civil services mains 2009

PHYSICS

Paper-I

Time Allowed Three Hours Maximum Marks 300

INSTRUCTIONS

Each question is printed both in Hindi and in English.

Answers rnust be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose. No marks will be given for the answers written in a medium other than that specified in the Admission Certificate.

Candidates should attempt Question Nos. :J.. and S which are compulsory, and any three of the remaining questions selecting at least one question from. each Section.

The number of marks carried by each question z.s indicated at the end of the question. Assume suitable data if considered necessary and indicate the same clearly. Symbols/notations carrytheir usual meanings, unless otherwise indicated.

Section-A

1. Show that the total energy per unit mass of liquid flowing from one point to another without any friction remains constant throughout the displacement. 10

Let EA E1 sin rot and E B E2 sin By using analytical method, obtain an expression to explain interference. Also show that intensity varies along the screen in accordance with the law of cosine square in interference pattern. 10

Two bodies of masses M1 and M 2 are placed at a distance d apart. Show that at this position where the gravitational field due to them is zero, the potential is given by V G/d (M1 M2 M2) 10

Show that a four-dimensional volume element dxdydzdt 1s invariant to Lorentz transformation. 10

Derive the condition for achromatism of two thin lenses separated by a finite distance and made up of same material. 10

What are the characteristics of stimulated emission? Show that in the optical region, stimulated emission is negligible compared to spontaneous em1ss1on. 10


2. Consider a spherical shell of mass M and radius R. Calculate the potential due to this shell at a point P when the point is outside the shell and inside the shell R). If the spherical shell is now replaced by a uniform solid sphere of same mass and radius, what will be its potential at the same external point? 15

Show that for any rigid body consisting of at least three particles, not arranged in one straight line, number of independent degrees of freedom is six. Define Euler's angles 0 and to describe the configuration of such a rigid body. Consider two frames of reference, one fixed to the body and the other to the space dermed as and S respectively. Show- that the angular momentum of the rigid body in the two frames are related by dL/DT=DL/DT+W*L where W is the angular velocity of rotation. 30

Obtain Poiseuille's equation for a viscous fluid flowing through a narrow tube of radius r and length l. If a spherical body of radius a is allO'wed to move at a speed V through the same fluid of viscosity Tl, show that the viscous force will increase with the speed linearly. 15

3. For stationary waves on a string whose ends are fixed, show that the energy density is maximum at antinodes and minimum at nodes. 20

Explain the phenomenon of interference in thin films. Why is the contrast better in brightness of fringes obtained from the interference of reflected light rays compared to the transmitted light rays? 20

Obtain the relativistic equation for aberration of light using velocity trans- formation equations. 20


Distinguish between high dispersive power and high resolving power. 5
Obtain an expression for the resolving power of a plane trans- mission grating. 10

Deduce the missing orders for a double-slit Fraunhofer pattern, if the slit widchs are O· 16 mm and 0·8 mm apart. 5

Show that the plane of polarisation 1s rotated through <img src='./qimages/46-4b1.jpg'> in optical rotation where symbols have their usual meanings. 15

A plane-polarised light is incident perpendicularly on a quartz plate cut with faces parallel to optic axis. Find the thickness of the quartz plate which introduces phase difference of 60° between and a-rays. 5

At what temperature are the rates of spontaneous and stimulated em1ss1on equal? (Assume lambda= 500 nm) 7

What are the important properties of a hologram? 6

Optical power of 1 mW is launched in to an optical fiber of length 100 m. If the pov.rer emerging from the other end is 0·3 mW, calculate the fiber attenuation. 7

Section-B

5. A cylindrical conductor 1s carrying a current along its axis which is assumed to be 1n z-direction. The current 1s uniformly distributed throughout its cross-section. Show that the vector potential A associated vvith the magnetic induction due to the currentcarrying cylindrical conductor 1s independent of z. 10

St:ate Gibbs phase rule. Show that for a 1-component closed thermodynamic system having two phases, the condition for equilibrium between the phases is that their specific Gibbs functions are equal. 10

A long solenoid of radius R and n turns per unit length cames a sinusoidal current 10 cos wt. Determine the magnitude of induced electric field outside the solenoid. 10
A series circuit 1s connected across a voltage source V l 00 sin 300t. If R 500 L l H and C 2 µF, calculate the average power delivered to the circuit.

Using Planck's radiation formula, deduce Wien's displace1nent 10

A thin dielectric cylindrical rod of cross-section A is situated along z-a.xis from z 0 to z L. The polarisation of the rod is along its length and it is gi ven by P (2z 2 S)z. Calculate bound volume charge density at each end of the rod. 10

6. For an arbitrary localised charged distribution, obtain an expression of electrostatic potential V m terms of multipole expansion. 25

State Biot-Sa vart law. Calculate the magnitude of axial magnetic induction due to a circular loop of area A carrying current I. 20

Consider in the region O z 1 m an infinite slab made of a material with relative permeability, µ r 3·5. If <img src='./qimages/46-6c.jpg'> 15
7. Consider the incidence of a planepolarised electromagnetic wave at the interface of two media having dielectric permittivity and magnetic permeability µ i and (E2 µ 2 respectively. The interface 1s chosen to be x 0 plane. and represent the propagator vectors associated with the incident, refracted and reflected waves respectively. Using the boundary conditions on them, establish the Snell's laws of refraction. 15

In a non-cha-.r ged current-free dielectric, p 0 and J 0. Show that m this medium, electric and magnetic fields satisfy three-dimensional wave equations <img src='./qimages/46-7b.jpg'> 20

A plane-polarised electromagnetic wave is incident on the interface of tvvo dielectrics having dielectric permittivity £ 1 and E 2 . Assume tl1at the electric vector E lies in the plane of incidence. Using the boundary conditions at the interface, obtain the expressions for the amplitude reflection coefficient (r1 1 and the a.Inplitude transmission constant (t1 1 Using the components of the Poynting vector E x H associated with the reflected and transmitted waves, obtain the expressions for reflection and transmission coefficients R11 and 7j 1 respect ively. Under hat condition, r11 0 and t1 l 25

8. Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S k log .Q, where Q 1s the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system <img src='./qimages/46-8a.jpg'> where the equality sign applies for internally reversible processes and the in equality for internally irreversible processes. 20

What are the limitations of Einstein's theory of specific heat of solids when compared with experiments at low temperature? Outline the assumptions made 1n Debye's theory and show that the specific heat at low temperature follows Cv T 3 la'v. What is the significance of Debye temperature, Tn 20

Energy distribution for ni particles in classical statistical mechanics is given <img src='./qimages/46-8c.jpg'> 20


Note English version of the Instructions is print ed on th e front cov er of this question paper.