Q3
(a) A homogeneous right triangular pyramid with the base side $a$ and height $\dfrac{3a}{2}$ is shown below. Obtain the moment of inertia tensor of the pyramid : 20 marks (b) Newton's rings are observed between a spherical surface of radius of curvature 100 cm and a plane glass plate. The diameters of 4th and 15th bright rings are 0·314 cm and 0·574 cm, respectively. Calculate the diameters of 24th and 36th bright rings and also the wavelength of light used. 15 marks (c) In He-Ne laser, what is the function of He gas? Explain the answer with the help of energy level diagram for He-Ne laser. 15 marks
हिंदी में प्रश्न पढ़ें
(a) एक समांगी समकोणीय पिरामिड, जिसका आधार पार्श्व $a$ और जिसकी ऊँचाई $\dfrac{3a}{2}$ है, नीचे चित्र में दिखाया गया है। पिरामिड का जड़त्व-आघूर्ण प्रदिश (टेंसर) ज्ञात कीजिए : 20 (b) न्यूटन के वलय (रिंग) 100 cm वक्रता-त्रिज्या की एक गोलाकार सतह और समतल कांच की प्लेट के मध्य देखे जाते हैं। 4वें और 15वें दीप्त वलयों के व्यास क्रमशः: 0·314 cm और 0·574 cm हैं। 24वें और 36वें दीप्त वलयों के व्यास और प्रयुक्त प्रकाश के तरंगदैर्ध्य की गणना कीजिए। 15 (c) He-Ne लेजर में He गैस की क्या भूमिका है? He-Ne लेजर के लिए ऊर्जा स्तर आरेख की सहायता से उत्तर स्पष्ट कीजिए। 15
Directive word: Derive
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How this answer will be evaluated
Approach
Derive requires systematic mathematical derivation with logical progression. Allocate approximately 40% time to part (a) for the moment of inertia tensor derivation using appropriate coordinate system and integration; 30% to part (b) for Newton's rings calculation with proper formula application and error propagation; 30% to part (c) for explaining He-Ne laser mechanism with energy level diagram. Structure: begin with coordinate setup for (a), proceed through integration, then solve (b) step-by-step showing diameter calculations, conclude with laser physics explanation and diagram.
Key points expected
- Part (a): Set up coordinate system with origin at apex or centroid, establish limits for triangular pyramid geometry, compute I_xx, I_yy, I_zz and off-diagonal elements using volume integral ∫ρ(r)(r²δ_ij - x_i x_j)dV
- Part (a): Apply parallel axis theorem if needed and present final 3×3 symmetric inertia tensor matrix with elements in terms of M and a
- Part (b): Use modified Newton's rings formula for bright rings D_n² = 4(n-½)λR, set up simultaneous equations using n=4 and n=15 data to find λ and verify consistency
- Part (b): Calculate D_24 and D_36 using derived λ, showing propagation of significant figures and physical reasonableness check (D_n ∝ √n)
- Part (c): Explain He as buffer gas providing efficient excitation transfer via resonant collision to Ne 2s and 3s levels, preventing depopulation of lower levels
- Part (c): Draw accurate energy level diagram showing He 2³S, 2¹S metastable states, collisional transfer to Ne 3s₂, 2s₂ upper laser levels, and transitions for 632.8 nm (3s₂→2p₄) and 1.15 μm, 3.39 μm lines with radiative decay to 1s ground state
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | Correctly identifies inertia tensor definition for continuous bodies; applies proper Newton's rings condition for bright fringes (2t = (m+½)λ); accurately describes resonant energy transfer mechanism in He-Ne laser and metastable state role | Basic understanding of inertia tensor but confuses discrete/continuous formulations; uses dark ring formula or incorrect interference condition; describes He as 'helper' gas without explaining resonant transfer physics | Confuses moment of inertia with angular momentum; applies lens maker's formula or thin film interference incorrectly; states He provides 'energy' or 'cooling' without specific atomic physics mechanism |
| Derivation rigour | 25% | 12.5 | For (a): complete volume integration with proper Jacobian, explicit limits 0≤z≤3a/2, 0≤y≤a(1-2z/3a), 0≤x≤y; clear stepwise algebra leading to symmetric tensor. For (b): systematic elimination to find λ, explicit propagation to D_24, D_36 with units. For (c): logical flow from discharge → He excitation → collisional transfer → population inversion → stimulated emission | Correct integration setup but algebraic errors in final tensor elements; correct λ calculation but arithmetic errors in subsequent diameters; describes processes sequentially but lacks explicit causal connections in laser mechanism | Missing integration steps or assumes point mass formula; plugs numbers without showing formula rearrangement; presents disconnected facts about laser without energy flow logic |
| Diagram / FBD | 15% | 7.5 | Clear 3D sketch of pyramid with coordinate axes labeled; Newton's rings apparatus diagram with R, t, r labeled; accurate Grotrian diagram for He-Ne showing energy scales, metastable states, laser transitions with wavelengths, and radiative decay arrows | 2D representation of pyramid without clear orientation; generic interference diagram without specific labels; energy level diagram with levels but missing transition arrows or wavelength labels | No geometric diagram for pyramid; no ray diagram for Newton's rings; text description only for energy levels or confused diagram showing wrong transitions (e.g., direct electrical excitation of Ne) |
| Numerical accuracy | 20% | 10 | λ calculated as ~589 nm (or precise value from data), D_24 ≈ 0.74 cm, D_36 ≈ 0.91 cm with 3 significant figures; self-consistency check using D_n²/D_m² = (n-½)/(m-½); proper unit conversion throughout (cm to m where needed) | Correct method but rounding errors in final values; inconsistent significant figures; correct order of magnitude for wavelengths and diameters | Order-of-magnitude errors (e.g., λ in mm or μm range); incorrect formula application giving imaginary diameters; unit confusion between cm and m in R value |
| Physical interpretation | 20% | 10 | Explains why tensor is symmetric and principal axes significance; interprets diameter progression showing √n dependence and validates against measurement precision; explains why 632.8 nm dominates (highest gain, selective cavity mirrors) and why He-Ne is low power/cw operation | States tensor symmetry without physical meaning; notes diameter increases with n without quantitative relationship; mentions 632.8 nm is 'common' red light without gain competition explanation | No interpretation of tensor physical meaning; no validation of calculated results against experimental expectations; no explanation of why specific wavelengths lase or role of cavity design |
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