Indian Institute of Technology Guwahati (IIT-G) 2006 JAM Physics - Question Paper

2.    Two electric dipoles P₁ and P₂ are placed at (0, 0, 0) and (1, 0, 0) respectively, with both of them pointing in the +z direction. Without changing the orientations of the dipoles, P₂ is moved to (0, 2, 0). The ratio of the electrostatic potential energy of the dipoles after moving to that before moving is

(C)

(D)

At a given point in space the total light wave is composed of three phasors P₁ = a,

a    a _ . q

e ¹ . The intensity of light at this point is

P₂ = e¹ and P₃ =

² 2 ³ 2

A small magnetic dipole is kept at the origin in the x-y plane. One wire Li is located at z = _ a in the x-z plane with a current I flowing in the positive x direction. Another wire L₂ is at z = + a in y-z plane with the same current I as in L_is flowing in the positive y direction. The angle <p made by the magnetic dipole with respect to the positive x-axis is

(A)    225

(B)    120

(C)    45

(D)    270

The ratio of the inner radii of two glass tubes of same length is V2. A fluid of viscosity 8.0cP flows through the first tube, and another fluid of viscosity 0.8cP flows through the second one when equal pressure difference is applied across both of them. The ratio of the flow rate in the first tube to that in the second tube is

The relation between angular frequency o and wave number k for given type of waves is (o² =ak + P k³. The wave number k₀ for which the phase velocity equals the group velocity is

(A)

(B)

(C)

^(D)

A neutron of mass m_n = 10 kg is moving inside a nucleus. Assume the nucleus to be a

A spring-mass system has undamped natural angular frequency o)₀ = 100 rad s^-1. The solution x(t) at critical damping is given by x(t) = x₀(1 + a>₀ t) exp(-₀1), where x₀ is a constant. The system experiences the maximum damping force at time

In an intrinsic semiconductor, the free carrier concentration n (in cm^- ) varies with temperature T (in Kelvin) as shown in the figure below. The band gap of the

semiconductor is (use Boltzmann constant k_B = 8.625 x 10 ⁵ eV K^-1)

^{2 3} 1000/T

11.    E(x, y, z, t) = A(3 i+ 4 j)exp\i(t - kz)] represents an electromagnetic wave. Possible directions of the fast axis of a quarter wave plate which converts this wave into a circularly polarized wave are

^(A)    72^[ [ +^{j ] and} 72^[-1 +⁷ j]

^(B)    72^[3 [ +^{4 j] and} 72^{[41 -} 3 j]

^(C)    72^[3 [ - 4 j ] and 72^[41 +^{3 j} ]

^(D)    72^{[7 [ - j] and} 72^[ i + ^7j]

12.    A particle of rest mass m₀ is moving uniformly in a straight line with relativistic velocity (c, where c is the velocity of light in vacuum and 0 < // < 1. The phase velocity of the de Broglie wave associated with the particle is

13. Electrons of energy E coming in from x = - ro impinge upon a potential barrier of width 2a and height V₀ centered at the origin with V₀ >E, as shown in the figure below. Let

Jim (V₀ - E ) . k = -. In the region - a < x < a, the wave function for the electrons is a

h

linear combination of

k

V

V0

-a

(A)    e^kx and e

(B)    e^lkx and e

(C)    e^lkx and e^- lkx

(D)    e-^lkx and e^kx

A solid melts into a liquid via first order phase transition. The relationship between the pressure P and the temperature T of the phase transition isP = -2T + P₀, where P₀ is a constant. The entropy change associated with the phase transition is 1.0 J mole^-1 K^-1. The

Clausius-Clapeyron equation for the latent heat is L = T| ^dP I Av. Here Av = v_liquid - v_solid

V dT J

is the change in molar volume at the phase transition. The correct statement relating the values of the volumes is

^{(A)    v}liquid = vsolid

^{(B)    v}liquid = ^vsolid 1

^{(C)    v}liquid = ^vsolid--

2

^{(D) v}liquid = ^vsolid + ²

The symmetric part of P = I (a 2 b) is

16. Consider a Body Centered Cubic (BCC) crystal with lattice constant a. Determine

(a)    the Miller indices for the (1, 0, 0) plane,

(b)    the number of atoms per unit area in the (1, 1, 1) plane.

= nRT, while during an adiabatic process

the gas obeysP V^Y = K, where a and K are positive constants. All other symbols have their usual meaning. Find the work done by the gas when it is expanded first isothermally

P

from (P, V) to (Pi, 2V) and then adiabatically from (Pi, 2V) to (-, Vj), where P₁ < P.

18. A conducting sphere of radius R_A has a charge Q. It is surrounded by a dielectric spherical shell of inner radius R_A and outer radius R_B (as shown in the figure below) having electrical permittivity s(r) = s₀ r .

(a)    Find the surface bound charge density at r = RA.    [12] (b)    Find the total electrostatic energy stored in the dielectric (region B).    [9]

19. For the transistor circuit shown below, evaluate V_E, R_B and Rc, given I_C =1 mA, Vce = 3.8 V, V_BE = 0.7 V and V_cc = 10 V. Use the approximation I_c ~ I_E

[21]

20.    For the vector field V = xz² i - yz²j + z (x² -y² )k ,

(a)    calculate the volume integral of the divergence of V over the region defined by

- a < x < a, - b < y < b and 0 < z < c .

_r [12]

(b)    calculate the flux of V out of the region through the surface at z = c. Hence deduce the net flux through the rest of the boundary of the region.    [9]

21.    The spherical surface of a plano-convex lens of radius of curvature R = 1m is gently placed on a flat plate. The space between them is filled with a transparent liquid of refractive index 1.55. The refractive indices of the lens and the flat plate are 1.5 and 1.6 respectively. The radius of the sixteenth dark Newtons ring in the reflected light of wavelength X is found to be V5mm.

(a)    Determine the wavelength X (in microns) of the light.    [12]

(b)    Now the transparent liquid is completely removed from the space between the lens and the flat plate. Find the radius (in mm) of the twentieth dark ring in the reflected light after this change.    [9]

22.    A resistor of 1 kQ and an inductor of 5 mH are connected in series with a battery of emf 4 V through a switch. The switch is closed at time t = 0. In the following, you may use e³ - 20.

(a)    Find the current flowing in the circuit at t = 15 micro-second.    [9]

(b)    Find the heat dissipated through the resistor during the first 15 micro-second.

[12]

23.    A photon of energy Ep_h collides with an electron at rest and gets scattered at an angle 60 with respect to the direction of the incident photon. The ratio of the relativistic kinetic energy T of the recoiled electron and the incident photon energy E_ph is 0.05.

(a)    Determine the wavelength of the incident photon in terms of the Compton

r h }

wavelength X_c =- , where h, m_e, c are Plancks constant, electron rest mass

I ^me^C)

and velocity of light respectively.    [12]

(b)    What is the total energy E_e of the recoiled electron in units of its rest mass ?

[9]

-    3

24.    A particle moves in a plane with velocity v = v_rr + v₀0 such that v_r = -4v₀. The time

dependence of the magnitude of the velocity | v | = 5t. It is given that r = 1, 0 = 0 and v_r > 0 at t = 0. (In the following, you may use e³ - 20.)

(a)    Determine the traj ectory r(0) of the particle.    [9]