Physics 2025 Paper I 50 marks Derive

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 at distance x from the wire. (20 marks) (b) As shown in the figure, a series circuit connected across a 200 V, 60 Hz line consists of a capacitor of capacitive reactance of 30 Ω, a non-inductive resistor of 44 Ω and a coil of inductive reactance 90 Ω and resistance 36 Ω. Determine: (i) Power factor of the circuit (ii) Power absorbed by the circuit (iii) Power dissipated in the coil (c) Consider a mixture of N_A molecules of a monatomic gas A and N_B molecules of a monatomic gas B. For this mixture, obtain the Helmholtz free energy and pressure. (The particle partition function for a monatomic gas is q = (2π mkT/h²)^(3/2) V).

हिंदी में प्रश्न पढ़ें

(a) वि�िद्युत धारा I प्रवाहित लम्बाई L के एक सीधे लम्बे तार को लीजिए। तार से दूरी x पर अवस्थित बिन्दु P पर चुंबकीय सदिश विभव $\vec{A}$ ज्ञात कीजिए। (20 अंक) (b) जैसा कि चित्र में दर्शाया गया है, एक 200 V, 60 Hz लाइन से संबद्ध एक श्रेणी परिपथ में 30 Ω की धारिता प्रतिघात का एक संधारित्र, 44 Ω का एक अप्रेरणिक प्रतिरोधक और 36 Ω प्रतिरोध तथा 90 Ω की प्रेरणिक प्रतिघात की एक कुंडली है। ज्ञात कीजिए: (i) परिपथ का शक्ति गुणांक (ii) परिपथ द्वारा अवशोषित शक्ति (iii) कुंडली में क्षतिग्रस्त शक्ति (c) एक एकपरमाणुक गैस A के N_A अणुओं और एक एकपरमाणुक गैस B के N_B अणुओं के एक मिश्रण को लीजिए। इस मिश्रण के लिए, हेल्महोल्ट्ज़ मुक्त ऊर्जा और दाब ज्ञात कीजिए। (एक एकपरमाणुक गैस के लिए कण संवितरण फलन है q = (2π mkT/h²)^(3/2) V).

Evaluate your answer to this question → See all 2025 Physics questions

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

Derive the magnetic vector potential for the finite wire in part (a) using proper integration limits and Coulomb gauge, then solve the AC circuit problem in part (b) by calculating impedance, phase angle, and power quantities stepwise, and finally derive the Helmholtz free energy for the gas mixture in part (c) using Maxwell-Boltzmann statistics. Allocate approximately 40% effort to part (a) given its 20-mark weight in the original scheme, 35% to part (b) for its three numerical sub-parts, and 25% to part (c) for the statistical mechanics derivation.

Key points expected

  • Part (a): Setup of integral for vector potential using A = (μ₀I/4π)∫(dl'/|r-r'|) with proper coordinate system and limits from -L/2 to +L/2
  • Part (a): Final expression A = (μ₀I/4π)ln[(L/2+√(x²+L²/4))/(-L/2+√(x²+L²/4))]ẑ or equivalent, with discussion of infinite wire limit
  • Part (b)(i): Calculation of total impedance Z = √[(R₁+R₂)²+(X_L-X_C)²] = √[80²+60²] = 100Ω, leading to power factor cosφ = 80/100 = 0.8 lagging
  • Part (b)(ii)-(iii): Power absorbed P = VIcosφ = 200×2×0.8 = 320W (or I²R_total = 4×80 = 320W), and power in coil = I²R_coil = 4×36 = 144W
  • Part (c): Derivation of Helmholtz free energy F = -kT[N_A ln(q_A/N_A) + N_B ln(q_B/N_B) + N_A + N_B] using Stirling's approximation
  • Part (c): Pressure derivation P = -(∂F/∂V)_T = (N_A+N_B)kT/V = nRT/V, showing ideal gas mixture law with Dalton's law implicit

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%2Correctly identifies Coulomb gauge (∇·A=0) for part (a), recognizes series RLC nature of part (b) with proper lagging/leading phase distinction, and correctly applies Maxwell-Boltzmann statistics with indistinguishability correction for part (c)Uses correct general formulas but makes minor errors in gauge choice, phase identification, or statistical counting; may confuse distinguishable vs indistinguishable particlesFundamental conceptual errors such as using Biot-Savart directly for A without integration, treating the circuit as parallel, or using classical Boltzmann statistics without Gibbs correction
Derivation rigour25%2.5Complete stepwise derivations: proper vector integration with dl' = dz'ẑ in (a), systematic impedance analysis with phasor diagram implied in (b), and rigorous combinatorial derivation with Stirling's approximation in (c)Correct final results but skips key steps or assumes results without proof; incomplete justification for limits or approximationsMissing derivations entirely or logically flawed steps; jumps from given data to final answer without showing work; algebraic errors in expansion
Diagram / FBD15%1.5Clear diagram for (a) showing wire geometry, field point P, coordinate system, and current direction; phasor diagram for (b) showing V, I, and component voltages; schematic circuit diagram with labeled elementsSketchy or incomplete diagrams; missing coordinate labels or incorrect phasor orientation; circuit diagram without valuesNo diagrams despite clear need; completely misleading geometry or phasor relationships; diagrams that contradict written solution
Numerical accuracy25%2.5Precise calculations: correct current I = 200/100 = 2A, power factor 0.8 lagging, total power 320W, coil power 144W; proper significant figures and units throughoutCorrect method but arithmetic errors; wrong power factor sign or magnitude; correct final answers with intermediate step errors that cancelGross calculation errors (e.g., impedance addition without squares); incorrect power formulas (using apparent power instead of real power); missing or wrong units
Physical interpretation15%1.5Discusses infinite wire limit for (a), explains lagging power factor meaning for inductive circuit in (b), and connects Helmholtz free energy to thermodynamic potentials with physical significance of mixture pressureBrief or superficial interpretation; states results without explaining physical meaning; generic statements about energy conservationNo physical interpretation provided; misinterprets results (e.g., claiming leading power factor for inductive circuit); irrelevant or incorrect physical statements

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