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(MAIN)
PHYSICS
PAPER -2004

Time Allowed: Three Hours Maximum Marks: 300
Candidates should attempt Questions 1 and 5 which are compulsory and any three the remaining quest' selecting at least one question from each Sectio
Assume suitable data if considered necessary .cate the same clearly.
SEcnON A
1. Answer any three of the following
A particle moves in space with the lagrangian
L 1/2 m (x2 y2
where B,C are given functions corresponding Hamiltonian in terms of coordinates and
momenta. .
A meson of rest comes to rest and disin­tegrates to a muon of rest mass mew and a neutrino of zero rest mass. Show th ·netic energy of motion of the muon is
.t=(pie-mew2)c2/2pie,
show that the ratio of the focal lengths of the two lenses in an achromatic doublet is given by
f1/f2=-w1/w2
where WI and w 1 are the dispersive powers of the tnses of focal lengths f1 f2 respectively.
A sire of of frequency 900 Hz is going towards a wall away froin an observer at a speed 10 m/sec& Deter­mine
Frequency of sound directly heard from the siren.
Frequency of sound reflected from the wall.
Number of beats per second heard by the observer.
(Velocity of· sound 330 m/sec)
2. What do you mean by the· moments and products of inertia? Show that the angular momentum vector is related to the angular velocity compon ts by linear tranSformation relations. 20
For a freely falling body from the height the surface of the earth. in the northern e e with a latitude O. show that the deviation of the body towards
east at the final stage is given
by 1/5d cos0(8h3/8)1/2,
where mew is the angular velocity of the earth and is the acceleration due to gravity; 20
If a single stage rocket fired vertically. from rest at the earth's surface bu el in a time of30 sec and their. relative velocity 3 km sec-I, what must be the mass ratio mo/m for a velocity u of 8 km sec-I? 20
3. write the expression for the relativistic mass of a part moving with a velocity u in terms of its· rest mass Establish from the above expression Einste mass-energy relation mc2 20
a thin converging lens and a thin diverging lens are placed coaxially at a distance of 5cm.. If the· focal length of each lens is·10cm find for the combination
the focal length the power, the positions of the principal points.. 20
Write down the equation of motion for a damped harmonic oscillator assuming the damping force proportional to. the velocity of the particle.. Obtain the general solution for its displacement as a function of time.
Discuss the cases of 'overdamping; '1,Jnder-damping' and critical damping'.. . 20
4. (a)·A quartz quarter wave plate is to be used with· the sodium light =.5896 . What . .. should be its thick­ness? 20
(b)Two microscope slides of length 10 em each form a wedge. At one end they arein.contaet and at ther end they are separated by a thin wire of diattle see the diagram below). Interference fringes tained . when illuminated vertically by·a monoch c light of lamda= 500 nanometers. The fringe spacing is found to be
1.25 cm. Estimate the diameter of the wire
<img src='./qimages/1128-4b.jpg'>
20
A narrow slit illummated by monochromatic light of A=6400 A is placed at a distance of 3 metres from straight edge.if the distance between the straight edge ·and the sere 6 metres, calculate the distance between th ill and the fourth dark bands.. 20
SECTION B .
5. any three of the following: 20 x 3 60
what is the volume density of the charge in· a reg of space, where the electrostatic potential is given
V y2) -·c In (x2 where c are constants?
A network ·PQRS is connected as shown in the figure Apply Kirchhoff's law and show that the current flowing through the 20 n resistor ·PR is 0.029 A
<img src='./qimages/1128-5b.jpg'>
Show that the entropy of one gram molecule of anideal gas is given by
S=Cp In V Cv In p So •
Given the temperature of the sun's surface T 5755 radius of the SUn 6.0into105 km, distance· between the Earth and the 5 x 108 km. Estimate the solar constant (i.e. t e. received per sec per unit area of the Eath"s surface). Assume sigma=5.7into10-8w/m2-k4.
A point large is held at a distance d in front of an infinite grounded conducting plane. What is the electric· pate' ront of the plane 20
Find magnetic field inside a long solenoid carrying.current i. and having n turns per unit length.20.
define the strength of a magnetic shell and calculate magnetic potential at any neighbouring point due to this shell. Does the potential depend on the shape shape of the shell . 20
What are the vector and scalar potentials Derive Maxwells wave equations in terms of these potentials. 20
What do you understand by a perfect black body? Can it be realised in practice? Show that the ratio of the
.emissive power to the absorptive power of all bodies at a given temperature is equal the emissive power of a perfectly black body. . 20
(c)Show that the Poynting vector INTO represents the energy·flow per unit area per unit time both in magnitude and direction in case of a plane electromagnetic wave.
A .gas expands isothermally from the pressure P1 and volume VI to the pressure p2 in the volume V2" Calculate
the change in internal energy
the change in ehtropy.
the change in enthaloy.
What will be the corresponding quantities when the gas expands adiabatioaly
Show that for a distribution of electrons, THEmumber of electrons NI in the energy state sigma1 are given by
<img src='./qimages/1128-8b.jpg'>
when the represents the number .of quantum states in the energy level sigma1. Further state under what conditions this .stribution law goes over to .Maxwell-Boltzmann
statistics.
Show by drawing curves how the fermi-Dirac distribution function varies with the energy at 0 and also at the other finite temperatures .30