iitk.ac.in Ph.D. Admission Test Physics Model Question Paper : Indian Institute of Technology Kanpur

Name of the Organisation : Indian Institute of Technology
Location : Kanpur
Exam : Ph.D. Admission Test
Document Type : Old Question Paper
Website : https://www.iitk.ac.in/phy/index.php
Download Model Question Paper :
May 2017 : https://www.pdfquestion.in/uploads/25023-May2017.pdf
Dec 2016 : https://www.pdfquestion.in/uploads/25023-Dec2016.pdf
May 2016 : https://www.pdfquestion.in/uploads/25023-May2016.pdf
Dec 2015 : https://www.pdfquestion.in/uploads/25023-DDec2015.pdf
May 2015 : https://www.pdfquestion.in/uploads/25023-May2015.pdf
May 2014 : https://www.pdfquestion.in/uploads/25023-May2014.pdf

IITK Ph.D. Admission Test Physics Model Question Paper

** Please click on the links above to download the Syllabus and Old Question Papers

Related : IIT Guwahati JAM 2015 Online Mock Test : www.pdfquestion.in/7059.html

Department of Physics :
May 23, 2017
Time : 9:30 – 11:30 AM
Maximum marks : 70

Sample Question

Question 1 :
(A) Consider the harmonic oscillator given by the Hamiltonian H = t — = — (n- t – – – Define the operators a = V2 (p – ix) and at = V2h (p + ix)
(i) The ground state |0) satisfies a 0) = 0. Use this to find the ground state wave-function tho (X), you do not have to normalize it. Also find the corresponding energy? 4+2 marks)

(ii) A perturbation V(x) = ex’ is added to the Hamiltonian. Calculate the first order correction to the ground state energy due to V(x). 4 marks

Useful formulae :
exp(-ax) dx = ; : X?”exp(-ax?) dx = (-1) i exp(-ax) dx

Question 2 :
(A) A spherical shell of radius R carries a surface charge distribution O (0) = Oocos8 (standard notation of spherical coordinates is used).
(i) Write the expansion of the electrostatic potential V(r, 6) inside and outside the shell using Legendre polynomials.
(ii) Write the appropriate boundary conditions satisfied by the potential.
(iii) Solve for the potential everywhere. 6 marks

You may find the following information helpful :
P (X) = 1; P (X) = X; P (X) = (3X: – 1)

(B) An infinitely long cylinder of radius R carries a magnetization M = ks?6 (standard notation of cylindrical coordinates is used).
(i) what is the divergence of the auxiliary field H = i. (B h M), where B is the magnetic field O due to the given magnetization?
(ii) Find the value of H and Beverywhere. 4 marks

Question 3 :
(A) A spherical ball of mass m falls under gravity in a viscous fluid. Find the position x(t) and velocity V(t) of the ball as a function of time t. Assume that the mass starts from rest at a heighth above the ground at t = 0. Solve for x(t) and v(t) assuming a turbulent drag of the form v, where y is a constant and V is the Velocity. 6 marks

(B) For a complex function f(e)=st perform the contour integral f f(z) dz over a circle of radius 2 with its center at the origin. 4 marks

Question 4 :
Consider a system with the Hamiltonian H = 1 – 2a cos2 cos3t of a particle of unit mass, generalized coordinate q, and generalized momentump. Write the Euler-Lagrange equation for the system in terms of q and solve it for the initial condition: g (0) = a and g (0) = 0. 10 marks

Question 5 :
(A) How many significant figures are there in the product of 0.007 and 1.2345? Express the product to the correct number of significant figures. 2 marks

(B) What is the maximum percentage uncertainty in x, if x = (23.381 + 0.007) – (23.178 + 0.006)? 2 marks

(C) A student wants to determine the acceleration due to gravity (g) by measuring the time-period of a simple pendulum. Determine the mean value and the sample standard deviation of the g values from the data given below. Calculate the best value of the uncertainty in g (standard deviation of mean). Uncertainties in the measurement of length of String and period of oscillation are not known. 6 marks Period of oscillation (in seconds)

Question 6 :
(A) A transistor circuit is shown in the figure on the right.
(i) If the input voltage = 0.2 V what will be the output voltage? (2 mark) input
(ii) If the input voltage can be expressed as Output Win = 2 sin(20t) (with t in seconds and Vin in volts) write 1 kg) an expression for the output voltage. 3 marks – -15 V.

(B) In the opamp circuit shown, assume that the opamp is 15 powered from +15 V supplies. RL = 1.0 k(). 5 marks)
(i) What is the output of the circuit if Win = +5 V’? Vin
(ii) What is the output of the circuit if Win = -5 V? Vout

Question 7 :
For a system of non-interacting electrons attemperature T and chemical potential LL :
(A) Show that the probability of finding an electron in a state with energy 6 above the chemical potential is the same as the probability of finding a hole at energy 6 below the chemical potential. 4 marks

(B) For the above system, suppose that the density of states g(s) is given by : (e – 8a) , for a > & g(e) = O , for e > & > 0 W(-e) , for e < 0 where 8 is the energy of the electron and e is a constant. Find the value of the chemical potential Writing all steps clearly. (At T = 0 electrons occupy states up to a = 0. However, at finite Tsome electrons are excited to higher energies). 6 marks

Syllabus

Mathematical Physics :
Linear vector space; matrices; vector calculus; linear differential equations; elements of complex analysis; complex integrals, Laplace transforms, Fourier analysis, elementary ideas about tensors.

Classical Mechanics:
Conservation laws; central forces, Kepler problem and planetary motion; collisions and scattering in laboratory and centre of mass frames; mechanics of system of particles; rigid body dynamics; moment of inertia tensor; noninertial frames and pseudo forces; variational principle; Lagrange’s and Hamilton’s formalisms; equation of motion, cyclic coordinates, Poisson bracket; periodic motion, small oscillations, normal modes; special theory of relativity – Lorentz transformations, relativistic kinematics, mass-energy equivalence.

Electromagnetic Theory:
Solution of electrostatic and magnetostatic problems including boundary value problems; dielectrics and conductors; Biot-Savart’s and Ampere’s laws; Faraday’s law; Maxwell’s equations; scalar and vector potentials; Coulomb and Lorentz gauges; Electromagnetic waves and their reflection, refraction, interference, diffraction and polarization. Poynting vector, Poynting theorem, energy and momentum of electromagnetic waves; radiation from a moving charge.