## IIT JAM PHYSICS 2008

Previous Year Question Paper with Solution.

1. The product PQ of any two real, symmetric matrices P and Q is:

(a) Symmetric for all P and Q

(b) Never symmetric

(c) Symmetric if PQ = QP

(d) Antisymmetric for all P and Q

Ans. (c)

Sol. P^{T} = P, Q^{T} = Q; (PQ)^{T} = Q^{T} P^{T} = QP

For PQ to be symmetric, PQ = QP should be satisfied.

2. The work done by a force in moving particle of mass 'm' from any point (x, y) to a neighbouring point (x + dx, y + dy) is given by dW = 2xydx + x^{2}dy. The work done for a complete cycle around a unit circle is:

(a) 0

(b) 1

(c) 3

(d)

Ans. (a)

Sol. Work done for a complete cycle around a unit circle is

3. EFGH is a thin square plate of uniform density and side 4a. Four point masses, each of mass m, are placed on the plate as shown in the figure. In the moment of inertia matrix 'I' of the composite system,

(a) Only I_{xy} is zero

(b) Only I_{xz} and I_{yz} are zero.

(c) All the product of inertia terms are zero

(d) None

Ans. (c)

Sol. Mass distribution is symmetric with respect to the given axes. Therefore, off diagonal terms in inertia tensor will be zero.

4. The chemical potential of an ideal Bose gas at any temperature is:

(a) Necessarily negative

(b) Either zero or negative

(c) Necessarily positive

(d) Either zero or positive

Ans. (b)

Sol.

(i) The chemical potential for classical gas is negative.

(ii) The chemical potential for ideal Bose gas is either zero or negative.

(iii) The chemical potential for ideal Fermi gas is positive or negative.

5. If the electrostatic potential at a point (x, y) is given by V = (2x + 4y) volts, the electrostatic energy density at that point (in J/m^{3}) is:

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

6. In an inertial frame S, a stationary rod makes an angle with the x-axis. Another inertial frame S´ moves with a velocity 'v' with respect of S along with the common x – x´ axis. As observed from S´ the angle made by the rod with the x´–axis is ´. Which of the following statements is correct?

(a)

(b)

(c) if v is negative and if v is positive

(d) if v is negative and if v is positive

Ans. (b)

Sol. Horizontal component along x-axis of length will get contracted in S´ it will be

7. Consider a doped semiconductor having the electron and the hole mobilities µ_{n} and µ_{p}, respectively. Its intrinsic carrier density is n_{1}. The hole concentration p for which the conductivity is minimum at a given temperature is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

8. Two coherent plane waves of light of equal amplitude, and each of wavelength 20p × 10^{–8} m, propagating at an angle of radian with respect to each other, fall almost normally on a screen. The fringe-width (i mm) on the screen is:

(a) 0.108

(b) 0.216

(c) 1.080

(d) 2.160

Ans. (b)

Sol.

9. A circular disc (in the horizontal xy-plane) is spinning about a vertical axis through it centre 'O' with a constant angular velocity . As viewed from the reference frame of the disc, a particle is observed to execute uniform motion, in the anticlockwise sense, centered at P. When the particle is at the point Q, which of the following figures correctly represents the directions of the Coriolis force and the centrifugal force ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. F_{efg} is always away from axis of rotation and perpendicular to the axis of rotation.

is in direction of .

10. Instantaneous position x(t) of a small block performing one-dimensional damped oscillation is x(t) = Ae^{–} cos . Here, is the angular frequency, the damping coefficient, A the initial amplitude and the initial phase. If = v, the values of A and (with n = 0, 1, 2, .....) are

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

11. A photon of wavelength is incident on a free electron at rest and is scattered in teh backward direction. The fractional shift in its wavelength in terms of the Compton wavelength of the electron is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Fractional shift in wavelength [where, is the compton wavelength]

For backward scattering,

12. The logic expression for the output Y of the following circuit is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

13. The activity of a radioactive sample is decreases to 75% of the initial value after 30 days. The half-life (in days) of the sample is approximately [You may use ln 3 1.1, ln 4 1.4]

(a) 38

(b) 45

(c) 59

(d) 69

Ans. (d)

Sol. Time variety of activity is as follows

14. The ratio of the second-neighbour distance of the nearest-neighbour distance to the nearest-neighbour distance in an fcc lattice is

(a)

(b) 2

(c)

(d)

Ans. (d)

Sol. First nearest distance in fcc is

Second nearest distance in fcc is = a

.

15. A thermodynamic system is maintained at constant temperature and pressure. In thermodynamic equilibrium, its

(a) Gibbs free energy is minimum

(b) Enthalpy is maximum

(c) Helmholtz free energy is minimum

(d) Internal energy is zero

Ans. (a)

Sol. Gibb's free energy is defined as

G = H – TS

Change in Gibb's free energy in terms of temperature and pressure

dG = VdP – SdT

at constant temperature (dT = 0) and pressure (dP = 0)

(condition for max and min)

In thermodynamics equilibrium entropy is max. So, Gibb's free energy (G = H – Ts) will be minimum.

16. A thin hollow cylinder of radius and length both equal to L is closed at the bottom. A disc of radius L/2 is removed from the bottom as shown in figure. This object carries a uniform surface-charge density . Calculate the electrostatic potential at the point P on the axis of the cylinder as shown in the figure. [You may use ].

Sol. Potential at point P = potential due to curved surface + potential due to base. Let us calculate potential at p due to curved surface.

Now, calculate potential due to base at point P

So, net potential at point P.

17. A particle of mass 1 kg is moving in a central force field given by

(a) Assuming that the particle is moving in a circular orbit with angular momentum 2 J-s, find the radius of the orbit.

(b) At t = 0, an additional force , where is the instantaneous velocity of the particle, is switched on. Show that the magnitude of its angular momentum after a time second is

Sol. Central force acting on 1 kg mass.

Where v is the central potential:

Let radius of circular orbit is r_{0}.

For circular orbit, V_{e} is minimum at r = r_{0}.

At t = 0, an addition force, starts acting on particle.

18. An incompressible fluid is enclosed between two horizontal surfaces located at z = 0 and z = d. The fluid motion is two dimensional, and the velocity field is given by are periodic functions of the horizontal coordinate x with wave number k.

(a) If the vertical velocity , find the horizontal velocity u(x, z, t) using the equation of continuity. What is the vorticity field ?

(b) Find the net fluid flux from a parallelepiped of size as shown in the figure, where .

Sol. (a) Equation of continuity for incompressible fluid flow is

Hence, the net flux from a parallelopiped is zero.

19. The wave function of a particle confined to a one-dimensional box of length 'L' with rigid walls is given by

(a) Determine the energy eigenvalues. Also, determine the eigenvalues and the eigenfunctions of the momentum operator.

(b) Show that the energy eigenfunctions are not the eigenfunctions of the momentum operator.

Sol.

Since, potential energy of the particle is zero.

So, Some kinetic energy in n^{th} eigen state:

Let be the eigen function of momentum operator.

So, energy eigen functions are not eigen functions of the momentum operator.

20. A mass and spring system consists of two blocks of mass 'M' and one block of mass m(<M). These blocks are connected with two identical springs of spring constant 'k' as shown in the figure. The system is constrained to move along a straight line on a frictionless horizontal surface. The spring follows Hooke's law. Find the angular frequencies of the independent oscillations (normal modes).

Now, the masses 'M' and 'm' are interchanged and the new arranged is shown in the following figure:

The ratio of the frequency of the new arrangement to that of the old arrangement, when the middle block remains stationary, is . Find the ratio of the frequencies in the two arrangement when the middle block oscillates.

Sol.

Let us define generalized coordinate as

q_{1} = x_{1} – x_{01}, q_{2} = x_{2} – x_{02}, q_{3} = x_{3} – x_{03}

_{ }

_{ }

_{ }The T and V matrices are

The secular equation,

If middle block is stationary

So, wen middle block oscillates.

In new arrangement,

21. A half-wave plate and a quarter-wave plate are placed between a polarizer P_{1} and at analyzer P_{2}. All of these are parallel to each other and perpendicular to the direction of propagation of unpolarized incident light (see figure). The optic-axis of the half wave plate makes an angle of 30º with respect to the pass-axis of P_{1} and that of the quarter wave plate is parallel to the pass-axis of P_{1}.

(a) Determine the state of polarization for the light after passing through (i) the half-wave plate and (ii) the quarter wave plate.

(b) What should be the orientation of the pass-axis P_{2} with respect to that of P_{1} such that the intensity of the light emerging from P_{2} is maximum?

Sol. Let us assume that pass axis of polariser and quarter wave plate is along x-axis.

Let the equation of polarised light be x = A sin , it will split into

After passing through half-wave plate. Extra ordinary wave will suffer an additional phase change with respect to ordinary wave

Equation of lissajous figure,

After passing through half wave plate, the light will remain plane polarises but, the plane of vibration will now shift from to with respect to optic axis.

Again, the plane polarises light will go under double refraction on a split into ordinary and extraordinary light.

The extra-ordinary light will suffer an additional phase

difference of with respect to ordinary light.

Lissagous figure in an ellipse with major axis along y-axis.

(b) The pass of axis of P_{2} should be along the major axis of ellipse i.e. y-axis. So as to get maximum intensity of light.

22. Consider a system N non-interacting distinguishable spin - ½ particles, each of magnetic moment . The system is at an equilibrium temperature T in a magnetic field such that n particles have their magnetic moments aligned parallel to .

(a) Find the energy E and the entropy S of the system.

(b) Using the relationship between E and S, find T. Hence determine the ratio n/N in terms of µ, B and T. [Use ln N! = N ln N – N].

Sol.

E_{1} = –µB (if magnetic moment is aligned parallel to B)

E_{2} = µB (if magnitude moment is aligned antiparallel to B)

Number of particle with magnetic moment aligned parallel to B = n.

Energy of the system E = nE_{1} + (N – n) E_{2}

= n(–µB) + (N – n) µB = (N – 2n) µB

23. (a) An ideal gas, kept in contact with a heat reservoir, undergoes a quasistatic process in which its pressure gets doubled. Obtain the Maxwell relation from the differential from dF = –SdT – PdV and evaluate the expression for the change in entropy of n modes of the gas.

(b) Using S = S(T, V), derive a general expression for , where U(S, V) is the internal energy. Evaluate it for the ideal gas as considered in part (a) Justify that the outcome is consistent with the expression for the average energy known from the kinetic theory.

Sol.

Temperature remains constant. So,

If pressure get doubled then its volume will get halved as T = constant.

The internal energy is the function of entropy and volume, U = U(S, V)

According to first law of Thermodynamics,

According to kinetic theory of gases, U is function of T only

So, outcome is consistent with expression for average energy known from kinetic theory.

24. Consider an ideal Fermi gas consisting of N non-relativistic spin-1/2 particles confined to a length 'L' in one dimensional at 0 K.

(a) Find an expression for the density of states an hence calculate the Fermi energy of the gas.

(b) Find the mean energy per particle in terms of the Fermi energy.

Sol. Allowed energy eigen values for particles confined to a length 'L' in 1-D is

Number of eigenstates for which energy lies between E to E + dE.

Density of states, F(E) = 1 for E < E_{F}

= 0 for E > E_{F}

Number of particles whose energy lies between E and E + dE.

N(E)dE = Z(E) F(E) dE

Number of gas molecules

25. A square loop of side 'L' and mass 'M' is made of a wire of cross-sectional area A and resistance R. The loop, moving with a constant velocity in the horizontal xy-plane, enters a region 0 < x < 2L having constant magnetic field .

(a) Find an expression for the x-component of the force acting on the loop in terms of its velocity , B, L and R.

(b) Find the speed of the loop as its side ad exists the field region at x = 2L and sketch its variation with x.

Sol. Let us take the variable 'x' as the distance of bc from the x = 0

** Case I:** 0 < x < L** **

Flux of magnetic field