Physics 2024 Paper I 50 marks Derive

Q4

(a) (i) Write down the system matrix for a combination of two thin lenses in paraxial approximation. Hence obtain the focal length of the combination and the positions of unit planes. (10 marks) (ii) Consider a thin lens combination of two convex lenses of focal lengths f₁ = + 10 cm and f₂ = + 20 cm, respectively, separated by 25 cm. Determine the focal length of the combination and the positions of unit planes. (10 marks) (b) The diameter of central zone of a zone plate is 2·4 mm. If a point source of light of wavelength 600 nm is placed at a distance of 5·0 m from the zone plate, calculate the position of the first image. (10 marks) (c) (i) Consider three inertial frames of reference O, O' and O''. Let O' move with a velocity V with respect to O and O'' move with a velocity V' with respect to O'. Both velocities are in the same direction. Write down the transformation equations relating x, y, z, t with x', y', z', t' and also those relating x', y', z', t' with x'', y'', z'', t''. Hence obtain the relations between x, y, z, t and x'', y'', z'', t''. (The direction of velocity is chosen along the x-axis as per convention) (15 marks) (ii) A galaxy in the constellation Ursa Major is receding from the Earth at 15000 km/s. If one of the characteristic wavelengths of light emitted by the galaxy is 550 nm, what is the corresponding wavelength measured by astronomers on the Earth? (5 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) उपाक्षीय सन्निकटन में दो पतले लेंसों के संयोजन के लिए निकाय आव्यूह (सिस्टम मैट्रिक्स) लिखिए। फिर संयोजन की फोकस दूरी और एकांक समतलों की स्थिति प्राप्त कीजिए। (10 अंक) (ii) 25 cm दूरी से पृथक और क्रमशः फोकस दूरियाँ f₁ = + 10 cm और f₂ = + 20 cm के दो पतले उत्तल लेंसों के संयोजन को लीजिए। संयोजन की फोकस दूरी और एकांक समतलों की स्थितियाँ निर्धारित कीजिए। (10 अंक) (b) एक जोन प्लेट के केंद्रीय जोन का व्यास 2·4 mm है। यदि 600 nm तरंगदैर्घ्य के प्रकाश के एक बिंदु स्रोत को जोन प्लेट से 5·0 m की दूरी पर रखा जाता है, तो प्रथम प्रतिबिंब की स्थिति की गणना कीजिए। (10 अंक) (c) (i) तीन जड़त्वीय निर्देश फ्रेमों O, O' और O" को लीजिए। O के सापेक्ष O' वेग V से और O' के सापेक्ष O" वेग V' से गतिमान है। दोनों वेग एक ही दिशा में हैं। x', y', z', t' के साथ x, y, z, t और x", y", z", t" के साथ x', y', z', t' से संबंधित रूपांतरण समीकरणों को लिखिए। फिर उसके बाद x, y, z, t और x", y", z", t" के बीच संबंधों को प्राप्त कीजिए। (प्रथानुसार वेग की दिशा x-अक्ष के अनुदिश ली जाती है) (15 अंक) (ii) तारामंडल उर्सा मेजर में एक आकाश-गंगा (गैलेक्सी) 15000 km/s की गति से पृथ्वी से दूर जा रही है। यदि गैलेक्सी द्वारा उत्सर्जित प्रकाश की अभिलक्षणिक तरंगदैर्ध्यों में से एक 550 nm है, तो पृथ्वी पर खगोलज्ञों द्वारा मापा गया संगत तरंगदैर्ध्य क्या है? (5 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

Derive the system matrix for two thin lenses using paraxial ray transfer matrices, then apply to the numerical case in (a)(ii). For (b), derive the zone plate focal length formula from constructive interference of half-period zones. For (c), derive the successive Lorentz transformations and compose them to demonstrate the velocity addition formula, then apply to the redshift calculation. Allocate ~35% time to (a) (20 marks), ~20% to (b) (10 marks), and ~45% to (c) (20 marks). Structure: systematic derivation → numerical application → physical interpretation for each part.

Key points expected

  • For (a)(i): System matrix as product of translation and lens matrices: M = T(d) × L(f₂) × T(d) × L(f₁), with correct ABCD elements and unit plane positions from B=0 and A=0 conditions
  • For (a)(ii): Numerical evaluation with f₁=10cm, f₂=20cm, d=25cm yielding effective focal length and unit plane locations (one real, one virtual configuration)
  • For (b): Zone plate focal length formula f = rₙ²/(nλ) with n=1, r₁=1.2mm, giving first image position at 6.0m from zone plate
  • For (c)(i): Lorentz transformation equations O→O' and O'→O'', matrix composition showing Einstein velocity addition: V'' = (V+V')/(1+VV'/c²)
  • For (c)(ii): Relativistic Doppler formula for receding source: λ_observed = λ_emitted × √[(1+β)/(1-β)] ≈ 577.5nm for v=15000 km/s

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies paraxial approximation limits, distinguishes converging/diverging zone plate behavior from conventional lens, and recognizes that three-frame composition in (c) must yield relativistic (not Galilean) velocity addition; cites Hecht or standard texts for matrix optics conventionsCorrect basic formulas but confuses sign conventions (Cartesian vs. real-is-positive) or states Galilean result for (c)(i) without correctionFundamental errors: treats zone plate as ordinary lens with 1/f=1/u+1/v, or applies non-relativistic Doppler shift for (c)(ii)
Derivation rigour20%10Explicit matrix multiplication with unit determinant verification for (a); clear zone construction showing path difference λ/2 per ring for (b); full Lorentz matrix multiplication preserving interval invariance for (c)(i)Correct final formulas but skips intermediate steps or assumes results without showing T(d)×L(f) multiplication explicitlyMissing derivations entirely—only states results; or algebraic errors in matrix multiplication leading to wrong focal length formula
Diagram / FBD20%10Ray diagram showing principal/unit planes with cardinal points for lens combination; zone plate schematic with alternating transparent/opaque rings; spacetime diagram or three-frame velocity schematic for relativistic partOne adequate diagram (typically only the lens system) with missing or poorly labeled unit planes; no zone plate figureNo diagrams despite explicit need for geometric visualization; or incorrect ray tracing through lens combination
Numerical accuracy20%10Precise calculation: (a)(ii) f_eff = -40/3 cm ≈ -13.3cm with unit planes at +10cm and -20cm from respective lenses; (b) f₁ = 2.4m; (c)(ii) λ' ≈ 577.5nm using exact relativistic formula (β=0.05, non-relativistic approximation gives 577.5nm anyway but full marks require showing exact treatment)Correct method but arithmetic errors (e.g., sign error in effective focal length, or unit plane position referenced to wrong origin)Order-of-magnitude errors; uses classical Doppler giving 577.5nm by coincidence but shows no relativistic understanding; or completely wrong substitution
Physical interpretation20%10Explains why negative focal length in (a)(ii) indicates diverging system despite two convex lenses; interprets zone plate as Fresnel diffraction device with multiple foci; connects (c) to cosmological redshift and Hubble's law context for Ursa Major galaxies; notes that at 15000 km/s, relativistic correction to classical Doppler is small (~0.1%) but conceptually essentialBrief comment on diverging system but no insight into multiple foci of zone plate or physical significance of velocity additionNo physical interpretation; treats all problems as purely mathematical exercises without connecting to optical instruments or astrophysical observations

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