UPSC Physics 2021

Solve each sub-part sequentially with clear section headers. For (a), derive the central force law using Binet's equation; for (b), apply Poiseuille's law for viscous flow; for (c), use Lorentz transformation for time intervals; for (d), apply thin film interference conditions; for (e), derive the equivalent spring constant for series combination. Allocate approximately 2-2.5 minutes per mark, ensuring all five parts receive proportional attention with brief physical interpretation after each numerical result.

• (a) Apply Binet's equation for central force motion: d²u/dθ² + u = -F(1/u)/(mh²u²) where u=1/r, substitute r=ke^(αθ), and derive F ∝ 1/r³
• (b) Use Poiseuille's formula Q = πpr⁴/(8ηL) with hydrostatic pressure p = ρgh, calculate volume flow rate, then find total volume in 10 minutes
• (c) Apply Lorentz transformation: in frame S' moving with velocity v where Δx'=0, find γ and then calculate Δt' = γ(Δt - vΔx/c²)
• (d) Apply thin film interference condition for reflected light: 2μtcosr = (2n+1)λ/2 for destructive interference or 2μtcosr = nλ for constructive, using Snell's law to find r from given angle of incidence
• (e) Show springs in series give 1/k_eq = 1/k₁ + 1/k₂, derive equation of motion md²x/dt² = -k_eqx, and obtain the given frequency expression
• For (c), identify that second observer sees explosions at same place implies finding proper time interval using spacetime interval invariance

This question demands rigorous derivation and proof across three distinct physics domains. Begin with conservation laws for the explosion problem, apply static equilibrium and elasticity theory for the suspended rod, and conclude with wave optics derivation showing modulation of interference and diffraction patterns. Structure as: (a) momentum-energy derivation with clear algebraic steps, (b) force analysis with two cases for stress and strain conditions, (c) Fraunhofer setup with mathematical demonstration of intensity modulation and conditions for extrema.

• Conservation of linear momentum (initial zero momentum implies equal and opposite momenta of fragments) and expression of total kinetic energy in terms of reduced mass
• Static equilibrium conditions: sum of vertical forces equals weight, and torque balance about suspension point for both stress and strain cases
• Correct application of Young's modulus relation: stress = Y × strain, and proper unit conversion from CGS to SI or consistent handling of cm and m²
• Fraunhofer setup with plane wavefront, lens arrangement, and mathematical derivation showing I(θ) = I₀(sinβ/β)²cos²α where β and α relate to single slit and double slit parameters
• Clear identification that (sinβ/β)² represents single slit diffraction envelope modulating the cos²α interference term
• Conditions for maxima: dsinθ = nλ (interference) and bsinθ = mλ (diffraction minima), with missing orders explained

Begin with clear definitions and derivations for each sub-part, allocating approximately 40% time to part (a) given its 20 marks, 30% to part (b) for 15 marks, and 30% combined to both parts of (c). Structure as: (a) define pulse/material dispersion with formulas → numerical calculation; (b) define chromatic aberration → derive achromatism condition → discuss same-material limitation; (c)(i)-(ii) apply relativistic mass, momentum and energy formulas with collision dynamics. Include ray diagrams for (b) and state all assumptions explicitly.

• Part (a): Distinguish pulse dispersion (intermodal/intramodal) from material dispersion (wavelength-dependent refractive index); calculate material dispersion Δtₘ = Dₘ × L × Δλ and estimate pulse broadening using numerical aperture
• Part (a) numerical: Apply NA = √(n₁² - n₂²) for intermodal contribution, combine with material dispersion for total pulse dispersion at 1 km
• Part (b): Define chromatic aberration as focal length variation with wavelength; derive 1/f = 1/f₁ + 1/f₂ and ω/f₁ + ω'/f₂ = 0 for achromatism condition
• Part (b) analysis: Prove same-material doublet cannot be achromatic (ω/ω' = f₂/f₁ requires different dispersive powers), citing crown-flint glass combination
• Part (c)(i): Calculate relativistic mass m = γm₀, momentum p = γm₀v, then apply conservation of relativistic momentum and energy for inelastic collision to find final velocity
• Part (c)(ii): Use K = (γ-1)m₀c² and E = γm₀c² with K = E/2 to find γ = 2, hence v = √3c/2 ≈ 0.866c and relativistic mass = 2m₀

Begin with precise definitions for (a)(i), then rigorously prove the Parallel Axis theorem with clear mathematical steps. For (a)(ii), apply rolling motion kinetic energy formula (translational + rotational). In (b), derive gravitational potential using Earth's mass from density data. For (c), derive the damped harmonic oscillator equation, solve the characteristic equation, and classify damping regimes. Conclude with logarithmic decrement calculation. Allocate time proportionally: ~30% to (a)(i), ~10% to (a)(ii), ~20% to (b), and ~40% to (c) given its 20 marks.

• For (a)(i): Correct definition of moment of inertia as I = Σmᵢrᵢ² and radius of gyration as k where I = Mk²; complete proof of Parallel Axis theorem I = I_cm + Md² with clear diagram showing axis displacement
• For (a)(ii): Correct application of rolling kinetic energy E = ½Mv² + ½Iω² with I = (2/5)MR² for solid sphere and v = Rω condition, yielding E = (7/10)Mv² = 3.15 J
• For (b): Derivation of gravitational potential V = -GM/R using M = (4/3)πR³ρ, yielding V ≈ -6.25 × 10⁷ J/kg or equivalent calculation with proper sign convention
• For (c): Correct equation of motion m(d²x/dt²) + b(dx/dt) + kx = 0 or equivalent; general solution derivation with characteristic roots; classification of underdamped (ω' = √(ω₀²-γ²)), critically damped (γ = ω₀), and overdamped/dead beat (γ > ω₀) cases
• For (c) continued: Correct calculation of logarithmic decrement δ = (1/n)ln(A₀/Aₙ) = (1/20)ln(30/3) = 0.115 or equivalent using damping factor relations

Calculate requires precise numerical solutions with full derivations. Structure: begin with stating relevant formulas for each sub-part, show step-by-step calculations with proper unit handling, and conclude with final numerical answers. Allocate ~20% time each to (a), (c), (d), (e) which involve multi-step derivations, and ~20% to (b) which is direct combinatorics. For (e), explicitly construct the impedance expression before solving for resonant frequency.

• For (a): Apply E = -∇V in spherical coordinates, convert to Cartesian components at point (4,2,3) where r = √29, obtaining Ex, Ey, Ez values
• For (b): Apply Bose-Einstein statistics for indistinguishable particles: ways = (n+k-1)!/(n!(k-1)!) = 13!/(8!×5!) = 1287
• For (c): Derive motional EMF using ε = ∫(v×B)·dl with v = ωr, integrating from 0 to l to get ε = ½Bωl²
• For (d): Calculate critical constants using Vc = 3b, Tc = 8a/(27Rb), Pc = a/(27b²), then Boyle temperature TB = a/(Rb)
• For (e): Set up parallel LC circuit admittance Y = 1/(R+jωL) + jωC, find ω where Im(Y) = 0 for resonance

Solve this multi-part numerical and theoretical problem by allocating approximately 40% time to part (a) for relativistic field transformations, 20% to part (b) for LR circuit energy dissipation, and 40% to part (c) for deriving thermodynamic work expressions. Begin with clear statements of governing equations, show systematic derivations with proper vector decomposition for (a), apply energy conservation for (b), and present characteristic equations with P-V diagrams for each process in (c). Conclude with physical interpretations of results.

• Part (a): Apply Lorentz transformation for electromagnetic fields with E parallel and perpendicular components to velocity; calculate E' = γ(E∥ + v×B) and B' = γ(B⊥ - v×E/c²) with proper angle decomposition at α = 45°
• Part (a): Compute γ = 1/√(1-0.36) = 1.25; resolve E into E∥ = Ecosα and E⊥ = Esinα; find E' magnitude and induced B' field
• Part (b): Apply energy conservation in RL circuit; calculate total energy stored in inductor (½LI²) where I = E/R₀ at steady state; determine heat distribution between resistances
• Part (b): Recognize that when switch opens, stored magnetic energy dissipates through R and R₀; compute heat in coil resistance R specifically
• Part (c): State defining conditions for each process (dT=0, dQ=0, dP=0, dV=0) and derive work integrals: W_isothermal = nRTln(V₂/V₁), W_adiabatic = (P₁V₁-P₂V₂)/(γ-1), W_isobaric = PΔV, W_isochoric = 0
• Part (c): Include P-V diagram sketches showing hyperbolic isotherm, steeper adiabat, horizontal isobar, and vertical isochore with proper labeling
• Part (c): Relate γ = Cp/Cv and connect to degrees of freedom for perfect gas; mention relevance to Carnot cycle and Indian power plant thermodynamics

Calculate the required quantities systematically across all three parts, allocating approximately 40% of effort to part (a) given its 20 marks, 20% to part (b), and 40% to part (c). Begin each part by stating the relevant governing equations, show step-by-step derivations with proper unit handling, and conclude with physically meaningful interpretations of the numerical results.

• Part (a): Apply boundary conditions for dielectric interface—tangential E continuous (E₁ₜ = E₂ₜ) and normal D continuous (D₁ₙ = D₂ₙ) with σ = 0; decompose D⃗₁ into normal (z) and tangential (x-y) components
• Part (a): Calculate D⃗₂ using ε₂/ε₁ ratio for tangential components and equality for normal component; then find P⃗₂ = D⃗₂ − ε₀E⃗₂ = D⃗₂(1 − 1/εᵣ₂)
• Part (b): Use skin depth formula δ = √(2/ωμσ) = 1/√(πfμσ) for good conductor; substitute f = 10⁶ Hz, μ = 4π×10⁻⁷ H/m, σ = 5.8×10⁷ S/m
• Part (c): Apply Wien's displacement law λₘT = b to verify T ≈ 5772 K, then use Stefan-Boltzmann law P = σAT⁴ for total radiated power
• Part (c): Calculate mass loss rate using E = mc² equivalence (ṁ = P/c²) and time for 1% mass loss as Δt = 0.01M☉/ṁ
• Explicit unit conversions: μm to m, MHz to Hz, nC to C, and proper handling of SI prefixes throughout

This is a multi-part numerical problem demanding precise calculation across thermodynamics, statistical mechanics and electrostatics. Allocate approximately 30% time to part (a) covering entropy change and diesel cycle efficiency, 40% to part (b) on Fermi-Dirac distribution with its conceptual plots and probability calculation, and 30% to part (c) on vector superposition of electric fields from line and sheet charges. Begin each part with stated assumptions and relevant formulas, show step-by-step calculations with proper units, and conclude with physical interpretation of results.

• For (a)(i): Calculate entropy change in three stages—heating solid tin from 100°C to 232°C, phase change at 232°C, and heating liquid tin from 232°C to 300°C using ΔS = ∫dQ/T and ΔS = mL/T
• For (a)(ii): Apply diesel cycle efficiency formula η = 1 - (1/γ)[(ρ^γ - 1)/(r^(γ-1)(ρ - 1))] where r = compression ratio, ρ = cutoff ratio = r/r_expansion = 13.8/6
• For (b)(i): State Fermi-Dirac distribution f(E) = 1/[exp((E-E_F)/k_BT) + 1], sketch three curves showing step function at T=0 and thermal broadening at T₁ > T₂ > 0, define Fermi level as E where f(E)=0.5 or as chemical potential
• For (b)(ii): Calculate f(E_F + 0.02 eV) using given values, showing substitution of k_BT at 300K ≈ 0.0259 eV
• For (c): Calculate electric field from infinite line charge E_line = λ/(2πε₀r) perpendicular to line, and from sheet E_sheet = σ/(2ε₀) perpendicular to sheet, then vectorially sum at point (10,10,10)m with proper distance calculations

Calculate numerical values for all five sub-parts with systematic derivations. Allocate approximately 20% time each to parts (a), (b), (c), (d), and (e) as they carry equal marks. Begin each sub-part by stating the relevant formula, show substitution with proper units, and conclude with the final numerical answer rounded as specified. For part (a), explicitly state the resolution criterion; for (d), show both Larmor frequency and wave number splitting.

• Part (a): Apply the criterion that Zeeman splitting must exceed spectrometer resolution; use Δλ = (eλ²B)/(4πmₑc²) or equivalent relation with Δλ = 0.010 nm to find B_min ≈ 0.43 T → round to 1 T (nearest high integer)
• Part (b): Calculate ⟨V⟩ = ⟨-e²/r⟩ using ψ₁₀₀; evaluate ∫ψ*(-e²/r)ψ d³r = -e²/πa₀³ ∫e^(-2r/a₀)(1/r)4πr²dr = -e²/a₀; final answer -27.2 eV or -2e²/a₀
• Part (c): Apply WKB or exact formula for rectangular barrier: T ≈ exp[-2κL] where κ = √[2m(V₀-E)]/ℏ; with E=12 eV, V₀=25 eV, L=0.05 nm, compute κ and transmission coefficient ≈ 10⁻⁴ to 10⁻⁵
• Part (d): Calculate Larmor frequency ω_L = eB/2mₑ = 4.40×10¹⁰ rad/s (or ν_L = 7.0×10⁹ Hz); then Zeeman splitting Δṽ = eB/4πmₑc = 0.467 cm⁻¹ (normal triplet separation)
• Part (e): Use rigid rotor formula E_J = BJ(J+1) where first line is 2B = 21.18 cm⁻¹; find B = 10.59 cm⁻¹, then I = ℏ/4πcB; calculate reduced mass μ = m_H m_Cl/(m_H+m_Cl); finally r₀ = √(I/μ) ≈ 1.27 Å

Begin with explicit statement of Pauli spin matrices, then systematically prove all three identities in part (a) showing anti-commutation relations, ladder operator properties, and cyclic permutation; for part (b) apply Heisenberg uncertainty principle with proper unit conversions from cm to meters; for part (c) set up the infinite square well wavefunction, identify n=1 ground state, and integrate probability density over the specified interval at box center. Allocate approximately 40% time to part (a) [20 marks], 30% to part (b) [15 marks], and 30% to part (c) [15 marks], ensuring all numerical answers carry proper units and significant figures.

• Explicit definition of Pauli matrices σₓ, σᵧ, σᵤ with standard matrix forms and their anti-commutation relations {σᵢ,σⱼ}=2δᵢⱼ
• Proof of (a)(i): σₓσᵧ+σᵧσₓ=0 etc. by direct matrix multiplication showing off-diagonal cancellation
• Proof of (a)(ii): σ₊σ₋=2(1+σᵤ) using σ₊=σₓ+iσᵧ, σ₋=σₓ−iσᵧ and σₓ²=σᵧ²=I
• Proof of (a)(iii): σₐσᵦ=iσᵧ (cyclic) using commutation [σᵢ,σⱼ]=2iεᵢⱼₖσₖ and anti-commutation results
• Part (b): Δp≥ℏ/(2Δx) with Δx=0.02 cm=2×10⁻⁴ m; electron and α-particle velocity uncertainties using mₑ=9.11×10⁻³¹ kg, mₐ=6.64×10⁻²⁷ kg
• Part (c): ψₙ(x)=√(2/L)sin(nπx/L) for 0<x<L, L=50Å=5×10⁻⁹ m; probability P=∫|ψ₁|²dx from 17.5Å to 32.5Å using sin² integral identity

This is a calculation-heavy question demanding precise numerical work for (a) and (c), plus conceptual explanation for (b). Allocate approximately 40% time to part (a) for Raman shift calculations and wavelength conversions, 30% to part (b) for explaining spin-orbit coupling mechanism and H-atom fine structure, and 30% to part (c) for L-S and j-j coupling term calculations. Begin with clear statements of formulas, show all intermediate steps, and conclude with physical significance of results.

• Part (a): Correct application of Raman shift formula Δν̃ = (1/λ₀ - 1/λs) in m⁻¹; calculation of Stokes and anti-Stokes wavelengths for new exciting line using the same Raman shift
• Part (b): Explanation of spin-orbit coupling as interaction between electron's spin magnetic moment and orbital magnetic field; derivation of fine structure splitting in H-atom showing ΔE ∝ j(j+1) - l(l+1) - s(s+1)
• Part (c)(i): L-S coupling: L = |l₁ - l₂| to l₁ + l₂ = 2,3,4,5,6; S = 0,1; J values from |L-S| to L+S for each combination
• Part (c)(ii): j-j coupling: j₁ = l₁ ± ½ = 7/2, 9/2; j₂ = l₂ ± ½ = 3/2, 5/2; J from |j₁-j₂| to j₁+j₂
• Physical interpretation: Term symbols notation ²ˢ⁺¹L_J for L-S coupling; comparison of coupling schemes showing L-S dominates for light atoms, j-j for heavy atoms

Begin with the relativistic energy-momentum relation to derive the de Broglie wavelength formula in part (a), then transition to quantum mechanical probability calculations for the harmonic oscillator in part (b), and conclude with a systematic exposition of Zeeman effects and degeneracy lifting in part (c). Allocate approximately 30% effort to part (a) due to its dual derivation-cum-calculation demand, 30% to part (b) for its integration-heavy probability analysis, and 40% to part (c) given its higher mark weightage and conceptual depth requiring clear energy level diagrams.

• Part (a): Derivation using E² = p²c² + m₀²c⁴ with E = K + m₀c², leading to λ = hc/√[K(K+2m₀c²)]; calculation of electron wavelength at 2 MeV (≈ 0.0056 Å or similar); non-relativistic limit reduction to λ = h/√(2m₀K)
• Part (b): Classical turning points for ground state SHO at x = ±√(ℏ/mω); probability integral P = ∫|ψ₀|²dx between -x₀ to +x₀ using Gaussian integral; demonstration that P ≈ 0.84 inside, hence ~16% outside classical limits
• Part (c): Normal Zeeman effect (spin singlet, orbital only, triplet splitting) vs anomalous Zeeman effect (spin-orbit coupling, Paschen-Back transition); role of g-factor (Landé g-factor for anomalous case); lifting of l-degeneracy in hydrogen via m_l quantum number and selection rules Δm = 0, ±1
• Clear distinction between strong and weak field regimes for Zeeman effects, with mention of Indian physicist S. Panck or relevant experimental context where applicable
• Proper handling of units throughout (MeV, Å, natural constants) and dimensional consistency checks

Solve each sub-part systematically with clear physical reasoning. For (a), apply Bragg's law correctly with order considerations; for (b), derive the specific heat expression rigorously and apply low-T approximation; for (c), explain decay mechanisms using strong vs. weak interaction; for (d), identify hadron properties from quark content; for (e), define chain reaction concepts with criticality condition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e).

• For (a): Correct application of Bragg's law 2d sinθ = nλ to both cases, equating interplanar spacing d to find λ₁ = 0.168 nm or 1.68 Å
• For (b): Proper differentiation of U with respect to T to obtain C, and valid exponential approximation e^(ℏω/k_BT) >> 1 at low T showing C ∝ e^(-ℏω/k_BT)
• For (c): Explanation that ρ⁰ decays via strong interaction (resonance, ~10⁻²³s) while K⁰ decays via weak interaction (strangeness violation, ~10⁻¹⁰s), both to π⁺π⁻ final state
• For (d): Identification of (a) π⁺ (uđ), (b) π⁻ (ūd), (c) Ξ⁰ or neutron-like (dds = -1 charge, strangeness -1), (d) Ξ⁻ (uss: charge -1, strangeness -2, baryon number 1)
• For (e): Definition of chain reaction as self-sustaining neutron-induced fission sequence; critical size as minimum dimension where neutron multiplication factor k=1 (production = loss)

Begin with a brief introduction on nuclear structure studies in Indian context (Bhabha's contributions). For part (a), derive the Schrödinger solution using square well potential, showing binding energy ~2.2 MeV proves loose binding; allocate ~40% time. For (b), derive the oscillator spacing using nuclear radius relation R=r₀A^(1/3) and equating Fermi energy to ℏω; allocate ~30% time. For (c), enumerate three neutrino flavors with mass hierarchy and mention Indian experiments like INO; allocate ~30% time. Conclude with significance for nuclear astrophysics.

• Part (a): Importance of deuteron as only two-nucleon bound system, simplest test of nuclear forces; solution of radial Schrödinger equation for l=0 with finite square well; boundary conditions at r=R; obtain transcendental equation; show binding energy E_b≈2.224 MeV << typical nuclear energy scale (~8 MeV/nucleon) proving loose binding
• Part (a): Mention deuteron's large size (r_d≈4.3 fm), no excited bound states, electric quadrupole moment indicating non-spherical shape and tensor force importance
• Part (b): Derivation starting from 3D isotropic harmonic oscillator with Hamiltonian H=p²/2m + ½mω²r²; energy levels E_N=(N+3/2)ℏω where N=2(n-1)+l; major shells at N=0,1,2,3...
• Part (b): Relate ℏω to nuclear size using R=r₀A^(1/3) and Fermi momentum; equate ℏ²k_F²/2m to ℏω scaling; derive ℏω≈41A^(-1/3) MeV with r₀≈1.2 fm
• Part (c): Three types—electron neutrino (ν_e), muon neutrino (ν_μ), tau neutrino (ν_τ); mass differences from neutrino oscillation experiments; solar neutrino puzzle and KamLAND, Super-Kamiokande evidence
• Part (c): Mass hierarchy: normal (m₁<m₂<m₃) vs inverted; Δm²₂₁≈7.5×10⁻⁵ eV², |Δm²₃₁|≈2.5×10⁻³ eV²; mention upper limits (~eV scale) and cosmological constraints; Indian Neutrino Observatory relevance

Calculate requires systematic numerical problem-solving across all three sub-parts. Allocate approximately 40% of effort to part (a) as it carries 20 marks and involves DC biasing analysis with feedback; 35% to part (b) for quantum mechanical angular momentum calculations; and 25% to part (c) for the FET pinch-off voltage derivation. Begin each part with the relevant formula, show complete derivation with substituted values, and conclude with proper units and physical significance.

• Part (a): Correct application of KVL to base-collector feedback loop with Ic = βIb and proper handling of the 100 kΩ feedback resistor to establish quiescent operating point
• Part (a): Recognition that VBE = 0 simplifies analysis to finding IB from Vcc = IB·RB + IE·RL relationship with proper current relationships
• Part (b): Correct determination of L, S, J values for 3d¹ configuration of Scandium (L=2, S=½, J=3/2, 5/2) and selection of lowest energy state using Hund's third rule for less than half-filled shell
• Part (b): Accurate calculation of Landé g-factor using g = 1 + [J(J+1) + S(S+1) - L(L+1)]/[2J(J+1)] and subsequent Zeeman energy splitting ΔE = g·μB·B·MJ
• Part (c): Proper application of pinch-off voltage formula VP = (q·ND·a²)/(2ε) with correct unit conversion from cm to m and use of relative permittivity ε = εr·ε0
• Part (c): Recognition that channel width 2a = 6×10⁻⁴ cm gives a = 3×10⁻⁶ m for the depletion region calculation

Begin with the directive 'calculate' for part (b) which carries the highest marks (20), then address 'sketch' for part (a) (10 marks) and 'explain/discuss' for part (c) (20 marks). Allocate approximately 35% time to part (b) for rigorous derivation of magnetic susceptibility using Hund's rules and Curie law, 25% to part (a) for accurate load line construction with proper intercepts, and 40% to part (c) for comprehensive explanation of internal conversion with coefficient definition and conditions. Structure as: (a) circuit analysis → load line sketch; (b) quantum mechanical derivation → numerical substitution; (c) phenomenon explanation → coefficient definition → conditions discussion.

• Part (a): Correct identification of DC load line endpoints (cut-off and saturation points) from circuit parameters, proper axes labeling (V_CE vs I_C), and accurate linear plot showing Q-point placement
• Part (b): Application of Hund's rules to determine total angular momentum J for Nd³⁺ (4f³ configuration), calculation of Landé g-factor, derivation of effective magnetic moment, and correct substitution into Curie law formula χ = μ₀Nμ_eff²/(3k_BT) with proper unit handling
• Part (b): Correct numerical evaluation yielding χ ≈ 4.5 × 10⁻³ (dimensionless SI) or equivalent, showing all intermediate steps for μ_eff, μ_B conversion, and temperature dependence
• Part (c): Clear explanation of internal conversion as radiationless nuclear de-excitation with energy transfer to orbital electron, proper definition of α_IC = N_e/N_γ (conversion electrons to gamma photons ratio)
• Part (c): Discussion of conditions favoring internal conversion: high Z nuclei, low transition energy, large change in nuclear angular momentum (ΔL ≥ 2), and Mössbauer spectroscopy relevance for Indian nuclear physics research