Physics 2024 Paper I 50 marks Derive

Q8

(a) (i) Explain the T-s diagram for the reversible Carnot cycle and hence obtain the expression for the efficiency of the Carnot engine. (10 marks) (ii) The specific heat of a solid at low temperatures is given by the relation Cᵥ = AT³, where A is a constant and T is the absolute temperature. How much heat will be required to raise the temperature of m gm of the solid from 300 K to 500 K? (5 marks) (b) Obtain the general boundary conditions for fields E, B, D and H at a boundary between two different media carrying charge density σ or a current density K. (15 marks) (c) A uniform plane wave with $\vec{E} = E_x \hat{a}_x$ propagates in a lossless medium ($\epsilon_r = 4$, $\mu_r = 1$, $\sigma = 0$) in the $z$-direction. Assume that $E_x$ is sinusoidal with a frequency 100 MHz and has a maximum value of $10^{-4}$ (V/m) at $t = 0$ and $z = \frac{1}{8}$ (m). (i) Write the expression for instantaneous $E$ for any $t$ and $z$. (ii) Write the expression for instantaneous $H$. (iii) Determine the locations where $E_x$ is a positive maximum, when $t = 10^{-8}$ (s). (20 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) उत्क्रमणीय कार्नो चक्र के लिए T-s रेखाचित्र की व्याख्या कीजिये और फिर कार्नो इंजन की दक्षता के लिए व्यंजक प्राप्त कीजिये। (10 अंक) (ii) निम्न तापक्रमों पर एक ठोस की विशिष्ट ऊष्मा सामर्थ्य Cᵥ = AT³ द्वारा व्यक्त की जाती है, जहाँ A एक स्थिरांक है और T परम ताप है। m gm के ठोस का तापक्रम 300 K से 500 K तक बढ़ाने में आवश्यक ऊष्मा की गणना कीजिये। (5 अंक) (b) आवेश घनत्व σ या धारा घनत्व K के दो भिन्न माध्यमों के बीच एक परिसीमा पर क्षेत्रों E, B, D और H के लिए व्यापक परिसीमा प्रतिबंधों को प्राप्त कीजिये। (15 अंक) (c) एक एकसमान समतल तरंग $\vec{E} = E_x \hat{a}_x$ एक क्षयविहीन माध्यम ($\epsilon_r = 4$, $\mu_r = 1$, $\sigma = 0$) में z-दिशा में संचरित है। मान लीजिये कि $E_x$, आवृत्ति 100 MHz के साथ ज्यावक्रीय है और t = 0 तथा z = 1/8 (m) पर उसका उच्चतम मान 10⁻⁴ (V/m) है। (i) किसी भी t और z के लिए तात्क्षणिक E हेतु व्यंजक लिखिये। (ii) तात्क्षणिक H के लिए व्यंजक लिखिये। (iii) जब t = 10⁻⁸ (s) है, उन अवस्थितियों को निर्धारित कीजिये, जहाँ $E_x$ धनात्मक अधिकतम है। (20 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

Begin with (a)(i) by sketching the T-s diagram for Carnot cycle, then rigorously derive efficiency η = 1 - T₂/T₁ using ∮dQ/T = 0; for (a)(ii) integrate Cᵥ = AT³ to find Q = mA(T₂⁴ - T₁⁴)/4. Allocate ~30% time to part (b): apply Maxwell's equations with Gaussian pillbox and Amperian loop to derive all four boundary conditions (Eₜ, Bₙ, Dₙ, Hₜ discontinuities). Spend ~40% on part (c): calculate ω, β, vₚ, η, write instantaneous E(z,t) and H(z,t) using correct phase from initial conditions, then solve for z when Eₓ is maximum at given t. Conclude by verifying consistency across all derived expressions.

Key points expected

  • T-s diagram for Carnot cycle showing two isotherms (T₁, T₂) and two adiabats with correct orientation; derivation of efficiency using area under curves or entropy relations
  • Integration of Cᵥ = AT³ from 300K to 500K yielding Q = mA(500⁴ - 300⁴)/4 = 1.36 × 10¹¹ mA/4 in appropriate units
  • Boundary conditions: E₁ₜ = E₂ₜ, B₁ₙ = B₂ₙ, D₂ₙ - D₁ₙ = σ, (H₂ - H₁) × n̂ = K; derivation using ∮E·dl = 0, ∮B·dA = 0, ∮D·dA = Q_free, ∮H·dl = I_free + dΦ_D/dt
  • For (c): calculation of β = ω√(με) = 4π/3 rad/m, vₚ = 1.5 × 10⁸ m/s, η = √(μ/ε) = 60π Ω; instantaneous E(z,t) = 10⁻⁴ cos(2π × 10⁸t - 4πz/3 + π/6) âₓ
  • H(z,t) = (10⁻⁴/60π) cos(2π × 10⁸t - 4πz/3 + π/6) âᵧ; locations z = (3/4)(n + 1/12) m for positive maxima at t = 10⁻⁸ s
  • Physical interpretation: Carnot efficiency as temperature-limited bound; boundary conditions as manifestation of field continuity; plane wave as TEM with E ⊥ H ⊥ propagation direction

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies reversible vs irreversible processes in Carnot cycle; distinguishes free surface charge σ from bound charge; recognizes TEM wave nature with orthogonal E, H, k vectors; correctly applies Debye T³ law regime validityMinor errors in identifying process directions in T-s diagram; confuses D with E or H with B in boundary conditions; correct wave type but wrong polarization or propagation directionFundamental misunderstanding of Carnot cycle as irreversible; treats all boundary fields as continuous; fails to recognize plane wave as transverse electromagnetic
Derivation rigour25%12.5Complete step-by-step derivation: Carnot efficiency from ∮dQ/T = 0 or Q₁/T₁ = Q₂/T₂; boundary conditions from Maxwell's equations with proper limiting procedures (h → 0); wave equations from ∇²E = με∂²E/∂t² with explicit phase constant determinationCorrect final formulas but skips key steps (e.g., omits justification for adiabatic Q = 0 in Carnot); boundary conditions stated without derivation from Maxwell equations; wave solution assumed without showing wave equation reductionCircular reasoning or missing essential steps; incorrect application of thermodynamic relations; boundary conditions memorized without physical basis; algebraic errors in wave number calculation
Diagram / FBD15%7.5Clear T-s diagram with labeled axes, four processes (1→2 isothermal expansion, 2→3 adiabatic expansion, 3→4 isothermal compression, 4→1 adiabatic compression), arrows showing direction, and Q₁, Q₂ indicated; Gaussian pillbox and Amperian loop sketches for boundary conditions with field orientations labeledT-s diagram drawn but missing labels or incorrect process sequencing; boundary condition diagrams present but lacking field orientation or normal vector indicationNo diagrams despite explicit requirement in (a)(i); confusing or incorrect sketches that misrepresent physical situations; diagrams unrelated to question
Numerical accuracy25%12.5Correct numerical values: Q = 2.72 × 10¹⁰ mA J (or equivalent symbolic); β = 4π/3 rad/m ≈ 4.19 rad/m; phase φ₀ = π/6 from initial condition; λ = 1.5 m; z = 0.0625, 0.8125, 1.5625... m for maxima at t = 10⁻⁸ s; proper unit handling throughoutCorrect formulas but arithmetic errors (e.g., 500⁴ - 300⁴ miscalculated); correct β but wrong phase determination; correct approach to maxima locations but algebraic slip in final z valuesOrder of magnitude errors; incorrect unit conversions (MHz not converted to rad/s); fundamental errors like treating β = ω/c instead of ω√(με); no numerical evaluation where required
Physical interpretation15%7.5Explains why Carnot efficiency depends only on temperature ratio (second law implication); interprets boundary conditions as charge/current source effects; relates wave impedance η = √(μ/ε) to medium properties; explains phase velocity vₚ < c due to εᵣ > 1; connects Debye T³ law to phonon excitations in solidsStates formulas without explaining physical significance; mentions but doesn't elaborate on why certain fields are continuous vs discontinuous; notes vₚ ≠ c without explaining whyNo physical interpretation provided; purely mathematical treatment; misinterprets physical meaning (e.g., claims Carnot efficiency can exceed 1)

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