(a) Find the energy stored in a system of four charges Q₁ = 1 nC, Q₂ = 2 nC, Q₃ = 3 nC and Q₄ = 4 nC placed at the cartesian coordinates R₁(1, 1), R₂(2, 1), R₃(1, 4) and R₄(2, 2), respectively. Assume free space. 10 marks

(b) Derive the expression f

Question

(a) Find the energy stored in a system of four charges Q₁ = 1 nC, Q₂ = 2 nC, Q₃ = 3 nC and Q₄ = 4 nC placed at the cartesian coordinates R₁(1, 1), R₂(2, 1), R₃(1, 4) and R₄(2, 2), respectively. Assume free space. 10 marks

(b) Derive the expression for the inductance per unit length of two long parallel wires each of radius a, separated by distance d from their axes and carrying equal and opposite current I. 10 marks

(c) Show that Continuity equation is embedded in Maxwell's equations. 10 marks

(d) Using Zeroth law of thermodynamics, introduce the concept of temperature. Explain how the isotherms of two different systems can be drawn. 10 marks

(e) Write down the expressions for the Fermi-Dirac distribution and the Bose-Einstein distribution. Plot the distributions as a function of the energy. 10 marks

UPSC Answer Check · Accepted Answer

This question requires solving five distinct problems spanning electrostatics, magnetostatics, electrodynamics, thermodynamics, and statistical mechanics. Allocate approximately 15-20% time to each sub-part, with slightly more attention to (b) and (c) due to their derivation demands. Structure your answer by clearly labeling each sub-part, showing all intermediate steps for calculations, and presenting derivations with logical flow from first principles. For (e), ensure plots are qualitatively accurate with proper labeling of axes and key features.
- For (a): Calculate pairwise distances between all four charges using Cartesian coordinates, then apply superposition principle for electrostatic potential energy using U = (1/4πε₀)Σᵢ<ⱼ QᵢQⱼ/rᵢⱼ
- For (b): Derive inductance per unit length by calculating magnetic flux linkage between two parallel wires, accounting for both external flux (between axes) and internal flux (within wire radius)
- For (c): Take divergence of Ampère-Maxwell law and substitute Gauss's law to obtain ∇·J + ∂ρ/∂t = 0, explicitly showing charge conservation
- For (d): State Zeroth law's transitive property (A~B and B~C implies A~C), define empirical temperature via thermal equilibrium, and sketch isotherms for ideal gas (hyperbolic) vs. van der Waals gas (with critical point)
- For (e): Write Fermi-Dirac f_FD = 1/[e^(E-μ)/kT + 1] and Bose-Einstein f_BE = 1/[e^(E-μ)/kT - 1], plot showing step-like FD at low T and singular BE divergence at E→μ

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies that (a) requires pairwise summation not simple addition; recognizes internal flux contribution in (b); correctly applies vector calculus identities in (c); distinguishes thermal equilibrium from temperature in (d); identifies physical significance of ±1 in denominators for (e)	Minor conceptual errors such as missing internal inductance in (b), confusing Zeroth with First law in (d), or incorrect asymptotic behavior in distribution plots	Fundamental misconceptions like treating total potential energy as sum of self-energies, omitting displacement current in (c), or equating FD and BE distributions
Derivation rigour	20%	10	Complete step-by-step derivations: for (b) explicit integration of B-field from surface to surface plus internal flux; for (c) clear vector calculus operations with Maxwell equations stated; logical progression from empirical temperature to isotherm construction	Derivations with gaps or skipped steps, such as assuming flux formula without integration, or stating continuity equation without showing divergence operations	Missing derivations entirely, or circular reasoning with conclusions assumed as premises; no attempt at mathematical justification
Diagram / FBD	15%	7.5	Clear coordinate system for charge positions in (a); cross-sectional diagram showing field regions for (b); annotated Maxwell equation relationships for (c); schematic of thermal equilibrium experiment for (d); accurately sketched FD/BE curves with T=0, T<<TF, T>TF cases labeled	Diagrams present but lacking labels or key features, such as unlabeled axes or missing asymptotic behavior in distribution plots	No diagrams where required, or seriously misleading sketches (e.g., BE distribution showing negative values)
Numerical accuracy	25%	12.5	Precise calculation in (a): distances r₁₂=1, r₁₃=3, r₁₄=√2, r₂₃=√10, r₂₄=1, r₃₄=√5; correct substitution with ε₀ = 8.854×10⁻¹² F/m; final answer in appropriate units (nJ or μJ); correct numerical factors in (b) including μ₀/π term	Correct method but arithmetic errors in distance calculations or unit conversions; correct formula in (b) with algebraic errors	Order-of-magnitude errors, incorrect distance formulas, or missing factors of 4π or 1/2 in energy calculation
Physical interpretation	20%	10	Interprets (a) result as work to assemble system; explains (b) result reduces to L ≈ (μ₀/π)ln(d/a) for d>>a; discusses charge conservation implications in (c); connects Zeroth law to temperature measurement via thermometry; contrasts fermion/boson quantum statistics and their macroscopic consequences (degeneracy pressure, BEC)	Brief or superficial interpretation, stating formulas without explaining physical significance	No physical interpretation provided, or incorrect interpretation (e.g., negative energy interpreted as repulsion)

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