Physics UPSC PYQ 2024

Answer the following: Briefly discuss the Kepler's laws of planetary motion. [5M] Show that the escape velocity Ve on the surface of the Earth is given by Ve = √2 g R, where g = 9·8 m s^−2 and R is the radius of the Earth. [5M] Two satellites A and B of same mass are orbiting the Earth at altitudes R and 5 R, respectively, where R is the radius of the Earth. Assuming their orbits to be circular, calculate the ratios of their kinetic and potential energies. [5M] <!--qid:MAINS_2024_Physics-I_Q2-->

(a) Prove that: (i) [L², Lz] = 0 (ii) [Lz, L+] = ħL+ (iii) [L+, L–] = 2ħLz (iv) L+ Ly = L² – Lz² + ħLz where ħ = h / 2π (h is Planck’s constant). (b) The ground-state wave function of a harmonic oscillator is ψ₀(x) = (mω / πħ)^{1/4} exp(–mωx² / 2ħ). (i) At which point is the probability density maximum? (ii) What is the value of the maximum probability density? (c) (i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid-wall potential of length 10⁻¹⁵ m, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of a neutron bound within the nucleus as described above. Can an electron be confined in a nucleus? Explain. <!--qid:MAINS_2024_Physics-II_Q2-->

(a) A particle limited to the x-axis has the wave function φ(x) = bx² between x = 0 and x = 2; the wave function φ(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1·0 and x = 1·5. (ii) Find the expectation value < x > of the particle position. (b) Show that the square of the orbital angular momentum operator (L²) commutes with any of the components of angular momentum operator L. Is it possible to measure L², Lx, Ly and Lz simultaneously? Give reasons for your answer. (c) How is Rydberg constant related to emission wavelength of hydrogen spectrum? (d) Explain how the hydrogen spectrum is used for imaging the universe. (e) Find the energy of the particle of mass m moving in a potential field V(x) = 2ħ²b²x² / m for which the time-independent wave function is ψ(x) = exp(–bx²). Here b is a constant. <!--qid:MAINS_2024_Physics-II_Q1-->

Answer any three of the following: <!--qid:MAINS_2024_Physics-I_Q1-->

Answer the following: Write down the system (transfer) matrix for a combination of two thin lenses in paraxial approximation. Hence obtain the focal length of the combination and the positions of unit planes. [10M] Consider a thin-lens combination of two convex lenses of focal lengths f1 = +10 cm and f2 = +20 cm, respectively, kept separated by 25 cm. Determine the focal length of the combination and the positions of unit planes. [10M] Consider three inertial frames O, O′ and O″. Frame O′ moves with velocity V relative to O and frame O″ moves with velocity V′ relative to O′, both velocities being along the same direction. Write down the transformation relations connecting the coordinates x, y, z, t with x′, y′, z′, t′ and those connecting x′, y′, z′, t′ with x″, y″, z″, t″. (Assume the velocities are along the x-axis.) [10M] <!--qid:MAINS_2024_Physics-I_Q4-->

(i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume that sodium has one free electron per atom and its density is 0·97 gm cm⁻³. (ii) Draw the energy level diagram and write the mathematical expressions for the following: I. En of an electron confined in a one-dimensional box, II. Linear harmonic oscillator. Make a qualitative comparison of the above two cases. Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume sodium has one free electron per atom and its density is 0·97 gm cm⁻³. [10M] Draw the energy level diagram and give mathematical expressions for (I) En of an electron confined in a one-dimensional box, and (II) a linear harmonic oscillator. Make a qualitative comparison of these two cases. [10M] <!--qid:MAINS_2024_Physics-II_Q4-->

(a) How do Stokes lines appear in Raman spectrum as per classical and quantum theory of Raman effect? (b) What is Lamb shift in the fine structure of hydrogen spectrum? Discuss its theory based upon second quantization. (c) Describe Electron Paramagnetic Resonance. Highlight its differences with NMR and discuss its applications. <!--qid:MAINS_2024_Physics-II_Q3-->

Answer the following: Explain the phenomenon of double refraction. What are positive and negative crystals? Give their examples. [5M] What do you understand by optical activity? A linearly polarized light is propagating along the optic axis of a quartz crystal of thickness 0·2 cm. If the difference in the refractive indices corresponding to right circularly polarized and left circularly polarized beams is 7 × 10^−5 and the wavelength of the light is 0·5 μm, calculate the angle of polarization. [10M] What do you understand by attenuation in optical fibres? What are the factors responsible for the attenuation? [5M] Consider a 10 mW laser beam passing through a 50 km fibre link of attenuation 0·5 dB km^−1. Calculate the power of the laser at the end of the link. [10M] State and explain Hooke's law of elasticity. Briefly discuss the features of the stress–strain diagram for the behaviour of a wire undergoing increasing stress. [10M] Explain the Poiseuille's equation for the rate of flow of a liquid through a capillary tube. From this, show that if two capillary tubes of radii r1 and r2 having lengths l1 and l2, respectively, are connected in series, the rate of flow of the liquid is given by Q = (π P)/(8 η) ( l1/r1^4 + l2/r2^4 )^−1, where P is the pressure across the arrangement and η is the coefficient of viscosity of the liquid. [10M] <!--qid:MAINS_2024_Physics-I_Q3-->