(a) Two inductors having inductances L₁ and L₂ are connected in parallel. The inductors have a mutual inductance M. Derive the expression for the effective inductance. Assume the inductors have negligible resistances. 15 marks

(b) (i) Define Joule-K

Question

(a) Two inductors having inductances L₁ and L₂ are connected in parallel. The inductors have a mutual inductance M. Derive the expression for the effective inductance. Assume the inductors have negligible resistances. 15 marks

(b) (i) Define Joule-Kelvin coefficient. Write it in its mathematical form. 5 marks

(ii) Determine the Joule-Kelvin coefficient for a van der Waals gas. Hence, obtain an expression for temperature of inversion. Discuss the conditions under which heating or cooling is produced. 10 marks

(c) Consider the interaction of an electromagnetic wave at the interface of two dielectric media. If electric field E⃗ is parallel to the plane of incidence, obtain Fresnel's equations and Brewster's law of polarization. 20 marks

UPSC Answer Check · Accepted Answer

Begin with a concise introduction stating the three physical contexts: coupled inductors, throttling processes, and electromagnetic boundary conditions. Allocate approximately 30% effort to part (a) deriving the parallel inductance formula with mutual inductance, 30% to part (b) covering Joule-Kelvin coefficient definition and van der Waals analysis, and 40% to part (c) for Fresnel's equations and Brewster's law with proper diagrams. Conclude by briefly connecting the unifying theme of energy transformations across electromagnetic and thermodynamic systems.
- Part (a): Correct application of Kirchhoff's laws to coupled parallel inductors, proper handling of mutual inductance sign (aiding/opposing), and final expression L_eff = (L₁L₂ - M²)/(L₁ + L₂ ∓ 2M)
- Part (b)(i): Precise definition of Joule-Kelvin coefficient as (∂T/∂P)_H and its thermodynamic relation μ_JK = (1/C_p)[T(∂V/∂T)_P - V]
- Part (b)(ii): Expansion of van der Waals equation, derivation of μ_JK ≈ (1/C_p)[(2a/RT) - b], inversion temperature T_i = 2a/Rb, and conditions for heating/cooling
- Part (c): Application of boundary conditions (continuity of E_tangential and B_normal), derivation of Fresnel equations for p-polarization: r_∥ = tan(θ₁-θ₂)/tan(θ₁+θ₂) and t_∥ = 2sinθ₂cosθ₁/sin(θ₁+θ₂)cos(θ₁-θ₂)
- Part (c): Derivation of Brewster's law tan θ_B = n₂/n₁ with physical explanation of complete polarization
- Clear distinction between series-aiding and series-opposing configurations in mutual inductance problems
- Physical interpretation of inversion temperature in terms of intermolecular forces (a) and molecular size (b)
- Diagram showing incident, reflected, transmitted rays with polarization directions and angles for p-polarization case

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates flawless understanding across all parts: correctly identifies dot convention for mutual inductance in (a), recognizes enthalpic constancy in Joule-Kelvin expansion for (b), and accurately applies electromagnetic boundary conditions at dielectric interfaces for (c); no conceptual conflations between isothermal, adiabatic, and throttling processes	Shows generally correct concepts with minor errors: occasional sign confusion in mutual inductance terms, imprecise statement of constant enthalpy condition, or incomplete boundary condition application; may conflate s and p polarization cases	Fundamental misconceptions evident: treats parallel inductors as simple resistive parallel combination, confuses Joule-Kelvin with isothermal expansion, or applies Snell's law without electromagnetic boundary conditions; serious errors in physical principles
Derivation rigour	25%	12.5	Mathematically rigorous derivations with all steps explicit: proper handling of coupled differential equations for inductors, systematic expansion of van der Waals equation retaining appropriate orders, and complete derivation of Fresnel coefficients from Maxwell's equations with clear identification of p-polarization geometry	Derivations mostly complete but with gaps: skips intermediate algebraic steps, assumes results without justification, or presents final formulas without showing how boundary conditions yield them; mathematically acceptable but lacks pedagogical clarity	Derivations fragmented or missing: states results without derivation, contains algebraic errors affecting final expressions, or confuses derivation with dimensional analysis; inability to connect starting principles to final results
Diagram / FBD	15%	7.5	High-quality diagrams essential for (c): clear ray diagram showing plane of incidence, incident/reflected/transmitted rays with θ₁, θ₁', θ₂ angles, E-field vectors perpendicular to plane of incidence for p-polarization, and wavefront construction; optional but helpful coupled inductor schematic for (a) showing winding directions and dot convention	Adequate diagrams present but lacking precision: angles not clearly marked, polarization directions ambiguous, or missing essential elements like wavefronts; diagram for (c) present but does not clearly distinguish p-polarization from s-polarization geometry	Diagrams absent or seriously flawed: no diagram for (c) despite its 20-mark weight, incorrect geometry (e.g., angles measured from wrong reference), or diagrams that contradict the derived equations; failure to recognize the visual nature of optical problems
Numerical accuracy	15%	7.5	Precise algebraic manipulation throughout: correct expansion of (V-b) terms in van der Waals equation, accurate partial derivative calculations for μ_JK, and exact trigonometric identities in Fresnel equation simplification; final expressions dimensionally consistent and reduce to known limits	Generally correct algebra with minor slips: arithmetic errors in coefficients, incorrect handling of small-parameter expansions, or sign errors in trigonometric identities that are partially corrected later; final answers qualitatively correct	Serious numerical/algebraic errors: incorrect binomial expansions, wrong partial derivatives, or fundamental errors in trigonometric identities; final expressions dimensionally inconsistent or physically implausible (e.g., negative definite quantities)
Physical interpretation	25%	12.5	Rich physical insight across all parts: explains why mutual inductance reduces effective inductance (energy storage perspective), connects Joule-Kelvin coefficient sign to competition between attractive forces (cooling) and work done against repulsive core (heating), and explains Brewster's angle as condition where dipole radiation perpendicular to reflected direction vanishes; references practical applications like gas liquefaction (Linde-Hampson cycle) and anti-reflection coatings	Some physical interpretation present but limited: states results without explaining underlying mechanisms, provides formula-centered explanations, or gives generic statements about 'energy conservation' without specificity; misses opportunity to connect parts through energy themes	Purely formal manipulation with no physical insight: treats derivations as mathematical exercises, unable to explain why inversion temperature exists or what Brewster's angle represents physically; no awareness of experimental significance or technological applications

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