Physics 2025 Paper II 50 marks Compulsory Derive

Q5

(a) Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0). 10 marks (b) ρ⁰ and K⁰ mesons both decay mostly to π⁺ and π⁻. Why the mean lifetime of ρ⁰ is 10⁻²³ s, whereas that of K⁰ is 0·89 × 10⁻¹⁰ s? 10 marks (c) Find the radius of the interstitial sphere which can just fit into the void at the body centre of the fcc structure coordinated by the facial atoms. 10 marks (d) In powder diffraction method pattern for lead with radiation of wavelength λ = 1·54 Å, the (220) Bragg reflection angle is θ = 32°. Find the radius of the atom. 10 marks (e) (i) What are the differences in electrical characteristics of FET (JFET) and MOSFET? 7 marks (ii) How does n-channel FET differ from p-channel FET? 3 marks

हिंदी में प्रश्न पढ़ें

(a) दर्शाइए कि त्रिक अवस्था (S = 1) की ऊर्जा, एकल अवस्था (S = 0) की ऊर्जा के बराबर नहीं होती है। 10 (b) ρ⁰ और K⁰ मेसॉन दोनों ही अधिकतर π⁺ और π⁻ में विघटित हो जाते हैं। ρ⁰ का औसत जीवनकाल 10⁻²³ s क्यों है, जबकि K⁰ का औसत जीवनकाल 0·89 × 10⁻¹⁰ s है? 10 (c) उस अंतरालीय गोलक की त्रिज्या ज्ञात कीजिए, जो फलक परमाणुओं द्वारा समन्वित fcc संरचना के अंतःकेंद्र पर रिक्ति में समायोजित हो सकता है। 10 (d) तरंगदैर्ध्य λ = 1·54 Å के विकिरण से सीसा (लेड) के लिए पाउडर विवर्तन विधि प्रतिरूप में (220) श्रेण परावर्तन कोण θ = 32° है। परमाणु का अर्धव्यास ज्ञात कीजिए। 10 (e) (i) FET (JFET) व MOSFET के वैद्युत अभिलक्षणों में क्या अंतर हैं? 7 (ii) n-चैनल का FET, p-चैनल के FET से किस प्रकार भिन्न है? 3

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Directive word: Derive

This question asks you to derive. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

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How this answer will be evaluated

Approach

Begin with a brief introduction acknowledging the diverse physics domains covered (quantum mechanics, particle physics, solid state, and electronics). For part (a), derive the energy splitting using exchange interaction and spin wavefunctions; for (b), explain using strong vs. weak decay selection rules; for (c), derive the octahedral void geometry in FCC; for (d), apply Bragg's law and unit cell calculation; for (e)(i)-(ii), tabulate comparative characteristics. Allocate approximately 20% time to (a), 15% to (b), 20% to (c), 20% to (d), and 25% to (e) combined, reflecting mark distribution and derivation complexity.

Key points expected

  • (a) Derivation of triplet-singlet energy splitting using symmetric/antisymmetric spin wavefunctions and exchange integral J, showing E_triplet = E_0 - J and E_singlet = E_0 + J for two-electron system
  • (b) Explanation of ρ⁰ decay via strong interaction (OZI-allowed, resonant, Γ ~ 150 MeV) versus K⁰ decay via weak interaction (ΔS = 1, strangeness changing, CP violation context with K_S and K_L)
  • (c) Geometric derivation: octahedral void radius r = 0.414R where R is atomic radius, using FCC geometry with face-center to body-center distance relationship
  • (d) Application of Bragg's law nλ = 2d sinθ, calculation of d_220 = a/√8, determination of lattice parameter a, and atomic radius r = a√2/4 for FCC lead
  • (e)(i) Distinction between JFET (depletion-mode only, pn-junction gate, higher input impedance ~10⁹ Ω) and MOSFET (enhancement/depletion modes, insulated gate, higher input impedance ~10¹² Ω, threshold voltage concept)
  • (e)(ii) Carrier type (electrons vs holes), mobility differences, threshold voltage polarity, and drain current direction in n-channel versus p-channel FETs

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies exchange interaction mechanism for (a), strong vs weak interaction hierarchy and OZI rule for (b), void geometry in FCC for (c), Bragg diffraction and FCC structure for (d), and device physics principles for (e); no conceptual errors across any sub-partMajor concepts correct but some confusion between interaction types or device modes; minor errors in identifying selection rules or gate structuresFundamental misconceptions such as attributing ρ⁰ decay to weak interaction, confusing tetrahedral with octahedral voids, or equating JFET and MOSFET operation principles
Derivation rigour20%10Complete mathematical derivations: explicit spin wavefunction construction with ± notation for (a), proper lifetime-width relation τ = ℏ/Γ for (b), step-by-step geometric proof for (c), full algebraic manipulation from Bragg's law to atomic radius for (d)Correct final formulas but skips critical steps; assumes results without showing symmetric/antisymmetric construction or omits geometric justificationMissing derivations entirely; states results without proof; incorrect algebraic manipulation leading to wrong numerical factors
Diagram / FBD15%7.5Clear FCC unit cell diagram showing octahedral void with labeled atomic and void radii for (c); energy level diagram for singlet-triplet splitting for (a); device structure diagrams comparing JFET and MOSFET cross-sections for (e)Diagrams present but poorly labeled or missing critical features; generic crystal structure without specific void indicationNo diagrams where essential; confusing or incorrect diagrams that misrepresent geometry or device structure
Numerical accuracy20%10Precise calculation: r_void = 0.414R exactly for (c); correct atomic radius of lead ~1.75 Å for (d); proper order-of-magnitude verification for lifetime ratio ~10¹³ in (b)Correct method but arithmetic errors; wrong significant figures; approximate values without unit conversionOrder-of-magnitude errors; incorrect formula substitution; missing units; no numerical verification of reasonableness
Physical interpretation25%12.5Insightful connection: Pauli exclusion and ferromagnetism relevance for (a); CP violation and particle physics standard model context for K⁰; materials science implications of void size for interstitial alloys; semiconductor industry relevance of FET comparisonSome physical context but limited to immediate problem; misses broader implications or connections to other physics domainsPurely mathematical treatment with no physical meaning; unable to explain why results matter or relate to observable phenomena

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