1. For the Op-Amp circuit shown below, choose the correct output waveform corresponding to the input V_in = 1.5 sin (in Volts). The saturation voltage for this circuit is V_sat = ±10 V.

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given circuit is an Schmitt Trigger circuit. So, in this configuration the output will always be limited between + V_sat to –V_sat.

2. Match the order of -decays given in the left column to appropriate clause in the right column. Here and are nuclei with intrinsic spin I and parity .

(a) 1–i, 2–ii, 3–iii, 4–iv

(b) 1–iv, 2–i, 3–ii, 4–iii

(c) 1–i, 2–iii, 3–ii, 4–iv

(d) 1–iv, 2–ii, 3–iii, 4–i

Ans. (b)

Sol. (1) = 0 = No which is allowed by -decay

(2) = 2 = Yes which indicates the first forbidden -decay

(3) = 3 = No no second forbidden -decay

(4) = 4 π= Yes third forbidden -decay

3. What is the maximum number of free independent real parameters specifying an n-dimensional orthogonal matrix?

(a) n(n – 2)

(b) (n – 1)²

(c)

(d)

Ans. (c)

Sol. Consider a 2 × 2 orthogonal matrix A =

Since, A is a matrix is orthogonal. Thus, it will satisfy the following relation

A^TA = I

From the above equation, we will have following constraint a² + c² = 1, b² + d² = 1

These are the two constraints on diagonal elements.

Also, ac + bd = 0 is one constraint on the value of diagonal elements. This will make one more component dependent.

Thus, the independent component of A is 4 – 3 = 1

For n × n matrix.

The total component is n².

The number of constraint along diagonal is n

The number of constraint along off diagonal is ⁿC₂ =

Thus, the total number of independent component n² – n –

4. An excited state of Ca atom is [Mg]3p⁵4s²3d¹. The spectroscopic terms corresponding to the total orbital angular momentum are

(a) S, P, and D

(b) P, D, and F

(c) P and D

(d) S and P

Ans. (b)

Sol. It is worthy to mentioned here that the core shell electrons and 4s electrons of Mg should be ignored to calculate its spectroscopic term. We have to consider only the p electron, (l₁ = 1) and d electron (l₂ = 2). Thus, total orbital angular momentum can be written as

L = |l₁ + l₂|....|l₁ – l₂| = 3, 2, 1 = D, P, S

5. On the surface of a spherical shell enclosing a charge free region, the electrostatic potential values are as follows: One quarter of the area has potential , another quarter has potential 2 and the rest has potential 4. The potential at the centre of the shell is

(You can use a property of the solution of Laplace's equation.)

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Thus potential at centre is only decided by A₀; A₀ = ?

Also, it can be shown that surface area of sphere from 0 = 0 is respectively

Let's apply the boundary condition, which is

As we only want A₀

6. A point charge q is performing simple harmonic oscillations of amplitude A at angular frequency . Using Larmor's formula, the power radiated by the charge is proportional to

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

7. Which of the following relationship between the internal energy U and the Helmholtz's free energy F is true?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The free energy can be written as follows

F = U – TS

dF = –SdT – PdV [Since, dU = TdS – PdV]

Now, the differentiation of above equation with respect to temperature at constant volume will provide entropy S as follows

8. If nucleons in a nucleus are considered to be confined in a three-dimensional cubical box, then the first four magic numbers are

(a) 2, 8, 20, 28

(b) 2, 8, 16, 24

(c) 2, 7, 14, 20

(d) 2, 10, 16, 28

Ans. (c)

Sol. The energy of cubial box having side L can be written as follows

where n_x = 1, 2, 3, 4, ....

n_y = 1, 2, 3, 4, ..... n_z = 1, 2, 3, 4, ....

So on....

9. Consider the ordinary differential equation

y" – 2xy' + 4y = 0

and its solution y(x) = a + bx + cx². Then

(a) a = 0, c = –2b 0

(b) c = –2a 0, b = 0

(c) b = –2a 0, c = 0

(d) c = 2a 0, b = 0

Ans. (b)

Sol. y" – 2xy' + 4y = 0 ...(1)

Given solution y(x) = a + bx + cx²

y' = b + 2cx

y" = 2c

Put the value of y, y' and y"m (1), we get

2c – 2x[b + 2cx] + 4[a + bx + cx²] = 0

2c – 2bx – 4cx² + 4a + 4bx + 4cx² = 0

2bx + (2c + a) = 0

b = 0 and 2c + 4a = 0

c = – 2a

Thus, b = 0 and c = – 2a

10. For an Op-Amp based negative feedback, non-inverting amplifier, which of the following statements are true?

(a) Closed loop gain < Open loop gain

(b) Closed loop bandwidth < Open loop bandwidth

(c) Closed loop input impedance > Open loop input impedance

(d) Closed loop output impedance < Open loop output impedance

Ans. (a, c, d)

Sol. Correct options are (a, c, d).

11. From the pairs of operators given below, identify the ones which commute. Here l and j correspond to the orbital angular momentum and the total angular momentum, respectively.

(a) l², j²

(b) j², j_z

(c) j², l_z

(d) l_z, j_z

Ans. (a, b, d)

Sol. Correct options are (a, b, d).

12. For normal Zeeman lines observed || a and to the magnetic field applied to an atom, which of the following statements are true?

(a) Only -lines are observed || a to the field

(b) -lines to the field are plane polarized

(c) -lines to the field are plane polarized

(d) Only ⊥-lines are observed || a to the field

Ans. (b, c, d)

Sol. The following diagram clearly shows that

-lines to the field are plane polarized

-lines to the field are plane polarized

Only -lines are observed || to the field.

13. Pauli spin matrices satisfy

(a)

(b)

(c)

(d)

Ans. (b, d)

Sol. General anti commutator relation.

Let us verify option (b)

Also

Let us verify the relation in option (d)

But, option (a) and (c) will not satisfy the following relation.

Thus, option (b) and (d) are correct option.

14. For the refractive index n = n_r() + in_im() of a material, which of the following statements are correct?

(a) n_r can be obtained from n_im and vice versa

(b) n_im could be zero

(c) n is an analytic function in the upper half of the complex ω plane

(d) n is independent of for some materials

Ans. (a, c)

Sol. Correct options are (a, c).

15. Complex function f(z) = z + |z – a|² (a is a real number) is

(a) continuous at (a, a)

(b) complex-differentiable at (a, a)

(c) complex-differentiable at (a, 0)

(d) analytic at (a, 0)

Ans. (a, c)

Sol. f(z) = z + |z – a|²

= x+ iy + |x – a + iy|²

= x + (x – a)² + y² + iy

Continuity to (a, a)

Parallel to x-axis

Parallel to y-axis

Along line having slope (m) passing through (a, a)

Equation of lines is y = a + m(x – a)

= a + 0 + (a + m0)² + i(a + m0)

= a + a² + ia ... ...(3)

As (1) = (2) = (3)

Hence function is continuous at (a, a)

Differentiability at (a, 0)

Be definition, the derivative of the function at (a, 0) is

Since f(z) = z + |z – a|²

Hence function is continuous at (a, a)

By definition, the derivative of the function at (a, 0) is

Since f(z) = z + |z – a|²

f(a) = a + |a – a|² = a

Along line having slope 'm' passing through (a, 0)

Equation of line is y = m(x – a)

= 1 + 0 – im0 = 1 ...(6)

As (4) = (5) = (6)

Hence function is differentiable at (a, 0)

Thus 'a' and 'c' are correct options.

16. If g(k) is the Fourier transform of f(x), then which of the following are true?

(a) g(–k) = +g*(k) implies f(x) is real

(b) g(–k) = –g*(k) implies f(x) is purely imaginary

(c) g(–k) = +g*(k) implies f(x) is purely imaginary

(d) g(–k) = –g*(k) implies f(x) is real

Ans. (a, b)

Sol. According to the given statement

taking complex conjugate of (1)

Replacing k by –km (1)

Now if g(–k) = g*(k) condition 'a'

Hence f(x) must be real and not purely imaginary

{Condition 'b'}

g(–k) = –g*(k)

f(x) = –f*(x). Thus f(x) must be purely imaginary and not real.

Thus 'a' and 'b' are correct options.

17. The ordinary differential equation

(1 – x²)y" – xy' + 9y = 0

has a regular singularity at

(a) –1

(b) 0

(c) +1

(d) no finite value of x

Ans. (a, c)

Sol. (1 – x²)y" – xy' + 9y = 0

Dividing by 1 – x²

Compare with y" + p(x)y' + Q(x)y = 0

At 1 and –1 both P(x) and Q(x) diverge first condition satisfied.

At x = 1

At x = –1

Thus (x – x₀)P(x) remains finite.

At x = 1

At x = –1

Thus (x – x₀)²Q(x) remains finite.

Thus both 1 and –1 are regular singular points.

18. For a bipolar junction transistor, which of the following statements are true?

(a) Doping concentration of emitter region is more than that in collector and base region

(b) Only electrons participate in current conduction

(c) The current gain depends on temperature

(d) Collector current is less than the emitter current

Ans. (a, c, d)

Sol. In order to explain the conduction in transistor both minority and majority charge carries are considered.

19. Potassium metal has electron concentration of 1.4 × 10²⁸m^–3 and the corresponding density of states at Fermi level is 6.2 × 10⁴⁶ Joule^–1 m^–3. If the Pauli paramagnetic susceptibility of Potassium is n × 10^–k in standard scientific form, then the value of k (an integer) is __________ (Magnetic moment of electron is 9. 3 × 10^–24 Joule T^–1; permeability of free space is 4 × 10^–7 T m A^–1)

Ans. 6

Sol. Given that

n_e = 1.4 × 10²⁸ m^–3, µ₀ = 4 × 10^–7 TmA^–1, D(E_F) = 6.2 × 10⁴⁶ Jm^–3, µ_B = 9.3 × 10^–24 JT^–1

We know that the susceptibility

= 4 × 10^–7 × (9.3 × 10^–2)² × 6.2 × 10⁴⁶

= 4 × (9.3)² × 6.2 × 10^{–7 – 48 + 46} = 6.735 × 10^–6

k = 6

20. A power supply has internal resistance R_S and open load voltage V_S = 5 V. When a load resistance R_L is connected to the power supply, a voltage drop of V_L = 4 V is measured across the load. The value of is __________ (Round off to the nearest integer)

Ans. 4

Sol. We know that

21. Electric field is measured along the axis of a uniformly charged disc of radius 25 cm. At a distance d from the centre, the field differs by 10% from that of an infinite plane having the same charge density. The value of d is _______cm.

(Round off to one decimal place)

Ans. 2.4 to 2.6

Sol.

Given, E_Disc = 90% E_{Infinite sheet}

22. In a solid, a Raman line observed at 300 cm^–1 has intensity of Stokes line four times that of the anti-Stokes line. The temperature of the sample is _______K.

(Round off to the nearest integer) (1 cm^–1 1.44 K)

Ans. 311 to 312

Sol.

Taking in both side, we can write

= 31.17 × 10¹ K = 311.7 K

Hence, T = 311 to 312

23. An electromagnetic pulse has a pulse width of 10^–3 s. The uncertainty in the momentum of the corresponding photon is of the order of 10^–N kg m s^–1, where N is an integer. The value of N is ____________ (speed of light = 3 × 10⁸ m s^–1, h = 6.6 × 10^–34 J s)

Ans. 39 to 40

Sol. Given that = 10^–3 sec, h = 6.6 × 10^–34, c = 3 × 10⁸ m/s

By using Heisenberg uncertainty principle we can write

The uncertainty in the momentum is given by

24. The wavefunction of a particle in a one-dimensional infinite well of size 2a at a certain time is . Probability of finding the particle in n = 2 state at that time is ______% (Round off to the nearest integer)

Ans. 33 to 34

Sol. Given to wavefunction as follows

The gs, 1^st excited and 2^nd excited state wavefunction can be written as follows

The normalization constant is obtained as follows.

Thus, the normalized wave function is given by

The probability of finding the particle in state n = 2 is

25. A spectrometer is used to detect plasma oscillations in a sample. The spectrometer can work in the range of 3 × 10¹² rad s^–1 to 30 × 10¹² rad s^–1. The minimum carrier concentration that can be detected by using this spectrometer is n × 10²¹ m^–3. The value of n is _____________ (Round off to two decimal places)

(Charge of an electron = –1.6 × 10^–1 C, mass of an electron = 9.1 × 10^–31 kg and = 8.85 × 10^–12 C² N^–1 m^–2)

Ans. 2.70 to 2.96

Sol.

= 283.1 × 10¹⁹ m^–3

n₀ = 2.83 × 10²¹ m^–3

2.83 × 10²¹ m^–3

26. Consider a non-interacting gas of spin 1 particles, each with magnetic moment µ, placed in a weak magnetic field B, such that . The average magnetic moment of a particle is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. In quantum mechanical treatment, single-dipole partition function is

Where x = β(gµ_BJ)B. The mean magnetic moment of the system is then given by

Where B_J(x) is the Brillouin function. For

Therefore,

Where, last term is written for J = 1

27. Water at 300 K can be brought to 320 K using one of the following processes.

Process 1: Water is brought in equilibrium with a reservoir at 320 K directly.

Process 2: Water is first brought in equilibrium with a reservoir at 310 K and then with the reservoir at 320 K.

Process 3: Water is first brought in equilibrium with a reservoir at 350 K and then with the reservoir at 320 K.

The corresponding changes in the entropy of the universe for these processes are and , respectively. Then

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Here initial temperature (T_i) of water is 300K & final temperature (T_f) is 320K.

Process 1:

= 0.06454C_W – 0.0625C_W

= 0.00204C_W

Process 2: will be same as initial and final equilibrium states are sa

= – C_W [0.03226 + 0.03125]

= – 0.06351C_W

= –0.142857C_W + 0.09375C_W

= –0.04911 C_W

= 0.0645C_W – 0.04911C_W

= 0.01543C_W

28. A student sets up Young's double slit experiment with electrons of momentum p incident normally on the slits of width w separated by distance d. In order to observe interference fringes on a screen at a distance D from the slits, which of the following conditions should be satisfied?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

More accurately

29. Consider a particle in three different boxes of width L. The potential inside the boxes vary as shown in figures (i), (ii) and (iii) with . The corresponding ground-state energies of the particle are E₁, E2 and E₃, respectively. Then

(a) E₂ > E₁ > E₃

(b) E₃ > E₁ > E₂

(c) E₂ > E₃ > E₁

(d) E₃ > E₂ > E₁

Ans. (a)

Sol. The ground state wave function is given by

Let us determine the correction in energy due to the potential in first box.

The ground state energy correction in first order is

Similarly, let us determine the correction in energy due to potential in

The ground state energy correction in first order is

Similarly let us determine the correction in energy due to potential in third box

The ground state energy correction in first order

Thus, the order of ground state energy in three boxes is given by

E₂ > E₁ > E₃

30. In cylindrical coordinates (s, , z), which of the following is a Hermitian operator?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The operator A is called Hermitian if it satisfy the following relation.

Option (c) is only following the above condition

Let us choose operator given is option (c).

31. A particle of mass 1 kg is released from a height of 1 m above the ground. When it reaches the ground, what is the value of Hamilton's action for this motion in J s? (g is the acceleration due to gravity; take gravitation potential to be zero on the ground)

(a)

(b)

(c)

(d)

Ans. (d)

Sol. At point B

Time taken to reach the point C

32. If is a constant of motion of a two-dimensional isotropic harmonic oscillator with Lagrangian

then is

(a)

(b)

(c)

(d) 0

Ans. (a)

Sol. The quantity A is constant of motion if

[A, H] = 0

[A, H] = 0

33. In a two-dimensional square lattice, frequency of phonons in the long wavelength limit changes linearly with the wave vector k. Then the density of states of phonons is proportional to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We know that the density of states can be written as follows

For phonon s = 1 and in two dimensional d = 2

34. At T = 0 K, which of the following diagram represents the occupation probability P(E) of energy states of electrons in a BCS type superconductor?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. There exist a band gap at fermi level at T = 0 K for BCS superconductor.

35. For a one-dimensional harmonic oscillator, the creation operator (a^†) acting on the n^th state , where n = 0, 1, 2, ..., gives . The matrix representation of the position operator for the first three rows and columns is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The position operator is can be written as follows

The expectation value of position operator can be written as follows

The matrix representation of position vector is

and general from is

36. A piston of mass m is fitted to an airtight horizontal cylindrical jar. The cylinder and piston have identical unit area of cross-section. The gas inside the jar has volume V and is held at pressure P = P_atmosphere. The piston is pushed inside the jar very slowly over a small distance. On releasing, the piston performs an undamped simple harmonic motion of low frequency. Assuming that the gas is ideal and no heat is exchanged with the atmosphere, the frequency of the small oscillations is proportional to

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Let initial pressure is P₁ = P

Initial volume is V

When pressure changes slightly by , i.e. piston is pushed in side, the volume is reduced by

Further given that no heat exchange is there

Applied external force F, that caused a displacement x (a volume change c given by

Applied external force F, that caused a displacement x (a volume change of gas by = Ax) is given by

Acceleration produced in piston is

37. A paramagnetic salt of mass m is held at temperature T in a magnetic field H. If S is the entropy of the salt and M is its magnetization, then dG = –SdT – MdH, where G is the Gibbs free energy. If the magnetic field is changed adiabatically by 0 and the corresponding infinitesimal changes in entropy and temperature are and , then which of the following statements are correct

(a)

(b)

(c)

(d)

Ans. (b, c)

Sol. The magnetic interaction energy = –M.dH... ...(1)

dU = TdS – M.dH... ...(2)

dG = SdT –M.dH... ...(3)

This is indeed the process of adiabatic unitization.

Where negative ΔH implies negative = 0.

38. A particle of mass m is moving inside a hollow spherical shell of radius a so that the potential is

The ground state energy and wavefunction of the particle are E₀ and R(r), respectively. Then which of the following options are correct?

(a)

(b)

(c)

(d)

Ans. (a, b, d)

Sol. The Schrodinger equation for a particle moving in radial potential is given by

Thus Schrodinger equation to

Defining R(r) = and substituting in above equation, we get

The solution of above equation is

U(r) = A sin kr + Bcos kr

Applying Boundary condition

U(r = 0) = 0; U(r = a) = 0

Thus the wave function is given by

U(r) = Asin kr

Applying boundary condition U(r = a) = 0

and the radial wave function is given by

For ground state the energy and wave function of the particle are

39. A particle of unit mass moves in a potential . If the angular momentum of the particle is , then which of the following statements are true?

(a) There are two equilibrium points along the radial coordinate

(b) There is one stable equilibrium point at r₁ and one unstable equilibrium point at r₂ > r₁

(c) There are two stable equilibrium points along the radial coordinate

(d) There is only one equilibrium point along the radial coordinate

Ans. (a, b)

Sol.

Here r₁ < r₂

These equilibrium points are corresponding to circular orbits of radius r₁ and r₂ respectively.

40. In a diatomic molecule of mass M, electronic, rotational and vibrational energy scales are of magnitude E_e, R_R and E_V, respectively. The spring constant for the vibrational energy is determined by E_e. If the electron mass is m then

(a)

(b)

(c)

(d)

Ans. (a, c)

Sol.

41. Electronic specific heat of a solid at temperature T is C = T, where is a constant related to the thermal effective mass (m_eff) of the electrons. Then which of the following statements are correct?

(a)

(b) m_eff is greater than free electron mass for all solids

(c) Temperature dependence of C depends on the dimensionality of the solid

(d) The linear temperature dependence of C is observed at T << Debye temperature

Ans. (a, d)

Sol. Lattice heat capacity depends on dimensionality of solid. But, electronic specific heat does not depends on dimensionality. The effective mass is not always greater than the free electron mass for all solids.

42. In a Hall effect experiment on an intrinsic semiconductor, which of the following statements are correct?

(a) Hall voltage is always zero

(b) Hall voltage is negative if the effective mass of holes is larger than those of electrons

(c) Hall coefficient can be used to estimate the carrier concentration in the semiconductor

(d) Hall voltage depends on the mobility of the carriers

Ans. (d)

Sol. Hall coefficient for intrinsic semiconductor generally negative. Because, electron mobility is higher than the hole mobility.

43. A parallel plate capacitor with spacing d and area of cross-section A is connected to a source of voltage V. If the plates are pulled apart quasistatically to a spacing of 2d, then which of the following statements are correct?

(a) The force between the plates at spacing 2d is

(b) The work done in moving the plates is

(c) The energy transferred to the voltage source is

(d) The energy of the capacitor reduces by

Ans. (a, c, d)

Sol.

(c) Energy tranferred to source must be equal to energy decrease of the capacitor.

(d) Initial energy = , final energy = change

44. A system with time independent Hamiltonian H(q, p) has two constants of motion f(q, p) and g(q, p). Then which of the following Poisson brackets are always zero?

(a) {H, f = g}

(b) {H, {f, g}}

(c) {H + f, g}

(d) {H, H + fg}

Ans. (a, b, d)

Sol. {H, f} = 0, {H, g} = 0

(a) {H, f + g} = {H, f} + {H, g} = 0

(b) {H, {f, g}} = – {f, {g, H}} – {g, {H, f}} = – {f, 0} – {g, 0} = 0

(c) {H + f, g} = {H, g} + {f, g} = {f, g}

(d) {H, H + f g} = {H, H} + {H, f g} = {H, f}g + f{H, g} = 0

45. In the action-angle variables (I₁, I₂, ), consider the Hamiltonian H = 4I₁I₂ and . Let . Which of the following are possible plots of the trajectories with different initial conditions in plane?

(a)

(b)

(c)

(d)

Ans. (b, c)

Sol. H = 4I₁I₂

Equation of motion

46. A particle of mass m in the x-y plane is confined in an infinite two-dimensional well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle are . If perturbation of the form V = Cxy, where C is a real constant, is applied, then which of the following statements are correct for the first excited state?

(a) The unperturbed energy is

(b) The unperturbed energy is

(c) First order energy shift due to the applied perturbation is zero

(d) The shift in energy due to the applied perturbation is determined by an equation of the form , where a and b are real, non-zero constants

Ans. (b, d)

Sol. We have

and its corresponding energies are,

For ground state n_x = n_y = 1, the ground state energy is given by

The first order correction in ground state energy is

as the energy state energy is odd function.

The first excited state has energy,

The wave function of the particle are

The perturbed matrix for this Hamiltonian is given by

The values of inner product are

Thus the eigen value of perturbed matrix is determined ground secular equation.

47. A junction is formed between a metal on the left and an n-type semiconductor on the right. Before forming the junction, the Fermi level E_F of the metal lies below that of the semiconductor. Then which of the following schematics are correct for the bands and the I-V characteristics of the junction?

(a)

(b)

(c)

(d)

Ans. (a, c)

Sol. For a metal and an n-type semiconductor rectifying Schottky contact is formed when work function of metal is greater than the work function of semiconductor.

48. A plane polarized electromagnetic wave propagating in y-z plane is incident at the interface of two media at Brewster's angle. Taking z = 0 as the boundary between the two media, the electric field of the reflected wave is given by

then which among the following statements are correct?

(a) The angle of refraction is

(b) Ratio of permittivity of the medium of refraction with respect to the medium on incidence

(c) The incident wave can have components of its electric field in y-z plane

(d) The angle of reflection is

Ans. (a, b, c)

Sol.

49. The minimum number of two-input NAND gates required to implement the following Boolean expression is ________________

Ans. 3

Sol.

50. In a nucleus, the interaction is responsible for creating spin-orbit doublets. The energy difference between p_1/2 and p_3/2 states in units of is ____________ (Round off to the nearest integer)

Ans. 3

Sol.

For p_1/2 : s = 1/2; l = 1; j = 1/2

For p_3/2 : s = 1/2; l = 1; j = 3/2

51. Two identical particles of rest mass m₀ approach each other with equal and opposite velocity v = 0.5c, where c is the speed of light. The total energy of one particle as measured in the rest frame of the other is E = m₀c². The value of is _____________ (Round off to two decimal places)

Ans. 1.65 to 1.70

Sol. v_AE = 0.5c

v_BE = –0.5c

52. In an X-Ray diffraction experiment on a solid with FCC structure, five diffraction peaks corresponding to (111), (200), (220), (311) and (222) planes are observed using 1.54 Å X-rays. On using 3 Å X-rays on the same solid, the number of observed peaks will be ________________

Ans. 1

Sol. Bragg's Law

Corresponding to maximum value of sin (= 1), the expression of has maximum values for the sin . From this condition we can find out the value of lattice parameter (a) from the peak corresponding to (222) plane. So

The maximum value of sin will be 1. So for wavelength = 3Å only (111) peak observed.

53. For 1 mole of Nitrogen gas, the ratio of entropy change of the gas in processes (I) and (II) mentioned below is ______________ (Round off to one decimal place)

(I) The gas is held at 1 atm and is cooled from 300 K to 77 K.

(II) The gas is liquified at 77 K.

(Take C_p = 7.0 cal mol^–1 K^–1, Latent heat L = 1293.6 cal mol^–1)

Ans. 0.5 to 0.7

Sol.

54. Frequency bandwidth of a gas laser of frequency v Hz is

where = 3.44 × 10⁶ m² s^–2 at room temperature and A is the atomic mass of the lasing atom. For ⁴He - ²⁰Ne laser (wavelength = 633 nm), = n × 10⁹ Hz. The value of n is ________ (Round off to one decimal place)

Ans. 1.2 to 1.4

Sol. Frequency bandwidth Δv of a He-Ne laser is given by,

The lasing atom is Ne for which atomic mass is 20 amu.

55. A current of 1 A is flowing through a very long solenoid made of winding density 3000 turns/m. As shown in the figure, a parallel plate capacitor, with plates oriented parallel to the solenoid axis and carrying surface charge density 6C m^–2, is placed at the middle of the solenoid. The momentum density of the electromagnetic field at the midpoint X of the capacitor is n × 10^–13 N s m^–3. The value of n is ______________ (Round off to the nearest integer)

(speed of light c = 3 × 10⁸ m s^–1)

Ans. 2

Sol.