CMI PhD Physics Entrance Exam Question Paper 2016 : Chennai Mathematical Institute
Name of the Organisation : Chennai Mathematical Institute
Exam : Entrance Exam
Subject : PhD Physics
Year : 2016
Document Type : Previous Years’ Question Papers
Website : http://www.cmi.ac.in/admissions/syllabus.php
Download Model/Sample Question Paper :
PhD Physics 2011 : https://www.pdfquestion.in/uploads/13302-pgphysics2011.pdf
PhD Physics 2012 : https://www.pdfquestion.in/uploads/13302-pgphysics2012.pdf
PhD Physics 2013 : https://www.pdfquestion.in/uploads/13302-pgphysics2013.pdf
PhD Physics 2014 : https://www.pdfquestion.in/uploads/13302-pgphysics2014.pdf
PhD Physics 2015 : https://www.pdfquestion.in/uploads/13302-pgphysics2015.pdf
PhD Physics 2016 : https://www.pdfquestion.in/uploads/13302-pgphysics2016.pdf
CMI Ph. D. Physics Entrance Exam 2016 :
Instructions :
** Enter your Registration Number here
** Enter your Examination Centre here
Related : Chennai Mathematical Institute MSc/PhD Mathematics Question Papers : www.pdfquestion.in/13299.html
** The time allowed is 3 hours.
** Total Marks: 100
** Each question carries 20 marks.
** Answer all questions.
Rough Work :
** The coloured blank pages are to be used for rough work only.
(1) A particle is constrained to move on a parabola y = x2 in the plane where is a constant of dimension inverse length. Let there be a constant gravitational force in negative z direction.
(a) Write down the suitable generalized co-ordinates for this system. [1 Mark]
(b) Write down the Lagrangian and obtain Lagrangian equations of Motion. [1 Mark]
(c) Find the equilibrium position of the particle and write the equation for small oscillations about this equilibrium. [1 Mark]
(d) By solving the above equation show that particle executes simple harmonic motion about the equilibrium position. [2 Marks]
(2) Consider three Newtonian particles of masses m1;m2;m3 which interact with each other only through Newtonian gravitational force. Let their position vectors be ~r1; ~r2, and ~r3 respectively with respect to the center of mass of the system.
(a) Write down the equations of motion for the system. [4 Marks]
(b) Assuming that the distance d between any pair of masses is equal and is in fact constant, show that the system rotates with a constant angular velocity in it’s plane and about the center of mass. [6 Marks]
(3) Consider a uniform hoop of mass M and radius R which hangs in a vertical plane supported by a hinge at a point on the hoop as shown in the figure. Calculate the natural frequency of small oscillations. [5 Marks]
(4) Recall that under a gauge transformation in magneto statics, the vector potential is transformed to A0(r) = A(r)-> r(r) where (r) is a scalar function. [20 Marks]
(a) Show that Ampere’s law (in differential form) takes the same form for both A and A0.[2 Marks]
(b) Give the differential equation that must satisfy so that A0 is in Coulomb gauge. Write an integral expression for its solution, assuming A ! 0 sufficiently fast as jrj ! 1. Proceed by first writing Poisson’s equation for the electrostatic potential (r) due to a localized charge distribution (r) and its solution via an integral. Give an appropriate analogy between the two problems. [5 Marks]
CMI Ph. D. Physics Entrance Exam 2015 :
(1) A particle of mass m moves under the influence of an attractive central force f(r)^r.
(a) Show that the motion is planar. [3 mks]
(b) The motion of the particle can be treated as one dimensional with the introduction of an appropriate effective potential which involves the angular momentum. Write down the effective potential and total energy of the particle. [4 mks]
(c) Using the result obtained above, show that by proper choice of the initial conditions, you can always get a circular orbit. [5 mks]
(d) If you subject the above circular orbit to a small radial perturbation, obtain the relation between f(r); r and @f @r for the orbit to be stable. [4 mks]
(e) If the force is of the form f(r) = ??K=rn, obtain the condition on n for the circular orbit to be stable. [4 mks]
(2) Problem (c) is compulsory. Attempt only one of the two problems (a) and (b). Maximum available points are 20.
(a) A ring of radius r has charge q uniformly distributed over it. The ring is placed in the x??y plane with its center at the origin.
(i) What is the electric eld at any point along the z-axis. [5 mks]
(ii) Use the above result to show the following :
Consider a charge -q which is constrained to move on the z-axis that is perpendicular to the plane of the ring, passing through it’s center.
Show that the charge will execute simple harmonic motion when placed along the z-axis at a distance z << r from the center of the ring. [5 mks]
(b) Consider two plane innite sheets with uniform charge densities which intersect at right angles. Find the magnitude and direction of E everywhere and sketch the electric field lines. [10 mks]
(c) A very long solenoid of radius r and having N turns per unit length carries a current I = I0 sin(!t) in the counter-clockwise direction. Assume that the axis of the solenoid coin cides with the z-axis.
(i) Write the expression for ?B(t) inside the solenoid. [3 mks]
(ii) Assuming that ?B is zero outside the solenoid, write down the expression for electric field +E
(t) inside and outside the solenoid. [7mks]
(3) Problems (a) and (b) are not related: both are compulsory.
(a) Consider a free particle of mass m in a 1-dimensional box. The potential has the form V (x) = 1; x < 0; x > b; V (x) = 0; 0 < x < a; V (x) = V0; a < x < b. Find the quantization condition for stationary state energy levels for energies E > V0, and the corresponding wave- function. In the limit of E much larger than V0, show that the resulting quantization condition you nd reduces to the familiar one for a particle in a box with no internal \step” (i.e. with V0 0). [10 mks]
(b) Consider a system of two non-interacting harmonic oscillators with Hamiltonian
(4) The rotational energy levels of a diatomic molecule made of two identical atoms with spin half nuclei is given by
In answering the following questions you need to consider only the rotational energy levels and the fact that the two nuclei are identical. In particular you need not consider electronic or vibrational levels.
(a) What are the allowed spin states? [2 mks]
(b) How does the spin wave function change under the exchange of the two spins? [2 mks]
(c) How would the rotational wave function change under the exchange of the two spatial co- ordinates of the nuclei? [2 mks]
(d) Which spin states are allowed for J = 0 and J = 1 ?[4 mks]
(e) How many states ( including spin wave function) are there with energy E = 0 ?[2 mks]
(f) How many states are there with energy ( including spin states) E = 2 X ?[2 mks]
(h) What will be relative population of the two states (J = 0; J = 1), if the system is in equilibrium at temperature T? [6 mks]