PYQ 2000 - VedPrep

GATE PHYSICS 2000
Previous Year Question Paper with Solution.

1.This question consists of FIFTEEN sub-questions (1.1 to 1.15) of ONE mark each. For each of these sub-questions, four possible answers (a, b, c and d) are given, out of which only one is correct.

1.1.A square matrix A is unitary if:

(a) A^† = A

(b) A^† = A^–1

(d) det (A) = 1

Ans. (b)

Sol. If matrix A is unitary, then A^†A = I and we know that for any square matrix A^–1 A = I

Hence, A^†A = I

1.2.A planet moves around the Sun in an elliptical orbit with semi-major axis a and time period T. T is proportional to

(a) a²

(b) a^1/2

(d) a³

Ans. (c)

Sol. From kepler's third law, we have

Where T is period of revolution of a planet about the sun and a is semi-major axis of the elliptical path.

1.3.A particle moves in a central force field , where k is a constant, r, the distance of the particle from the origin and is the unit vector in the direction of position vector . Closed stable orbits are possible for:

(a) n = 1 and n = 2

(b) n = 1 and n = –1

(d) n = 1 and n = –2

Ans. (d)

Sol. Bertrand's theorem states that among central force potentials with bound orbitals, there are only two types of central force potentials with the property that all bound orbits are also closed orbital.

(1) An inverse-square central force such as the gravitational or electrostatic potential

, and

(2) The radial harmonic oscillator potential

Corresponding forces,

Comparing F₁ and F₂ with the form given, , we get n = –2 and n = 1

1.4.The space between the plates of a parallel plate capacitor is filled with two dielectric slabs of dielectric constants k₁ and k₂ as shown in the figure. If the capacitor is charged to a potential V, then at the interface between the two dielectrics.

(a) is discontinuous and is continuous

(b) is discontinuous and is discontinuous

(d) is continuous and is discontinuous

Ans. (a)

Sol. The electric field at the region (1) is

, A to B direction

and the electric field at the region (2) is

, A to B direction

The displacement vector does not depend of permittivity. It depends only on the free charge density on the plate.

D₁ = , A to B direction

And D₂ = , A to B direction

Since . So, is discontinuous and is continuous.

1.5.Two large parallel plates move with a constant speed v in the positive y-direction as shown in the figure. If both the plates have a surface charge density > 0, the magnetic field at the point P just above the top plate will have:

(a) Larger magnitude than the field at the mid-point between the plates and point towards

(b) Smaller magnitude than the field at the mid-point between the plates and point towards

(d) Smaller magnitude than the field at the mid-point between the plates and point towards

Ans. (c)

Sol. Since both the plate moving with same speed and having surface charge density .

Therefore, the surface current density,

Therefore, the magnetic field at P due to both plate will be

And the magnetic field at the mid-point Q is

Therefore, the magnetic field at P is larger than the field at the mid-point Q between the plates and points towards .

1.6.Which of the following is an example of a first order phase transition?

(a) A liquid-gas phase transition at the critical point.

(b) A paramagnet-ferromagnet phase transition

(d) A liquid-gas phase transition away from the critical point.

Ans. (a)

Sol. A liquid-gas phase transition at the critical point first order phase transition

A parmagnet - ferromagnet phase transition second order phase transition

A normal metal-superconductor phase transition second order phase transition

A liquid-gas phase transition away from the critical point second order phase transiton

1.7.A tungsten wire of uniform cross section and high resistance is supported by two copper supports of low resistance in vacuum. The length of the wire is l. A constant current I is sent through the wire. The temperature profile (Tvs. x) of the wire will look like.

(a)

(b)

(c)

(d)

Ans. (b)

Sol. When current is passed through the tungsten wire heat develops in it. At the ends heat is transferred to the copper support whereas at the middle conduction of heat is less. Therefore, temperature at the middle is higher and at the ends is lower.

1.8.The energy E of X-rays emitted from targets of different atomic number Z varies as

(a) Z²

(b) Z^2/3

(d) Z^1/2

Ans. (a)

Sol. According to Moseley's law. The square root of frequency of X-ray is given by

1.9.In a Stern-Gerlach experiment the atomic beam whose angular momentum state is to be determined, must travel through

(a) homogeneous radio frequency magnetic field

(b) homogeneous static magnetic field

(d) inhomogeneous radio frequency magnetic field

Ans. (c)

Sol. In Stem-Gerlach experiment the transverse deflection suffered by atoms will be

at a constant temperature

If field is homogeneous then the , So, d = 0. but if field is inhomogeneous then the . So, it will produce a finite transverse separation between up and down spin atom.

1.10.Let E₁, E₂, E₃ be the respective ground state energies of the following potentials: Which one of the following is correct?

(a) E₁ < E₂ < E₃

(b) E₃ < E₁ < E₂

(d) E₂ < E₁ < E₃

Ans. (c)

Sol. If we remove the perturbation part from all boxes, particle will have same ground state energy say E₃.

For positive perturbation, E₁ > E₃ and for negative perturbation E₂ < E₃.

So, E₂ < E₃ < E₁

1.11.The mean momentum of a nucleon in a nucleus with mass number A varies as

(a) A

(b) A²

(d) A^–1/3

Ans. (d)

Sol. For a nucleon to be inside nucleus, maximum uncertainty in position = Radius of nucleus

So,

And also from uncertainty principle,

And also R = R₀A^1/3 (where, R₀ is a constant and A is the mass number of the nucleus)

So,

Mean momentum of a nucleon =

So, mean momentum

1.12.A particle is scattered by a central potential. If the dominant contribution to the scattering is from the p-wave, the differential cross-section is

(a) isotropic

(b) proportional to cos²

(d) proportional to sin² sin²

Ans. (b)

Sol. According to partial wave analysis the scattering amplitude is given by

For p wave

Therefore, the differential cross-section

1.13.Magnons in ferromagnets

(a) decrease the magnetization

(b) increase the magnetization

(d) cause critical magnetic fluctuations

Ans. (c)

Sol. At absolute zero, Heisenberg ferromagnet reaches state of lowest energy. As T increases, more and more spins deviate from alignment, increasing the internal energy and reducing the net magnetization.

where, T_c is the (material dependent) critical temperature, M₀ is the magnitude of spontaneous magnetization.

1.14.The Fermi energy of a free electron as depends on the electron density as

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The density of states is

Since, electron has spin half,

The Fermi-Dirac distribution is

The total number of particles is

At 0 K, where, = electron density

Clearly,

1.15.If the output of the logic circuit shown in the figure is 1, the input could be

(a) A = 1, B = 1, C = 1, D = 0

(b) A = 1, B = 1, C = 0, D = 0

(d) A = 0, B = 1, C = 1, D = 1

Ans. (a)

Sol.

2.This question consists of THIRTY sub-questions (2.1 to 2.30) of TWO marks each. For each of these sub-questions, four possible answers (a, b, c and d) are given, out of which only one is correct.

2.1.If Z₁ and Z₂ are complex numbers, which of the following is always true?

(a) Z₁* Z₂ + Z₂* Z₁ = 2 Z₁*Z₂

(b) Z₁*Z₂ + Z₂*Z₁ = 2Z₁Z₂

(d) Z₁*Z₂ + Z₂*Z₁> 2|Z₁| |Z₂|

Ans. (c)

Sol.

2.2.The eigenvalues of the matrix are

(a) 1, 0

(b) 1, 1

(d) 0, 2

Ans. (d)

Sol. Eigenvalue equation of a matrix A,

2.3.The Dirac delta function can be represented by:

(a)

(b)

(c)

(d)

Ans. (b)

Sol. A Dirac-delta function has the property that

For option (b), we have

2.4.If is a vector of constant magnitude, which of the following is true?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Differentiating w.r.t. t, we get

2.5.A particle constrained to move along the x-axis in a potential V = kx², is subjected to an external time-dependent force . Here k is a constant, x is the distance from the origin, and t the time. At some time T, when the particle has zero velocity x = 0, the external force is removed. The particle will then,

(a) execute simple harmonic motion

(b) move along +x direction

(d) remain at rest

Ans. (d)

Sol. Given: V = kx²

Force acting on the particle due to the potential

At time t = T, f(t) is removed and v = 0, x = 0.

Force acting on the particle F = 0.

Therefore, the particle remains at rest.

2.6.Consider two particles with position vectors and . The force exerted by particle 2 on particle 1 is, . The force is:

(a) Central and conservative

(b) Non-central and conservative

(d) Non-central and non-conservative.

Ans. (d)

Sol. A central force must have the mathematical form as shown below,

is a function of r only.

A central force is always a conservative force. The given force depends on velocity and thus, it is non-central. This is also implies that it is a non-conservative force.

2.7.A closed tall jar containing air and a fly is placed on a sensitive weighing machine. When the fly is stationary, the reading of the weighing machine is W. If the fly starts flying with some upwards acceleration, the reading of the machine will be

(a) W

(b) > W

(d) directly proportional to the acceleration

Ans. (b)

Sol. A fly flies by pushing air down. In this case, the jar is closed. So, the air can't go through the bottom of the jar. This downward motion of air will exert a force on the bottom of the jar and will reflect in the weighing machine reading. The fly starts flying with some upwards acceleration, which means it has to produce a thrust more than its weight. Therefore, the reading on the machine will be greater than fly's weight W.

2.8.A solenoid with an iron core is connected in series with a battery of emf V and it is found that a constant current I₀ passes through the solenoid. If at t = 0, the iron core is pulled out from the solenoid quickly in a time , which one of the following could be a correct description of the current passing through the solenoid?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The magnetic field inside solenoid is given by

Since, µ > µ₀, so if we remove core material from the solenoid, the magnetic flux will decrease. According to Lenz's law, the emf induced in the solenoid will oppose the decrease in magnetic flux. As a result, the current in the solenoid will increase first and then it will decrease to its initial constant value.

2.9.A point charge q is kept at the mid-point between two large parallel grounded conducting plates. Assume no gravity. The charge is displaced a little towards the right plate. The charge will now.

(a) Stay where it is

(b) Move towards the right plate

(d) Oscillate between the plates

Ans. (b)

Sol. If we give a small displacement to the charge particle towards right plate, net force on the charge particle due to plates will along right plate. So, particle will move towards right plate.

2.10.A very long solenoid with n turns per unit length carries a current I. The magnetic field at a point, which is on its axis and its end face, is

(a) µ₀nI

(b) (2/3)µ₀nI

(d) (1/22)µ₀nI

Ans. (d)

Sol. The magnetic field at p due to the elementary part of solenoid is

2.11.Three plane waves are given as

, where and A₁, A₂, A₃ are constants. If these waves are superposed pairwise, which superposition will lead to interference?

(a)

(b)

(c)

(d) No interference in any pair

Ans. (c)

Sol. We know that the superposition of two plane waves will give interference if

(a) They are monochromatic

(b) They have same state of polarization

Since, are orthogonal polarized light. So, they will not give interference but have same state of polarization they will give interference.

2.12.If a spin 1 particle is in the state with respect to a quantization axis , which of the following is correct?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. For a spin 1 particle, assume the quantization axis to be z-axis.

Given: state of the particle is will be eigenstate of operator.

In an eigenstate of operator,

2.13.Let a particle move in a potential field given by

The allowed energies of this particle are

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have potential,

The wave function of harmonic oscillator is given by

Now, the boundary condition as x = 0

So,

We know that,

for n is odd

And at x = 0 for n is even

Therefore, the allowed wave function is

Therefore, the energy of the n^th state of the particle is

2.14.A spin ½ particle with g > 0, is subjected to a magnetic field . If the quantization axis is along , then the minimum energy eigenstate is given by,

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We know that the magnetic moment due to spin

Therefore, Hamiltonian of the system,

So, the minimum energy eigenstate is

2.15.Bound state eigenfunctions of an attractive finite range smooth potential behave for larger r as:

(a) exp(–r/r₀) where r₀ is a positive constant.

(b) (I/rⁿ) where n > 0

(d) exp(ikr)/r

Ans. (a)

Sol. For bound state wave function at and wave function and its first derivative will be continuous, only option (a) will satisfy the both conditions.

2.16.The order of magnitude of the number of air molecules in a room of volume 50 m³ at STP is

(a) 10²⁷

(b) 10²⁴

(d) 10²⁰

Ans. (a)

Sol. Using ideal gas equation,

PV = Nk_BT, where P = 1 atm and T = 273 K at STP

2.17.An amount of heat Q is transferred from a heat reservoir at temperature T_A to another heat reservoir at temperature T_B. What is the change in the entropy of the combined system?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The change in the entropy of the combined system is

2.18.In a canonical ensemble

(a) the energy and the temperature are constants.

(b) the entropy and the energy are constants.

(d) the density and the entropy are constants.

Ans. (c)

Sol. The canonical ensemble is the ensemble in which all elements will have same macrostate represented by same density and temperature for all elements.

2.19.What is the second nearest-neighbour distance in a face centred cubic lattice whose conventional unit cell parameter is a?

(a)

(b) a/2

(d)

Ans. (c)

Sol. For FCC lattice:

6 atoms are at six faces.

Nearest neighbour distance,

Second nearest neighbour is on the opposite face which is at a distance 'a'

2.20.Magnetic long range order is typically exhibited by

(a) noble metals

(b) alkali metals

(d) transition metals

Ans. (d)

Sol. Long range magnetic ordering is a consquence of spin polarisation.

Spin only magnetic moments are usually observed for the first row transition metal ions such as Sc³⁺, Ti⁴⁺, Cr²⁺, Mn³⁺, Fe³⁺, Co³⁺, Co²⁺, Ni²⁺, Cu²⁺ etc.

Hence, transition metal will exhibit long range magnetic order.

2.21.X-rays with a wave vector are scattered from a simple cubic lattice with lattice spacing a = . The scattered X-rays have wave vector . The possible values of = K_x – K'_x for which there are peaks in the scattered intensity are:

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

From Bragg's law analysis, we know = G

where, = change in wave vector

G = Reciprocal lattice vector.

2.22.Under the LS coupling scheme, the possible spectral terms ^2s+1L_J for the electronic configuration 2s3s are

(a) ²S_1/2, ²P_3/2, ²P_1/2

(b) ¹S₀, ³P₁

(d) ³S₀, ³S₁

Ans. (c)

Sol. 2s 3s

s = |s₁ – s₂|......|s₁ + s₂| = 0, 1

Therefore, the possible spectral terms are

¹s₀, ³s₁

2.23.Which of the following is the spectroscopic ground state (^2s+1L_J) (for Mn³⁺ ions of electronic configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁴ predicted by Hund's rule?

(a) ⁵D₀

(b) ⁵D₄

(d) ⁵D₂

Ans. (a)

Sol.

Since, orbit is less than half filled, the J value of the ground state is

j = j_min = 0

Therefore, the ground state ^2S+1T_J = ⁵D₀

2.24.The Bohr model gives the value for the ionisation potential of Li²⁺ ion as

(a) 13.6 eV

(b) 27.2 eV

(d) 122.4 eV

Ans. (d)

Sol. According Bohr model, the energy of the hydrogen like atom is given by

Therefore, the ground state energy of Li⁺²(z = 3) is

E₁ = –(3)² × (13.6)eV = – 122.4 eV

Therefore, the value of the ionization energy of Li²⁺ ion is 122.4 eV

2.25.An admissible potential between the proton and the neutron in a deutron is

(a) Coulomb

(b) Harmonic oscillator

(d) Infinite square well.

Ans. (d)

Sol. Since, deuteron has 1 proton and 1 neutron in it. The interaction between neutron and proton cannot be electrical i.e. of Coulombic type because neutron has no charge; also not of gravitational type because masses of neutron and proton are very small and also not of magnetic type as magnetic moments are very small.

Here, we must accept that there exist nuclear force in the deutron that holds neutron and proton together in nucleus. This force is short range, attractive and acts along the line joining the two particles. This force can be derived from a potential. Since the force is attractive, therefore, V(r) will be negative and decreases with decreasing r.

Since it is short range

Therefore, potential between proton and neutron in deutron is of infinite square well type.

2.26.The deutron is known to be in a state with S = 1, J = 1, I= 0 where S, J, I refer to spin, total angular momentum and isospin quantum numbers respectively. The allowed values of the orbital angular momentum quantum number L are

(a) 0

(b) 0, 1, 2

(d) 0, 2

Ans. (b)

Sol. We know that

J = |L – S| to |L + S| with +1 gap

Given: S = 1 and J = 1

Therefore, the allowed value of L are 0, 1, 2

2.27.Identify the reaction which has the same transition probability as

(a)

(b)

(c)

(d)

Ans. (b)

Sol. In case of mesons are scattered more strongly in the backward direction than in the forward direction. This type of transition probability exists in option (b) , where meson are scattered more strongly in backward direction than in forward direction.

While in others, [(a) (c) , (d) ] strong scattering occurs in forward direction.

2.28.In the circuit shown below, R_B = 1 k and R_C = 100. If the transistor β(h_FE) is 100, the current through R_C will be

(a)

(b)

(d) oscillating between 0 and 50 mA.

Ans. (a)

Sol. From figure, we can write,

V_CC = I_CR_C + V_CE and V_CC = I_BR_B + V_BE

2.29.The output of the circuit on the right will be

(a) 1V

(b) 11V

(d) 0V

Ans. (a)

Sol. from virtual ground concept

V_out = V_–; V_out = 1V

Note: Voltage drop across both resistor is zero.

Since, no current in op-amp.

Note: Circuit is voltage follower

2.30.The circuit in the figure on the right shows a constant current source charging in a capacitor. The initial voltage across the capacitor is V₀. The switch is closed at time t = 0. The voltage across the capacitor is best described by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. A very small time interval (dt) after switch on the change of voltage across the capacitor is

Answer any FIFTEEN questions. Each question carries 5 marks.

3.Evaluate the following integral by using the method of residues:

Sol.

Residue of f(z) at z = ia is

4.The height of a hill at a point (x, y) in metres is given by

h(x, y) = exp [2xy – x² – 2y² – 4x + 8y +1]

where x and y are in km with respect to a certain origin.

(a) where is the top of the hill located?

(b) what is the unit vector in the direction of steepest ascent at the origin?

Sol. h(x, y) = exp(2xy – x² – 2y² – 4x + 8y + 1)

At the top of the hill

On solving (i) and (ii),

x = 0, y = 2

Hence, top of hill is located at (0, 2).

Therefore, unit vector along the direction of steepest ascent is

5.A particle of mass m is constrained to move on the surface of a sphere of radius R. The sphere is resting on ground.

(a) Set up the Lagrangian for the particle by clearly identifying the kinetic energy and potential energy

(b) What are the constants of the motion for the particle?

Sol.

Here, (x, y, z) are Cartesian coordinates

In this case, r = R,

Lagrangian of the particle,

Therefore, the momentum conjugate to is conserved (constant of motion)

Here, energy is also a constant of motion since Lagrangian does not explicitly depend on time.

6.A particle of mass m and velocity collides elastically with another particle of the same mass at rest in laboratory frame. The scattering angle in the centre of mass frame is found to 90º. Find the velocities of the scattered particles in the centre of mass frame and the laboratory frame.

Sol.

Before collision:

Velocities in centre of mass frame,

Velocities in lab frame,

7.A pion (rest mass m₀ = 135 MeV/c²) is moving with a velocity . If it decays by emitting two photons both of which move in z-direction, find their energies and frequencies in units of MeV and MeV/h respectively.

Sol. Given: Rest mass of pion

In lab frame, total energy of pion,

Two photons (with momentum p₁ and p₂) are emitted along the pion's momentum direction, with one going forward and the other backwards.

Using conservation of linear momentum,

Using E = pc for each photon,

… (i)

Using conservation of energy,

E = 225 MeV = E₁ + E₂

… (ii)

Solving equation (i) and (ii),

E₁ = 202.5 MeV

E₂ = 22.5 MeV

8.Let and denote respectively the ground state and second excited state energy eigenfunctions of a particle moving in harmonic oscillator potential with frequency . If at time t = 0 the particle has the wavefunction.

(a) Find

(b) Determine the expectation value of the energy as a function of time

Sol. (a) We have wave function at t = 0 is given by

Therefore, the wave function at t > 0 can be written as

9.Let be the unperturbed Hamiltonian and

be a small perturbation. Determine the first order correction to the ground state energy. What is the condition on for the first order perturbation theory to be valid? The ground state wave function is given as

Sol. We have kperturbed potential,

The ground state wave function of 1-D SHM where,

The first order energy correction to the ground state

First order perturbation will be valid if is much less than one, << 1.

10.A helium-neon laser beam ( = 632 nm) of intensity 1.0 W/cm² is travelling along x-axis in vacuum. Find:

(a) Amplitudes of E and B fields associated with the laser beam

(b) Expression for and if the E-field is polarized along direction.

Sol. We know intensity of electromagnetic wave is given,

According to Maxwell's equation,

Therefore, the electric field associated with the laser light is given by

And magnetic field,

11.Two wires of the same cross sectional area and electrical conductivity and are connected as shown in the figure. If a constant current I is made to pass through the wires, find the induced charge density at the junction between the two wires.

Sol. We know the current density,

According to boundary condition,

12.The ends of a circular coil of radius r(=2 cm) and N(=100) turns are connected to a resistor. the coil is placed such that a uniform magnetic field of strength 1.0 Tesla is perpendicular to the plane of the coil. If the coil is rotated by 180º, find the charge that will flow through the resistor.

Sol. Consider sec time will take to rotate the ring by an angle 180º.

Therefore, the change of flux is

Therefore, the current passing through the resistance,

Therefore, the charge will flow through the resistance,

13.Consider the stellar atmosphere consisting of hydrogen atoms, which is in thermal equilibrium. Let the average kinetic energy of hydrogen atoms be 1.0 eV. Find,

(a) the temperature of the atmosphere

(b) the ratio of the number of atoms in the second excited state to those in the ground state

Sol. (a) The average kinetic energy of hydrogen atoms is

Temperature of the atmosphere

(b) The required ratio is

14.Consider a system with ground state energy E₀ and an excited state with energy E₁. Determine the partition function and internal energy of the at a temperature T. Also, find the specific heat of the system in the limit T 0.

Sol. The partition function is

The internal energy of the system is

where, E₀ < E₁

At low temperature, is very small.

15.Consider the two branches of a phonon spectrum of a cubic lattice, and . In the Debye approximation give the phonon dispersion relations and the density of phonon levels for each branch.

Sol.

Debye approximation low temperature (long wavelength region)

Long wavelength small k

For first branch:

For second branch:

16.The band structure of an electron in a one-dimensional periodic potential is given by E₁(k) = A[1 – cos(k)] and E₂(k) = B. Find the elective mass of the electron for each band (for ). Mention which branch could contribute to electrical conduction.

Sol. E₁(k) = A[1 – cosk]; E₂(k) = B

17.The first two energies and spin particles for Er¹⁶⁶ is shown in the figure. Using the rotational model, determine the spin, parity and energy of the next two levels. Also, determine the moment of inertia of the nucleus in the units (h²/keV)

Sol. The energy of the Er¹⁶⁶ according to rotational model

and the allowed value of

J = 0, 2, 4, 6, 8 .............

Given: E₂ = 80.85 KeV

18.For the processes given below identify those that are allowed and those disallowed. State the intersection responsible for the former and the symmetry that forbids the latter.

(1)

(2)

(3)

(4)

(5)

Sol.

Therefore, it is allowed reaction.

Therefore, the section is forbidden

Therefore, the reaction is allowed.

19.Draw the circuit diagram of an inverting amplifier of gain –10 and input impedance 10 . The circuit should work for dc as well as ac signals. You are allowed to use only one op-amp and two resistors. If the circuit should amplify signals in the range –1.6 V to +1.6 V, what should be the op-amp supply voltages?

Sol. We have gain, A = –10. Since, A_OL is not given, A_OL

Virtual ground concept is valid.

Since, output voltage can't greater than supply voltage. So, supply voltage must be greater than the +1.6 × (–10) = +16V

20.The circuit on the right is given a 3V amplitude triangle wave input. The switch is open to begin with. What are the maximum and minimum output voltages? Sketch the output voltage vs. time, for two cycles, marking the voltage axis clearly.

Also sketch the output voltage when the switch is closed.

Sol.

If switch is closed and V_in > 0

If V_in < –1.2

21.The moment of inertia of HBr molecule is 3.30 × 10^–47 kg m². Find the wave number (in cm^–1) of absorption lines involving transitions between the rotational ground state and the first two excited states of the molecule under electric dipole approximation.

Sol. The rotational energy levels of HBr molecule can be expressed as

= 0.016 × 10⁵ m^–1 = 16 cm^–1

And due to transition from ground state to second excited state

22.Consider Zeeman effect in alkali metal spectra:

(a) Sketch the Zeeman split components of the terms ²P_3/2 and ²S_1/2 and find the energy difference in units of µ_BB between each Zeeman component and the unperturbed position of the term. The Lade g factor for ²P_3/2 and ²S_1/2 is 4/3 and 2 respectively. Here µ_B and B are Bohr magneton and magnetic field respectively.

(b) How many separate lines occur in the multiple arising from ²P_3/2²S_1/2 transition in the presence of weak magnetic field?