Physics 2022 Paper I 50 marks Derive

Q6

(a) Write down Maxwell's equations in a non-conducting medium with constant permeability and susceptibility (ρ = j = 0). Show that E⃗ and B⃗ each satisfies the wave equation, and find an expression for the wave velocity. Write down the plane wave solutions for E⃗ and B⃗, and show how E⃗ and B⃗ are related. (15 marks) (b) (i) One mole of gas obeys van der Waals equation of state. If its molar internal energy is given by u = cT – a/V (in which V is the molar volume, a is one of the constants in the equation of state and c is a constant), calculate the molar heat capacities Cv and Cp. (10 marks) (ii) A compressor designed to compress air is used instead to compress helium. It is found that the compressor overheats. Explain this effect, assuming that the compression is approximately adiabatic and the starting pressure is same for both the gases. [γHe = 5/3, γAir = 7/5] (10 marks) (c) A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions $$p(T, V) = aT^{1/2} + bT^3 + cV^{-2}$$ $$C_v(T, V) = dT^{1/2} + eT^2V + fT^{1/2}$$ where $a$ through $f$ are constants which are independent of $T$ and $V$. Find the differential of the internal energy $dU(T, V)$ in terms of $dT$ and $dV$. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) मैक्सवेल के समीकरण को अचालक माध्यम में नियत पारगम्यता और सुग्राहिता (ρ = j = 0) के साथ लिखिए। दर्शाइए कि E⃗ और B⃗ दोनों तरंग समीकरण को संतुष्ट करते हैं, और तरंग वेग के लिए एक व्यंजक प्राप्त कीजिए। E⃗ और B⃗ के लिए समतल तरंग हल लिखिए और दर्शाइए कि E⃗ और B⃗ किस प्रकार संबंधित हैं। (15 अंक) (b) (i) गैस का एक मोल वान्डर वाल्स अवस्था समीकरण का पालन करता है। यदि इसकी मोलर आंतरिक ऊर्जा u = cT – a/V है (जिसमें V मोलर आयतन, a अवस्था समीकरण में एक स्थिरांक और c एक स्थिरांक है), तो मोलर उष्मा धारिताओं Cv और Cp की गणना कीजिए। (10 अंक) (ii) हीलियम को संपीड़ित करने के लिए हवा को संपीड़ित करने हेतु अभिकल्पित एक संपीड़ित्र का उपयोग किया गया है। यह पाया गया कि संपीड़ित्र ज्यादा गरम होता है। इस प्रभाव की व्याख्या कीजिए, यह मानते हुए कि संपीड़न लगभग रुद्धोष्म है और दोनों गैसों के प्रारंभिक दाब समान हैं। [γHe = 5/3, γहवा = 7/5] (10 अंक) (c) परस्पर क्रिया करने वाले परमाणुओं की एक गैस की अवस्था और स्थिर आयतन पर ऊष्मा धारिता के समीकरणों के व्यंजक निम्नानुसार हैं : $$p(T, V) = aT^{1/2} + bT^3 + cV^{-2}$$ $$C_v(T, V) = dT^{1/2} + eT^2V + fT^{1/2}$$ जहाँ $a$ से $f$ नियतांक हैं जो $T$ और $V$ से स्वतंत्र हैं। आंतरिक ऊर्जा $dU(T, V)$ का अवकल मान $dT$ और $dV$ के पदों में ज्ञात कीजिए। (15 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Correctly states all four Maxwell's equations in appropriate form; accurately identifies van der Waals constants and their physical meaning; correctly applies γ values for monoatomic vs diatomic gases; properly identifies exact differential form for internal energyStates most equations correctly but misses one Maxwell equation or makes minor errors in thermodynamic identities; confuses γ values or their origins; partial understanding of exact differential conditionsMajor errors in fundamental equations; incorrect identification of gas types; fundamental misunderstanding of thermodynamic potentials or Maxwell relations
Derivation rigour25%12.5Complete step-by-step derivation of wave equations using ∇×(∇×A) identity; rigorous derivation of Cp-Cv relation with proper Jacobian manipulation; clear derivation of adiabatic temperature relation; systematic derivation of dU with all partial derivatives explicitly calculatedSkips some intermediate steps but reaches correct results; uses shortcut formulas without derivation; minor algebraic errors that don't affect final structureMissing crucial steps; incorrect vector identities; invalid algebraic manipulations; circular arguments or unsupported claims
Diagram / FBD10%5Clear diagram showing E⃗, B⃗, and k⃗ as mutually perpendicular vectors with proper labeling; PV diagram sketch for adiabatic compression comparing air and helium; schematic of compressor process if relevantBasic diagram showing field vectors but missing some labels; minimal or no diagrams for thermodynamic partsNo diagrams where clearly needed; incorrect vector relationships shown; confusing or misleading sketches
Numerical accuracy20%10Correct final expressions: wave velocity 1/√(με); Cv = c and Cp = c + R/[1-2a/RTV] or equivalent; explicit comparison of temperature exponents (0.4 for air vs 0.67 for helium); simplified correct dU expression with properly combined termsCorrect final forms but with algebraic errors in coefficients; correct γ values used but arithmetic errors in exponents; partially simplified dU expressionIncorrect numerical coefficients; wrong final expressions; calculation errors that propagate through; missing key terms in final answers
Physical interpretation20%10Explains why E⃗ and B⃗ are in phase and transverse; interprets 'a' as intermolecular attraction reducing Cp-Cv; physically explains why helium overheats (fewer degrees of freedom → higher γ → more work converts to internal energy); discusses integrability conditions for U being state functionSome physical insight but limited connection to molecular behavior; basic explanation of overheating without reference to degrees of freedom; minimal discussion of state function requirementsPurely mathematical treatment with no physical insight; incorrect physical explanations; misses key conceptual connections between parts

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