Physics 2023 Paper I 50 marks Derive

Q8

(a) A circular ring of radius R lying on the x-y plane and centred at the origin, carries a uniform line charge λ. Find the first three terms (monopole, dipole and quadrupole) of the multipole expansion of potential V(r, θ). 20 marks (b) Two charges Q₁ = 3 nC and Q₂ = 4 nC are placed at the cartesian points (0, 2, 2) m and (0, – 2, 4) m, respectively. The z = 0 plane is connected to the ground. Calculate the electric potential and the electric field at the point (3, 2, 4) m using the method of images. 15 marks (c) Use the Maxwell-Boltzmann distribution to find the number of oxygen molecules whose velocities lie between 195 m/s and 205 m/s at 0°C. The given mass of oxygen gas is 0·1 kg. (Assume mass of proton to be 1·66 × 10⁻²⁷ kg) 15 marks

हिंदी में प्रश्न पढ़ें

(a) x-y तल में स्थित और मूल-बिंदु पर केंद्रित एक अर्ध्व्यास R के वृत्ताकार वलय पर एक एकसमान रैखिक आवेश λ धारित है । विभव V(r, θ) के बहुध्रुव प्रसार के प्रथम तीन पद (एकध्रुव, द्विध्रुव और चतुर्ध्रुव) ज्ञात कीजिए । 20 (b) दो आवेश Q₁ = 3 nC और Q₂ = 4 nC कार्तीय बिंदुओं (0, 2, 2) m और (0, – 2, 4) m पर क्रमशः रखे गए हैं । z = 0 तल भू-संपर्कित है । प्रतिबिंब विधि का प्रयोग करते हुए बिंदु (3, 2, 4) m पर विद्युत विभव और विद्युत-क्षेत्र की गणना कीजिए । 15 (c) मैक्सवेल-बोल्ट्ज़मान बंटन का प्रयोग करते हुए उन ऑक्सीजन अणुओं की संख्या ज्ञात कीजिए जिनका 0°C पर वेग 195 m/s और 205 m/s के बीच है । ऑक्सीजन गैस का दिया गया द्रव्यमान 0·1 kg है । (प्रोटॉन का द्रव्यमान 1·66 × 10⁻²⁷ kg मान लीजिए) 15

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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

Derive the multipole expansion for part (a) using Legendre polynomials and spherical harmonics, spending ~40% of effort on this highest-weight section. For part (b), apply the method of images systematically with proper image charge placement and superposition, allocating ~30% of time. For part (c), derive the number density from Maxwell-Boltzmann distribution with proper integration limits and molecular mass calculation, using remaining ~30%. Structure: state key formulas → step-by-step derivation → substitution → final numerical result with units.

Key points expected

  • Part (a): Multipole expansion of charged ring potential with monopole term V₀ = λR/(4πε₀r), dipole term zero by symmetry, and quadrupole term involving P₂(cosθ)
  • Part (a): Correct use of generating function for Legendre polynomials 1/|r-r'| = Σ(r'<r) (r'/r)^l P_l(cosγ) with γ being angle between r and ring element
  • Part (b): Image charges placement: Q₁' = -Q₁ at (0,2,-2) and Q₂' = -Q₂ at (0,-2,-4) due to grounded z=0 plane
  • Part (b): Superposition of four contributions (two real + two image charges) for potential and field at (3,2,4)
  • Part (c): Maxwell-Boltzmann speed distribution f(v) = 4π(m/2πkT)^(3/2) v² exp(-mv²/2kT) with m = 32×1.66×10⁻²⁷ kg
  • Part (c): Number of molecules N = nN_A = (0.1/0.032)×6.022×10²³, then dN = N·f(v)·Δv with Δv = 10 m/s
  • Part (c): Proper temperature conversion to 273 K and evaluation of Gaussian integral approximation for narrow velocity range

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies multipole expansion framework for (a), image charge method for grounded plane in (b), and Maxwell-Boltzmann statistics for (c); recognizes oxygen is diatomic (O₂) with molecular mass 32 amuBasic formulas correct but misses symmetry arguments in (a) or image charge sign in (b); treats oxygen as atomic or uses wrong molecular massFundamental misconceptions: uses point charge formula for ring, ignores grounding condition, or applies wrong statistical distribution
Derivation rigour25%12.5Complete derivations: (a) integrates over ring angle using addition theorem, shows dipole cancellation explicitly; (b) derives image positions systematically; (c) sets up proper integral with exact differentialsCorrect final formulas but skips key steps: assumes dipole zero without proof, states image positions without derivation, or jumps to approximation in (c)Missing derivations, logical gaps, or mathematically invalid steps like incorrect expansion orders or wrong integration limits
Diagram / FBD15%7.5Clear 3D sketch for (a) showing ring, field point with angles; accurate diagram for (b) with coordinate axes, real charges, image charges below z=0, and field point; labeled geometry with distances2D sketches or incomplete diagrams missing one component; adequate but cluttered visualizationNo diagrams, or seriously misleading sketches showing wrong charge positions or incorrect coordinate system
Numerical accuracy25%12.5Precise calculations: correct distance evaluations in (b) using 3D distance formula, proper handling of vector components for E-field; (c) accurate exponent evaluation and final molecule count with correct powers of 10Correct method but arithmetic errors, unit inconsistencies, or significant figure mishandling; partial credit for correct setupMajor calculation errors, wrong orders of magnitude, missing units, or incorrect substitutions
Physical interpretation15%7.5Interprets (a) quadrupole as leading anisotropic term; explains (b) image method as enforcing boundary condition V=0 at z=0; discusses (c) result relative to most probable speed and Maxwell-Boltzmann peakBrief comments on results without deep insight; mentions boundary conditions or distribution shape superficiallyNo physical interpretation, purely mathematical manipulation without understanding significance of results

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