Indian Institute of Science 2009 Ph.D Physics Int - Question Paper

3. A Youngs double slit apparatus has a slit separation of 0.5 mm and a screen placed at a distance of 100cm from the slits. When illuminated by a:632nm_ Helium-Neon laser, the position of the first intensity minimum with respect to ' ' the central maximum will'be

4.    A half-wave plate is an anisotropic optical element which introduces a retardation of A/2 in the optical path length for the electric field component parallel to its optical axis' where A is the wavelength of the incident radiation. If linearly polarized light is incident on'a half-wave plate with its polarization at 45 to the optical axis, the transmitted light is (compared to the initial polarization)

(A)    linearly polarized and rotated by 45

(B)    linearly polarized and rotated by 90

(C)    left circularly polarized .

(D)    right circularly polarized

5.    The dimensionless fine structure constant a is the ratio of the electron speed in the Bohr model of the hydrogen atom tolhe speed of-light. If e is the electron charge, h = /i/27r, Cq the permittivity of vacuum, and c the speed of light, then a is given by

(A)

(B)

(C)

(D).

At the resonant frequency / = l/27r\/LC,

(A)    the current through R is infinite

(B)    the current through R is zero

(C)    the current through L is zero

(D)    the current through C is zero

8. An RC circuit is connected to a DC voltage source through a switch as shown below.

The switch, is initially open and the capacitor uncharged. At time t = 0, the switch is closed. The current through-the circuit for t > 0 is given by

(^A)    exp(-f R/C)

(^B)    [l~^exp (~tR/C)]

(C)    7f^exP (~t/^RC)

(D)    [1-exp {-t/RC)}    '

9. An optical pyrometer gives an output voltage proportional to the incident radiation intensity. For a black body at 600 K, it reads 1.5 V. If the temperature changes by 2K, the change in the output signal is

10. A particle of mass M which is at rest in the laboratory, decays into two particles, one of which has a mass M/2 while the other is massless. What is the magnitude of the momentum of each of the particles in the laboratory frame?

11.    Given that j is the total angular momentum, which set of quantum numbers is allowed for an election state in the hydrogen atom?

(A)    77 = 3, / = 2, j = 5/2

, (B)    n = 3, / = 2, j = 3

(C)    n = 3, Z = 3, j = 5/2    '

(D)    n = 3, I = 3, j = 3

13. The ratio oF energies of the Balmer-7 (n = 5 n == 2) to the Lyman-7 (n = 4 * 77,= 1) is

(A)    28 : 125

(B)    3:5

(C)    3 : 10

(D)    21 : 10

14.    What is the ground state energy of six non-interacting electrons in a 2-dimenslonal, isotropic, simple harmonic oscillator characterized by u?

(A)    3fiu

<{B)    5 hu

(C)    10hu

(D)    12 fiw \    '

15.    Given that the binding energy per nucleon is 1.1 MeV for deuteron and 7.0 MeV for an cv-particle, how much energy is released in deuteron fusion?

(A)    5.9 MeV

(B)    ' 11.8 MeV

(C)    25.8 MeV

(D)    23.6 MeV

16.    What is the maximum electron energy in neutron /3-decay?

. (A)    783 eV

(B)    783 keV

(C)    783 GeV

(D)    783 TeV - *

17. Which of these is a doubly magic nucleus?

(A)    c

(B)    . 12C

(C)    M0

(D) 160

18. Given that when light is shone ori an aluminium target (work function for Aluminium is 4.08 eV) leads to the emission of electrons with maximum kinetic energy of 5.0 eV, what was the frequency of the light?

(A)    1.2 x 10¹⁵ Hz

(B)    9.9x10^mHz

(C)    2.2 x 10¹⁵ Hz

(D)'    2.2 x JO¹⁴ Hz

19.    Replace in a hydrogen atom, the electron with a muon, which has the same charge, but is 210 times as heavy. The ionization potential of this atom is

approximately    ' .

(A)    13.6 eV

(B)    2.9 keV

- (C) 65.7 meV.

(D) 27.2 eV

20.    Consider a collection of molecules of an ideal gas at temperature T. The ratio of the rms speed to the most probable speed of the molecule is

(A)    v/3 : \/2 . (B) v/3: y/8

(C)    3 : 8

i

(D)    3:2

\ ^;

21.    A monoatomic ideal gas of N atoms undergoes isothermal reversible expansion from volume Vi to V2. The change in entropy of the gas is    ' -

(A^} 0

(B)    MRln '    '

(C)    2NAr_aln '

V\    ' /

. (D) Nk_B lnj|

22. Two vessels separately contain two ideal gases A and B at the same temperature. the. pressure of A being twice that of B. Under these conditions, the density of A is found to be one and half times the density of B. The ratio of molecular weights of A and B is

(A)	.1/2
(B)	2/3
(C)	3/4
(D)	2

23. A system has two energy levels 0 and e > 0. The energy levels are populated according to the Boltzmanns distribution law. In the limit T > oo the entropy

per;	particle is
(A)	+oo
(B)	kg In 2
(C)	2 kB
(D)	ks In c2

24.    Keeping density and sound speed constant, doubling the pressure implies that the ratio of'specific heats CpjCv    '

(A).    doubles

(B)    . halves    .

'I    %

(C)    remains constant

(D)    quadruples    

25.    Considering fermionic particles A and B with chemical potentials // and p. respectively moving in a box at temperature T. The number density of particle A is    '

' (A) greater than that of B    ., '

(B) less than that of B . (C) equal to that of B .    ' ':

(D) zero

26.    Let z\ and z₂ be two non-zero complex numbers. If

| Zi + z₂1 =?\zi\ + \Z2\,

which of the following is true

(A)    Re(zif2) < 0, and Im) = 0

(B)    Re(ziz₂) > 0, and Im(ziz2) = 0

(C)    Re(ziz₂) > 0, and Im) > 0

(D)    Re(zi₂) < 0, and Im(ziz₂) < 0

27.    Let M be a 3 x 3 Hermitian matrix which satisfies the matrix equation

M² -5M + 6/ = 0

where I refers to the identity matrix. Which of the following are possible

eigenvalues of M
(A)	{1, 2, 3 }
(B)	{2, 2. 3}
(C)	{2, 3. 5}.
(D)	{5, 5, 6}

28.    Given two dice of which one is unbiased while the other is such that the probability of obtaining an even number is 1/4 and the probabilities of obtaining odd numbers are the same. In a simultaneous throw of both the dice what is the probability of obtaining an outcome of 4

(A)    

(B)    i .

(C)    h t

(D)    T2 \

' :

29.    The number of times that a circle of radius tt, centered at the origin intersects the curve y = tan a: is    .

r (A) 2 .    . .

.(B). 4    :

(C) 6 ' (15) 8'

+ -2 = o

dx 3x

with the condition y = 2 when x = 3 is given by

(A).    x²y^s = 72

(B)    iV = 108

(C)    ,rV = 108-

(D)    x*y² = 72

31. A point particle is moving in the (:r,y) plane on a trajectory given in polar coordinates by the equation

r²

The trajectory of the particle is

(A)    a parabola

(B)    a straight line

(C)    a circle

(D)    a hyperbola

32. The n-th term of a sequence is given by

-    ^tn=(^a:+n).

where x, y are real numbers. Which.of the following is true -

(A)    If -c = 1 then lim t_n =

fl

(B)    If x > 1 then lim t = e^y/l '

rtcc

(C)    If z = 1 then lim f₍₁ = e^y

^V '    XI *oo

(D)    If x^v< 1 then lim

n'OO

n

, 1 1 1 ^{1 1}~2 ⁺ 3~l ⁺ l ⁺

which of the following is true?

(A)    The series is divergent

(B)    The series is absolutely qonvergent

(C)    The series converges to In 2

(D)    The series converges to 7r/2

34. The normal to the surface given by the equation

z = cos x cosh y

at the point x = j and y . O lies in

(A)    (x,y)' plane

(B)    [x,z) plane

(C)    {y,z) plane

(D)    On the plane given by the equation x + y + z = + 1

35. Consider the wave function of a bound state in the one dimensional potential shown below.

V(i)

'-2a

2 V. -

Which of the following is true (A) /_ⁿJ4-(z₎i)|²rfx<l ;

I'l'fx, t)\²dx is time dependent

CB)

(C)

(D)

/2q    \n*,t)\²dx=i

at

for a given Hamiltonian H. Which of the following can never be a solution or the above equation?

(A)    /) +    i). where 0 is a constant

(B)     (i(x.t) + 'M-t, ))

(C)    72 (l 0 ^_ 0)

(D)    |tfiMl² + l*20r,*)|²

37. A hydrogen beam is prepared in the state

Mr* *) = -y ^exP (^,:x) ⁺ \ ^exp (*ir)

Where Ei and ₂ are the energies of the ground state and the first excited state, of the hyrogen atom and i(r) and (r) are their wave functions respectively. The beam is incident on a detector which measures their energy. Let Eq be the ionization energy of the hydrogen atom- The average energy measured by the detector is given by

(A) -E0

(B)    -gb-    '

.(P)    -}/*

(b) -{

38. The nncertainity in the measurement of the momentum of a particle of mass m confined to move in one dimension at time t is given l\y

, >. where p is the momentum pf the particle. Which of the following values can'

be a possible nncertainity in measurement of the position x at the same time

.)

t.
(A)	Air =	s
(B)	A.v =	1 fih-2 V m
(C)	Ax	1 flh 2 V 2m
(D)	Ax =	HE V 3fri

39.    A plane parallel capacitor is charged up to a surface charge density a. What is the attractive force per unit area between the two plates?

(A)

OB)

<^C> * .

40.    A spherically symmetric charge distribution p(r) has zero net charge. For an arbitrarily chosen coordinate system, which ofthe following statements is tine?

(A)    Dipole moment is the first non-zero moment

(B)    Quadrupoie moment is always non-zero

(C)    All moments are zero

. (D) The monopole and the dipole moments are always zero

41.    An ensemble of identical harmonic oscillators with a. Hamiltonian

H = -mv² + -kx² 2 2 -

have energies between E and E + AE. If the oscillators are initially, uniformly : distributed in the phase spate then which of the following is false?,

(A)'

()

(C).

(D)

More particles are at the maximum amplitude than at the origin ' The. mean displacement is zero

Mean kinetic energy is equal, to the mean potential energy More particles are at the origin than at maximum amplitude

42.    A particle is moved quasi-statically from the. point (3,0) to (3,0), along a . path y = i² 9, in an external force field given by F = yi + 3yj. Given that all physical quantitites are in SI units, the magnitude of the work done on the particle is given by

(A)

36 J 18 J 9 J 0

(B)

(C)

(D)

43. Consider the solution to the one dimensional wave equation:

= Poexpv'x + 147f) where x is in meters and t in seconds. The velocity of the wave is

(A)	7 m/s
(B)	-1/7 m/s
(C)	-7 m/s
(D)	1/7 m/s

44.    A charge is placed at the vertex of a cube. The electric flux through a face that does not touch the charge is given by

A,    -

(B>- 5    

<^C!

(D) 0    '     '

45.    The electric field due to an unknown charge distribution is given by

E =-\ exp (4r) r .

'    - T    

The total integrated charge over all space J pdV is given by

(A)    0

(?) .... .: ... .."(C) 9/4

(D) qf'K

46.    Ahorizontally placed hollow tube has a crossectional area A at the beginning of the tube that gradually tapers off to Ajl at the end. An incompressible, ideal-fluid of density p enters the tube¹ with a velocity v at the beginning of the tube. What is the difference in pressure at the two ends of the tube?

(ar - p/2.; . ' - ;    .    ;

(B)    3/7i;²/2

(C)    2/2

- (D). 4pt'/2 .     ,

- /

If the mass of the star is M and its radius is R, then what is the minimum value of b such that the asteroid will miss the star?

47. A small asteroid is approaching a massive star with a speed from a very large distance, at an impact parameter b as shown below.

(A)    Ryj 1 + 2GM/v²R - - _f.

(B)    R\j\ 2GM/v²R

(C)    R

(D)    R/2

48. A thin rod is inclined to the vertical at an angle $ = 45 degree as shown below

The length of the rod is L and the.angular velocity ui is in the vertical direction. What is the magnitude of the angular momentum?

(A)    ..MXV12    ' ' '

(B)    ML²uj( 12n/2.-

(C)    A/LV6 ;

(D).    ML²u/2A

49. A point particle at rest is released from the top of a sphere of radius R and slides down frictionlessly under gravity. At what angle From the vertical does it leave the sphere?

(A)    cos' ¹(l/3)

(B)    cojT¹)    :

(C)    shr^l(l/v/2)

(D)    cos~^l(l/V2)    .

A ball of mass m is dropped from a height H on a table with a restitution coefficient 77 <Si 1 . How much distance does the ball travel before it comes to a halt? (Hint: At each bounce the speed after the bounce is 77 times the initial speed)

(C) ()ff

(t)*

* End of the question paper *

1

An incoherent beam of light, strikes a glass plate at normal incidence. It undergoes multiple reflections at the two edges before exiting the plate. The intensity reflection coefficient at each interface is R and the glass has negligible absorption. Neglecting interference effects, the ratio of the total transmitted intensity to the incident intensity is given by

2

(D)    10    . ^;    .T

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