1. Choose the graph that best describes the variation of dielectric constant () with temperaure (T) in a ferroelectric material.

    (T_C is the critical temperature)

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    A ferromagnetic state is usually observed at low temperatures, but not at high temperatures, when there is no spontaneous polarization. The temperature at which ferroelectric properties are lost is referred to as the ferroelectric curie temperature T_c. And above this temperature, the material becomes Paraelectric. Most of these materials above T_c will lose the piezoelectric property as well. The variation of dielectric constant () by means of temperature (T) in the non-polar, paraelectric state is shown by Curie-Weiss law as given below

    

    Thus the correct option is (a).

2. A matter wave is represented by the wave function

        

    where A is a constant. The unit vector representing the direction of propagation of this matter wave is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Propagation vector is given by :

    The unit vector representing the direction of the propagation of this matter wave is

    

3. As shown in the figure, X-ray diffraction pattern is obtained from a diatomic chain of atoms P and Q. The diffraction condition is given by a cos = n, where n is the order of the diffraction peak. Here, a is the lattice constant and is the wavelength of the X-rays. Assume that atomic form factors and resolution of the instrument do not depend on . Then, the intensity of the diffraction peaks is

    (a)    zero for even values of n

    (b)    lower for odd values of n, when compared to even values of n

    (c)    lower for even values of n, when compared to odd values of n

    (d)    zero for odd values of n

Ans.    (b)

Sol.    Intensity for even values of n is I = 16|f_P + f_Q|²

    And the intensity for odd values of n is I = 16|f_P – f_Q|²

    Where f_P and f_Q are the atomic scattering factor of atom P and atom Q.

4. As shown in the figure, two metal-semiconductor junctions are formed between an n-type semiconductor S and metal M. The work functions of S and M are and , respectively with > .

    The I – V characteristics (on linear scale) of the junctions is best represented by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    The correct option is (a)

5. Consider a tiny current loop driven by a sinusoidal alternating current. If the surface integral of its time-averaged Poynting vector is constant, then the magnitude of the time – averaged magnetic field intensity, at any arbitrary position, , is proportional to

    (a)

    (b)

    (c)

    (d) r

Ans.    (c)

Sol.    If , formula for pointing vector :

    Time averaged Poynting vector :

    Then

6. Consider a solenoid of length L and radius R, where R << L. A steady-current flows through the solenoid. The magnetic field is uniform inside the solenoid and zero outside.

    Among the given options, choose the one that best represents the variation in the magnitude of the vector potential, at z = L/2, as a function of the radial distance (r) in cylindrical coordinates.

    Useful information: The curl of a vector , in cylindrical coordinates is

    

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

7. Assume that ¹³N (Z = 7) undergoes first forbidden decay from its ground state with spin – parity , to a final state . The possible value for and , respectively are

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

8. In an experiment, it is seen that an electric – dipole (E1) transition can cannot an initial nuclear state of spin – parity to a final state . All possible values of are

    (a)    1⁺, 2⁺

    (b)    1^–, 2^–

    (c)    1⁺, 2⁺, 3⁺

    (d)    1^–, 2^–, 3^–

Ans.    (d)

Sol.    In an electric dipole (E1) transition : = 0, +1 and Parity will change

    

    So, option (d) is correct.

9. Choose the correct statement from the following.

    (a)    Gallium Arsenide is an indirect band gap semiconductor.

    (b)    Conductivity of metals decreases with increase in temperature.

    (c)    Silicon is a direct band gap semiconductor.

    (d)    Conductivity of semiconductor decreases with increase in temperature.

Ans.    (b)

Sol.    Silicon is indirect and GaAs is a direct bandgap semiconductor. The conductivity of the semiconductor increases with temperature due to increase in carrier concentration. The conductivity of metals decreases with increase in temperature due to increase in number of phonons and hence electron-phonon scattering. Thus correct option is (b).

10. A two-dimensional square lattice has lattice constant a.

    k represents the wavevector in reciprocal space. The coordinates (k_x, k_y) of reciprocal space where band gap(s) can occur, are

    (a) (0, 0)

    (b)

    (c)

    (d)

Ans.    (b, c, d)

Sol.    The band structure of a two-dimensional photonic crystal with a square lattice shows zero band gap at (0, 0), while the band gap increases along (k_x, k_y) directions. Thus, correct options are (b), (c) and (d).

11. As shown in the figure, an electromagnetic wave with intensity I_I is incident at the interface of two media having refractive indices n₁ = 1 and n₂ = . The wave is reflected with intensity I_R and transmitted with intensity I_T. Permeability of each medium is the same. (Reflection coefficient R = I_R/I_I and Transmission coefficient T = I_T/I_I).

            

    Choose the correct statement(s).

    (a)    R = 0 if = 60º and polarization of incident light is perpendicular to the plane of incidence.

    (b)    T = 1 if = 60º and polarization of incident light is parallel to the plane of incidence.

    (c)    R = 0 if = 0º and polarization of incident light is parallel to the plane of incidence.

    (d)    T = 1 if = 60º and polarization of incident light is perpendicular to the plane of incidence.

Ans.    (b)

Sol.    Brewester angle at interface is

    In case of parallel polarization of Transverse magnetic polarization

    

12. A material is placed in a magnetic field intensity H. As a result, bound current density J_b is induced and magnetization of the material is M. The magnetic flux density is B. Choose the correct option(s) valid at the surface of the material.

    (a)

    (b)

    (c)

    (d)

Ans.    (b, d)

Sol.    

    

13. For a finite system of Fermions where the density of states increases with energy the chemical potential

    (a)    decreases with temperature

    (b)    increases with temperature

    (c)    does not vary with temperature

    (d)    corresponds to the energy where the occupation probability is 0.5

Ans.    (a, d)

Sol.    For fermionic system the chemical potential is equal tothe fermi energy. The probability function is written as

    

    At infinite temperature the chemical potential and hence the fermi energy is written as

    

    With increasing temperature, the chemical potential decrease and from probability distribution function it is clear that is also corresponds to the energy where the occupation probability is 0.5. Thus correct options are (a) and (d).

14. Among the term symbols

    ⁴S₁, ²D_7/2, ³S₁ and ²D_5/2

    choose the option(s) possible in the LS coupling notation.

    (a)    ⁴S₁

    (b)    ²D_7/2

    (c)    ³S₁

    (d) ²D_5/2

Ans.    (c, d)

Sol.    (a)    

        So, ⁴S₁ is not possible.

    (b)    

        So, ³D₇₂ is not possible.

    (c)    

        So, ³S₁ is possible.

    (d)    

        So, ²D_5/2 is possible.

15. To sustain lasing action in a three – level laser as shown in the figure, necessary condition(s) is (are)

    (a)    lifetime of the energy level 1 should be greater than that of energy level 2

    (b)    population of the particles in level 1 should be greater than that of level 0

    (c)    lifetime of the energy level 2 should be greater than that of energy level 0

    (d)    population of the particles in level 2 should be greater than that of level 1

Ans.    (a, b)

Sol.    (a)    Level one should be metastable state for lasing action.

    (b)    It is necessary for population inversion.

16. If y_n (x) is a solution of the differential equation

        y" – 2xy' + 2ny = 0

    where n is an integer and the prime (' ) denotes differentiation with respect to x, then acceptable plot (s) of is (are)

    (a)

    (b)

    (c)

    (d)

Ans.    (b, c)

Sol.    It's a Hermite Differential Equation

    y" – 2xy' + 2ny = 0, y_n(x) is Hermite Polynomial

    H₀(x) = 1, H₁(x) = 2x

    

    

17. The donor concentration in s sample of n – type silicon is increased by a factor of 100. Assuming the sample to be non – degenerate, the shift in the Fermi level (in meV) at 300 K (rounded off to the nearest integer) is_______.

    (Given : k_BT = 25 meV at 300 K)

Ans.    115.15

Sol.    

    Thus shift is = kT ln (100) = 25 ln (100) meV = 115.15 meV.

18. wo observers O and O' observe two events P and Q. The observers have a constant relative speed of 0.5c. In the units, where the speed of light, c, is taken as unity, the observer O obtained the following coordinates:

    Event P: x = 5, y = 3, z = 5, t = 3

    Event Q: x = 5, y = 1, z = 3, t = 5

    The length of the space – time interval between these two events, as measured by O', is L. The value of |L| (in integer) is_____.

Ans.    2

Sol.    

19. A light source having its intensity peak at the wavelength 289.8 nm is calibrated as 10,000 K which is the temperature of an equivalent black body radiation. Considering the same calibration, the temperature of light source (in K) having its intensity peak at the wavelength 579.6 nm (rounded off to the nearest integer) is_______.

Ans.    5000

Sol.    By Wein's law : = b, b is known as Wein's constant.

    

    579.6 × T = 289.8 × 10000

    

20. A hoop of mass M and radius R rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass m slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is __________.

Ans.    2

Sol.    Two independent motion one is pure rotation and other is pure translation in one dimension. So degrees of freedom is two.

21. Three non – interacting bosonic particles of mass m each, are in a one – dimensional infinite potential potential well of width a. The energy of the third excited state of the system is The value of x (in integer) is_______.

Ans.    6

Sol.    Ground state is first excited state is

    Second excited state is

    Third excited state is

22. The spacing between two consecutive S – branch lines of the rotational Raman spectra of hydrogen gas is 243.2 cm^–1. After excitation with a laswer of wavelength 514.5 nm, the Stoke's line appeared at 17611.4 cm^–1 for a particular energy level. The wavenumber (rounded off to the nearest integer), in cm^–1, at which stoke's line will appear for the next higher energy level is ______.

Ans.    17368.2

Sol.    The spacing between two consecutive S-branch lines of the rotational Raman spectra

    4B = 243.2 cm^–1

    The wavelength of a stoke's line for a particular energy level = 17611.4 cm^–1

    The wavelength of the stoke's line for the next higher energy level

        = 17611.4 – 4B = 17611.4 – 243.2 = 17368.2 cm^–1

23. The transition line, as shown in the figure, arises between ²D_3/2 and ²P_1/2 states without any external magnetic field. The number of lines that will appear in the presence of a weak magnetic field (in integer) is _________.

Ans.    6

Sol.    In the presence of a weak magnetic field : Anamolous zeeman effect

    E = (gm_j)(µ_BB)

    

24. Consider the atomic system as shown in the figure, where the Einstein A coefficients for spontaneous emission for the levels are = 2 × 10⁷ s^–1 and = 10⁸ s^–1. If 10¹⁴ atoms/cm³ are excited from level 0 to level 2 and a steady state population in level 2 is acheived, then the steady state population at level 1 will be x × 10¹³ cm^–3. The value of x (in integer) is______.

Ans.    2

Sol.    

    

25. In and are constant vectors, and are generalized positions and conjugate momenta, respectively, then for the transformation and to be canonical, the value of (in integer) is______.

Ans.    –1

Sol.    

26.

    The above combination of logic gates represents the operation

    (a)    OR    (b)    NAND    (c)    AND    (d)    NOR

Ans.    (b)

Sol.    

27. In a semiconductor, the ratio of the effective mass of hole to electron is 2:11 and the ratio of average relaxation time for hole to electron is 1:2. The ratio of the mobility of the hole to electron is

    (a)    4:9

    (b)    4:11

    (c)    9:4

    (d)    11:4

Ans.    (d)

Sol.    

28. Consider a spin particle in the state . The probability that a measurement finds the state with is

    (a)    15/18

    (b)    5/18

    (c)    11/18

    (d)    17/18

Ans.    (d)

Sol.     the eigen state corresponds to eigenv value is

    

29. An electromagnetic wave having electric field E = 8 cos V cm^–1 is incident at 90º (normal incidence) on a square slab from vaccum (with refractive index n₀ = 1.0) as shown in the figure. The slab is composed of two different materials with refractive indices n₁ and n₂. Assume that the permeability of each medium is the same. After passing through the slab for the first time, the electric field amplitude, in V cm^–1, of the electromagnetic wave, which emerges from the slab in region 2, is closest to

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Critical angle at the interface of two slab is

    

    Since incident angle at the interface is = 45° > = 30°, so there is total internal reflection at interface. Thus I₂ = I₁.

    

    

    

    

30. Consider a point charge +Q of mass m suspended by a massless, inextensible string of length l in free space (permittivity ) as shown in the figure. It is placed at a height d (d > l) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane.

    If q represents the angular position and its corresponding canonical momentum, then the correct Hamiltonian of the system is

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Potential energy due to gravitation mg

    Potential energy due to conducting plate is (we can use a concept of image problem)

    Kinetic energy is

    So Hamiltonian is

31. Consider two concentric conducting spherical shells as shown in the figure. The inner shell has a radius a and carries a charge +Q. The outer shell has a radius b and carries a charge –Q. The empty space between them is half-filled by a hemispherical shell of a dielectric having permittivity . The remaining space between the shells is filled with air having the permittivity .

    The electric field at a radial distance r from the center and between the shells (a < r < b) is

    (a)    everywhere

    (b)     on the air side and on the dielectric side

    (c)     on the air side and on the dielectric side

    (d)     everywhere

Ans.    (a)

Sol.    From Gauss Law :

    Let us find the electric field at a radial distance r from the centre and between the shells (a < r < b). Draw a Gaussian sphere of radius r, then

    

32.

    For the given sets of energy levels of nuclei X and Y whose mass numbers are odd and even, respectively, choose the best suited interpretation.

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Out of these two is rotational band and other is vibrational band I. For even-even nuclei, the ground state is always 0⁺ and first excited state is 2⁺ for vibrational and rotational bands. However, in case of vibrational band, next excited states are O⁺, 2⁺, 4⁺. So, set I represents vibrational band of even Y nuclei.

33. Consider a system of three distinguishable particles, each having spin S = 1/2 such that S_z = ±1/2 with corresponding magnetic moment µ_z = ±µ. When the system is placed in an external magnetic field H pointing the z – axis, the total energy of the system is µH. Let x be state where the first spin has S_z = 1/2. The probability of having the state x and the mean magnetic moment (in the +z direction) of the system in state x are

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    Accessible states 5, 6, 7. States 5 is the x state. Probability of having the state x and the mean magnetic moment (in the +z direction) of the system in state x are .

34. Consider a particle in a one – dimensinoal infinite potential well with its walls at x = 0 and x = L. The system is perturbed as shown in the figure.

    The first order correction to the energy eigenvalue is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

35. Consider a state described by (x,t) = ₂ (x, t) + ₄ (x, t), where ₂(x, t) and ₄ (x, t) are respectievely the second and fourth normalized harmonic oscillator wave functions and is the angular frequency of the harmonic oscillator. The wave function (x, t = 0) will be orthogonal to ψ(x, t) at the time t equal to

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

36. Consider a single one – dimensional harmonic oscillator of angular frequency ω, in equilibrium at temperature T = . The states of the harmonic oscillator are all non – degenerate having energy with equal probability, where n is the quantum number. The Helmholtz free energy of the oscillator is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    For 1D Harmonic Oscillator E_n =

    

    

    Helmholtz free energy

    

37. A sytem of two atoms can be in three quantum states having energies 0, and 2 3. The system is in equilibrium at temperature T = (k_B)^–1. Match the following satistics with the partition function.

    

    (a) CD : Z1, CI : Z2, FD : Z3, BE : Z4

    (b)    CD : Z4, CI : Z1, FD : Z2, BE : Z3

    (c)    CD : Z3, CI : Z4, FD : Z1, BE : Z2

    (d)    CD : Z2, CI : Z3, FD : Z4, BE : Z1

Ans.    (c)

Sol.    Two particles, Quantum states

    CD : Classical distinguishable (A, B)

    

    

    CD = Z3

    CI : Classical Indistinguishing Particles

    

    CI = Z4

    FD : Fermi Dirac, means they follow Pauli exclusive principle

    

    

    FD : Z1

    BE : Bose-Einstein (AA)

    

    BE = Z2

    

    [CD : Z3, CI : Z4; FD : Z1, BE : Z2]

38. The free energy of a ferromagnet is given by V = F₀ + a₀(T – T_C)M² + bM⁴, where F₀, a₀ and b are positive constants, M is the magnetization, T is the temperature, and T_C is the Curie temperature. The relation between M² and t is best depicted by

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    Minimising free energy of ferromagnetic occurs when = 0

        

    Since M can not be zero therefore, first part must be zero

        2a₀(T – T_C) + 4bM² = 0

        

    Thus graph (b) correctly represent the variation of M² vs temperature.

39. Consider a spherical galaxy of total mass M and radius R, having a uniform matter distribution. In this idealized situation, the orbital speed of a star of mass m (m << M) as a function of the distance r from the galactic centre is best described by

    (G is the universal gravitational constant)

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Gravitational field of galaxy can be given by

        

    

40. Consider the potential U(r) defined as

        

    where and U₀ are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering asmplitude calcualted with U(r) for a (wave – vector) momentum transfer q and 0, is proportional to

    (Usefule integral :

    (a)    q^–2

    (b)    q

    (c)    q^–1

    (d)    q²

Ans.    (a)

Sol.    Scattering amplitude is given by

    

    

    

41. As shown in the figure, inverse magnetic susceptibility (a/) is plotted as a function of temperature (T) for three different materials in paramagnetic states.

    (Curie temperature of ferromagnetic material = T_C

    Neel temperature of antiferromagnetic material = T_N)

    Choose the correct statement from the following

    (a)    Material 1 is paramagnetic, 2 is antiferromagnetic (T < T_N), and 3 is ferromagnetic (T < T_C).

    (b)    Material 1 is antiferromagnetic (T < T_N), 2 is paramagnetic, and 3 is ferromagnetic (T < T_C).

    (c)    Material 1 ferromagnetic (T < T_C), 2 is antiferromagnetic (T < T_N), and 3 is paramagnetic

    (d)    Material 1is ferromagnetic (T < T_C), 2 is paramagnetic and 3 is antiferromagnetic (T < T_N).

Ans.    (b)

Sol.    The magnetic susceptibility for various material is written as

    1. For paramagnetic : , thus graph 2 is for paramagnetic materials.

    2. For Ferromagnetic : , thus graph 3 is for ferromagnetic material.

    3. For antiferromagnetic : , thus the graph 1 is for antiferromagnetic material.

    The correct option is (b).

42. A function f (t) is defined only for t > 0. The Laplace transform of f(t) is

        

    whereas the Fourier transform of f(t) is

        

    The correct statement(s) is (are)

    (a)    The variable s is always real.

    (b)    The variable s can be complex.

    (c)    L(f ; s) and can never by made connected.

    (d)    L(f ; s) and can be made connected.

Ans.    (b, d)

Sol.    , where s is a complex number, can be complex or real.

    

    

    So, L(f, s) and can be related.

43. P and Q are two Hermitian matrices and there exists a matrix R, which diagonalizes both of them, such that RPR^–1 = S₁ and RQR^–1 = S₂, where S₁ and S₂ are diagonal matrices. The correct statement(s) is(are)

    (a)    All the elements of both matrices S₁ and S₂ are real

    (b)    The matrix PQ can have complex eigenvalues.

    (c)    The matrix QP can have complex eigenvalues.

    (d)    The matrices P and Q commute

Ans.    (a, d)

Sol.    P⁺ = P,  Q⁺ = Q

    S₁ = RPR^–1, S₂ = RQR^–1

    S₁, S₂ are diagonal matrices of P and Q

    As P and Q are Hermitian matrices, their eigenvalues are always real. Hence their diagonal matrix will only have real elements.

    S₁, S₂ All elements of S₁, S₂ are real.

    P and Q are simultaneously diagonal, they both commute

    P = R^–1S₁R, Q = R^–1S₂R

    PQ = R^–1S₁RR^–1S₂R = R^–1S₁S₂R as S₁S₂ = S₂S₁

    QP = R^–1S₂RR^–1S₁R = R^–1S₂S₁R

    PQ = QP [PQ] = 0

    P and Q commute.

    As PQ = R^–1S₁S₂R, QP = R^–1S₂S₁R

    S₁S₂, S₂S₁ are diagonal matrix of PQ, QP

    S₁S₂ have real elements then S₁S₂ and S₂S₁ also have real elements.

    Hence, PQ and QP have real eigenvalues not complex.

44. A uniform block of mass M slides on a smooth horizontal bar. Another mass m is connected to it by an inextensible string of length l of negligible mass, and is constrained to oscillate in the X – Y plane only. Neglect the size of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as x and , as shown in the figure.

 

    If p_x and are the generalized momenta corresponding to x and , respectively, then the correct option(s) is(are)

    (a)

    (b)

    (c) p_x is conserved

    (d) is conserved

Ans.    (a, c)

Sol.    

    

45. The Gell – Mann – Okuba mass formula defines the mass of baryons as M = M₀ + aY + b. Where . M₀, a and b are constants, I represents the isospin and Y represents the hypercharge. If the mass of Σ hyperson is same as that of A hyperons, then the correct option (s) is(are)

    (a)    M I (I + 1)

    (b)    M Y

    (c)    M does not depend on I

    (d)    M does not depend on Y

Ans.    (b, c)

Sol.    Isospin I = 1 for hypersons; Y = 0 Isospin I = 0 for Λ ; Y = 0

    As isospin for Λ and hypersons are different, but theri mass is same. So according to given formula, mass of Baryons is independent of I.

    As hypercharge Y = 0 for both Λ and hyperons and their mass is also same. So, one can conclude that mass M is dependent of Y only according to given formula.

46. The time derivative of a differentiable function g(q_i, t) is added to a Lagrangian such that

        

    where are the generalized coordinates, generalized velocities and time, respectively. Let p_i be the generalized momentum and H the Hamiltonian associated with . If and H' are those associated with L', then the correct option(s) is(are)

    (a)    Both L and L' satisfy the Euler-Lagrange's equations of motion

    (b)

    (c) If p_i is conserved, then is necessarily conserved

    (d)

Ans.    (a, b)

Sol.    

    

    

47. A linear charged particle accelerator is driven by an alternating voltage source operating at 10 MHz. Assume that it is used to accelerate electrons. After a few drift-tubes, the electrons attain a velocity 2.9 × 10⁸ m s^–1. The minimum length of each drift-tube, in m, to accelerate the electrons further (rounded off to one decimal place) is _________.

Ans.    14.0 to 15.0

Sol.    The length of the drift tube should be

    

48. The Coulomb energy component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is _________.

    (Given: )

Ans.    8

Sol.    

49. For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices for particles 1 and 2, respectively. The value of (in integer) is _________.

Ans.    3

Sol.    

    

50. A contour integral is defined as

        

    where n is a positive integer and C is the closed contour, as shown in the figure, consisting of the line from – 100 to 100 and the semicircle traversed in the counter – clockwise sense.

    The value of (in integer) is______.

Ans.    5

Sol.    

    

    

    

    

51. The normalized radial wave function of the second excited state of hydrogen atom is

        

    where a is the Bohr radius and r is the distance from the center of the atom. The distance at which the electron is most likely to be found is y × a. The value of y (in integer) is______.

Ans.    4

Sol.    Probability density = |R(r)|²r²

    For most probable distance

52. Consider an atomic gas with number density n = 10²⁰ m^–3, in the ground state at 300 K. The valence electronic configuration of atoms is f ⁷. The paramagnetic susceptibility of the gas x = m × 10^–11. The value of m (rounded off to two decimal places) is______.

    (Given:    Magnetic permeability of free space µ₀ = 4π × 10^–7 H m^–1

            Bohr magneton µ_B = 9.274 × 10^–24 A m²

            Boltzmann constant k_B = 1.3807 × 10^–23 J K^–1)

Ans.    5.48

Sol.    The magentic susceptibility of paramagnetic gas is , where , is called effective number of Bohr magneton and g is Lande-g factor.

    According to Hund's rule

    

    L = – 3 – 2 – 1 + 0 + 1 + 2 + 3 = 0

    

    

    µ_B = 9.274 × 10^–24 Am² and k_B = 1.3807 × 10^–23 JK^–1

    

53. Consider a cross-section of an electromagnet having an air-gap of 5 cm as shown in the figure. It consists of a magnetic material (µ = 20000µ₀) and is driven by a coil having NI = 10⁴ A, where N is the number of turns and I is the current in Ampere.

    Ignoring the fringe fields, the magnitude of the magnetic field (in Tesla, rounded off to two decimal places) in the air-gap between the magnetic poles is ________.

Ans.    0.25

Sol.    

    Let B_c = B_g

    l_c and l_g are core and gap length µ_c = 20000µ₀ and µ_g = µ₀

    Thus

    

54. The spin and orbital angular momentum of an atom precess about , the total angular momentum. precesses about an axis fixed by a magnetic field , where B₀ is a constant. Now the magnetic field is change to . Given the orbital angular momentum quantum number l = 2 and spin quantum number s = 1/ 2, is the angle between and for the largest possible values of total angular quantum number j and its z – component j_z. The value of (in degree rounded off to the nearest integer) is________.

Ans.    92 to 93

Sol.    Correct answer is (92 to 93)

55. The spin-orbit effect splits the ²P ²Stransition (wavelength, = 6521 Å) in Lithium into two lines with separation of = 0.14 Å. The corresponding positive value of energy difference between the above two lines, in eV, is m × 10^–5. The value of m (rounded off to the nearest integer) is _________.

    (Given: Planck's constant, h = 4.125 × 10^–15 eV s, Speed of light, c = 3 × 10⁸ m s^–1)

Ans.    4.08

Sol.