(a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    a to b should be a constant temperature so B and D is in correct C is incorrect because b to c should be adiabatic a constant entropy curve.

2. Which one of the following is a dimensionless constant?

    (a)    Permittivity of free space

    (b)    Permeability of free space

    (c)    Bohr magneton

    (d)    Fine structure constant

Ans.    (d)

Ans.    Fine structure constant

    

3. Choose the most appropriate matching of the items in Column 1 with those in Column 2.

        Column - 1                Column - 2

    (i)    PIN diode            P.    Voltage regulation

    (ii)    Tunnel diode            Q.    Radio frequency and microwave devices

    (iii)    Zener diode            R.    Optoelectronic detection

    (iv)    Photo diode            S.    Oscillator

    (a)    (i) - Q; (ii) - S; (iii) - P; (iv) – R

    (b)    (i) - R; (ii) - Q; (iii) - P; (iv) - S

    (c)    (i) - R; (ii) - S; (iii) - P; (iv) – Q

    (d)    (i) - P; (ii) - Q; (iii) - R; (iv) - S

Ans.    (a)

Ans.    Correct option is (a).

4. The atomic number of an atom is 6. What is the spectroscopic notation of its ground state, according to Hund's rules?

    (a)    ³P₀

    (b)    ³P₁

    (c)    ³D₃

    (d)    ³S₁

Ans.    (a)

Ans.    Electronic configuration of atomic number 6 (carbon)

    1s²2s²2p²

    Optical electrons are equivalent with less than half filled, these are 2p²

    2p²l₁ = 1, l₂ = 1

    l₁ = 1, l₂ = 1

    

    Maximize L = |m_l1| + |m_l2| = 1 + 0 = 1, Maximize S = |m_s1| + |m_s2| = 1

    Less than half filled J = L – S = 0

    G.S ³P₀

5. H is the Hamiltonian, the orbital angular momentum and L_Z is the z-component of . The 1S state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?

    (a)    H, L² and L_Z

    (b)    H, , L² and L_Z

    (c)    L² and L_Z only

    (d)    H and L_Z only

Ans.

Ans.

6. The Hall experiment is carried out with a non-magnetic semiconductor. The current I is along the x-axis and the magnetic field B is along the Z-axis. Which one of the following is the CORRECT representation of the variation of the magnitude of the Hall resistivity as a function of the magnetic field?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    

7. Consider a two dimensional Cartesian coordinate system in which a rank 2 contravariant tensor is represented by the matrix . The coordinate system is rotated anticlockwise by an acute angle with the origin fixed. Which one of the following matrices represents the tensor in the new coordinate system?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    For rotational matrix det A = 1.

8. A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy of the tetrahedral voids and the Y ions occupy of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?

    (a)    XY₂Z₄

    (b)    XYZ₃

    (c)    XYZ₂

    (d)    XYZ₄

Ans.    (b)

Ans.    Effective lattice point in FCC unit cell = 4,

    No. of Z ions in the unit cell = 4

    No. of tetrahedral voids = 8

    No. of X ions in the unit cell =

    No. of octahedral voids = 4.

    No. of Y ions in the unit cell =

    Chemical compound ratio of ions, = 1:1:3

    Formula of Chemical compound XYZ₃

9. For a non-magnetic metal, which one of the following graphs best represents the behaviour of , where C is the heat capacity and T is the temperature?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    For non-magnetic metal C = AT + VT³ = A + BT²

10. For nonrelativistic electrons in a solid, different energy dispersion relations (with effective masses and ) are schematically shown in the plots. Which one of the following options is correct?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Ans.    

11. The figure schematically shows the M (magnetization) - H (magnetic field) plots for certain types of materials. Here M and H are plotted in the same scale and units. Which one of the following is the most appropriate combination?

    (a) (Q) - Paramagnet; (R) - Type-I Superconductor; (S) - Antiferromagnet

    (b)    (P) - Paramagnet; (Q) - Diamagnet; (R) - Type-I Superconductor

    (c)    (P) - Paramagnet; (Q) - Antiferromagnet; (R) - Type-I Superconductor

    (d)    (P) - Diamagnet; (R) - Paramagnet; (S) - Type-I Superconductor

Ans.    (b)

Ans.    For type I superconductor

    For diamagnetic material is small & negative

    For paramagnetic material is small & positive

12. Graphene is a two dimensional material, in which carbon atoms are arranged in a honeycomb lattice with lattice constant a. As shown in the figure, and are two lattice vectors. Which one of the following is the area of the first Brillouin zone for this lattice?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    Area of the cell A = 6 ×

    Area of the 1^st Brillouin 300

    

13. A ⁶⁰Co nucleus emits a -particle and is converted to ⁶⁰Ni^* with J^P = 4⁺, which in turn decays to the ⁶⁰Ni ground state with J^P = 0⁺ by emitting two photons in succession, as shown in the figure. Which one of the following statements is CORRECT?

    (a)    4⁺ 2⁺ is an electric octupole transition

    (b)    4⁺ 2⁺ is a magnetic quadrupole transition

    (c)    2⁺ 0⁺ is an electric quadrupole transition

    (d)    2⁺ 0⁺ is a magnetic quadrupole transition

Ans.    (c)

Ans.     = No change & L = 2, 3, 4, 5, 6

    So, elects 4⁺2⁺ transition will be of electric quadrupole (E2) type other magnetic multipole transitions are possible here. Thus, contribution/Intensity of E2 will be less. Here, 2⁺0⁺ ( = No change, L = 2). So, 2⁺0⁺ transition will only be of electric quadrupole (E2) type. Therefore, it will be highly intense.

14. Which one of the following options is correct for the given logic circuit?

    (a) P = 1, Q = 1; X = 0

    (b)    P = 1, Q = 0; X = 1

    (c)    P = 0, Q = 1; X = 0

    (d)    P = 0, Q = 0; X = 1

Ans.    (d)

Ans.    

15. An atom with non-zero magnetic moment has an angular momentum of magnitude . When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?

    (a)    3

    (b)    7

    (c)    9

    (d)    25

Ans.    (b)

Ans.    s(s + 1) = 12 s = 3 number of states is 2s + 1 = 2.3 + 1 = 7

16. A 4 × 4 matrix M has the property and M⁴ = 1, where 1 is the 4 × 4 identity matrix. Which one of the following is the correct set of eigenvalues of the matrix M?

    (a)    (1,1,–1,–1)

    (b)    (i, i, –i, –i)

    (c)    (i, i, i, –i)

    (d)    (1, 1, –i, –i)

Ans.    (b)

Ans.    It is a skew Hermitian matrix. So, Eigen values are imaginary or zero. Moreover, Trace should be zero. Also

17. The particle is a member of the Baryon decuplet with isospin state and strangeness quantum number –2. In the quark model, which one of the following is the flavour part of the . wavefunction?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    Quark composition of is uus . The wavefunction will be mixing of quark state will be uss, ssu,usu. The flavour part of decouplet wavefunction is symmetric. Thus, it will be . But, the color part of decouplet wavefunction is anti-symmetric.

18. Which of the following is(are) the correct option(s) for the Joule-Thomson effect?

    (a)    It is an isentropic process

    (b)    It is an isenthalpic process

    (c)    It can result in cooling as well as heating

    (d)    For an ideal gas it always results in cooling

Ans.    (b, c)

Ans.    JT effect is the change in fluid temperature when it goes from higher to lower pressure. Above inversion temperature it warns µ_JT < 0; Below inversion temperature it cools µ_JT > 0.

19. The deuteron is a bound state of a neutron and a proton. Which of the following statements is(are) correct?

    (a)    The deuteron has a finite value of electric quadrupole moment due to non-spherical electronic charge distribution

    (b)    The magnetic moment of the deuteron is equal to the sum of the magnetic moments of the neutron and the proton

    (c)    The deuteron state is an admixture of ³S₁ and ³D₁ states

    (d)    The deuteron state is an admixture of ³S₁ and ³P₁ states

Ans.    (a, c)

Ans.    A is correct because

    Due to non-spherical and even parity of deuteron. The ground state is mixing of ³S₁ and ³D₁.

20. The Geiger-Muller counter is a device to detect and radiations. It is a cylindrical tube filled with monatomic gases like argon, and polyatomic gases such as ethyl alcohol. The inner electrode is along the axis of the cylindrical tube and the outer electrode is the tube. Which of the following statements is(are) correct?

    (a)    Argon is used so that ambient light coming from the surroundings do not produce any signal in the detector

    (b)    Ethyl alcohol is used as a quenching gas

    (c)    The electric field strength decreases from the axis to the edge of the tube and the direction of the field is radially outward

    (d)    The electric field increases from the axis to the edge of the tube and the field direction is radially inward

Ans.    (a, b, c)

Ans.    The basic construction and working principle of Geiger-Muller demonstrated that Argon is used so that ambient light coming from the surroundings do not produce any signal in the detector. Ethyl alcohol is used as quenching vapour. Electric field at a distance from r of the center E = .

21. Consider an isolated magnetized sphere of radius R with a uniform magnetization along the positive z direction, with the north and south poles of the sphere lying on the z axis. It is given that the magnetic field inside the sphere is , where µ₀ is the permeability of vacuum. Which of the following statements is(are) correct?

    (a)    The bound volume current density is zero

    (b)    The bound surface current density has maximum magnitude at the equator, where this magnitude equals

    (c)    The auxiliary field

    (d)    Far from the sphere, the magnetic field is due to a dipole of moment , where

Ans.    (a, b, d)

Ans.    

    

    (d)    , To related dipole moment, take theta zero then B =

22. Which of the following options represent(s) linearly independent pair(s) of functions of a real variable x?

    (a)    e^ix and e^–ix

    (b)    x and e^x

    (c)    2^x and 2^–3+x

    (d)    e^ix and sin x

Ans.    (a, b, d)

Ans.    Let us check the Wronskian of the solutions

    

    

23. In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive z axis for a 2p electron?

Ans.    45°

Ans.    

24. For a transistor amplifier, the frequency response is such that the mid band voltage gain is 200. The cutoff frequencies are 20 Hz and 20 kHz. What is the ratio (rounded off to two decimal places) of the voltage gain at 10 Hz to that at 100 kHz?

Ans.    2.28

Ans.    For 10 Hz

    

    For 100 Hz

    

    The ratio of gain will be = = 2.28

25. An electric field as a function of radial coordinate r has the form . where is a constant. Assume that dimensions are appropriately taken care of. The electric flux through a sphere of radius , centered at the origin, is . What is the value of (rounded off to two decimal places)?

Ans.    0.38

Ans.

26. It is given that the electronic ground state of a diatomic molecule X₂ has even parity and the nuclear spin of X is 0. Which one of the following is the correct statement with regard to the rotational Raman spectrum (J is the rotational quantum number) of this molecule?

    (a)    Lines of all J values are present

    (b)    Lines have alternating intensity in the ratio of 3:1

    (c)    Lines of only even J values are present

    (d)    Lines of only odd J values are present

Ans.    (c)

Ans.    For given molecule, the nuclear wavefunction corresponding to odd J is anti-symmetric and it is symmetric for even J. Also, the degeneracy of anti-symmetric (Odd J) (2I + 1)I = 0. Also, the degeneracy of anti-symmetric (2I + 1) (I + 1) = 1. Thus, odd J states will be absent. Only, even J states will be present. Thus, lines for even J will be present.

27. An input voltage in the form of a square wave of frequency 1 kHz is given to a circuit, which results in the output shown schematically below. Which one of the following options is the correct representation of the circuit?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    Input signal will be differentiated.

    For differentiated output

    RC > 5T

    RC = 5 × 10³ × 1 × 10^–6 = 5 msec

    

28. A simple harmonic oscillator with an angular frequency is in thermal equilibrium with a reservoir at absolute temperature T, with . Which one of the following is the partition function of the system?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    

29. Which one of the following options is the most appropriate match between the items given in Column 1 and Column 2?

        Column - 1                Column - 2

    (i)    Visible light            P.    Transition between core energy levels of

                        atoms

    (ii)    X-rays            Q.    Transition between nuclear energy levels

    (iii)    Gamma rays            R.    Pair production

    (iv)    Thermal neutrons            S.    Crystal structure determination

                    T.    Photoelectric effect

    (a)    (i) - T; (ii) - P,S,T; (iii) - Q,R; (iv) – S

    (b)    (i) - P,T; (ii) - S; (iii) - R,S; (iv) - S,T

    (c)    (i) - T; (ii) - R,S; (iii) - Q,R; (iv) – S

    (d)    (i) - S,T; (ii) - P,S; (iii) - R,T; (iv) - S

Ans.    (a)

Ans.    Visible light can only do Photoelectric effect because it has very less energy which cannot do the other given process. X-Ray has enough energy which can make transition from inner core electron, Crystal structure, Photoelectric effect. Gamma ray can do transition between electronic level and pair production, these are well known from basic concept of radioactivity. Thermal neutron can do crystal structure determination.

30. A rod PQ of proper length L lies along the x-axis and moves towards the positive x direction with speed with respect to the ground (see figure), where c is the speed of light in vacuum. An observer on the ground measures the positions of P and Q at different times t_P and t_Q respectively in the ground frame, and finds the difference between them to be . What is the value of t_Q – t_P?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Ans.    

    It is also given

    

31. A symmetric top has principal moments of inertia about a set of principal axes 1, 2, 3 respectively, passing through its center of mass, where is a positive constant. There is no force acting on the body and the angular speed of the body about the 3-axis is rad/s. With what angular frequency in rad/s does the angular velocity vector precess about the 3-axis?

    (a)    2

    (b)    3

    (c)    5

    (d)    7

Ans.

Ans.

32. A particle of mass m is free to move on a frictionless horizontal two dimensional (r, ) plane, and is acted upon by a force with k being a positive constant. If p_r and are the generalised momenta corresponding to r and respectively, then what is the value of ?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Ans.    

    If motion is in two-dimensional plane, then p_r and are generalized momentum for generalized coordinates conjugate to r and . If V(r) is potential then Hamiltonian in polar coordinate is given by H = +V(r). The Hamiltonian equation of motion is given by

    

33. Consider two real functions

        U(x, y) = xy(x² – y²),

        v(x, y) = ax⁴ + by⁴ + cx²y² + k,

    where k is a real constant and a, b, c are real coefficients. If U(x, y) + iV(x, y) is analytic, then what is the value of a × b × c?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Ans.    U_x = V_y, 3x²y – y³ = 4by³ + 2cx²y, 4b = –1, 2c = 3, U_y = –V_x

    

34. Young's double slit experiment is performed using a beam of C₆₀ (fullerene) molecules, each molecule being made up of 60 carbon atoms. When the slit separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from the slits. Now, the experiment is repeated with C₇₀ molecules with a slit separation of 92.5 nm. The kinetic energies of both the beams are the same. The position of the 4^th bright fringe for C₆₀ will correspond to the n^th bright fringe for C₇₀. What is the value of n (rounded off to the nearest integer) ?

    (a)    5

    (b)    6

    (c)    7

    (d)    8

Ans.    (d)

Ans.    YDSE with C₆₀ beam

    

    YDSE with C₇₀ beam

    

    

35. A neutron beam with a wave vector and an energy 20.4 meV diffracts from a crystal with an outgoing wave vector . One of the diffraction peaks is observed for the reciprocal lattice vector of magnitude 3.14 Å^–1. What is the diffraction angle in degrees (rounded off to the nearest integer) that makes with the plane? (Use mass of neutron = 1.67 × 10^–27 Kg)

    (a)    15

    (b)    30

    (c)    45

    (d)    60

Ans.    (b)

Ans.    Energy of neutron beam = 20.4 meV

    Corresponding de Broglie wavelength

    

    Diffraction condition

    

36. In the first Brillouin zone of a rectangular lattice (lattice constants a = 6 Å and b = 4 Å), three incoming phonons with the same wave vector interact to give one phonon. Which one of the following is the correct wave vector of the resulting phonon?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    

    

    Thus,

    kx = 1.2 + 1.2 + 1.2 – 1.047 = 2.56 A^{0 – 1} & ky = 0.6 + 0.6 + 0.6 – 1.57 = 0.23 A^{0 – 1}

37. For a covalently bonded solid consisting of ions of mass m, the binding potential can be assumed to be given by

        

    where and r₀ are positive constants. What is the Einstein frequency of the solid in Hz ?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    From the concept of stability analysis

    

    

    

38. In a hadronic interaction, 's are produced with different momenta, and they immediately decay into two photons with an opening angle between them. Assuming that all these decays occur in one plane, which one of the following figures depicts the behaviour of as a function of the momentum p?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    The conservation of linear momentum revealed that when decay at rest then angle of produced photon will be 180° . However, when the decayed photon is not at rest then the laws of conservation of momentum again says that that the angle of produced photon will be smaller than 1800. This property is only observable in option (a).

39. A particle has wavefunction

        ,

    where N is a normalization constant and α is a positive constant. In this state, which one of the following options represents the eigenvalues of L² and L_Z respectively?

    Some value of are:

    

    (a) 0 and 0

    (b)

    (c)

    (d)

Ans.    (c)

Ans.    

    

    So l = 1, m = 0 measurement of L² is and measurement of L_Z is

40. The wavefunction of a particle in one dimension is given by

        

    Here M and a are positive constants. If is the corresponding momentum space wavefunction, which one of the following plots best represents ?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Ans.    

    

    

41. Consider a particle in a two dimensional infinite square well potential of side L, with 0 < x < L and 0 < y < L. The wavefunction of the particle is zero only along the line , apart from the boundaries of the well. If the energy of the particle in this state is E, what is the energy of the ground state?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Ans.    For two-dimensional box

    For given information n_x = 1, n_y = 2, So E_{1, 2} = (1² + 2²)

    Ground state is E_{1, 1} = (1² + 1²)

42. Consider two non-identical spin particles labelled 1 and 2 in the spin product state . The Hamiltonian of the system is

        ,

    where and are the spin operators of particles 1 and 2, respectively, and is a constant with appropriate dimensions. What is the expectation value of H in the above state?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Ans.    The concept of GC coefficient demonstrated that the above state can be written in the following form

    

    

    

43. A spin particle is in a spin up state along the x-axis (with unit vector ) and is denoted as . What is the probability of finding the particle to be in a spin up state along the direction , which lies in the xy-plane and makes an angle with respect to the positive x-axis, if such a measurement is made?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Ans.     eigen value is with corresponding eigen state and

    The along which is making an angle θwith the x-axis in the x-y plane is

    

    So, operator the eigenvalue of the eigenstate of the operator with eigenstate and

    The probability that state in eigen state is

    

    The probability that state in eigen state is

    

44. Different spectral lines of the Balmer series (transitions n 2, with n being the principal quantum number) fall one at a time on a Young's double slit apparatus. The separation between the slits is d and the screen is placed at a constant distance from the slits. What factor should d be multiplied by to maintain a constant fringe width for various lines, as n takes different allowed values?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Ans.    

45. Under parity and time reversal transformations, which of the following statements is(are) TRUE about the electric dipole moment p and the magnetic dipole moment µ?

    (a)    p is odd under parity and µ is odd under time reversal

    (b)    p is odd under parity and µ is even under time reversal

    (c)    p is even under parity and µ is odd under time reversal

    (d)    p is even under parity and µ is even under time reversal

Ans.    (a)

Ans.    

46. Consider the complex function

        .

    At z = , which of the following options is(are) correct?

    (a)    The order of the pole is 4

    (b)    The order of the pole is 3

    (c)    The residue at the pole is

    (d)    The residue at the pole is

Ans.    (b)

Ans.    

    

47. Consider the vector field consisting of the velocities of points on a thin horizontal disc of radius R = 2 m, moving anticlockwise with uniform angular speed = 2 rad/sec about an axis passing through its center. If , then which of the following options is(are) correct? (In the options, and are unit vectors corresponding to the plane polar coordinates r and ).

    You may use the fact that in cylindrical coordinates (s, , z) (s is the distance from the z-axis), the gradient, divergence, curl and Laplacian operators are:

    

    

    (a)

    (b)

    (c) , where is a unit vector perpendicular to the (r, ) plane

    (d)

Ans.    (a, c, d)

Ans.    

    

    

    

48. A slow moving particle is captured by a deuteron (d) and this reaction produces two neutrons (n) in the final state, i.e., . Neutron and deuteron have even intrinsic parities, whereas has odd intrinsic parity. L and S are the orbital and spin angular momenta, respectively of the system of two neutrons. Which of the following statements regarding the final two-neutron state is(are) correct?

    (a)    It has odd parity

    (b)    L + S is odd

    (c)    L = 1, S = 1

    (d)    L = 2, S = 0

Ans.    (a, c)

Ans.    

    The slow-moving means L = 0 for . The deuteron is having which represents L = Even for it. Also, it is important to tbe mentioned here that deuteron is fermion having intrinsic parity +1.

    From the conservation of Parity in strong interaction

    

    (–)(+) = (+)(+)(–1)^L    (Since L' = Even)

    (–) = (–1)^L

    Thus, L = Odd = 1, 3, 5, ......

    The conservation of parity itself says the product is having odd parity. Thus, option (a) is correct.

    From spin conservation

    

    

    Resultant S of product will be 1.

    So, spin of final two neutron stale option (c) is correct and option (d) is wrong. L + S will be even.

49. Two independent electrostatic configurations are shown in the figure. Configuration (I) consists of an isolated point charge q = 1 C, and configuration (II) consists of another identical charge surrounded by a thick conducting shell of inner radius R₁ = 1m and outer radius R₂ = 2m, with the charge being at the center of the shell. and , where E_I and E_II are the magnitudes of the electric fields for configurations (I) and (II) respectively, is the permittivity of vacuum, and the volume integrations are carried out over all space. If , what is the value of the integer n?

Ans.    2

Ans.    

    R₁ = 1 m, R₂ = 2 m

    

    

50. In pion nucleon scattering, the pion and nucleon can combine to form a short lived bound state called the particle . The masses of the pion, nucleon and the particle are 140 MeV/c², 938 MeV/c² and 1230 MeV/c², respectively. In the lab frame, where the nucleon is at rest, what is the minimum energy (in MeV/c², rounded off to one decimal place) of the pion to produce the particle?

Ans.    340 MeV

Ans.    

51. Consider an electromagnetic wave propagating in the z-direction in vacuum, with the magnetic field given by . If B₀ = 10^–8 T, the average power passing through a circle of radius 1.0 m placed in the xu plane is P (in Watts). Using , what is the value of (rounded off to one decimal place)?

Ans.    13.5

Ans.    

    

52. An -particle is emitted from the decay of Americium (Am) at rest, i.e., . The rest masses of and are 224.544 GeV/c², 220.811 GeV/c² and 3.728 GeV/c² respectively. What is the kinetic energy (in MeV/c², rounded off to two decimal places) of the -particle?

Ans.    4.917

Ans.    

    

53. Consider 6 identical, non-interacting, spin atoms arranged on a crystal lattice at absolute temperature T. The z-component of the magnetic moment of each of these atoms can be ±µ_B. If P and Q are the probabilities of the net magnetic moment of the solid being 2µ_B and 6µ_B respectively, what is the value of (in integer)?

Ans.    15

Ans.    For total moment to be 2µ_B the arrangement is like

    For total moment to be 6µ_B the arrangement is like

    

54. Two identical, non-interacting ⁴He₂ atoms are distributed among 4 different non-degenerate energy levels. The probability that they occupy different energy levels is p. Similarly, two ³He₂ atoms are distributed among 4 different non-degenerate energy levels, and the probability that they occupy different levels is q. What is the value of (rounded off to one decimal place)?

Ans.    0.6

Ans.    Bosons, p = 6/10

    

    Fermions, p = 6/6

    

    p/q = 0.6

55. Two identical bodies kept at temperatures 800 K and 200 K act as the hot and the cold reservoirs of an ideal heat engine, respectively. Assume that their heat capacity (C) in Joules/K is independent of temperature and that they do not undergo any phase change. Then, the maximum work that can be obtained from the heat engine is n × C Joules. What is the value of n (in integer)?

Ans.    200

Ans.    Q₁ = C[T₁ – T'] & Q₂ = C[T' – T₂]

    W = Q₁ – Q₂ = C[T₁ – T' – T' + T₂] = C[T₁ + T₂ – 2T']