(a) (i) Explain the phenomenon of double refraction. What are positive and negative crystals? Give their examples. (5 marks)

(ii) What do you understand by optical activity? A linearly polarized light is propagating along the optic axis of a quartz

Question

(a) (i) Explain the phenomenon of double refraction. What are positive and negative crystals? Give their examples. (5 marks)

(ii) What do you understand by optical activity? A linearly polarized light is propagating along the optic axis of a quartz crystal of thickness 0·2 cm. If the difference in the refractive indices corresponding to right circularly polarized and left circularly polarized beams is 7×10⁻⁵ and the wavelength of the light is 0·5 μm, calculate the angle of polarization. (10 marks)

(b) (i) What do you understand by attenuation in optical fibers? What are the factors responsible for the attenuation? (5 marks)

(ii) Consider a 10 mW laser beam passing through a 50 km fiber link of attenuation 0·5 dB/km. Calculate the power of the laser at the end of the link. (10 marks)

(c) (i) State and explain the Hooke's law of elasticity. Briefly discuss the features of stress-strain diagram for the behaviour of a wire undergoing increasing stress. (10 marks)

(ii) Explain the Poiseuille's equation for the rate of flow of a liquid through a capillary tube. From this, show that if two capillary tubes of radii r₁ and r₂ having lengths l₁ and l₂, respectively, are connected in series, the rate of flow of the liquid is given by

Q = (πP/8η)(l₁/r₁⁴ + l₂/r₂⁴)⁻¹

where P is the pressure across the arrangement and η is the coefficient of viscosity of the liquid. (10 marks)

UPSC Answer Check · Accepted Answer

Begin with clear definitions for each sub-part, allocating approximately 30% effort to (a) on crystal optics, 30% to (b) on fiber optics, and 40% to (c) on elasticity and fluid mechanics given its higher mark weightage. Structure as: (a) explain double refraction with crystal classification and solve optical rotation numerically; (b) define attenuation with mechanisms and calculate power loss; (c) state Hooke's law with stress-strain diagram features, then derive Poiseuille's equation and prove the series combination formula. Include labeled diagrams for crystal structures, fiber cross-section, stress-strain curve, and capillary flow geometry.
- (a)(i) Double refraction explained via anisotropic crystals; distinction between positive (n_e > n_o, e.g., quartz) and negative (n_e < n_o, e.g., calcite) crystals with correct examples
- (a)(ii) Optical activity as rotation of plane of polarization; correct application of θ = (πd/λ)(n_L - n_R) or equivalent formula yielding θ ≈ 25.2° or 0.44 rad
- (b)(i) Attenuation defined as power loss per unit length (dB/km); factors: absorption (intrinsic/extrinsic), scattering (Rayleigh, Mie), bending losses
- (b)(ii) Correct use of P_out = P_in × 10^(-αL/10) or equivalent logarithmic relation yielding P_out ≈ 0.316 mW or -5 dBm
- (c)(i) Hooke's law within proportional limit (σ = Eε); stress-strain diagram showing: proportional limit, elastic limit, yield point, ultimate stress, breaking stress, and plastic region
- (c)(ii) Poiseuille's equation derived from viscous force balance and pressure gradient; correct derivation of series combination showing equivalent resistance analogy with Q = πP/8η(l₁/r₁⁴ + l₂/r₂⁴)⁻¹

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Precise definitions: double refraction as birefringence with ordinary/extraordinary rays; optical activity as molecular asymmetry effect; attenuation mechanisms clearly distinguished; Hooke's law stated with proportional limit caveat; all physical concepts applied correctly across (a), (b), and (c)	Basic definitions correct but missing nuances (e.g., conflates elastic and proportional limits, or lists attenuation factors without explanation); minor errors in crystal classification	Fundamental misconceptions: confuses double refraction with optical activity, omits proportional limit in Hooke's law, or misidentifies attenuation as dispersion
Derivation rigour	20%	10	Poiseuille's equation derived from first principles (viscous drag = pressure force, velocity profile integration); series combination derived by equating pressure drops and adding resistances with clear algebraic steps; logical flow with explicit assumptions (steady, incompressible, laminar flow, no-slip)	Poiseuille's equation stated correctly but derivation skips key steps (e.g., assumes parabolic profile without proof); series combination result obtained but with unclear resistance analogy or algebraic gaps	Formula stated without derivation; incorrect approach (e.g., adds flow rates instead of equating them in series); major gaps in mathematical reasoning
Diagram / FBD	20%	10	Four quality diagrams: (a) calcite crystal showing double refraction with O-ray and E-ray; (b) optical fiber structure with core/cladding and attenuation schematic; (c) complete stress-strain diagram with all labeled regions and points; (d) capillary tube with velocity profile and force balance FBD; all properly labeled with symbols	Two to three adequate diagrams with basic labels; missing some features (e.g., no yield point marked, or no velocity profile in capillary); rough sketches without proper scaling	One or no diagrams; poorly labeled or irrelevant diagrams; missing critical diagrams for stress-strain or capillary flow derivation
Numerical accuracy	20%	10	Both calculations correct with proper unit handling: (a)(ii) θ = (π × 0.2 × 7×10⁻⁵)/(0.5×10⁻⁶) ≈ 25.2° (or 0.44 rad); (b)(ii) P_out = 10 × 10^(-0.5×50/10) = 10 × 10⁻²·⁵ ≈ 0.316 mW; all steps shown with SI unit conversions	One calculation correct with minor errors in the other (e.g., wrong exponent in dB conversion, or degree/radian confusion); correct formulas but arithmetic slips	Both calculations incorrect or missing; major unit errors (e.g., uses dB as linear factor); no working shown for numerical parts
Physical interpretation	20%	10	Clear physical insight: explains why crystals become positive/negative based on molecular structure; connects optical activity to enantiomeric molecular arrangement; relates attenuation to fiber quality and communication range; interprets stress-strain regions in terms of atomic bonding; explains why series combination uses inverse sum of resistances (flow analogy to electrical circuits)	Some physical interpretation present but superficial; mentions applications without explaining underlying physics; limited connection between formula and physical meaning	Purely mathematical/formulaic approach with no physical insight; fails to explain what derived results mean in practical contexts (e.g., no comment on significance of power loss in fibers)

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