Physics 2024 Paper I 50 marks Derive

Q6

(a) Using Maxwell's equations, obtain Poisson's equation and Laplace's equation. The region $-\frac{\pi}{2} < \frac{z}{z_0} < \frac{\pi}{2}$ has a charge density $\rho = 10^{-8} \cos\left(\frac{z}{z_0}\right)$ (C/m³). Elsewhere the charge density is zero. Find the electric potential $V$ and electric field $E$ from the Poisson's equation. (15 marks) (b) (i) What is anomalous dispersion? How does the phenomenon of dispersion lead to the separation of white light into its constituent colours? (5 marks) (ii) Consider a uniformly magnetized sphere of radius $a$ and magnetization $\vec{M} = M_0\hat{z}$ surrounded by a vacuum region. Obtain an expression for scalar magnetic potential for $r < a$. (10 marks) (c) Define internal energy U, Helmholtz's function F, enthalpy H, Gibbs' potential G and hence obtain the four Maxwell's thermodynamic relations. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) मैक्सवेल समीकरणों का प्रयोग करते हुए प्वासों समीकरण और लाप्लास समीकरण प्राप्त कीजिये। $-\frac{\pi}{2} < \frac{z}{z_0} < \frac{\pi}{2}$ क्षेत्र में आवेश घनत्व $\rho = 10^{-8} \cos\left(\frac{z}{z_0}\right)$ (C/m³) है तथा अन्य स्थानों पर आवेश घनत्व शून्य है। प्वासों समीकरण से विद्युत विभव $V$ और विद्युत क्षेत्र $E$ ज्ञात कीजिये। (15 अंक) (b) (i) असंगत विष्केपण क्या है? विष्केपण की परिघटना से किस प्रकार श्वेत प्रकाश का उसके संघटक रंगों में पृथक्करण होता है? (5 अंक) (ii) अर्धव्यास $a$ और चुंबकन $\vec{M} = M_0\hat{z}$ के समान रूप से चुंबकित एक गोले को लीजिये, जिसके चारों ओर निर्वात क्षेत्र है। अदिश चुंबकीय विभव का व्यंजक, $r < a$ के लिए प्राप्त कीजिये। (10 अंक) (c) आंतरिक ऊर्जा U, हेल्महोल्ट्ज फलन F, एन्थैल्पी H, गिब्स विभव G को परिभाषित कीजिये और फिर मैक्सवेल के चार उष्मागतिकी संबंधों को प्राप्त कीजिये। (20 अंक)

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Directive word: Derive

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How this answer will be evaluated

Approach

Derive the required equations systematically, spending approximately 30% time on part (a) due to its computational complexity, 20% on (b)(i) for conceptual explanation, 25% on (b)(ii) for boundary value problem, and 25% on (c) for thermodynamic derivations. Begin with clear definitions, proceed through step-by-step derivations with proper mathematical justification, and conclude with physical interpretations of each result.

Key points expected

Derive Poisson's equation (∇²V = -ρ/ε₀) and Laplace's equation (∇²V = 0) from Maxwell's equations, specifically using Gauss's law and the electrostatic condition E = -∇V
Solve the 1D Poisson equation for the given charge density ρ = 10⁻⁸cos(z/z₀) with appropriate boundary conditions, obtaining V(z) = (ρ₀z₀²/ε₀)cos(z/z₀) + C₁z + C₂ and E = -dV/dz ẑ
Explain anomalous dispersion as occurring near absorption resonances where dn/dλ > 0, contrasting with normal dispersion, and describe how wavelength-dependent refractive index separates white light
Obtain scalar magnetic potential Φₘ for r < a using ∇²Φₘ = 0 with boundary conditions, yielding Φₘ = -(M₀/3)r cosθ, and relate to bound surface currents
Define U, F = U - TS, H = U + pV, G = H - TS and derive all four Maxwell relations: (∂T/∂V)ₛ = -(∂p/∂S)ᵥ, (∂T/∂p)ₛ = (∂V/∂S)ₚ, (∂S/∂V)ₜ = (∂p/∂T)ᵥ, (∂S/∂p)ₜ = -(∂V/∂T)ₚ using exact differentials

Evaluation rubric

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies all physical conditions: electrostatic approximation for (a), resonance absorption for anomalous dispersion in (b)(i), magnetostatic scalar potential validity for (b)(ii), and exact differential conditions for thermodynamic potentials in (c); no conceptual conflation between scalar electric and magnetic potentials	Identifies most conditions correctly but may miss subtle points like the range of validity for scalar magnetic potential or confuse anomalous with normal dispersion; minor errors in thermodynamic variable dependencies	Fundamental misconceptions such as applying Laplace equation where ρ ≠ 0, confusing F with G, or treating magnetic scalar potential as always valid without noting ∇·B = 0 requirement
Derivation rigour	25%	12.5	Complete mathematical rigour: explicit statement of Maxwell equations used, clear vector calculus steps (curl of curl identity for Poisson), proper integration constants with boundary condition justification, Legendre transformation logic for thermodynamic potentials, and systematic derivation of all four Maxwell relations	Correct overall flow but skips key steps like justifying boundary conditions for the cosine potential, omits intermediate steps in Legendre transformations, or derives only 2-3 Maxwell relations completely	Missing crucial steps, incorrect vector identities, unjustified dropping of integration constants, or purely memorized statements without derivation structure
Diagram / FBD	10%	5	Clear diagram for part (a) showing charge distribution ρ(z) vs z and resulting V(z), E(z) profiles; spherical coordinate diagram for (b)(ii) showing M, θ, and boundary; thermodynamic potential hierarchy diagram for (c) showing Legendre transformation pathways	At least one relevant diagram present, possibly missing labels or with incomplete coordinate systems; may lack the thermodynamic potential relationship diagram	No diagrams or completely irrelevant sketches; failure to visualize the 1D geometry of the charge slab or spherical geometry of the magnetized sphere
Numerical accuracy	20%	10	Correct final expressions with proper constants: V(z) = (10⁻⁸z₀²/ε₀)cos(z/z₀) for the particular solution, correct field E = (10⁻⁸z₀/ε₀)sin(z/z₀)ẑ, and proper handling of boundary conditions at z = ±πz₀/2; correct numerical factors in magnetic potential Φₘ = -(M₀/3)r cosθ	Correct functional forms but errors in numerical prefactors (e.g., missing factor of 1/3 in magnetic potential, sign errors in integration), or incorrect handling of boundary conditions leading to wrong integration constants	Major errors in solving the differential equation, wrong functional dependence, or completely omitted numerical evaluation where required
Physical interpretation	25%	12.5	Insightful interpretation: for (a), explains why V is maximum where ρ is maximum and E is zero; for (b)(i), relates anomalous dispersion to material resonances and group velocity; for (b)(ii), interprets result as uniform B field inside sphere; for (c), explains physical meaning of each potential (F for isothermal processes, G for phase equilibrium) and significance of Maxwell relations in connecting measurable quantities	Some physical interpretation present but superficial; may state results without explaining their significance, or miss the connection between Maxwell relations and experimental thermodynamics	Purely mathematical treatment with no physical insight; failure to interpret what the derived potentials represent or why the Maxwell relations are useful; no discussion of practical implications

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