Physics 2023 Paper I 50 marks Compulsory Derive

Q1

(a) A force F⃗ is given by F⃗ = x²y x̂ + zy² ŷ + xz² ẑ. Determine whether or not the force is conservative. 10 marks (b) Calculate the gravitational self-energy of the Earth. Given : Mass of Earth Mₑ = 6 × 10²⁴ kg and the Radius of Earth Rₑ = 6·4 × 10⁶ m 10 marks (c) What are the consequences of Lorentz transformations on length and time when observed from a frame moving at relativistic velocities ? 10 marks (d) Using Huygens' principle for a plane wave travelling from rarer medium 1 to a denser medium 2, show that $$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{\mu_2}{\mu_1},$$ where i and r are the angles of incidence and refraction, respectively. $v_1, \mu_1$ and $v_2, \mu_2$ are the velocities and refractive indices in media 1 and 2, respectively. 10 marks (e) What are three and four level pumping schemes ? Explain the lasing action in these with schematic diagrams. 10 marks

हिंदी में प्रश्न पढ़ें

(a) एक बल F⃗ = x²y x̂ + zy² ŷ + xz² ẑ से दिया गया है । ज्ञात कीजिए कि बल संरक्षी है या नहीं । 10 (b) पृथ्वी की गुरुत्वीय नैज-ऊर्जा की गणना कीजिए । दिया गया है : पृथ्वी का द्रव्यमान Mₑ = 6 × 10²⁴ kg और पृथ्वी की त्रिज्या Rₑ = 6·4 × 10⁶ m 10 (c) अपेक्षिक वेग पर गतिशील फ्रेम से प्रेक्षित होने की दशा में लोरेंट्ज़ रूपांतरण का लंबाई और समय पर क्या प्रभाव पड़ेगा ? 10 (d) हाइगेंस के नियम का प्रयोग करते हुए, एक समतल प्रगामी तरंग के लिए, जो कि विरल माध्यम 1 से सघन माध्यम 2 में जा रही है, दर्शाइए कि $$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{\mu_2}{\mu_1},$$ जहाँ कि i और r क्रमशः आपतन कोण और अपवर्तन कोण हैं । $v_1, \mu_1$ और $v_2, \mu_2$ माध्यम 1 और माध्यम 2 में क्रमशः वेग और अपवर्तनांक हैं । 10 (e) तीन और चार स्तरीय पम्पिंग योजनाएँ क्या हैं ? इनमें लेज़िंग क्रिया को योजित रेखाचित्र सहित समझाइए । 10

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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

This question demands rigorous derivation and calculation across five distinct physics domains. Structure your answer by addressing each sub-part sequentially: for (a) apply the curl test for conservative forces; for (b) integrate gravitational potential energy for a uniform sphere; for (c) derive length contraction and time dilation from Lorentz transformations; for (d) construct wavefront diagrams using Huygens' construction; for (e) draw energy level diagrams and explain population inversion mechanisms. Allocate approximately equal time (~20%) to each 10-mark sub-part, ensuring complete derivations with clear physical reasoning.

Key points expected

  • For (a): Compute ∇ × F⃗ and show it equals (2yz - z²)x̂ + (z² - x²)ŷ + (y² - x²)ẑ ≠ 0, proving the force is non-conservative
  • For (b): Derive U = -3GMₑ²/5Rₑ and calculate U ≈ -2.24 × 10³² J, showing integration steps for uniform density sphere
  • For (c): Derive length contraction L = L₀/γ and time dilation Δt = γΔt₀ from Lorentz transformations, defining γ = 1/√(1-v²/c²)
  • For (d): Apply Huygens' principle with wavefront construction at interface, using equal time travel to derive Snell's law and refractive index relation
  • For (e): Contrast three-level (Ruby laser: E₁→E₃→E₂→E₁) and four-level (He-Ne laser: E₁→E₃→E₂→E₁ with E₂→E₁ fast) pumping schemes with population inversion requirements

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies curl condition for (a), uses proper self-energy integral for (b), applies correct Lorentz transformation equations for (c), states Huygens' principle accurately for (d), and distinguishes metastable state lifetimes for (e); no conceptual errors across any sub-partCorrectly solves 3-4 sub-parts with minor errors in one; may confuse conservative force conditions, misapply Lorentz factor, or conflate three/four level schemesFundamental misconceptions in 2+ sub-parts; treats force as conservative without curl test, uses point-mass formula for Earth's self-energy, or shows no understanding of population inversion
Derivation rigour20%10Complete mathematical derivations: explicit curl calculation with partial derivatives, step-by-step gravitational integration from 0 to Rₑ, full Lorentz transformation algebra, geometric wavefront construction with equal-time argument, and rate equation analysis for laser schemesDerivations present but with skipped steps or missing justification; correct final results but incomplete intermediate steps in 1-2 sub-partsMissing derivations entirely or logically flawed; states results without proof, uses dimensional arguments instead of proper integration, or presents circular reasoning
Diagram / FBD20%10Clear Huygens' construction with incident/refracted wavefronts and secondary wavelets for (d); accurate energy level diagrams showing pump transitions, laser transitions, and fast/slow decay paths for (e); properly labeled axes and anglesDiagrams present but lacking detail; missing secondary wavelet arcs in (d) or unclear labeling of metastable states in (e); rough sketches without proper scalingNo diagrams for (d) and (e) despite explicit requirement; or completely incorrect diagrams showing wrong physics (e.g., equal spacing of levels, wrong wavefront geometry)
Numerical accuracy20%10Precise calculation for (b): U = -3(6.67×10⁻¹¹)(6×10²⁴)²/(5×6.4×10⁶) ≈ -2.24×10³² J with correct units and sign; proper handling of significant figures; correct γ expressions in (c) with sample numerical valuesCorrect formula with arithmetic errors; wrong power of ten or sign error in (b); order-of-magnitude correct but imprecise; missing unitsOrder-of-magnitude errors; completely wrong numerical substitution; treats all quantities as order-unity; no numerical evaluation where required
Physical interpretation20%10Explains why non-conservative force implies path-dependence; interprets negative self-energy as binding energy; discusses twin paradox implications for (c); explains why four-level systems achieve easier population inversion; connects to Indian applications (INSAT lasers, LIGO-India relativistic corrections)Some physical insight present but superficial; mentions binding energy without elaboration, or states laser efficiency without explaining why four-level is betterPurely mathematical treatment with no physical meaning; no interpretation of results or connection to real-world phenomena; fails to explain significance of any derived quantity

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