Physics 2024 Paper I 50 marks Solve

Q7

(a) How does Planck's law resolve the ultraviolet catastrophe predicted by classical physics? Calculate the average energy ε̄ of an oscillator of frequency 0·60×10¹⁴ s⁻¹ at T = 1800 K, treating it as (i) classical oscillator and (ii) Planck's oscillator. (15 marks) (b) (i) What do you understand by macrostates and microstates? Briefly explain. (5 marks) (ii) A three-level laser system emits laser light at a wavelength of 550 nm. If the population of the upper level exceeds that of the lower level by 25%, determine the negative temperature characterizing the system. (10 marks) (c) Consider a situation shown in the figure below. The wire PQ has mass m, resistance r and can slide on the smooth, horizontal parallel rails separated by a distance l. The resistance of rails is negligible. A uniform magnetic field B exists in the rectangular region and a resistance R connects the rails outside the field region. At t = 0, the wire PQ is pushed towards right with a speed V₀. Find (i) the current in the loop at an instant when the speed of the wire PQ is V and (ii) the acceleration of the wire at this instant. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) चिरप्रतिष्ठित (क्लासिकी) भौतिक विज्ञान द्वारा प्रागुक्त पराबैंगनी विपद (अल्ट्रावायलेट कैटास्ट्रॉफी) को प्लांक का नियम किस प्रकार सुलझाता है? तापक्रम T = 1800 K पर आवृत्ति 0·60×10¹⁴ s⁻¹ के एक दोलक की औसत ऊर्जा ε̄ की गणना कीजिये, यह मानकर कि यह एक (i) क्लासिकी दोलक है और (ii) प्लांक का दोलक है। (15 अंक) (b) (i) स्थूल अवस्थाओं और सूक्ष्म अवस्थाओं से आप क्या समझते हैं? संक्षेप में समझाइये। (5 अंक) (ii) एक तीन-स्तरीय लेजर तंत्र 550 nm के तरंगदैर्ध्य के लेजर प्रकाश का उत्सर्जन करता है। यदि ऊपर के स्तर की जनसंख्या, निम्न स्तर की जनसंख्या से 25% अधिक है, तो तंत्र का अभिलाक्षणिक ऋणात्मक तापक्रम निर्धारित कीजिये। (10 अंक) (c) निम्न चित्र में दर्शाई गई स्थिति पर गौर कीजिये। द्रव्यमान m और प्रतिरोध r का तार PQ, दूरी l से पृथक्कृत चिकनी क्षैतिज समांतर पट्टियों पर फिसल सकता है। पट्टियों का प्रतिरोध नगण्य है। एक एकसमान चुंबकीय क्षेत्र B आयताकार क्षेत्र में विद्यमान है और एक प्रतिरोध R चुंबकीय क्षेत्र से बाहर पट्टियों को जोड़ता है। t = 0 समय पर, तार PQ को गति V₀ के साथ दाहिनी ओर धकेला जाता है। (i) जब तार PQ की गति V है, उस क्षण लूप में धारा और (ii) उसी क्षण तार का त्वरण ज्ञात कीजिये। (20 अंक)

Evaluate your answer to this question → See all 2024 Physics questions

Directive word: Solve

This question asks you to solve. 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 explanation of Planck's quantum hypothesis resolving the ultraviolet catastrophe, then systematically solve all numerical parts: (a) calculate classical and quantum average energies with clear formula substitution (~30% time), (b)(i) define macro/microstates with statistical examples (~10% time), (b)(ii) solve for negative temperature using population inversion (~15% time), and (c) derive induced current and acceleration with proper FBD analysis (~45% time). Conclude by interpreting the physical significance of negative temperature and electromagnetic damping.

Key points expected

  • Explanation of Rayleigh-Jeans law divergence and Planck's energy quantization E = nℏω resolving UV catastrophe
  • Classical equipartition result ε̄ = k_BT and Planck's result ε̄ = ℏω/(e^(ℏω/k_BT) - 1) with correct numerical substitution for ν = 0.60×10¹⁴ Hz at T = 1800 K
  • Clear distinction: macrostate (thermodynamic variables P,V,T) vs microstate (specific particle configurations Ω); relation S = k_B lnΩ
  • Negative temperature calculation: using N₂/N₁ = 1.25 = exp(-ℏω/k_BT) → T < 0, with correct wavelength-to-frequency conversion
  • EM induction setup: motional emf ε = Blv, total resistance (R+r), induced current I = Blv/(R+r) opposing motion via Lenz's law
  • Acceleration derivation: F = IBl = ma → a = -B²l²v/[m(R+r)] showing exponential velocity decay
  • Free body diagram showing velocity v→, magnetic field B↓, induced current direction, and opposing magnetic force F←

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Planck's quantization as resolving UV catastrophe; accurate definitions of macro/microstates with statistical weight; proper understanding of population inversion and negative temperature; correct application of Faraday's law and Lenz's law for motional emfBasic understanding of quantization but confuses classical/quantum regimes; acceptable macro/microstate definitions but missing Ω relation; partial grasp of negative temperature; correct emf formula but confused about current directionMisidentifies UV catastrophe cause; conflates macro/microstates or omits entirely; treats negative temperature as 'below absolute zero'; fundamental errors in electromagnetic induction concepts
Derivation rigour20%10Complete step-by-step derivation of Planck's average energy from partition function; explicit Boltzmann factor derivation for laser levels; rigorous force balance leading to differential equation dv/dt = -λv with proper integration constantsStates correct final formulas with partial derivation steps; acceptable algebra for negative temperature; shows F = IBl but skips explicit substitution for accelerationJumps to answers without derivation; missing critical steps like partition function or force balance; incorrect algebraic manipulation leading to wrong final expressions
Diagram / FBD15%7.5Clear labeled diagram for part (c) showing: rails PQ, magnetic field region (shaded/arrow), resistance R, velocity vector v₀, induced current direction (anticlockwise), magnetic force opposing motion, and all dimensions l, with proper Fleming's rules indicatedBasic sketch of rail system with B field indicated; shows current direction but may have arrow errors; missing some labels or force directionsNo diagram provided; or completely incorrect diagram with wrong current/force directions; fails to show essential elements of the electromagnetic setup
Numerical accuracy25%12.5Accurate calculations: (a) classical ε̄ = 2.484×10⁻²⁰ J, quantum ε̄ = 2.06×10⁻²⁰ J (or similar with ℏω/k_BT ≈ 1.59); (b)(ii) T ≈ -2.3×10⁴ K with correct sign; proper unit handling throughout; 2-3 significant figuresCorrect formulas with minor calculation errors (power of 10, factor of 2); acceptable negative temperature magnitude but wrong sign; mostly correct substitution with arithmetic slipsOrder of magnitude errors; incorrect unit conversions (nm to Hz); wrong constants used; completely wrong numerical answers or missing calculations
Physical interpretation20%10Interprets Planck's result showing energy saturation at high frequencies; explains negative temperature as hotter than +∞ (population inversion); describes electromagnetic braking as energy dissipation in R leading to exponential decay; connects to real applications like laser cooling and railgun physicsBasic interpretation of results without deeper insight; mentions braking but not energy transformation; limited connection to physical applicationsNo physical interpretation provided; misinterprets negative temperature as colder; fails to explain why current opposes motion or where energy goes

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 2024 Paper I