Physics 2025 Paper I 50 marks Derive

Q7

(a) A ternary system consists of three components (A, B and C) in equilibrium with two phases. Determine the number of degrees of freedom using the Gibb's phase rule and discuss the effect of pressure and temperature variations on the phase equilibrium. (b) Discuss briefly the considerations which led Van der Waals to modify the gas equation. What are the critical constants of a gas ? Calculate the values of these constants in terms of the constants of the Van der Waals equation. (15 marks) (c) Consider a conducting sphere of radius 'a' in a uniform electric field $\vec{E}$. Find the induced surface charge density on the sphere and determine the electric field $\vec{E}$ at a point P characterized by radius vector $\vec{r}$. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) एक त्रिभुजी निकाय में दो प्रावस्थाओं के साथ संतुलन में तीन घटक (A, B और C) हैं । गिब्स के प्रावस्था नियम का प्रयोग करके स्वतंत्रता की कोटियों की संख्या निर्धारित कीजिए और प्रावस्था संतुलन पर दाब तथा तापक्रम के विचरणों के प्रभाव की विवेचना कीजिए । (b) उन निमित्तियों/विचारों की संक्षेप में चर्चा कीजिए जिन्होंने वान्डर वाल्स को गैस समीकरण को संशोधित करने के लिए प्रेरित किया। एक गैस के क्रांतिक नियतांक क्या हैं ? वान्डर वाल्स समीकरण के नियतांकों के पदों में इन नियतांकों के मानों की गणना कीजिए। (15 अंक) (c) एक एकसमान विद्युत-क्षेत्र $\vec{E}$ में अर्धव्यास 'a' के एक चालक गोले को लीजिए। गोले पर प्रेरित पृष्ठीय आवेश घनत्व ज्ञात कीजिए और त्रिज्या सदिश $\vec{r}$ द्वारा अभिलक्षित बिन्दु P पर विद्युत-क्षेत्र $\vec{E}$ निर्धारित कीजिए। (20 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

This multi-part question requires deriving key results across thermodynamics and electrostatics. Allocate approximately 15% time to part (a) on Gibbs phase rule, 35% to part (b) on Van der Waals equation and critical constants, and 50% to part (c) on the conducting sphere problem which carries the highest marks. Structure with clear headings for each sub-part, present derivations step-by-step with justified assumptions, and conclude with physical interpretations of each result.

Key points expected

  • Part (a): Correct application of Gibbs phase rule F = C - P + 2 for ternary system (C=3, P=2) yielding F=3 degrees of freedom; discussion of how fixing temperature and pressure reduces variance
  • Part (b): Physical reasoning for Van der Waals modifications (finite molecular volume via 'b', intermolecular attractions via 'a'); derivation of critical constants T_c = 8a/27Rb, V_c = 3b, P_c = a/27b² from inflection point conditions (∂P/∂V)_T=0 and (∂²P/∂V²)_T=0
  • Part (c): Setup using superposition of uniform field and induced dipole potential; boundary condition V=constant on sphere surface; derivation of induced surface charge density σ = 3ε₀E₀cosθ; expression for total field at arbitrary point P using Legendre expansion or method of images
  • Clear statement of assumptions: ideal solution behavior for (a), single-phase fluid for (b), perfectly conducting isolated sphere for (c)
  • Dimensional consistency checks and limiting case verification (e.g., field reduces to applied field far from sphere)
  • Physical interpretation: screening effect of conductor, dipole moment of induced distribution p = 4πε₀a³E₀

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%2Correctly identifies F=3 for part (a); accurately states both Van der Waals corrections with physical origins for (b); properly applies uniqueness theorem and boundary conditions for conducting sphere in (c); no conceptual confusion between intensive/extensive variables or electrostatic boundary conditionsMinor errors in phase rule application (e.g., forgetting +2) or incomplete physical reasoning for Van der Waals corrections; correct general approach for sphere but confused about whether field inside is zero or constantFundamental misunderstanding of phase rule (e.g., F=C-P), confuses critical point with triple point, or applies dielectric boundary conditions instead of conductor conditions for the sphere
Derivation rigour20%2Step-by-step derivations with explicit mathematical justification: shows (∂P/∂V)_T = (∂²P/∂V²)_T = 0 for critical point; solves Laplace's equation with azimuthal symmetry for sphere; justifies why only l=0,1 terms survive; clear logical flow from potential to field to charge densityCorrect final formulas but skips key steps (e.g., states critical conditions without showing derivatives, jumps to solution for potential without separation of variables); some algebraic manipulation shown but gaps remainFinal answers stated without derivation; or serious mathematical errors in differentiation, algebraic manipulation, or solving differential equations; incorrect application of chain rule or partial derivatives
Diagram / FBD20%2Clear diagram for part (c) showing: uniform external field E₀, induced dipole field lines, equipotential spherical surface, coordinate system with θ angle labeled, and field line distortion pattern; may include P-V isotherms sketch for part (b) showing critical isotherm with horizontal inflectionBasic diagram present for sphere but missing key elements (e.g., no angle θ labeled, no indication of field asymmetry); or diagram only for one sub-part when multiple could benefit from visualizationNo diagrams despite physical situations that clearly warrant them; or misleading diagrams showing field lines penetrating conductor or incorrect isotherm shapes
Numerical accuracy20%2Exact symbolic expressions for all quantities: F=3 for (a); T_c, V_c, P_c in terms of a,b,R with correct numerical factors (8/27, 3, 1/27); σ(θ) = 3ε₀E₀cosθ and correct field expression E(r,θ) showing 1/r³ dipole falloff for (c)Correct symbolic forms but wrong numerical prefactors (e.g., factor of 2 errors in critical constants); or correct final answers with intermediate algebraic slips that cancelMissing factors entirely, wrong powers of variables (e.g., a² instead of a³ for dipole moment), or dimensional inconsistency (e.g., charge density with wrong units); critical constants not expressed in terms of a and b
Physical interpretation20%2Insightful discussion: for (a) explains why ternary systems need 3 variables controlled for phase specification; for (b) connects critical point to liquid-gas indistinguishability and critical opalescence; for (c) explains dipole moment, field enhancement at poles, shielding effect, and connection to polarizability α = 4πε₀a³Some physical interpretation present but superficial or formulaic; mentions 'dipole' but doesn't explain origin or magnitude; states facts without explaining significancePurely mathematical answer with no physical discussion; or incorrect physical interpretations (e.g., claims field inside conductor is zero rather than constant, confuses induced charge with polarization charge)

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