civils mains 2013 PHYSICS Paper I

Time allowed: Three Hours

Maximum Marks: 250

Question Paper Specific Instructions

Please read each ofthe following instructions carefully before attempting questions:
There are EIGHT questions divided in two SECTIONS and printed both in HINDI and in ENGLISH.
Candidate has to attempt FIVE questions in all.
Questions no. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at least ONE from each section.
The number ofmarks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer Booklet in the space provided. No marks will be given for answers written in a medium other than the authorized one.
Assume suitable data, ifconsidered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
Attempts of questions shall be counted in chronological order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion ofthe page left blank in the answer book must be clearly struck of

SECTION A

Answer all the five parts given below 10x5=50
Show that the kinetic energy and aniular momentum of the torque free motion of a rigid body is constant. 10

Suppose that an S'-frame is rotating with respect to a fixed frame having the same origin. Assume that the angular velocity r;t of the S'-frame is given by <img src='./qimages/194-1b.jpg'> where t is time and the position vector? of a typical particle at time t as assumed in S'-frame is given by <img src='./qimages/194-1b1.jpg'> Calculate the Coriolis acceleration at t second.

Show that a particle of rest mass rna, total energy E and linear momentum It satisfies the relation E2 c2p 2 m20c4 where c is the velocity of light in free space.

During an earthquake, a horizontal shelf moves vertically. If its motion can be regarded simple harmonic, calculate the maximum value of amplitude of oscillation so that the books resting on it stay in contact with it always. Take g =9·8 ms-2 and T =0·5 s. 10

Explain why information carrying capacity of an optical fibre can be enhanced by reducing the pulse dispersion. How does one minimize pulse dispersion using a graded index optical fibre? 10

Q2. If the forces acting on a particle are conservative, show that the total energy of the particle which is the sum of the kinetic and potential energies is conserved. 20

Prove that as a result of an elastic collision of two particles under non-relativistic regime with equal masses, the scattering angle will be 900 • Illustrate your answer with a vector diagram. 5

Calculate the horizontal component of the Coriolis force acting on a body of mass 0·1 kg moving northward with a horizontal velocity of 100 ms-1 at 30° N latitude on the Earth. 15
Derive the relativistic length contraction using Lorentz transformation. 10

Q.3(a) The dispersion relation for deep water waves is given by w2 gk ak3 where g and a are constants. Obtain expressions for phase velocity and group velocity in terms of the wavelength lambda,w and k represent the angular frequency and wave number, respectively. 5+10=15

A parallel beam oflight from a He Ne laser =630 nm) is made to fall on a narrow slit of width 0·2 x 10-3 m. The Fraunhofer diffraction pattern is observed on a screen placed in the focal plane of a convex lens of focal length 0·3 m. Calculate the distance between the first two minima and first two maxima on the screen. 15

The displacement associated with a three-dimensional plane wave lS given by <img src='./qimages/194-3c.jpg'> Calculate the angles made by the propagating wave with the y and z-axes. 10

Explain the physical significance of resolving power of a grating with relevant mathematical expression. 10

Q4. A particle describes a circular orbit under the influence of an attractive central force directed towards a point on the circle. Show that the force varies as the inverse fifth power of distance. 15

A particle of rest mass M x 10-27 kg, disintegrates into two particles of rest masses M1 3 x 10-27 kg and M2 1 x 10-27 kg. Show that the energies E 1 and E2 of these two parts after disintegration satisfy the condition E 1 E2 while moving in opposite directions with equal linear momenta. Give necessary mathematical derivation. 15

Show that the operator<img src='./qimages/194-4c.jpg'> is invariant under Lorentz transformations.

SECTIONB

Q5.Answer all the five parts given below 10x5=50
in the circuit diagram shown below,calculate the current passing through the milliammeter. <img src='./qimages/194-5a.jpg'>

Consider the equation for a series RLC circuit and compare this to the parallel resonant circuit shown below: <img src='./qimages/194-5b.jpg'> Calculate the value of Rp if a series RLC circuit and the parallel RLC circuit are to have same equations for the potential of capacitance while they both have the same C and Q with Q being the total charge. 10

A thermally insulated ideal gas is compressed quasi-statically from an initial state with volume Vo and pressure Po to a final state of volume Vf and pressure Pf- Show that the work done on the gas in the process is given by W=Cv/R(PfVf-P0V0) where Cv and R having standard meanings.

In a tungsten filament lamp, thermionic emISSIOn takes place at 1·2 x 103 K. Calculate the ratio of spontaneous emission to stimulated emission for non-degenerate energy levels. Interpret your result physically. Take A 550 nm, kB 1·38 x 10-23 h 6·67 X 10--34 Js and c x 108 ms-I . 10

The electric field of a plane e.m. wave travelling along the z-axis is <img src='./qimages/194-5e.jpg'> Determine the magnetic field.10

Q6.(a) A series LCR circuit has resonant frequency 000 and a large quality factor Q. Write down in terms of 00, 000 and its impedance at resonance, impedance at half-power points and the approximate forms of its impedance at low and high frequencies. 15

Consider the following coupled inductor capacitor circuit: <img src='./qimages/194-6b.jpg'> <br><br>Calculate the ratio of the frequencies of the anti-symmetric and symmetric modes wa/ws (given
Using the fundamental concepts of electromagnetism, determine the electric field of an electric dipole It at a distance f and its energy in an electric field . 15
ABCD is a rectangle in which charges of 10-11 2 x 10-11 C and 10-11 C are placed at corners C and respectively. <img src='./qimages/194-6d.jpg'> Calculate the potential at the corner A and the work done in carrying a charge of 2 coulombs to A. 10

Q7. Considering an isotropic, linear, non-conducting, non-magnetic and inhomogeneous dielectric medium with <img src='./qimages/194-7a.jpg'> Write down the scalar equation for Ex from the above equation. 5 Interpret physically the situation if we move from homogeneous to an inhomogeneous medium. 5 Obtain the similar vector equation for the magnetic field H in inhomogeneous medium. 15

For a uniform wire oflength L and radius a having a potential difference V between the ends and a current I along it, calculate the energy per unit time delivered to the wire by Poynting vector. 10

Q8. The vapour pressure, in mm of Hg, of a substance in solid state is given by the relation In p 23·03 3754 where T is in Kelvin. The vapour pressure, in rom of Hg, of the substance in liquid state is given by the relation In p 19·49 3063 . Calculate the coordinates of the triple point, and the latent heat of vaporisation at the triple point. Take Gas constant R =8·314 J mol-1 K-1. 15

In Leh, temperature of ice on a cold winter night is measured as 20°C. Calculate the change in entropy when 1 kg of ice is converted into steam at 100°C. Given specific heat capacity of ice is 500 cal kg-1 latent heat of ice is 3·36 x 105 J latent heat of steam IS 2·26 x 106 J kg-1 and J 4·2 J cal-1. 15

The coefficient of viscosity of helium at 27°C is 2 x 10-5 kg m-1 s-l. Calculate the average speed and the diameter of a helium molecule, if it is assumed that the gas obeys Maxwell Boltzmann distribution. Given Boltzmann constant kB 1·38 x 10-23 J K-1 and mass ofhelium atom =6·67 x 10-27 kg. 10

N particles obeying Classical Statistics are distributed among three states having energies £1 £2 kBT and £3 2 kBT, where kB is Boltzmann constant. If the total equilibrium energy of the system is 1000 kBT, calculate the value of N. 10