Physics 2023 Paper I 50 marks Calculate

Q4

(a) Write conditions for working of a step-index optical fiber. In a step-index fiber, the core and cladding materials have refractive indices 1·50 and 1·43, respectively. Find the following : (i) Critical propagation angle (ii) Acceptance angle (iii) Total time delay in 1 km length of the fiber (iv) Total dispersion in 50 km length of the fiber (b) Define streamline flow of a fluid. Using the equation of continuity for an isotropic fluid, find different components of total energy per unit volume. (c) (i) What is the difference between Fresnel diffraction and Fraunhofer diffraction ? (ii) What is resolving power of a telescope ? Why is the resolving power of microscope more with UV light than with visible light ?

हिंदी में प्रश्न पढ़ें

(a) स्टेप-इण्डेक्स प्रकाशिक तन्तु की कार्यविधि की शर्तों को लिखिए । एक स्टेप-इण्डेक्स तन्तु में, कोर और क्लैडिंग पदार्थों के अपवर्तनांक क्रमशः 1·50 और 1·43 हैं । निम्नलिखित को ज्ञात कीजिए : (i) कांतिक संचरण कोण (ii) स्वीकरण कोण (iii) 1 km लम्बाई के तन्तु में कुल समयान्तराल (iv) 50 km लम्बाई के तन्तु में कुल प्रकीर्णन (b) एक तरल के धाररेखी प्रवाह को परिभाषित कीजिए । सांतत्य के समीकरण का उपयोग करते हुए समदैशिक तरल के लिए प्रति एकांक आयतन की कुल ऊर्जा के विभिन्न घटकों को ज्ञात कीजिए । (c) (i) फ्रेनल विवर्तन और फ्राउनहोफर विवर्तन में क्या अंतर है ? (ii) एक दूरदर्शक की विभेदन क्षमता क्या होती है ? एक सूक्ष्मदर्शी की विभेदन क्षमता UV प्रकाश में दृश्य प्रकाश की अपेक्षा ज्यादा क्यों होती है ?

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Directive word: Calculate

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Approach

Begin by stating the conditions for total internal reflection in step-index fibers, then systematically calculate all four numerical parameters in part (a) showing each formula substitution. For part (b), define streamline flow precisely, then apply continuity equation to derive kinetic, potential, and pressure energy components. For part (c), use a comparative table for Fresnel vs Fraunhofer diffraction, then explain resolving power with Rayleigh criterion and justify UV advantage for microscopes through wavelength dependence. Allocate approximately 40% effort to part (a) due to heavy calculations, 30% each to (b) and (c).

Key points expected

  • Conditions for step-index fiber: n_core > n_cladding, total internal reflection at core-cladding interface, light launched within acceptance cone
  • Calculated values: critical propagation angle θ_c = sin⁻¹(n₂/n₁) ≈ 72.3°, acceptance angle θ_a = sin⁻¹(√(n₁²-n₂²)) ≈ 23.6°, time delay Δt = Ln₁²/(cn₂) ≈ 4.9 μs/km, total dispersion over 50 km
  • Streamline flow definition: velocity at each point remains constant in time, no eddies; Bernoulli derivation yielding ½ρv² + ρgh + P = constant representing kinetic, potential, and pressure energy density
  • Fresnel vs Fraunhofer distinction: source/screen at finite vs infinite distance, no lens vs lens used, spherical vs plane wavefronts, cylindrical vs uniform illumination
  • Resolving power of telescope: R = D/(1.22λ); microscope resolution higher with UV due to λ_UV < λ_visible giving smaller minimum resolvable distance d = 0.61λ/NA

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%2Correctly identifies TIR condition n₁>n₂, distinguishes near-field (Fresnel) from far-field (Fraunhofer) diffraction regimes, and accurately relates resolving power to wavelength and apertureStates basic TIR condition and defines diffraction types superficially; minor confusion between propagation angle and acceptance angle conceptsFundamental error like n_cladding > n_core, or confuses Fresnel with Fraunhofer conditions; incorrect resolving power formula
Derivation rigour20%2Complete derivation of acceptance angle from Snell's law at air-core and core-cladding interfaces; systematic application of continuity equation to obtain energy components; Rayleigh criterion properly statedAcceptance angle formula quoted without derivation steps; Bernoulli equation stated but energy components not explicitly identified; resolving power formula given without contextMissing critical steps like numerical aperture derivation; incorrect application of continuity equation; no derivation of time delay formula
Diagram / FBD20%2Clear ray diagram showing core-cladding interface with critical angle, acceptance cone geometry with θ_a, and labeled refractive indices; streamline pattern for fluid flow; diffraction geometry comparisonBasic fiber cross-section shown but angles not labeled; generic streamline drawing without velocity profile; no diffraction setup diagramsNo diagrams despite geometric nature of problem; or incorrect diagrams showing rays bending away from normal in denser medium
Numerical accuracy20%2All four calculations correct: θ_c ≈ 72.3°, θ_a ≈ 23.6° (or NA ≈ 0.40), Δt ≈ 4.9 μs/km, total dispersion ≈ 245 ns over 50 km; proper unit handling throughoutCorrect formulas but arithmetic errors in one or two parts; confusion between degrees and radians; order of magnitude correct but precise value wrongMultiple calculation errors; wrong formulas leading to nonsensical answers; missing units or incorrect unit conversions (e.g., km to m)
Physical interpretation20%2Explains modal dispersion origin in step-index fiber and its limitation for high-bandwidth communication; connects energy components to Bernoulli's principle applications; relates UV microscopy to semiconductor fabrication needsBrief mention of dispersion causing pulse broadening; states energy conservation without application context; notes shorter wavelength improves resolution without explaining Abbe limitNo physical interpretation provided; calculations presented as mechanical exercises without understanding why step-index fibers have limited bandwidth or why UV improves resolution

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