(a) In a partially conducting medium, $\varepsilon_r = 18.5$, $\mu_r = 800$ and $\sigma = 1$ S m⁻¹. Find α, β, η and the velocity u, for a frequency of 10⁹ Hz. Determine $\vec{H}(z, t)$. Given, $\vec{E}(z, t) = 50 e^{-\alpha z} \cos(\omega t - \beta

Question

(a) In a partially conducting medium, $\varepsilon_r = 18.5$, $\mu_r = 800$ and $\sigma = 1$ S m⁻¹. Find α, β, η and the velocity u, for a frequency of 10⁹ Hz. Determine $\vec{H}(z, t)$. Given, $\vec{E}(z, t) = 50 e^{-\alpha z} \cos(\omega t - \beta a_z) a_y$ V m⁻¹. (20 marks)

(b) What do you understand by negative temperature? Write and explain various restrictions on a system for the concept of negative temperature to be meaningful. (15 marks)

(c) Starting from the Laplace's equation in a cylindrical polar coordinate system and using the method of separation of variables, obtain the differential equations for the solutions of r, φ and z components of the potential. (15 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction acknowledging the three distinct domains: electromagnetic wave propagation in lossy media, statistical mechanics of negative temperature, and electrostatic boundary value problems. Allocate approximately 40% of effort to part (a) given its 20 marks and computational demands; 30% each to parts (b) and (c). For (a), systematically calculate loss tangent, then α, β, η, and u before deriving H(z,t). For (b), define negative temperature and enumerate the three key restrictions (finite energy levels, thermal isolation, and population inversion). For (c), present the separation of variables derivation clearly with proper handling of the radial equation. Conclude with a synthesis noting how these topics span classical electrodynamics, statistical mechanics, and mathematical physics.
- Part (a): Calculate loss tangent tan δ = σ/(ωε) to classify the medium as good conductor or lossy dielectric, then correctly apply formulas for α, β, η and phase velocity u = ω/β
- Part (a): Derive H(z,t) using the intrinsic impedance relationship η = E/H with proper vector orientation (aₓ × aᵧ = aᵧ, accounting for the phase of η in complex form)
- Part (b): Define negative temperature as T < 0 occurring when (∂S/∂U) < 0, occurring in systems with upper energy bound, not 'colder than absolute zero'
- Part (b): Enumerate restrictions: (i) system must have finite number of energy levels, (ii) thermally isolated (no energy exchange with reservoir), (iii) population inversion required; cite examples like nuclear spin systems or laser media
- Part (c): Write Laplace's equation in cylindrical coordinates: (1/r)∂/∂r(r∂V/∂r) + (1/r²)∂²V/∂φ² + ∂²V/∂z² = 0
- Part (c): Apply separation V(r,φ,z) = R(r)Φ(φ)Z(z) to obtain three ODEs: Bessel's equation for R, harmonic equation for Φ, and exponential/trigonometric equation for Z with separation constants
- Part (c): Identify the physical significance: R(r) involves Bessel functions Jₙ and Yₙ (or modified Bessel Iₙ, Kₙ), Φ(φ) requires single-valuedness giving integer n, Z(z) depends on boundary conditions

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	22%	11	Correctly identifies the medium as a good conductor (loss tangent >> 1) in (a); accurately defines negative temperature via entropy-energy relation in (b); properly recognizes Laplace equation form and separation validity in (c); no conceptual confusion between phase velocity and group velocity, or between negative temperature and low positive temperature	Minor errors in medium classification or partially correct definition of negative temperature; Laplace equation written correctly but some confusion about coordinate-specific terms; understands basic ideas but conflates key distinctions	Fundamental errors such as treating the medium as lossless dielectric, defining negative temperature as 'below absolute zero', or writing wrong form of Laplace equation; demonstrates confusion between unrelated concepts
Derivation rigour	22%	11	Complete step-by-step derivation in (c) with explicit separation of variables, clear identification of separation constants, and proper reduction to standard ODE forms; in (a), systematic derivation of H from E using Maxwell's equations or impedance relation with complex arithmetic handled correctly; in (b), rigorous thermodynamic derivation from S = k ln Ω	Derivations mostly complete but skips key steps or assumes results without justification; separation of variables applied but separation constants not clearly motivated; some algebraic steps omitted that would be needed for full marks	Missing derivations entirely or presenting only final formulas; separation of variables stated but not executed; logical gaps that prevent reconstruction of the result; circular reasoning or unsupported claims
Diagram / FBD	12%	6	Clear coordinate system diagram for (c) showing cylindrical geometry (r, φ, z); field orientation diagram for (a) showing E in aᵧ direction, propagation in aᵧ direction, and H perpendicular; energy level diagram for (b) illustrating population inversion and finite level system; all diagrams properly labeled with unit vectors	At least one relevant diagram present but missing labels or incomplete; coordinate system implied but not drawn; field directions stated verbally but no visual representation; diagrams support but do not enhance the answer	No diagrams where clearly needed, or diagrams that misrepresent the physics (e.g., Cartesian coordinates for cylindrical problem, wrong field orientations); diagrams that confuse rather than clarify
Numerical accuracy	22%	11	Correct numerical values for all quantities in (a): ε = εᵣε₀, μ = μᵣμ₀ calculated properly; ω = 2πf; loss tangent, then α, β, η, u all computed with correct units and significant figures; final H(z,t) with correct amplitude, phase, and polarization; explicit statement that a should be z in the given expression (likely typographical error in question)	Correct formulas but arithmetic errors in final values; correct order of magnitude but wrong units; one or two quantities calculated correctly but others wrong; confusion between α and β or between real and imaginary parts of η	Order-of-magnitude errors in results; wrong formulas leading to nonsensical values (e.g., velocity > c); missing calculations entirely; unit errors throughout; no recognition of the likely typo (βaᵧ should be βz)
Physical interpretation	22%	11	Interprets (a) results physically: large α means rapid attenuation, skin depth δ = 1/α, wave is strongly damped; explains why η is complex and nearly inductive; in (b), connects negative temperature to population inversion in lasers/masers and nuclear spin systems in solids (Indian context: work at IISc, TIFR); in (c), explains why Bessel functions appear and their asymptotic behavior, physical boundary conditions eliminating Yₙ at r=0	Some physical interpretation present but superficial; mentions applications without explaining connection to derived results; understands that α means attenuation but doesn't quantify skin depth; knows negative temperature is 'weird' but cannot explain experimental realization	Purely mathematical treatment with no physical insight; no connection to real-world phenomena; fails to explain why the results matter or what they imply about the physical systems; interpretation statements that are physically incorrect

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