Physics 2025 Paper I 50 marks Derive

Q4

(a) Consider a thick lens of thickness t made of a material of relative refractive index n. Let $R_1$ and $R_2$ be the radii of curvature of its two surfaces. Obtain the system matrix of the lens. (15 marks) (b) Consider multiple reflections from a plane parallel film of thickness h and refractive index $n_2$ and derive an expression for the total reflectivity from the surface of the film. (20 marks) (c) A solid shaft of mass M, length $l$ and radius r is to be replaced by a lighter hollow shaft of the same length $l$ and having the same ratings of $\tau/\theta$, where $\tau$ is the couple and $\theta$ is the angle of twist. Estimate the percentage reduction in mass of the hollow shaft if the outer radius of the shaft is twice the inner radius. Assume the material of the new shaft is same as that of the replaced shaft. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) सापेक्ष अपवर्तनांक n के एक पदार्थ से निर्मित मोटाई t के एक मोटे लेंस को लीजिए। मान लीजिए कि उसके दो पृष्ठों की वक्रता के अर्ध्व्यास $R_1$ और $R_2$ हैं। लेंस की निकाय मैट्रिक्स (आव्यूह) प्राप्त कीजिए। (15 अंक) (b) अपवर्तनांक $n_2$ और मोटाई h की एक समतल समांतर फिल्म से होने वाले बहुल परावर्तनों को लीजिए और फिल्म के पृष्ठ से होने वाली कुल परावर्तकता के लिए व्यंजक की व्युत्पत्ति कीजिए। (20 अंक) (c) एक द्रव्यमान M, लंबाई $l$ और अर्ध्व्यास r के ठोस कूपक (शाफ्ट) को समान लंबाई $l$ और समान $\tau/\theta$ की रेटिंग, जहाँ $\tau$ बल-युग्म और $\theta$ व्यावर्तन कोण है, के एक हल्के खोखले कूपक द्वारा प्रतिस्थापित किया जाना है। यदि खोखले कूपक का बाह्य अर्ध्व्यास उसके आंतरिक अर्ध्व्यास का दो गुना है, तो उसके द्रव्यमान में प्रतिशत कमी का आकलन कीजिए। मान लीजिए कि नए कूपक और प्रतिस्थापित कूपक का पदार्थ समान है। (15 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

Begin with a brief introduction distinguishing matrix optics from Gaussian optics. For part (a), derive the system matrix by multiplying refraction and translation matrices in correct order. For part (b), use the method of summing infinite geometric series of reflected amplitudes with proper phase considerations. For part (c), equate torsional rigidity C = τ/θ for both shafts and solve for mass ratio. Allocate approximately 30% time to (a), 40% to (b) as it carries highest marks, and 30% to (c). Conclude with brief remarks on practical applications in optical instruments and mechanical engineering.

Key points expected

  • Part (a): Correct identification of individual matrices — refraction at first surface (R1), translation through thickness t, and refraction at second surface (R2) with proper sign convention
  • Part (a): Proper matrix multiplication order R2 × T × R1 yielding final system matrix with elements A, B, C, D satisfying AD-BC=1 for unimodular property
  • Part (b): Application of Fresnel coefficients at each interface with correct amplitude reflection/transmission coefficients r12, t12, r23, t23
  • Part (b): Inclusion of phase factor δ = (4πn2h cosθ2)/λ and summation of infinite series leading to Airy formula for reflectivity
  • Part (c): Expression for torsional rigidity C = πGr⁴/(2l) for solid shaft and C = πG(r₂⁴-r₁⁴)/(2l) for hollow shaft
  • Part (c): Setting equal rigidity ratings, substituting r₂ = 2r₁, solving for r₁ in terms of r, then calculating mass ratio and percentage reduction
  • Clear statement of assumptions: paraxial approximation for (a) and (b), same material (same G, ρ) for (c), thin film interference conditions

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all fundamental concepts: for (a) understands that system matrix relates output ray to input ray; for (b) distinguishes between amplitude and intensity reflectivity with proper boundary conditions; for (c) correctly identifies that torsional rigidity depends on polar moment of inertia, not just cross-sectional areaIdentifies basic concepts but confuses some details: may mix up matrix order in (a), or uses intensity instead of amplitude coefficients in (b), or equates polar moments incorrectly in (c)Fundamental misconceptions: treats thick lens as thin lens in (a), ignores multiple reflections in (b), or uses bending stress formula instead of torsion formula in (c)
Derivation rigour25%12.5Step-by-step derivations with no logical gaps: explicit matrix multiplication with all elements shown for (a); clear geometric series summation with convergence justification for (b); systematic algebraic manipulation with explicit substitution r₂=2r₁ for (c)Derivations mostly correct but skips key steps or uses 'it can be shown that' excessively; minor algebraic errors that don't affect final result; partial credit for correct approach with execution errorsMissing crucial derivation steps; incorrect mathematical operations; no justification for approximations made; derivations that don't lead to requested expressions
Diagram / FBD15%7.5Clear labeled diagrams: for (a) shows thick lens with principal planes, curvature centers, and ray path; for (b) depicts film with incident, reflected, transmitted rays and path difference; for (c) shows cross-section comparison of solid vs hollow shafts with dimension labelsDiagrams present but inadequately labeled or missing some elements; rough sketches without proper ray directions or missing path difference indication in (b)No diagrams despite their necessity for understanding; or completely incorrect diagrams that misrepresent the physical situation; diagrams without any labels
Numerical accuracy20%10Exact final expression for (c) showing percentage reduction as 100×[1-(15r₁⁴/r⁴)] or equivalent simplified form with correct numerical evaluation; all algebraic manipulations preserve equality; proper handling of (r₂⁴-r₁⁴) with r₂=2r₁ giving 15r₁⁴Correct approach to numerical evaluation but arithmetic errors in final percentage; or leaves answer in terms of r₁ without eliminating it; partial credit for correct setupMajor numerical errors: incorrect expansion of (2r₁)⁴-r₁⁴, or wrong mass ratio formula, or percentage calculation errors; no numerical evaluation attempted where required
Physical interpretation20%10Insightful interpretation: for (a) explains significance of system matrix elements (A=1/f for focal length, B relates to principal planes); for (b) discusses conditions for constructive/destructive interference and fringe visibility; for (c) explains why hollow shafts are preferred in aerospace and automotive applications (IISc Bangalore research on lightweight structures)Brief mention of physical significance without elaboration; standard textbook interpretations without connecting to real-world applications; misses opportunity to discuss optimization in (c)No physical interpretation provided; or incorrect interpretation of results; purely mathematical treatment without understanding physical meaning of derived expressions

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