Physics UPSC PYQ 2023

What are the consequences of Lorentz transformations on length and time when observed from a frame moving at relativistic velocities?

(i) Prove that the separation of two colliding particles is same, when observed in centre of mass and laboratory systems. (ii) Determine the kinetic energy of a thin disc of mass 0.5 kg and radius 0.2 m rotating with 100 rotations per second around the axis passing through its centre and perpendicular to its plane.

A beam of hydrogen atoms emitted from an oven at 400 k is sent through a Stern-Gerlach experiment having magnet of length 1 m and a gradient field of 10 tesla/m. Calculate the transverse deflection of an atom at the point where the beam leaves the magnet.

(i) What is the difference between Fresnel diffraction and Fraunhofer diffraction ? (ii) What is resolving power of a telescope ? Why is the resolving power of microscope more with UV light than with visible light ?

Consider the potential V(x) = { (0, 0 < x < a), (infinity, elsewhere) }(a) Estimate the energies of the ground state as well as those of the first and the second excited states for(i) an electron enclosed in a box of size a = 10^-10 m.(ii) a 1 g metallic sphere which is moving in a box of size a = 10 cm.(b) Discuss the importance of the Quantum effects for both of these systems.(c) Estimate the velocities of the electron and the metallic sphere using uncertainty principle.

What is Lande's g factor ? Evaluate the Lande's g factor for the ^3P_1 level in the 2p3s configuration of the ^6C atom. Also calculate the splitting of the level when the atom is placed in an external magnetic field of 0.1 tesla.

What are three and four level pumping schemes ? Explain the lasing action in these with schematic diagrams.

By assuming the nucleus as a cubical box of length equal to the nuclear diameter 10^-12 cm, calculate the kinetic energy of the highest level occupied nucleon of iron-56 nucleus.

An operator P describing the interaction of two spin 1/2 particles is P = a + b vec(sigma)_1 . vec(sigma)_2, where a and b are constants, and vec(sigma)_1 and vec(sigma)_2 are Pauli matrices of the two spins. The total spin angular momentum vec(S) = vec(S)_1 + vec(S)_2 = (1/2)h_bar (vec(sigma)_1 + vec(sigma)_2). Show that P, S^2 and S_z can be measured simultaneously.

(i) Derive the expressions for gravitational potentials at a point (I) outside the spherical shell, (II) inside the spherical shell. (ii) Calculate the escape velocity of a body of mass 10 kg from the surface of Moon (g Moon = 1/6 g Earth). Mass of Moon = 7.3 x 10^22 kg, Radius of Moon = 1.7 x 10^6 m

A neutral atom consists of a point nucleus +q surrounded by a uniformly charged spherical cloud (-q) of radius r. Show that when such an atom is placed in a weak external electric field E, the atomic polarizability of the atom is proportional to the volume of the sphere.

What is Raman effect ? Explain Quantum theory of Raman effect and Rotational Structure of a Raman spectrum.

(i) Determine the input and output impedances of the amplifier in given figure. The op-amp datasheet gives Z_in = 2 MOhm, Z_out = 75 Ohm and A_ol = 200,000 (open loop voltage gain).(ii) Find the closed-loop voltage gain.

Obtain condition for achromatism of two thin lenses separated by a finite distance. If the dispersive powers of the materials of the two lenses are 0.020 and 0.028, their focal lengths are 10 cm and 5 cm, respectively. Calculate the separation between them in order to form achromatic combination.

Derive an expression for the specific heat of a solid based on the Debye theory and show how it agrees with the experimental values. What is the most important assumption of Debye theory in comparison to Einstein theory ? Is there any drawback of Debye theory ?

Calculate the zero point energy for a particle in an infinite potential well for the following cases :(i) a 100 g ball confined on a 5 m long line.(ii) an oxygen atom confined to a 2x10^-10 m lattice.(iii) an electron confined to a 10^-10 m atom.Why zero point energy is not important for macroscopic objects ? Comment.

Two charges Q1 = 3 nC and Q2 = 4 nC are placed at the cartesian points (0, 2, 2) m and (0, -2, 4) m, respectively. The z = 0 plane is connected to the ground. Calculate the electric potential and the electric field at the point (3, 2, 4) m using the method of images.

Consider a stream of particles of mass m each moving in the positive x-direction with kinetic energy E towards the potential barrierV(x) = 0 for x <= 0V(x) = (3E)/4 for x > 0Find the fraction of particles reflected at x = 0.

Derive the expression for the inductance per unit length of two long parallel wires each of radius a, separated by distance d from their axes and carrying equal and opposite current I.

(i) What are the requisite conditions for observation of interference pattern on a screen ? (ii) Derive the expression for fringe width and intensity at a point on the screen in a double slit experiment.

A circular ring of radius R lying on the x-y plane and centred at the origin, carries a uniform line charge lambda. Find the first three terms (monopole, dipole and quadrupole) of the multipole expansion of potential V(r, theta).

How could you establish that nu_e and bar(nu)_e are two different particles ?

Evaluate the most probable distance of the electron of the hydrogen atom in its 2p state. What is the radial probability density at that distance ?

Write equation for damped harmonic oscillations and obtain expression for logarithmic decrement. In a damped harmonic motion, the first amplitude is 10 cm, which reduces to 2 cm after 50 oscillations, each of period 4 seconds. Determine the logarithmic decrement. Also, calculate the number of oscillations in which the amplitude decreases to 25%.

(i) Define Joule-Kelvin coefficient. Write it in its mathematical form. (ii) Determine the Joule-Kelvin coefficient for a van der Waals gas. Hence, obtain an expression for temperature of inversion. Discuss the conditions under which heating or cooling is produced.

Establish the Rutherford's scattering cross section formula for alpha-particle by considering the standard assumptions and symbols.

Write down the expressions for the Fermi-Dirac distribution and the Bose-Einstein distribution. Plot the distributions as a function of the energy.

Using Huygens' principle for a plane wave travelling from rarer medium 1 to a denser medium 2, show that (sin i) / (sin r) = v1 / v2 = mu2 / mu1, where i and r are the angles of incidence and refraction, respectively. v1, mu1 and v2, mu2 are the velocities and refractive indices in media 1 and 2, respectively.

Using Zeroth law of thermodynamics, introduce the concept of temperature. Explain how the isotherms of two different systems can be drawn.

A particle constrained to move along x-axis in the domain 0 <= x <= L has a wave function psi(x) = sin((npi x)/L), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle.

If an atom is placed in a magnetic field of strength 0.1 weber/m^2, then calculate the rate of precession and torque on an electron with l = 3 in the atom.Given that the magnetic moment of the electron makes an angle of 30° with the magnetic field.

Find the energy stored in a system of four charges Q1 = 1 nC, Q2 = 2 nC, Q3 = 3 nC and Q4 = 4 nC placed at the cartesian coordinates R1(1, 1), R2(2, 1), R3(1, 4) and R4(2, 2), respectively. Assume free space.

What is superconductivity ? Explain Meissner effect. Why superconductors should be a diamagnetic material ?

Define streamline flow of a fluid. Using the equation of continuity for an isotropic fluid, find different components of total energy per unit volume.

What do you understand by nuclear forces ? Explain meson theory of exchange forces.

With a neat circuit diagram, explain the working of Wien-Bridge oscillator.

Two inductors having inductances L1 and L2 are connected in parallel. The inductors have a mutual inductance M. Derive the expression for the effective inductance. Assume the inductors have negligible resistances.

(i) The quantities of rotatory motion are analogous to those of translatory motion. Write the corresponding equations of translatory and rotatory motion. (ii) Describe the theorems of perpendicular and parallel axes in case of a plane lamina.

Name the interactions via which the above nuclear decays occur :(i) K^+ -> pi^+ + pi^+ + pi^-(ii) pi^+ + p -> pi^+ + pi^+ + n(iii) pi^+ + p -> Delta^++ -> pi^+ + p(iv) Sigma^0 -> lambda^0 + gamma(v) Sigma^+ -> lambda^0 + e^+ + nu_e(vi) K^- + p -> K^+ + K^0 + Omega^-(vii) pi^0 -> gamma + e^+ + e^-(viii) Sigma^- -> n + e^- + bar(nu)_e(ix) lambda^0 -> p + e^- + bar(nu)_e(x) e^+ + e^- -> gamma + gamma

Show that the Fermi level shifts upward, closer to the conduction band in an n-type semiconductor and shifts downward, closer to the valence band in a p-type semiconductor.

Derive diffraction conditions using reciprocal lattice concept. What are these conditions known as ?

A force F is given by F = x^2 y x-hat + zy^2 y-hat + xz^2 z-hat. Determine whether or not the force is conservative.

A spherical shell of radius R, carrying a uniform surface charge sigma, is set spinning at angular velocity omega about its axis. Find the vector potential it produces at point r.

What is vector atom model ? How the principal features of vector atom model were explained by Stern-Gerlach experiment ?

A particle of mass m is in a spherically symmetric attractive potential of radius a. Find the minimum depth of the potential needed to have two bound states of zero angular momentum.

Calculate the gravitational self-energy of the Earth. Given : Mass of Earth Me = 6 x 10^24 kg and the Radius of Earth Re = 6.4 x 10^6 m

Consider a particle of mass m and charge q moving under the influence of one dimensional harmonic oscillator potential. Assume it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = (p^2)/(2m) + (1/2)momega^2 X^2 - qEX. Obtain the energy expression and the wave function of the nth excited state of the particle.

Use the Maxwell-Boltzmann distribution to find the number of oxygen molecules whose velocities lie between 195 m/s and 205 m/s at 0°C. The given mass of oxygen gas is 0.1 kg. (Assume mass of proton to be 1.66 x 10^-27 kg)

Show that Continuity equation is embedded in Maxwell's equations.

Write conditions for working of a step-index optical fiber. In a step-index fiber, the core and cladding materials have refractive indices 1.50 and 1.43, respectively. Find the following : (i) Critical propagation angle (ii) Acceptance angle (iii) Total time delay in 1 km length of the fiber (iv) Total dispersion in 50 km length of the fiber

Consider the interaction of an electromagnetic wave at the interface of two dielectric media. If electric field E is parallel to the plane of incidence, obtain Fresnel's equations and Brewster's law of polarization.

A piston-cylinder device initially contains air at 150 kPa and 27°C. At this state, the piston is resting on a pair of stops, as shown in the figure, and the enclosed volume is 400 L. The mass of the piston is such that a 350 kPa pressure is required to move it. The air is now heated until the volume is doubled. Determine : (i) the final temperature, (ii) the work done by the air, and (iii) the total heat transferred to air. Given : U300K = 214 kJ/kg and Ufinal = 1113 kJ/kg. Gas constant of air, R = 0.287 kPa.m^3/kg.K

What is the age of a fossil that contains 6 g of carbon ^14C and has a decay rate of 27 decays per minute ?Given : Ratio (^14C)/(^12C) = 1.3 x 10^-13, Half life (T_(1/2)) of ^14C = 5730 yrs.

What do you understand by the critical size of a reactor ? Explain the main features of nuclear reactors.

What is nuclear magnetic resonance ? Explain its working principle and use in magnetic resonance imaging systems.

Explain classical theory of diamagnetism. Show that the susceptibility of diamagnetic substances is directly proportional to the atomic number. Why all the electrons in an atom contribute to diamagnetism ?