Physics 2022 Paper I 50 marks Compulsory Calculate

Q5

(a) Assume that the Earth's atmosphere is pure nitrogen in thermodynamic equilibrium at a temperature of 300 K. Calculate the height above sea level at which the density of the atmosphere is one-half its sea level value. (Molecular weight of N₂ is 28 gm/mole) (10 marks) (b) A body of constant heat capacity Cₚ and a temperature Tᵢ is put into contact with a reservoir at temperature Tƒ. Equilibrium between the body and the reservoir is established at constant pressure. Determine the total entropy change and prove that it is positive for either sign of [(Tƒ — Tᵢ)/Tƒ]. Consider |Tƒ — Tᵢ|/Tƒ < 1. (10 marks) (c) Consider two point particles of charge q each, separated by a distance d, and travelling at non-relativistic velocity v⃗. If the line joining the two charges is perpendicular to v⃗, then write an expression for the magnetic force between the two particles, and illustrate the direction of the force on each particle. (10 marks) (d) A cell of internal resistance 1 ohm, 1·5 volt e.m.f. and another cell of internal resistance 2 ohm, 2 volt e.m.f. are connected in parallel across the ends of an external resistance of 5 ohm. Find the current in each branch of the circuit. (10 marks) (e) Consider the R-L-C circuit shown here. Calculate the Q-factor of the circuit. Does the circuit have a resonant frequency? Justify your answer : (10 marks)

हिंदी में प्रश्न पढ़ें

(a) मान लीजिए कि पृथ्वी का वायुमंडल 300 K के तापमान पर ऊष्मागतिक साम्य में शुद्ध नाइट्रोजन है। समुद्र-तल से ऊँचाई की गणना कीजिए जिस पर वायुमंडल का घनत्व उसके समुद्र-तल मान का आधा है। (N₂ का आण्विक भार 28 gm/mole है) (10 अंक) (b) नियत ऊष्मा धारिता Cₚ और तापमान Tᵢ का एक पिंड तापमान Tƒ पर एक भंडार (जलाशय) के संपर्क में आता है। अचल दाब पर पिंड और भंडार के बीच साम्यावस्था स्थापित होती है। कुल एन्ट्रॉपी परिवर्तन का निर्धारण कीजिए और सत्यापित कीजिए कि यह [(Tƒ — Tᵢ)/Tƒ] के किसी भी चिह्न के लिए धनात्मक है। मान लीजिए कि |Tƒ — Tᵢ|/Tƒ < 1 है। (10 अंक) (c) विचारिए कि दो बिंदु कण जिनमें प्रत्येक का आवेश q है, d दूरी से पृथक्कृत और अनापेक्षिकीय वेग v⃗ से गति कर रहे हैं। यदि दोनों आवेशों को मिलाने वाली रेखा v⃗ के लम्बवत् है, तो दोनों कणों के बीच चुंबकीय बल के लिए व्यंजक लिखिए और प्रत्येक कण पर बल की दिशा का वर्णन कीजिए। (10 अंक) (d) 1·5 वोल्ट विद्युत वाहक बल तथा 1 ओम आंतरिक प्रतिरोध का एक सेल और 2 वोल्ट विद्युत वाहक बल तथा 2 ओम आंतरिक प्रतिरोध का दूसरा सेल 5 ओम के बाह्य प्रतिरोध के सिरों पर समांतर क्रम में जुड़े हुए हैं। परिपथ की प्रत्येक शाखा में धारा का मान ज्ञात कीजिए। (10 अंक) (e) चित्र में दिखाए गए R-L-C परिपथ पर विचार कीजिए। परिपथ के Q-गुणक की गणना कीजिए। क्या परिपथ में अनुनादी आवृत्ति होती है? उत्तर का औचित्य दीजिए : (10 अंक)

Evaluate your answer to this question → See all 2022 Physics questions

Directive word: Calculate

This question asks you to calculate. 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

This is a multi-part calculation-based question requiring precise derivations and numerical solutions. Allocate approximately 20% time to each sub-part (a)-(e), with slightly more attention to (b) for its entropy proof and (e) for the missing diagram interpretation. Begin each part with the relevant fundamental equation, show complete derivation steps, substitute values with units, and conclude with physical interpretation of results.

Key points expected

  • (a) Apply barometric formula/ Boltzmann distribution: ρ(h) = ρ₀exp(-mgh/kBT) or n(h) = n₀exp(-Mgh/RT); solve for h when ρ/ρ₀ = 1/2 using m = 28×10⁻³/NA kg
  • (b) Calculate entropy change of body: ΔS_body = Cₚln(Tf/Ti); entropy change of reservoir: ΔS_res = -Cp(Tf-Ti)/Tf; total ΔS = Cₚ[ln(Tf/Ti) - (Tf-Ti)/Tf]; expand ln(1+x) for |Tf-Ti|/Tf < 1 to prove positivity
  • (c) Use Biot-Savart law for moving charge: B = (μ₀/4π)qv×r̂/r²; magnetic force F = qv×B; derive F = (μ₀q²v²)/(4πd²) with direction along line joining charges (attractive for parallel currents)
  • (d) Apply Kirchhoff's laws: set up junction equation I = I₁ + I₂ and loop equations; solve simultaneous equations for I₁, I₂ and I through 5Ω resistor
  • (e) Calculate Q-factor = (1/R)√(L/C) or ω₀L/R; identify whether series or parallel RLC configuration; discuss resonant frequency ω₀ = 1/√(LC) existence based on circuit topology

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies: (a) isothermal atmosphere as exponential decay; (b) entropy as state function with reservoir at constant T; (c) magnetic field from moving charge and force between current elements; (d) parallel cell combination with internal resistance; (e) Q-factor definition and resonance condition for given topologyIdentifies most concepts correctly but confuses: isothermal vs adiabatic atmosphere, or treats reservoir entropy incorrectly, or misapplies Biot-Savart direction, or makes sign errors in KVLFundamental misconceptions: uses linear density variation for (a), ignores reservoir entropy for (b), applies Coulomb's law for magnetic force in (c), omits internal resistance in (d), or cannot define Q-factor
Derivation rigour20%10Complete step-by-step derivations: (a) integrates dp/p = -mgdz/kBT; (b) shows integral of dQrev/T and Taylor expansion to O(x²); (c) derives B field then cross product for force; (d) systematic KCL/KVL; (e) impedance derivation for Q-factorCorrect final formulas but skips key steps: jumps to exponential without integration, omits expansion details in entropy proof, states force formula without B-field derivation, or solves (d) by inspection without systematic approachMissing derivations or logically flawed: states results without justification, incorrect integration limits, algebraic errors in entropy expansion, or no attempt at derivation for any part
Diagram / FBD15%7.5Clear diagram for (c) showing v⃗, B-field circles, and force directions on both charges; circuit diagram for (d) with labeled nodes, currents, and polarities; annotated impedance diagram or phasor sketch for (e) if relevantBasic sketches present but incomplete: missing field directions in (c), unlabeled circuit elements in (d), or no response to missing diagram in (e)No diagrams where essential, or seriously misleading diagrams: wrong force directions, incorrect current arrows, or complete omission despite explicit 'shown here' reference in (e)
Numerical accuracy25%12.5Precise calculations: (a) h ≈ 5.9-6.0 km using correct molar mass conversion; (b) correct algebraic expression with proper series expansion; (c) correct force magnitude with proper constants; (d) I₁ ≈ 0.36A, I₂ ≈ 0.57A, I ≈ 0.93A or equivalent; (e) symbolic Q-factor correctCorrect method but arithmetic errors: wrong powers of 10 in (a), coefficient errors in expansion, or sign errors in currents; partial credit for correct setupGross numerical errors: uses 28 g/mole as kg, omits NA, wrong order of magnitude, or no numerical substitution where required
Physical interpretation20%10Insightful commentary: (a) compares to actual scale height ~8.5 km (O₂ lighter, temperature variation); (b) explains irreversibility and entropy increase; (c) relates to Ampère's force law and parallel currents; (d) discusses power dissipation; (e) analyzes bandwidth and selectivity implicationsBrief concluding statements without depth: mentions 'entropy always increases' without context, or states resonance exists without explaining sharpnessNo physical interpretation provided, or incorrect statements: claims entropy can decrease, or suggests magnetic force is repulsive when attractive

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 2022 Paper I