SECTION A
answer all of the following: 10x5=50

The density inside a solid sphere of radius a is given by p P0 a/r where Po is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.

If I and I be the Moments of Inertia of a body about an axis passing through an arbitrary origin and about a patallel axis through the centre of mass respectively, show that MR2+1, where R is the position vector of the centre of mass with respect to the arbitrary origin and M is the mass of the body.


A sphere of radius R moves with velocity u in an incompressible, non-viscous ideal fluid. Calculate, the pressure distribution over the surface of the' sphere. Do you think that'a: force is necessary to keep the sphere in uniform motion?


What is the physical significance of Einstein's A-coefficient explain, why it is more difficult to achieve Lasing action at X-ray wavelength than at infra-red wavelength.

For a multi mode step index optical fibre, the core refractive index is 1·5 .and fractional index difference is 0·001. Calculate the pulse broadening for 1 km'length of the fibre. Over a length of 2 km of the fibre, calculate the minimum pulse separation that can be transmitted without overlap.

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Q2. Consider a rigid body'rotating about an axis passing through a fixed point in the body with an angular velocity w.Determine the kinetic . energy of such a rotating body in a coordinate system of principal axis. If the Earth suddenly stops rotating, what will happen to the rotational kinetic energy? Comment in detail.

A body turns about a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.

Explain the working principle of a 3-1evellaser a specific example. Comment on why the third level is needed.

Q3. A mirror is moving through vacuum with a relativistic v in the A beam of light with frequency (OJ is normally incidqnt (from x on the mirror. (i)What is the frequency of the reflected light expressed in termsof wi, c and v What is the energy of each reflected photon? 25


In question if the average energy· flux of the incident beam is Pj what is the energy flux of the reflected beam? 15

In a certain engine, a piston undergoes vertical SHM with an amplitude of 10 em. A washer rests on the top of the piston. As the motor .is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston? 10

Q4. Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression? 25

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A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time. period of the motion. 15
Considering a plane transmission diffraction grating, where d is the distance between two consecutive ruled lines, m. as the order number and 0 as the angle of diffraction for normal incidence, calculate the angular d0/dy for an incidentlight of wavelength y. 10
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SECTION B
Q5.
answer above the following: 10x5=50
In a Young double slit experiment, the first bright· maximum IS displaced by y 2 cmJrom'the central maximum. If the spacing between. slits and distance from the screen arc 0·1 mm and 1m respectively, find the wavelength of light. 10

define enthalpy and show that it remains constant in a throttling process. 10

How does h0lography differ from conventional photography? What are the requirements for the formation and reading of a hologram? 10

In deriving radiation laws; we consider a cubical container of volume V containing a photon gas in equilibrium. Calculate the differential number of allowed normal modes of frequency w. 10
10

Starting from Maxwell's equation, obtain the wave' equation for the electric field E in free space and appropriate wave equation for the . electric field E Ez z. 10
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Q6. Show that the group velocity is equal to particle veloCity. Also prove that the group velocity ofthe photons is equal to the velocity oflight. 15
for initial current conditions I=I0 and dt/dl=0 at show that the time dependent current in the critical damping case for an LCR circuit is given by <img src='./qimages/288-6-b.jpg'>
Using Ampere's Law and continuity equation, show that the divergence of the total current density is zero., 15
Q7. (a)State 'and explain Stefan-Boltzmann. Law. Show that log P log K 4 log where P is the power emitted by black body and R is the resistance of the black body, K is a constant. 10
One kg of water at 20°C is converted into ice at -10°C at· constant pressure. Heat' 'capacity of water is 4,200 J/kg.K and that of ice is 2,100 J/kg.K. Heat of fusion of ice at O°C is 335 X 103 Jlkg. Calculate the total change in entropy of the system. 15
When connected in series, LI, C1 have the !same resonant frequency as L2,C2 also connected in series. Prove that if all these circuit elements are connected in series, the new circuit will have the same resonant frequency as either of the circuits first mention.end.
Show that the ,energy flow due to a plane electromagnetic wave propagating along z-direction in a dielectric medium is given by <img src='./images/288-7-d.jpg'> .where k and·to are the propagation vector and angular' frequency, Eo is electric field amplitude, J..l is the relative permeability of the medium. 10
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Q8.(a)Consider a system of free gas particles having f degrees pf freedom. Use equipartition theorem to establish the relation <img src='./qimages/288-8-a.jpg'> .where Cp and Cy are molar specific heats at constant. pressure and constant volume respectively. Obtain the values of cp/cv for diatomic and triatomic gases. 15
Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by <img src='./qimages/288-8-b.jpg'> where, ftE) is much smaller than unity, J..l and kBT are the chemical .. potential and thermal energy of the atom.
Explain the four relations of Maxwell. US,using the same, obtain the Clausius-clapeyron equation <img src='./qimages/288-8-c.jpg'>
(d)using Maxwell.Boltzmann. distribution law prove that there cannot be any negative absolute temperature.