UPSC Physics 2022 — Paper I

This question demands solving five distinct 10-mark problems covering classical mechanics and special relativity. The answer should present each solution as a self-contained unit with clear problem identification, step-by-step working, and final boxed answers. Begin with the orbital mechanics problem (Kepler's first law derivation), followed by the SHM energy theorem, then the Doppler effect calculation, the relativistic kinetic energy approximation, and finally the quarter-wave plate wavelength determination. Conclude with a brief synthesis noting the unifying theme of oscillatory and wave phenomena across the problems.

• For the electron orbit: Apply Newton's second law with Coulomb force, use polar coordinates, and derive the orbit equation r(θ) = l/(1+εcosθ) showing ε<1 for bound states
• For SHM energies: Derive ⟨T⟩ and ⟨V⟩ over one complete period using x = Asin(ωt+φ), showing both equal ¼kA² or ¼mω²A²
• For Doppler effect: Identify this as a moving source problem (observer stationary), apply f' = f[v/(v±vs)] with proper sign convention for approaching/receding train
• For relativistic KE: Expand γ = (1-v²/c²)^(-½) using binomial theorem, keeping terms up to v²/c² to recover classical result
• For quarter-wave plate: Use condition (μe-μo)t = λ/4, solve for λ = 4(μe-μo)t, and verify result falls in visible range (400-700 nm)

Begin with a brief statement of the rotating frame transformation equations and collision kinematics principles. Allocate approximately 30% time to Part 1 (rotating frame forces with explicit vector calculations), 30% to Part 2 (collision geometry derivation with momentum conservation), and 40% to Part 3 (thick lens matrix method leading to thin lens limit). Present derivations step-by-step with clear identification of final expressions, and conclude with a verification check for the thin lens limit.

• Part 1: Correct identification that Euler's force is zero for uniform ω, and calculation of Coriolis force (-2mω⃗×v⃗') and centrifugal force (-mω⃗×(ω⃗×r⃗)) with proper vector cross products
• Part 1: Explicit computation showing v⃗' = v⃗ - ω⃗×r⃗ = 14aᵧ - 12aᵧ = 2aᵧ m/s, yielding Coriolis force = -12a_z N and centrifugal force = 84a_x + 48a_y N
• Part 2: Application of momentum conservation in x and y directions and kinetic energy conservation (elastic collision) to establish tanθ = sin(2φ)/(m₁/m₂ + cos(2φ))
• Part 2: Derivation of maximum θ condition leading to sinθ_max = m₂/m₁ when m₁ > m₂, with physical constraint that scattering is impossible if m₁ < m₂
• Part 3: Construction of system matrix M = R₂T(d)R₁ for thick lens with two refractions and translation, using refraction matrix R = [[1, 0], [(n₁-n₂)/n₂R, n₁/n₂]] and translation matrix T = [[1, d/n], [0, 1]]
• Part 3: Reduction to thin lens formula 1/f = (n-1)(1/R₁ - 1/R₂) by taking d→0 limit and identifying focal length from matrix element C = -1/f

Derive requires systematic mathematical derivation with logical progression. Allocate approximately 40% time to part (a) for the moment of inertia tensor derivation using appropriate coordinate system and integration; 30% to part (b) for Newton's rings calculation with proper formula application and error propagation; 30% to part (c) for explaining He-Ne laser mechanism with energy level diagram. Structure: begin with coordinate setup for (a), proceed through integration, then solve (b) step-by-step showing diameter calculations, conclude with laser physics explanation and diagram.

• Part (a): Set up coordinate system with origin at apex or centroid, establish limits for triangular pyramid geometry, compute I_xx, I_yy, I_zz and off-diagonal elements using volume integral ∫ρ(r)(r²δ_ij - x_i x_j)dV
• Part (a): Apply parallel axis theorem if needed and present final 3×3 symmetric inertia tensor matrix with elements in terms of M and a
• Part (b): Use modified Newton's rings formula for bright rings D_n² = 4(n-½)λR, set up simultaneous equations using n=4 and n=15 data to find λ and verify consistency
• Part (b): Calculate D_24 and D_36 using derived λ, showing propagation of significant figures and physical reasonableness check (D_n ∝ √n)
• Part (c): Explain He as buffer gas providing efficient excitation transfer via resonant collision to Ne 2s and 3s levels, preventing depopulation of lower levels
• Part (c): Draw accurate energy level diagram showing He 2³S, 2¹S metastable states, collisional transfer to Ne 3s₂, 2s₂ upper laser levels, and transitions for 632.8 nm (3s₂→2p₄) and 1.15 μm, 3.39 μm lines with radiative decay to 1s ground state

Begin with a concise introduction linking the three phenomena—fluid dynamics, wave optics, and relativistic kinematics—as exemplars of classical and modern physics. Allocate approximately 30% of effort to part (a) deriving the Venturi flow rate, 40% to part (b) discussing Fraunhofer diffraction with intensity derivation, and 30% to part (c) solving the relativistic velocity transformation. For (b), note that 'discuss' requires qualitative explanation before the mathematical proof, while (c) demands explicit calculation with proper velocity addition formula.

• Part (a): Apply Bernoulli's equation and continuity equation between the two cross-sections, incorporate the manometer height difference h = (p₁-p₂)/ρg, and derive Q = A₁A₂√(2gh/(A₁²-A₂²))
• Part (b): Explain Fraunhofer diffraction conditions (plane wave, distant screen/lens), derive intensity distribution I(θ) = I₀(sinβ/β)² where β = (πa sinθ)/λ, locate secondary maxima by tanβ = β approximation, and prove the intensity ratio 1 : 4/9π² : 4/25π² : 4/49π²
• Part (c): Apply Einstein velocity addition formula u' = (u-v)/(1-uv/c²), set up equations with v = ±V (Earth frame), relative speed 0.7c, and solve quadratic to obtain V ≈ 0.41c for each spaceship
• Explicit statement of assumptions: incompressible, steady, irrotational flow for (a); far-field approximation, monochromatic source for (b); inertial frames, isotropic c for (c)
• Dimensional verification of final expressions and physical reasonableness checks (e.g., Q→0 as h→0; intensity maxima decrease; V < c always)

This is a multi-part calculation-based question requiring precise derivations and numerical solutions. Allocate approximately 20% time to each sub-part (a)-(e), with slightly more attention to (b) for its entropy proof and (e) for the missing diagram interpretation. Begin each part with the relevant fundamental equation, show complete derivation steps, substitute values with units, and conclude with physical interpretation of results.

• (a) Apply barometric formula/ Boltzmann distribution: ρ(h) = ρ₀exp(-mgh/kBT) or n(h) = n₀exp(-Mgh/RT); solve for h when ρ/ρ₀ = 1/2 using m = 28×10⁻³/NA kg
• (b) Calculate entropy change of body: ΔS_body = Cₚln(Tf/Ti); entropy change of reservoir: ΔS_res = -Cp(Tf-Ti)/Tf; total ΔS = Cₚ[ln(Tf/Ti) - (Tf-Ti)/Tf]; expand ln(1+x) for |Tf-Ti|/Tf < 1 to prove positivity
• (c) Use Biot-Savart law for moving charge: B = (μ₀/4π)qv×r̂/r²; magnetic force F = qv×B; derive F = (μ₀q²v²)/(4πd²) with direction along line joining charges (attractive for parallel currents)
• (d) Apply Kirchhoff's laws: set up junction equation I = I₁ + I₂ and loop equations; solve simultaneous equations for I₁, I₂ and I through 5Ω resistor
• (e) Calculate Q-factor = (1/R)√(L/C) or ω₀L/R; identify whether series or parallel RLC configuration; discuss resonant frequency ω₀ = 1/√(LC) existence based on circuit topology

Derive the required expressions systematically across all four sub-parts, allocating approximately 30% time to part (a) for Maxwell's equations and wave derivations, 20% each to (b)(i) and (b)(ii) for thermodynamic derivations and physical explanations, and 30% to part (c) for the internal energy differential. Begin with stating fundamental equations, proceed through rigorous mathematical steps, and conclude with physical interpretations of results.

• Part (a): Write all four Maxwell's equations in source-free non-conducting medium; derive wave equations for E⃗ and B⃗ using vector identities; obtain wave velocity c = 1/√(με); write plane wave solutions E⃗ = E⃗₀exp[i(k⃗·r⃗ - ωt)] and B⃗ = B⃗₀exp[i(k⃗·r⃗ - ωt)]; show E⃗ and B⃗ are mutually perpendicular and perpendicular to propagation direction with |E|/|B| = c
• Part (b)(i): Apply first law and definition Cv = (∂u/∂T)v to get Cv = c; use thermodynamic identity Cp - Cv = T(∂p/∂T)v(∂v/∂T)p with van der Waals equation to derive Cp = c + R/(1 - 2a(v-b)²/RTv³) or appropriate approximation
• Part (b)(ii): Explain that for adiabatic compression T₂/T₁ = (P₂/P₁)^(γ-1)/γ; since γ_He = 5/3 > γ_Air = 7/5, helium has higher temperature rise for same pressure ratio; compressor designed for air overheats with helium due to greater heating
• Part (c): Apply fundamental relation dU = Cv dT + [T(∂p/∂T)v - p]dV; compute (∂p/∂T)v = a/(2T^(1/2)) + 3bT²; substitute to get dU = (d+eT²V+f)T^(1/2)dT + [(a/(2T^(1/2)) + 3bT²)T - aT^(1/2) - bT³ - cV^(-2)]dV and simplify

Solve this multi-part numerical problem by allocating approximately 40% of effort to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin with clear identification of governing equations for each sub-part, proceed through systematic derivation with proper unit handling, and conclude with boxed final answers and brief physical explanations. For (a), explicitly state when maximum emf occurs; for (b), apply energy conservation and ideal gas law; for (c), use magnetic boundary conditions with proper vector notation.

• Part (a): Apply motional emf formula ε = Blv using instantaneous velocity of vibrating string element; recognize velocity is maximum at equilibrium position (y=0) giving maximum emf; calculate second harmonic emf noting frequency doubles but amplitude typically reduces
• Part (a): Correctly integrate over string length with sinusoidal amplitude profile y(x,t) = y_max sin(πx/L)cos(ωt), finding v_max = ωy_max at antinode; emf_max = Bωy_maxL/π for fundamental
• Part (b): Apply first law of thermodynamics for isolated system (ΔU=0); use n₁CvΔT₁ + n₂CvΔT₂ = 0 for temperature equilibrium; apply ideal gas law with total volume constraint for final pressure
• Part (b): Recognize piston is heat-conducting so final temperatures equalize, and frictionless mechanical equilibrium gives equal final pressures; solve simultaneous equations for p_f and T_f
• Part (c): Apply magnetic boundary conditions: normal B continuous (B₁z = B₂z), tangential H discontinuous by surface current (aₙ × (H₂ - H₁) = K); correctly handle vector cross product with K = 9.0 a_y
• Part (c): Calculate H₁x and H₁z components separately using μ₁H₁ = μ₂H₂ for normal component and H₂x - H₁x = K for tangential; then find B₁ = μ₀μ_r₁H₁

Begin with a brief introduction acknowledging the three distinct domains: electromagnetic wave propagation in lossy media, statistical mechanics of negative temperature, and electrostatic boundary value problems. Allocate approximately 40% of effort to part (a) given its 20 marks and computational demands; 30% each to parts (b) and (c). For (a), systematically calculate loss tangent, then α, β, η, and u before deriving H(z,t). For (b), define negative temperature and enumerate the three key restrictions (finite energy levels, thermal isolation, and population inversion). For (c), present the separation of variables derivation clearly with proper handling of the radial equation. Conclude with a synthesis noting how these topics span classical electrodynamics, statistical mechanics, and mathematical physics.

• Part (a): Calculate loss tangent tan δ = σ/(ωε) to classify the medium as good conductor or lossy dielectric, then correctly apply formulas for α, β, η and phase velocity u = ω/β
• Part (a): Derive H(z,t) using the intrinsic impedance relationship η = E/H with proper vector orientation (aₓ × aᵧ = aᵧ, accounting for the phase of η in complex form)
• Part (b): Define negative temperature as T < 0 occurring when (∂S/∂U) < 0, occurring in systems with upper energy bound, not 'colder than absolute zero'
• Part (b): Enumerate restrictions: (i) system must have finite number of energy levels, (ii) thermally isolated (no energy exchange with reservoir), (iii) population inversion required; cite examples like nuclear spin systems or laser media
• Part (c): Write Laplace's equation in cylindrical coordinates: (1/r)∂/∂r(r∂V/∂r) + (1/r²)∂²V/∂φ² + ∂²V/∂z² = 0
• Part (c): Apply separation V(r,φ,z) = R(r)Φ(φ)Z(z) to obtain three ODEs: Bessel's equation for R, harmonic equation for Φ, and exponential/trigonometric equation for Z with separation constants
• Part (c): Identify the physical significance: R(r) involves Bessel functions Jₙ and Yₙ (or modified Bessel Iₙ, Kₙ), Φ(φ) requires single-valuedness giving integer n, Z(z) depends on boundary conditions