the scattering angle and E IS the energy of the incident particle. 30
Discuss the significance of the impact parameter and the scattering angle In the analysis of Rutherford scattering phenomena. 10

3 [Contd.]
3. Show how the momentum components of a moving particle transfonn under Lorentz

transformation. A particle of momentum rest
mass fil is incident upon a stationary particle of rest mass ffi2. Show that the velocity of the centre of mass system is given by

25
When an external sinusoidal force is applied to a
vibrating system, we have a situation like forced
vibration. Show that in the steady state, the
frequency of the forced vibration is the same as
that of the external force. 15
4. 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 ofview is then show
that the wavelength of light is given by A. =2d/n. 15
Show that the dispersion D and the resolving
power of a grating are given respectively by
dO -ciA m dcos6 and


6)...
where d is the grating element, m is the order number and N is the total number of rulings in the grating.
B-JGT-J-QIA 4 {Contd.l
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.
B-JGT-J-QIA 5 [Contd.}
SECTION B

,..
5. Answer any four questions from the following (Each answer must not be more than 150 words long)
4xl0=40
In an RC circuit a fully charged capacitor of 1 discharges through a resistor of 100 kn. After how much time will the energy stored in the
capacitor become th of its initial value?
16
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
VT3 =constant.
If the initial temperature is 2000 K and the 33
volume is increased from 10 cmto 1250 cm, reversibly and adiabatically, what would be the final temperature of the radiation
Consider a system of two particles and three quantum states. Discuss how the particles will distribute themselves among the three states if they obey

Maxwell -Boltzmann statistics


Bose -Einstein statistics


Fermi -Dirac statistics.
Ignore the spin degree of freedom.
Write a short note on Saha's ionization formula and discuss its applications.
B-JGT-J-QIA 6 [Contd.]
Calculate the radius of an oxygen molecule, if its coefficient of viscosity at 15° C is 19·6 x 10-6 Pl (Pl Poiseuille) and the mean speed of the oxygen molecules is 436 ros-I.
6. 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 charge 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. 20

State Poynting's theorem for a combined system of charges and electromagnetic fields in the form of continuity equation. Explain the physical significance of each term in the equation. 10


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. Derive Planck's law of radiation. Obtain its limiting forms for very low frequencies very high frequencies. 20

Consider a system of N non-interacting bosons, occupying a volume at a temperature T. Derive an expression for the average number of bosons occupying the energy state E. 16


What do you understand by Bose -Einstein condensation 4


B-JGT-J-QIA 7 [Contd.]
8. Derive an expression for the specific heat of a
solid based on Einstein's theory. Obtain the
limiting form of the specific heat at very low
temperatures. 16
For diamond, the Einstein temperature,
eE 1450.K Calculate the specific heat of
diamond at T 290 K 4
For a system of N non-interacting fermions,
enclosed in a volume at T find an
expression for the internal energy the
pressure P. 15
Cd) 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 O. 5

B-JGT-J-QIA 8