Physics 2023 Paper I 50 marks Derive

Q7

(a) A neutral atom consists of a point nucleus +q surrounded by a uniformly charged spherical cloud (-q) of radius r. Show that when such an atom is placed in a weak external electric field E⃗, the atomic polarizability of the atom is proportional to the volume of the sphere. 15 marks (b) A piston-cylinder device initially contains air at 150 kPa and 27°C. At this state, the piston is resting on a pair of stops, as shown in the figure, and the enclosed volume is 400 L. The mass of the piston is such that a 350 kPa pressure is required to move it. The air is now heated until the volume is doubled. Determine: (i) the final temperature, (ii) the work done by the air, and (iii) the total heat transferred to air. 20 marks Given: U₃₀₀ ₖ = 214 kJ/kg and U_final = 1113 kJ/kg Gas constant of air, R = 0·287 kPa.m³/kg.K (c) A spherical shell of radius R, carrying a uniform surface charge σ, is set spinning at angular velocity ω about its axis. Find the vector potential it produces at point r⃗ . 15 marks

हिंदी में प्रश्न पढ़ें

(a) एक उदासीन परमाणु में एकसमान रूप से आवेशित त्रिज्या r के गोलीय अभ्र (-q) से घिरा एक बिंदु नाभिक +q है । दर्शाइए कि जब इस प्रकार के परमाणु को एक दुर्बल बाह्य विद्युत-क्षेत्र E⃗ में रखा जाता है, तो परमाणु की परमाण्वीय ध्रुवणीयता गोले के आयतन के समानुपाती होती है । 15 (b) एक पिस्टन-सिलिंडर युक्ति प्रारंभ में 150 kPa और 27°C पर वायु धारण करती है। इस अवस्था में, पिस्टन दो अवरोधों पर, जैसा कि चित्र में दर्शाया गया है, स्थिर है और संलग्न आयतन 400 L है। पिस्टन का द्रव्यमान इस तरह से है कि इसको विस्थापित करने में 350 kPa दाब की आवश्यकता पड़ती है। अब वायु को तब तक गर्म करते हैं जब तक कि उसका आयतन दुगुना न हो जाए। ज्ञात कीजिए : (i) अंतिम तापमान, (ii) वायु के द्वारा किया गया कार्य, और (iii) वायु को स्थानांतरित की गई कुल ऊष्मा की मात्रा। 20 दिया गया है : U₃₀₀ K = 214 kJ/kg और Uअंतिम = 1113 kJ/kg वायु का गैस नियतांक, R = 0·287 kPa.m³/kg.K (c) एकसमान पृष्ठीय आवेश σ के एक R अर्ध्व्यास के गोलीय कोश को उसके अक्ष के परितः ω कोणीय वेग से घूर्णन कराया जा रहा है । उसके द्वारा r⃗ बिन्दु पर उत्पन्न सदिश विभव ज्ञात कीजिए । 15

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Directive word: Derive

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How this answer will be evaluated

Approach

Derive the atomic polarizability for part (a) using electrostatics and displacement of charge distribution; solve the thermodynamic cycle for part (b) identifying the constant-volume and constant-pressure stages with proper state calculations; derive the vector potential for part (c) using magnetic dipole moment and spherical harmonics or direct integration. Allocate approximately 30% time to (a), 40% to (b) due to its three numerical sub-parts, and 30% to (c).

Key points expected

  • Part (a): Calculate electric field inside uniformly charged sphere using Gauss's law, find displacement of nucleus relative to cloud center, express induced dipole moment p = qd, and show α = 4πε₀r³ ∝ volume
  • Part (b-i): Identify state 1 (150 kPa, 300K, 400L), state 2 (350 kPa, V=400L, isochoric heating), state 3 (350 kPa, 800L), apply ideal gas law to find T₃ = 1400K or 1127°C
  • Part (b-ii): Calculate work as W = PΔV for constant pressure process 2→3 only (W₁₂ = 0 for isochoric), yielding W = 350 kPa × 0.4 m³ = 140 kJ
  • Part (b-iii): Apply first law Q = ΔU + W, find mass m = P₁V₁/RT₁, calculate ΔU = m(u₃ - u₁), sum to get total heat transfer ≈ 766-770 kJ
  • Part (c): Recognize spinning charged shell creates magnetic dipole moment m = (4π/3)σωR⁴, derive vector potential A = (μ₀/4π)(m×r̂)/r² for r>R (dipole approximation) or exact solution using surface current K = σv
  • Part (c) alternative: Direct integration of A = (μ₀/4π)∫K(r')/|r-r'| da' with proper handling of azimuthal symmetry and Legendre polynomial expansion

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies: (a) the displacement mechanism creating induced dipole and linear restoring force; (b) the two-stage heating process with stops constraint and ideal gas behavior; (c) equivalence of rotating charged shell to magnetic dipole or current distribution with proper boundary conditionsIdentifies basic concepts but confuses: (a) treats cloud as point charge or misses linear restoring nature; (b) assumes single-stage process or misidentifies constant-pressure condition; (c) states dipole formula without justification from surface currentsFundamental errors: (a) equates polarizability to simple Coulomb force; (b) applies isothermal or adiabatic relations incorrectly; (c) confuses vector potential with scalar potential or magnetic field
Derivation rigour20%10Complete mathematical rigor: (a) explicit E-field integration, displacement d derivation, and α ∝ r³ proof; (b) clear state table with P, V, T for all states and explicit first law application; (c) proper surface current formulation K = σωRsinθ φ̂ and multipole expansion or exact integrationCorrect final formulas with gaps: (a) states E-field result without derivation; (b) correct numerical answer with unclear intermediate steps; (c) quotes dipole A-field without deriving m from rotating chargeMissing critical steps: (a) no Gauss's law application; (b) algebraic errors in ideal gas calculations; (c) dimensional inconsistencies or incorrect angular dependence in A-field
Diagram / FBD15%7.5Clear labeled diagrams: (a) nucleus displacement from cloud center with E-field vectors; (b) P-V diagram showing vertical line (1→2) and horizontal line (2→3), with piston-cylinder schematic; (c) spherical shell with ω, surface current direction, and field point rBasic sketches present but incomplete: (a) simple sphere without displacement shown; (b) P-V diagram without state labels or missing piston diagram; (c) sphere with ω arrow but no coordinate systemNo diagrams or seriously misleading figures: unlabeled axes, wrong process directions on P-V diagram, or missing geometric setup for integration limits
Numerical accuracy25%12.5Precise calculations: (b-i) T₃ = 1400 K exactly; (b-ii) W = 140 kJ with proper sign convention; (b-iii) m ≈ 0.697 kg, ΔU ≈ 626 kJ, Q ≈ 766 kJ with unit consistency throughout; proper handling of given internal energy valuesCorrect method with minor errors: temperature in Celsius not Kelvin for intermediate steps, arithmetic errors within 10%, correct final formula with substitution errorsMajor numerical errors: wrong mass calculation (forgetting R value), confusing kPa with Pa, temperature not converted to Kelvin, or order-of-magnitude errors in work/heat
Physical interpretation20%10Insightful physical commentary: (a) explains why α ∝ volume indicates atomic polarizability scales with electron cloud size, connecting to dielectric theory; (b) interprets work done against piston weight, explains why Q > ΔU due to boundary work; (c) relates rotating shell to magnetic dipole radiation and atomic analogy (electron spin)Minimal interpretation: states results without physical significance, or generic statements about electromagnetic/thermodynamic principles without question-specific insightNo physical interpretation provided, or incorrect physical significance assigned (e.g., claiming process is reversible when irreversible, or confusing vector potential with measurable field)

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