## IIT JAM PHYSICS 2005

Previous Year Question Paper with Solution.

1. A solid sphere of mass 'm' and radius 'a' is rolling with a linear speed 'v' on a flat surface without slipping. The magnitude of the angular momentum of the sphere with respect to a point along the path of the sphere on the surface is:

(a)

(b)

(c) m av

(d)

Ans. (b)

Sol. Angular momentum of solid sphere about point O.

I = The moment of inertia about an axis passing through the CM of sphere.

2. The susceptibility of a diamagnetic material is:

(a) Positive and proportional to temperature

(b) Negative and inversely proportional to temperature

(c) Negative and independent of temperature

(d) Positive and inversely proportional to temperature

Ans. (c)

Sol. The susceptibility of a diamagnetic material is given by

where, N is the number of atoms per m^{3}

M is the mass of an electron.

is the average distance of electron from the nucleus.

3. The molar specific heat of a gas as given from the kinetic theory is . If it is not specified whether it is C_{P} or C_{V}, one could conclude that the molecules of the gas.

(a) Are definitely monoatomic

(b) Are definitely rigid diatomic

(c) Are definitely non-rigid diatomic

(d) Can be monoatomic or rigid diatomic

Ans. (d)

Sol. We know, the specific heat at constant volume of a gas is given by and C_{P}– C_{V} = R

where, is the average energy.

According to the kinetic theory of gas,

for monoatomic gas.

for non rigid diatomic gas.

for rigid diatomic gas.

For monoatomic,

For rigid diatomic,

4. The value of entropy at absolute zero of temperature would be

(a) zero for all the materials

(b) Finite for all the materials

(c) zero for some materials and non-zero for others

(d) Unpredictable for any material

Ans. (c)

Sol. According to the third law of thermodynamics – "The entropy of any system vanishes at the absolute zero temperature"

And according to Nernst's heat theorem: "Any entropy changes in an isotherm at reversible process approach zero as the temperature approaches zero".

For an example, in crystal of CO has finite entropy at absolute zero temperature.

5. A circuit and the signal applied at its input terminals (V_{i}) are shown in figure below. Which one of the options correctly describes the output waveform (V_{0}). (Assume all the devices used are ideal)

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Let us begin with second half cycle, the input voltage is –2V, the diode will get forward biased and capacitor will charged, to +2V and output voltage will be zero. In the next half cycle, the input voltage will be +2V. The diode will be reversed biased but capacitor will hold charge 2 eV.

So, the output voltage is V_{0} = 2V + 2V = 4V.

6. Consider a beam of light of wavelength incident on a system of a polarizer and an analyzer. The analyzer is oriented at 45º to the polarizer. When an optical component is introduced between them, the output intensity becomes zero. (Light is incident normally on all components). The optical component is:

(a) A full-wave plate

(b) A half-wave plate

(c) A quarter-wave plate

(d) An ordinary glass plate

Ans. (b)

Sol. When half wave plate is introduced, the plane of polarization is rotated by . Since the intensity of light after analyser is zero. Plane of polarization rotates by an angle of 45º. Then

7. A small loop of wire of area A = 0.01 m^{2}, N = 40 turns and resistance R = is initially kept in a uniform magnetic field B in such a way that the field is normal to the plane of the loop. When it is pulled out of the magnetic field, a total charge of Q = 2 × 10^{–5} C flows through the coil. The magnitude of the field B is:

(a) 1 × 10^{–3} T

(b) 4 × 10^{–3} T

(c) Zero

(d) Unobtainable, as the data is insufficient

Ans. (a)

Sol. Flux change in magnetic field,

8. If M_{e}, M_{p} and M_{H} and the rest masses of electron, proton and hydrogen atom in the ground state (with energy –13.6 eV), respectively, which of the following is exactly true? (c is the speed of light in free space)

(a) M_{H} = M_{p} + M_{e}

(b)

(c)

(d)

Ans. (b)

Sol. We know that, the binding energy of an electron in the ground state of hydrogen atom is +13.6 eV.

Therefore, according to mass energy law,

9. An observer is sitting on a horizontal platform which is rotating with a constant angular velocity. He puts an object on the smooth frictionless floor of the platform, away from the axis of rotation, with zero initial velocity with respect to him. Let the time at this instant be t = 0. In the frame of the platform, the object would

(a) Remain at rest for all t > 0

(b) Accelerate purely in a radial direction outwards for all t > 0

(c) Accelerate purely in a tangential direction for all t > 0

(d) Accelerate radially in the outward direction at t = 0, however the direction of acceleration changes for t > 0.

Ans. (d)

Sol. Initially the body will accelerate under the centrifugal force (F_{c} = mω^{2}r) in the radially outward direction at t = 0, but at t > 0. The direction of force as well as acceleration changes due to the coriolis force .

10. Which of the following is Incorrect for the matrix ?

(a) It is its own inverse

(b) It is its own transpose

(c) It is non-orthogonal

(d) It has eigen values +1.

Ans. (c)

Sol.

11. A combination of two thin convex lenses of equal focal lengths, is kept separated along the optic axes by a distance of 20 cm between them. The combination behaves as a lens system of infinite focal length. If an object is kept at 10 cm from the first lens, its image will be formed on the other side at a distance x from the second lens. The value of x is:

(a) 10 cm

(b) 20 cm

(c) 6.67 cm

(d) Infinite

Ans. (a)

Sol.

12. Two points charges +q_{1} and +q_{2} are fixed with a finite a distance d between them. It is desired to put a third charge q_{3} in between these two charges on the line joining them so that the charge q_{3} is in equilibrium. That is:

(a) Possible only if q_{3} is positive

(b) Possible only if q_{3} is negative

(c) Possible irrespective of the sign of q_{3}

(d) Not possible at all.

Ans. (c)

Sol. Let q_{3} is positive charge and

Let q_{3} is negative charge

13. A periodic function can be expressed in a Fourier series of the form, . The functions f_{1}(x) = cos^{2} x and f_{2}(x) = sin^{2} x are expanded in their respective Fourier series. If the coefficients for the first series are and , and the coefficients for the second series are and , respectively, then which of the following is correct?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

14. Which of the following statements is correct for NaCl crystal structure?

(a) It is a simple cubic lattice with one atom basis.

(b) It is a face-centered cubic lattice with one atom basis

(c) It is a simple cubic lattice with two atom basis

(d) It is a face centered cubic lattice with two atom basis

Ans. (d)

Sol. Basis: The group of identical atoms is called basis of crystal structure. In NaCl, basis atoms are Na^{+} and Cl^{–}.

15. Which of the following is incorrect?

(a) Indistinguishable particles obey Maxwell-Boltzmann statistics

(b) All particles of an ideal Bose gas occupy a single energy state at T = 0

(c) The integral spin particles obey Bose-Einstein statistics

(d) Protons obey Fermi-Dirac statistics.

Ans. (a)

Sol. Correct option is (a)

16. (a) Consider a constant vector field . Find any one of the many possible vectors , for which

(b) Using Stoke's theorem, evaluate the flux associated with the field through the curved hemispherical surface defined by x^{2} + y^{2} + z^{2} = r^{2}, z > 0.

Sol.

(Using Stokes theorem)

17. A logic circuit and time varying logic levels applied at its A and B inputs are shown below. Sketch the corresponding output waveform at points C, D, E, F and G in the space given below the waveforms A and B.

Sol.

18. (a) Determine whether the force represented by is conservative or not. Here k = 1 Nm^{–2}.

(b) Calculate the work done by his force in moving a particle from the origin O(0, 0, 0) to the point D(1, 1, 0) on the z = 0 plane along the paths OABD and OD as shown in figure, where the coordinates are measured in metres.

Sol.

Therefore, work done by a conservative field in moving a particle from 0 to D, is independent of path i.e.

Work done in moving the particle from 0 to D

19. A rod is moving with a speed of 0.4c along its length in the positive x-direction, and a particle is moving along the negative x-direction with a speed 0.8c as shown in the figure below. Both the speeds are measured in an inertial frame S, and c is the velocity of light in free space. The length of the rod as measured in the S-fram is 3.6 m.

(a) Find the relative velocity of the rod (in terms of c in the rest frame of the particle).

(b) Find (i) the time taken for the particle to cross the rod in the S-frame and in the rest frame of the rod, and (ii) time taken by the rod to cross the particle in the rest frame of the particle.

Sol. Relativity velocity of rod with respect to particle is velocity of rod, in the frame attached to the particle.

(b) (i) Time taken for the particle to cross the rod in s frame

Let the proper length of rod be *l*_{0}.

According to length contraction

Time taken by particle to cross the rod in rest frame of rod

(ii) Length of rod in rest frame of particle.

Time taken by rod to cross the particle in the rest frame of particle.

20. A particle of mass m and energy E moving in the positive x-direction, encounters a one-dimensional potential barrier at x = 0. The barrier is defined by

V = 0 for x < 0

V = V_{0} for x > 0 (V_{0} is positive and E > V_{0})

If the wave function of the particle in the region x < 0 is given as A e^{ikx} + B e^{–ikx}

(a) Find the ratio B/A

(b) If , find , and the transmission and reflection coefficients.

Sol. E > V_{0
}

_{
}

_{
}

_{
}

_{
}

_{
}

_{ }From (1) and (2),

Transmission coefficient, T = 1 – R = 1 – 0.16 = 0.84.

21. (a) Establish the equation , given that dU = TdS – PdV and , where U, P, T, V and S are, respectively, the internal energy, pressure, temperature, volume and entropy of the system.

(b) If the specific heat is taken to be independent of T, utilize the above equation to derive an expression for U(T, V) for one mole of a van der Waals gas and then obtain the corresponding expression for an ideal gas.

Sol. (a) T – dS equation, TdS = dU + PdV

dU = TdS – PdV ... (1)

Let entropy S be the function of T and V

S = S(T, V); ... (2)

Putting into (1)

(b) For vander waals gas.

From (3)

22. Solve the differential equation with the initial condition y = 2 when x = 1.

Sol.

Putting y = 2 when x = 1, we get c = 9

Solution: 2y^{2} + x^{2} = 9x^{6}

23. A beam of light is incident normally on a diffraction grating of width 2 cm. It is found that at 30º, then n^{th} order diffraction maximum for = 5000Å is super-imposed on the (n + 1)^{th} order for = 4000Å. How many lines per cm does the grating have? Find out the whether the first order spectrum from such a grating can be used to resolve the wavelengths = 5800 Å and = 5802Å?

Sol.

Resolving power of grating = Nm

For m = 1

Resolving power = 2500 × 1 = 2500

Resolving power required to resolve to wavelengths l_{3} and l_{4} is given by

Since required resolving power is larger than resolving power of grating, these two wavelengths can not be resolved in first order spectrum.

24. Consider two electromagnetic plane waves propagating in vacuum with their electric field vectors and .

(a) Evaluate the magnetic field vector corresponding to the superposition of these two waves.

(b) Calculate the time-averaged energy density as well as the time-averaged Poynting vector for the resultant wave. (The time average is carried over one period of oscillation).

Sol. From principle of superposition we get,

To calculate magnetic field we use equation,

Integrating w.r.t. time we get,

(b) Energy density,

25. A 150 resistor a 10 µF capacitor and a 0.1 H inductor are connected in series to an a.c. source operating at an angular frequency .

(a) Find the value of for which the combination acts as a pure resistive load.

(b) The a.c. source is operated at a peak voltage of and a frequency equal to half the resonance frequency of the circuit. Find the peak value of the current in the circuit and the phase difference between the current and voltage. Also, find the peak voltage across the inductor.

Sol.

For Z to be purly resistive

(b) For AC voltage is operated at half the resonant frequency,

Voltage applied,

So, peak value of current in the circuit is I_{0} = 2A and phase differences between current and voltage .

Voltage across inductor =

Peak voltage across inductor = 100 V.

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