PHYSICS


Time Allowed: 3 Hours M=imumMarks: 300
Candidates should attempt Question 1 and 5 which are compulsory, and any three of the remammg questions selecting at least one question from each &Jetton All questions carry equal marks

SECTIONNA
1 Answer any three of the following
x 20 60)
Write down the Lagrangian of a lIee particle in rectangular carteSian coordinates Identify all the cyclic coordinates. Show that the constants of motion obtained 110m the equations for the conSidered form of the Lagrangian are not exactly the same as those which follow 110m the concept of a free particle

What are chromatic and spherical aberrations Explain Calculate the longitudinal aberration for rays at a height of 0.01 m from the principal axiS and
incident on the cowed surface (radius of curvature 0.1 of a thin piano-convex lens made of glass (refractive index 1
A body of rest mass mo is mOving in the positive direction at a velocity of 06 C relative to the laboratory frame. Calculate the components of the four dimensIOnal momentum vector in the laboratory frame and in the frame of an observer who is travelling in the positive x-direction at a speed of 0.8 C relative to the laboratory frame
(d)ConSider super
position of two plane polarized electromagnetic waves
Ey y-cos(kx and
Ez =z-cos
Discuss the conditions for the resultant WeNe to be left circularly and night Circularly polarized adopting the convention as seen by an observer travelling with the wave
2. Derive the relation into r,where v0 is the velocity of a particle located at r in a fixed frame of reference S and v that as observed in a frame rotating with angular veloCity with respect to S but having the common origin
Show, using the above relation, that the equation of motion of the particle in S gets modified in giving me to variOUS fictitious forces Identify the coriolis force and describe e its effect on the flow of rivers
The anlHlar momentum M of a rigid body compnSlng of N particles and rotating with angular veloCity . is given by
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where the origin comCldes with the centre of mass. Express the components of !VI interms of
components of the inertia tensor Hence, show that the most general free rotation of a spherical iS
a uniform rotation about an axlS fixed m space(15)
State the postulates of the special theory of relativity and based on these obtain Lorentz as well as inverse Lorentz transformations. Hence, obtain an expressiOn to conclude that a moving clock runs more slowly than a stationary clock
An unstable particle has a l1fetime of 5 is in its own frame of reference and lS moving towards the
earth at a speed of 0.8 C. What w111 be the lifetime 0f the particle to an observer on the earth

Establish the relationship between the phase velocity Vp and the group velocity Vg ofweNes Under
what phYSical conditions Vg Vp, Vg V and Vg Vp can be possible?
4. Describe the working 0 f a Fabry-Perot interferometer Determine the intenS! ty 0 f the fringes 0 f the transmitted light Why the fringes obtained in the Fabry-Perot interferometer are comparatively sharper than these obtained from the Michelson interferometer?
The radius 0 f the first zone in a zone pl ate is 2 nun What will be the position 0f the first image of a
point source of light of wavelength 1c 500 nm placed at a distance of 5 in from the zone plate?
Define resolving power and dispersive power 0 f a grating Two spectral lines of wavelengths 500 nm and 500.5 nm are seen clearly resolved in second order spectrum of a grating If the grating has 250 lines per cm, what should be the null1mum Width of the grating?
SECTION B
Answer any three of the folloWlng
(20x3=60)
Prove the law of increase 0 f entropy Show that for a system at fixed temperature and pressure to be
in equilibrium, its Gibbs free energy should be minimum.
A system at temperature lS brought in contact with a reservOlr at temperature T When the
system and the reservOlr reach thermal equil1bnum, calculate the change in entropy of the Ull1verse
assuming the heat capacity of Cp of the system to be constant Discuss whether the conSidered
change lS positive or not
In an ac. circuit, areSlStance 100 and a capaCItance 100 pie in series are connected to an
ac. source 200 Sm (100 pie t). Calculate the current through the circuit and the voltages across R and C. Draw a vector diagram representing the magnitudes and phases of the voltages
State Amperes law of magneto statics. Using this law, find the magnetic field at a point due to an
in finitely long filamentary current
Derive POlsson equation starting from the Coulomb's law for a set of point charges(20)
Obtain the solution of the Laplace equation in cylindrical coordinates
WhatiS a magnetic shell? Define the strength 0f a magnetic shell
A 2 nun thick magnetic shell Weighing 100 gm has magnetic moment 0f 100 0 units The density of the shell material,iS 10 gm/cc Calculate the intensity of magnetisation and the strength 0f the shell
7.(a)Explain the Rayleigh Scattering of light. Show that the energy denSity of light scattered from an lSotroplc homogenous medium of a gas lS inversely proportional to the fourth power of the wavelength 0f the incident light
State Rayleigh-Jeans law. Show that the intensity of emlSSions at a particular wave-length is proportional to the temperature T. Discuss the limitations of this law is describing the lntenSity distribution 0f emlSSion spectrum 0f a blackbody
8 Establish the relation
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Use it to find out an expression for Cp of one mode of a gas whose internal energy lS given by U ct-a/v and which satisfies the equation of state Here a,b and c are constants
Derive a relation between the total number of Eermions in terms of Fermi momentum and hence obtain the expreSSiOn for the total energy E of the system at absolute zero. Combine this expression.with the equation of state PV =2/3Eto show that the pressure of an ideal Fermi gas at T 0 is proportional to 5/3 power of its number denSity