Q8
(a) (i) In free space, an electric field ($\vec{E}$) is given by the following expression : $$\vec{E} = 10 \cos (\omega t - 100 x)\hat{j} \text{ V/m}$$ Find the angular frequency $\omega$ and the displacement current. (10 marks) (ii) An electromagnetic wave has its magnetic field $|\vec{B}| = 55 \times 10^{-8}$ T. Determine the magnitude of the Poynting vector. (5 marks) (b) Explain why, at equilibrium, the chemical potential of a component must be the same in all coexisting phases. Derive the equilibrium condition for a binary liquid-vapour system in terms of chemical potential. (15 marks) (c) Derive the Planck's radiation law for blackbody radiation using the Bose-Einstein distribution function. Explain how results from quantum statistics differ from classical results derived from the Rayleigh-Jeans law. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) (i) मुक्त आकाश में एक विद्युत-क्षेत्र ($\vec{E}$) निम्नलिखित व्यंजक द्वारा व्यक्त किया गया है : $$\vec{E} = 10 \cos (\omega t - 100 x)\hat{j} \text{ V/m}$$ कोणीय आवृत्ति $\omega$ और विस्थापन धारा ज्ञात कीजिए। (10 अंक) (ii) एक विद्युत-चुंबकीय तरंग का चुंबकीय क्षेत्र $|\vec{B}| = 55 \times 10^{-8}$ T है। पॉइंटिंग सदिश का परिमाण ज्ञात कीजिए। (5 अंक) (b) व्याख्या कीजिए कि क्यों, साम्यावस्था पर एक घटक का रासायनिक विभव सभी सहविद्यमान प्रावस्थाओं में एकसमान होना चाहिए। एक द्वयी द्रव-वाष्प निकाय के लिए रासायनिक विभव के पदों में साम्यावस्था प्रतिबंध की व्युत्पत्ति कीजिए। (15 अंक) (c) बोस-आइंस्टाइन बंटन फलन का प्रयोग करके कृष्णिका विकिरण के लिए प्लांक विकिरण नियम की व्युत्पत्ति कीजिए। रेले-जीन्स नियम से व्युत्पन्न क्लासिकी परिणामों से किस प्रकार क्वांटम सांख्यिकी परिणाम भिन्न हैं, इसकी व्याख्या कीजिए। (20 अंक)
Directive word: Derive
This question asks you to derive. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.
See our UPSC directive words guide for a full breakdown of how to respond to each command word.
How this answer will be evaluated
Approach
Begin with a concise introduction stating the electromagnetic wave parameters in (a), then systematically solve for angular frequency using c = ω/k, displacement current via Maxwell's equations, and Poynting vector magnitude. For (b), explain chemical potential equality using entropy maximization/Gibbs free energy minimization, then derive the binary liquid-vapor equilibrium condition μₗ = μᵥ for each component. For (c), derive Planck's law by applying Bose-Einstein statistics to photon gas, obtaining energy density u(ν,T), then explicitly contrast with Rayleigh-Jeans divergence at high frequencies (ultraviolet catastrophe). Allocate approximately 25% time to (a), 35% to (b), and 40% to (c) based on mark distribution.
Key points expected
- For (a)(i): Calculate ω = ck = 3×10⁸ × 100 = 3×10¹⁰ rad/s, and displacement current density J_d = ε₀(∂E/∂t) = 10ε₀ω sin(ωt-100x) A/m² with correct magnitude
- For (a)(ii): Apply Poynting vector magnitude S = B²c/μ₀ = (55×10⁻⁸)² × 3×10⁸/(4π×10⁻⁷) ≈ 72.4 W/m² with proper unit conversion
- For (b): Explain that at equilibrium, dS = 0 requires equal chemical potentials to prevent particle flow; derive μᵢˡ = μᵢᵛ for binary system using Gibbs-Duhem or equality of fugacities
- For (c): Derive Planck's law starting from Bose-Einstein distribution ⟨n⟩ = 1/(e^(hν/kT)-1), obtaining u(ν,T) = (8πhν³/c³)/(e^(hν/kT)-1) and energy density integration
- For (c) contrast: Explicitly show Rayleigh-Jeans u(ν,T) = 8πν²kT/c³ diverges as ν→∞ (ultraviolet catastrophe) while Planck's law converges, introducing quantum h
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 2 | Correctly identifies wave number k=100 m⁻¹, uses c=3×10⁸ m/s; properly defines chemical potential μ=(∂G/∂n)_{T,P} and its equality across phases as equilibrium condition; accurately states photon spin-1 Bose-Einstein statistics | Identifies basic relationships but confuses phase/group velocity, or states μ equality without thermodynamic justification; mixes up BE with FD statistics or omits photon zero chemical potential | Fundamental errors: uses wrong wave speed, treats chemical potential as mechanical potential, or applies Maxwell-Boltzmann statistics to photons |
| Derivation rigour | 20% | 2 | Complete step-by-step derivations: Maxwell-Ampère law for displacement current, Gibbs free energy minimization for phase equilibrium, and full BE-to-Planck derivation with density of states g(ν) = 8πν²/c³ | Derivations present but with gaps: skips from ∇×B to J_d, asserts μ equality without proof, or jumps from BE distribution to final formula without showing intermediate steps | Missing derivations entirely or logically flawed: states results without justification, circular reasoning, or mathematically incorrect steps |
| Diagram / FBD | 20% | 2 | Clear diagram showing E, B, k propagation directions with proper orthogonal relationships; schematic of binary liquid-vapor equilibrium with μ vs T or P curves; spectral energy density plot comparing Planck vs Rayleigh-Jeans with Wien's law peak | Basic sketches without labels or incomplete diagrams; missing key features like direction arrows or axis labels | No diagrams where essential, or completely incorrect representations (e.g., parallel E and B fields) |
| Numerical accuracy | 20% | 2 | Precise calculations: ω = 3×10¹⁰ rad/s, J_d magnitude with ε₀ = 8.85×10⁻¹² F/m, S ≈ 72 W/m²; correct Stefan-Boltzmann constant integration σ = 2π⁵k⁴/(15h³c²) ≈ 5.67×10⁻⁸ Wm⁻²K⁻⁴ | Correct order of magnitude but calculation errors: wrong powers of 10, incorrect constants, or arithmetic mistakes in final values | Gross numerical errors: wrong formulas, unit confusion (e.g., T in Celsius), or answers off by orders of magnitude |
| Physical interpretation | 20% | 2 | Explains displacement current as source of B-field changing E-field; interprets μ equality as balance of particle exchange; clearly explains ultraviolet catastrophe resolution via energy quantization and its impact on quantum mechanics development in Indian context (S.N. Bose's contribution) | States physical meanings without depth: mentions 'changing electric field' for J_d, 'same in both phases' for μ, 'quantum solves problem' without specifics | No physical interpretation or incorrect understanding: treats J_d as real current, μ as spatially varying at equilibrium, or misses quantum-classical distinction entirely |
Practice this exact question
Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.
Evaluate my answer →More from Physics 2025 Paper I
- Q1 (a) Consider a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger c…
- Q2 (a) A body moves about a point 'O' under no force, the principal moments of inertia at 'O' being 3A, 5A and 6A. The components of the initi…
- Q3 (a) Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for t…
- Q4 (a) Consider a thick lens of thickness t made of a material of relative refractive index n. Let $R_1$ and $R_2$ be the radii of curvature o…
- Q5 (a) Consider a point charge of 5 nC placed at a distance of 1 m from a perfect conducting plane (z = 0) of infinite extent. Find the electr…
- Q6 (a) Consider a long straight wire of length L carrying a current I. Determine the magnetic vector potential $\vec{A}$ at a point P located…
- Q7 (a) A ternary system consists of three components (A, B and C) in equilibrium with two phases. Determine the number of degrees of freedom u…
- Q8 (a) (i) In free space, an electric field ($\vec{E}$) is given by the following expression : $$\vec{E} = 10 \cos (\omega t - 100 x)\hat{j} \…