Physics 2021 Paper I 50 marks Calculate

Q3

(a) In a step-index optical fiber system, explain the terms pulse dispersion and material dispersion. An optical fiber having refractive indices of core and cladding n₁ = 1·463 and n₂ = 1·444 respectively, uses a Laser diode with λ₀ = 1·50 μm with a spectral width of 2 nm. At this wavelength if the material dispersion coefficient, Dₘ is 18·23 ps/km.nm, then calculate the pulse dispersion and material dispersion for 1 km length of the fiber. (20 marks) (b) What is chromatic aberration? Obtain the condition for achromatism using combination of two thin lenses placed in contact to each other. Can this system work as achromatic doublet if both are of same material? Justify your answer. (15 marks) (c) (i) Calculate the mass and momentum of a proton of rest mass 1·67 × 10⁻²⁷ kg moving with a velocity of 0·8c, where c is the velocity of light. If it collides and sticks to a stationary nucleus of mass 5·0 × 10⁻²⁶ kg, find the velocity of the resultant particle. (8 marks) (ii) Calculate the mass of the particle whose kinetic energy is half of its total energy. Find the velocity with which the particle is travelling. (7 marks)

हिंदी में प्रश्न पढ़ें

(a) एक स्टेप-इंडेक्स ऑप्टिकल फाइबर निकाय में स्पंद (पल्स) प्रकीर्णन और पदार्थ प्रकीर्णन पदों को समझाइए । एक ऑप्टिकल फाइबर, जिसके कोर और क्लैडिंग पदार्थ का अपवर्तनांक क्रमशः: n₁ = 1·463 और n₂ = 1·444 है, एक लेसर डायोड, जिसका λ₀ = 1·50 μm और स्पेक्ट्रल चौड़ाई 2 nm, का उपयोग करता है । इस तरंगदैर्ध्य पर यदि पदार्थ का प्रकीर्णन गुणांक Dₘ = 18·23 ps/km.nm है, तो 1 km लम्बे फाइबर के लिए स्पंद (पल्स) प्रकीर्णन और पदार्थ प्रकीर्णन की गणना कीजिए । (20 अंक) (b) वर्ण विपथन क्या है ? दो एक-दूसरे से सटे हुए पतले लेंसों को उपयोग में लाते हुए अवर्णकता की शर्त को प्राप्त कीजिए । यदि दोनों लेंस एक ही पदार्थ के बने हों, तो क्या यह निकाय अवर्णक द्विक की तरह कार्य कर सकता है ? अपने उत्तर का औचित्य बताइए । (15 अंक) (c) (i) एक प्रोटॉन के द्रव्यमान और संवेग की गणना कीजिए जिसका स्थिर द्रव्यमान 1·67 × 10⁻²⁷ kg है तथा यह 0·8c के वेग से गति कर रहा है, जहाँ c प्रकाश की गति है। यदि यह प्रोटॉन एक स्थिर नाभिक जिसका द्रव्यमान 5·0 × 10⁻²⁶ kg है, से टकराता है और उससे चिपक जाता है, तो परिणामी कण का वेग ज्ञात कीजिए। (8 अंक) (ii) एक कण के द्रव्यमान की गणना कीजिए जिसकी गतिज ऊर्जा उसकी कुल ऊर्जा की आधी है। कण जिस वेग से गति कर रहा है उस वेग को ज्ञात कीजिए। (7 अंक)

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Approach

Begin with clear definitions and derivations for each sub-part, allocating approximately 40% time to part (a) given its 20 marks, 30% to part (b) for 15 marks, and 30% combined to both parts of (c). Structure as: (a) define pulse/material dispersion with formulas → numerical calculation; (b) define chromatic aberration → derive achromatism condition → discuss same-material limitation; (c)(i)-(ii) apply relativistic mass, momentum and energy formulas with collision dynamics. Include ray diagrams for (b) and state all assumptions explicitly.

Key points expected

  • Part (a): Distinguish pulse dispersion (intermodal/intramodal) from material dispersion (wavelength-dependent refractive index); calculate material dispersion Δtₘ = Dₘ × L × Δλ and estimate pulse broadening using numerical aperture
  • Part (a) numerical: Apply NA = √(n₁² - n₂²) for intermodal contribution, combine with material dispersion for total pulse dispersion at 1 km
  • Part (b): Define chromatic aberration as focal length variation with wavelength; derive 1/f = 1/f₁ + 1/f₂ and ω/f₁ + ω'/f₂ = 0 for achromatism condition
  • Part (b) analysis: Prove same-material doublet cannot be achromatic (ω/ω' = f₂/f₁ requires different dispersive powers), citing crown-flint glass combination
  • Part (c)(i): Calculate relativistic mass m = γm₀, momentum p = γm₀v, then apply conservation of relativistic momentum and energy for inelastic collision to find final velocity
  • Part (c)(ii): Use K = (γ-1)m₀c² and E = γm₀c² with K = E/2 to find γ = 2, hence v = √3c/2 ≈ 0.866c and relativistic mass = 2m₀

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Precisely distinguishes pulse dispersion (modal/delay distortion) from material dispersion (group velocity dispersion); correctly identifies chromatic aberration origin in lens dispersion; applies relativistic mass-velocity relation and conservation laws accurately for (c)Basic definitions correct but confuses intermodal/intramodal contributions; states achromatism condition without explaining dispersive power significance; minor errors in relativistic energy-momentum relationsFundamental confusion between dispersion types; incorrect statement that same-material lenses can form achromatic doublet; treats relativistic collision classically or uses wrong γ factor
Derivation rigour20%10Complete derivation of achromatism condition from lensmaker's equation with wavelength dependence; explicit derivation of ω/f₁ + ω'/f₂ = 0; clear relativistic momentum-energy conservation steps with Lorentz factor manipulationStates final formulas correctly but skips intermediate steps; assumes achromatism condition without derivation; correct algebra but missing physical justification for stepsMissing derivations entirely or mathematically incorrect; no attempt to derive condition for achromatism; algebraic errors in solving for relativistic velocity
Diagram / FBD15%7.5Clear ray diagram for chromatic aberration showing focal point variation with color; labeled diagram of achromatic doublet with crown and flint elements; schematic of step-index fiber structure with ray paths for different modesBasic ray diagram present but poorly labeled; fiber structure mentioned without diagram; adequate but not exemplary visual representationNo diagrams despite explicit need for (b); or completely incorrect ray diagrams showing no understanding of dispersion effects
Numerical accuracy25%12.5Accurate calculation: NA = 0.234, material dispersion = 36.46 ps/km, pulse dispersion ≈ 50 ns/km (intermodal) + material component; relativistic mass = 2.78×10⁻²⁷ kg, momentum = 6.68×10⁻¹⁹ kg·m/s, final velocity = 0.133c; γ = 2, v = 0.866c, mass = 2m₀Correct formulas but arithmetic errors; order of magnitude correct but significant figures mishandled; partial credit for correct setup with wrong final valuesMajor calculation errors or wrong formulas used; nonsensical results (v > c, negative dispersion); no unit consistency checks
Physical interpretation20%10Explains why material dispersion limits fiber bandwidth- distance product; discusses practical implications for Indian fiber-optic networks (BharatNet); explains why achromatic doublets require different glasses; interprets relativistic collision result in center-of-mass frameMentions practical applications superficially; states results without explaining physical significance; limited connection to real-world optical systemsNo physical interpretation provided; purely mathematical treatment; fails to explain why same-material doublet fails or why dispersion matters in fibers

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