All 8 questions from UPSC Civil Services Mains Physics
2025 Paper I (400 marks total). Every stem reproduced in full,
with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2025Year
Paper IPaper
Topics covered
Classical mechanics, gravitation, relativity, optics (1)Rigid body dynamics, damped harmonic oscillator, special relativity (1)Laser physics and rotational dynamics (1)Optics and elasticity (1)Electromagnetism and thermodynamics (1)Electromagnetism, AC circuits and statistical mechanics (1)Gibbs phase rule, Van der Waals equation, conducting sphere in electric field (1)Electromagnetic waves, chemical potential equilibrium, Planck's radiation law (1)
(a) Consider a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger cylinder. Determine the equation of motion of the smaller cylinder. 10
(b) Derive the expression for the gravitational self-energy of a uniform solid sphere of mass M and radius R. 10
(c) A particle of rest mass 1 kg and velocity of magnitude 0·9c collides with a particle of mass 2 kg at rest. After collision the two particles coalesce and form a single particle of mass M and velocity V. Determine M and V. 10
(d) In a double slit Fraunhofer diffraction experiment, the slit width is 0·12 mm and the spacing between the two slits is 0·48 mm. The distance of the screen from the slits is 1·5 m. If the wavelength of the light used is 600 nm, determine (i) the missing orders of the interference maxima, and (ii) the distance between the central maxima and the first minima. 10
(e) A light beam of wavelength 600 nm produced by a 20 mW laser source is incident on a plane mirror. Determine :
(i) number of photons per second striking the surface of the mirror.
(ii) force exerted by the light beam on the mirror. 10
हिंदी में पढ़ें
(a) आन्तरिक अर्द्धव्यास R के एक बहुत स्थिर बेलन (सिलिंडर) को लीजिए। अर्द्धव्यास r का एक छोटा ठोस बेलन बहुत बेलन के अन्दर बिना फिसले लुढ़कता है। छोटे बेलन की गति का समीकरण निर्धारित कीजिए। 10
(b) द्रव्यमान M और अर्द्धव्यास R के एक एकसमान ठोस गोले की गुरुत्वीय नेज-ऊर्जा के लिए व्यंजक व्युत्पन्न कीजिए। 10
(c) विरामावस्था द्रव्यमान 1 kg और 0·9c परिमाण के वेग का एक कण विरामावस्था में द्रव्यमान 2 kg के एक कण से टकराता है। संघटन के पश्चात दोनों कण संलयित हो जाते हैं और द्रव्यमान M तथा वेग V का एक एकल कण बनाते हैं। M और V निर्धारित कीजिए। 10
(d) एक द्वि-स्लिट फ्राउनहोफर विवर्तन प्रयोग में, स्लिट चौड़ाई 0·12 mm है और दोनों स्लिटों के बीच पार्थक्य 0·48 mm है। स्लिटों से स्क्रीन की दूरी 1·5 m है। यदि प्रयुक्त प्रकाश का तरंगदैर्ध्य 600 nm है, तो (i) व्यतिकरण उच्चिष्ट के लुप्त क्रम, और (ii) केन्द्रीय उच्चिष्ट और प्रथम निम्निष्ट के बीच की दूरी ज्ञात कीजिए। 10
(e) एक 20 mW लेजर स्रोत द्वारा उत्पन्न तरंगदैर्ध्य 600 nm का एक प्रकाश पुंज एक समतल दर्पण पर आपतित है। निर्धारित कीजिए :
(i) दर्पण की सतह से टकराते प्रति सेकंड फोटॉनों की संख्या।
(ii) दर्पण पर प्रकाश पुंज द्वारा डाला गया बल। 10
Answer approach & key points
Begin with clear statement of physical principles for each sub-part. For (a), set up Lagrangian with constraint; for (b), integrate gravitational potential energy; for (c), apply relativistic energy-momentum conservation; for (d), combine single-slit diffraction envelope with double-slit interference; for (e), use photon energy and radiation pressure concepts. Allocate approximately 20% time each to (a), (b), (c), (d), and (e) combined, with (e)(i) and (e)(ii) sharing that final 20%. Present derivations step-wise with final boxed answers for numerical parts.
(a) Rolling constraint: arc length relation (R-r)θ = rφ; correct Lagrangian with kinetic energy of CM plus rotational energy; equation of motion as simple harmonic oscillator with period depending on √(R-r)/g
(b) Gravitational self-energy: assemble sphere shell by shell; integration of -GM(r)dm/r from 0 to R; final result U = -3GM²/5R with correct handling of negative sign
(c) Relativistic collision: calculate γ = 1/√(1-0.9²) ≈ 2.294; conserve total energy and momentum; solve for M and V with M > 3kg due to kinetic energy conversion to mass
(d) Missing orders: condition d/a = 4 implies interference maxima at n=4,8,12... coincide with diffraction minima; first minima distance using y = λD/d for interference pattern
(e)(i) Photon flux: N = Pλ/(hc) ≈ 6.03 × 10¹⁶ photons/second; (e)(ii) Force: F = 2P/c = 2×20mW/c ≈ 1.33 × 10⁻¹⁰ N for perfect reflection
50MsolveRigid body dynamics, damped harmonic oscillator, special relativity
(a) A body moves about a point 'O' under no force, the principal moments of inertia at 'O' being 3A, 5A and 6A. The components of the initial angular velocity about the principal axes are ω₁ = n, ω₂ = 0 and ω₃ = n. Find the components ω₁, ω₂ and ω₃ for large values of time t. 20
(b) A harmonic oscillator is represented by the equation
m d²x/dt² + γ dx/dt + kx = 0;
where m = 0·25 kg, γ = 0·07 kg s⁻¹ and k = 85 Nm⁻¹. Determine (i) the period of oscillation, and (ii) the number of oscillations in which its amplitude will become half of its original value. 15
(c) Show that the electromagnetic wave equation is invariant under Lorentz transformations. 15
हिंदी में पढ़ें
(a) एक पिण्ड बिना किसी बल के अधीन एक बिन्दु 'O' के परितः गतिमान है। 'O' पर जड़त्व के मुख्य आघूर्ण 3A, 5A और 6A हैं। मुख्य अक्षों के परितः आरम्भिक कोणीय वेग के घटक ω₁ = n, ω₂ = 0 और ω₃ = n हैं। समय t के बहुत मानों के लिए घटकों ω₁, ω₂ और ω₃ को ज्ञात कीजिए। 20
(b) एक सरल आवर्ती दोलक निम्नलिखित समीकरण द्वारा निरूपित है
m d²x/dt² + γ dx/dt + kx = 0;
जहाँ m = 0·25 kg, γ = 0·07 kg s⁻¹ और k = 85 Nm⁻¹ है। निर्धारित कीजिए (i) दोलन का आवर्तकाल, और (ii) दोलनों की संख्या जिनमें उसका आयाम उसके प्रारम्भिक मान का आधा हो जाएगा। 15
(c) दर्शाइए कि लोरेन्ट्ज रूपान्तरणों के अधीन विद्युत-चुम्बकीय तरंग समीकरण निश्चर है। 15
Answer approach & key points
This is a multi-part problem requiring analytical solutions: spend approximately 40% of effort on part (a) given its 20 marks, using Euler's equations for force-free rigid body motion; allocate 30% each to parts (b) and (c). For (b), solve the damped harmonic oscillator differential equation and extract numerical values; for (c), apply Lorentz transformation to electromagnetic wave equation and demonstrate invariance. Present derivations step-by-step with clear final boxed answers.
Part (a): Apply Euler's equations for force-free rotation, identify that motion occurs in 1-3 plane with I₁=3A, I₃=6A, use conservation of energy and angular momentum to find asymptotic behavior where ω₂→0 and ω₁, ω₃ approach constant values
Part (b): Identify underdamped regime (γ² < 4mk), calculate damped angular frequency ω' = √(k/m - γ²/4m²), find period T = 2π/ω', and determine logarithmic decrement to find number of oscillations for amplitude halving
Part (c): State Lorentz transformation equations, transform ∂²/∂x² - (1/c²)∂²/∂t² using chain rule, show wave operator remains invariant (∂²/∂x'² - (1/c²)∂²/∂t'² = ∂²/∂x² - (1/c²)∂²/∂t²)
Correct identification of intermediate axis instability in part (a) — rotation about axis with intermediate moment of inertia (I₂=5A) is unstable, causing ω₂ to decay
Numerical calculation in (b): ω' ≈ 18.44 rad/s, T ≈ 0.34 s, and n ≈ 31 oscillations for amplitude to halve using ln(2)/ln(A₁/A₂) relationship
Explicit demonstration that phase velocity c remains invariant under Lorentz transformation in part (c), connecting to Einstein's second postulate
(a) Consider a laser system consisting of an active medium placed between a pair of mirrors forming a resonator. Obtain an expression for the threshold population inversion required for the oscillations of laser. (20 marks)
(b) For a He – Ne laser system, what will be the magnitude of $\Delta\omega_D$ which represents FWHM of the line shape function g($\omega$), if resonant frequency $\omega_0 = 3 \times 10^{15}$ s$^{-1}$ and temperature T = 300 K ? (10 marks)
(c) A cube of mass M and side 'a' is rotating with angular velocity $\omega$ around one of its edges, which is, say, along the x-axis. Obtain the expressions for its angular momentum and kinetic energy.
(Given that the $I_{XX} = \frac{2}{3} Ma^2$, $I_{YX} = -\frac{1}{4} Ma^2$ and $I_{ZX} = -\frac{1}{4} Ma^2$) (20 marks)
हिंदी में पढ़ें
(a) एक अनुनादक बनाते दर्पणों के एक युग्म के बीच रखे एक सक्रिय माध्यम के एक लेजर निकाय को लीजिए। लेजर के दोलनों के लिए आवश्यक देहली (थ्रेशोल्ड) जनसंख्या व्युत्क्रमण के लिए एक व्यंजक प्राप्त कीजिए। (20 अंक)
(b) एक He – Ne लेज़र निकाय के लिए, यदि अनुनादी आवृत्ति $\omega_0 = 3 \times 10^{15}$ s$^{-1}$ और तापक्रम T = 300 K है, तो $\Delta\omega_D$ का परिमाण क्या होगा जो रेखा आकृति फलन g($\omega$) के FWHM को निरूपित करता है? (10 अंक)
(c) द्रव्यमान M और भुजा 'a' का एक घन x-अक्ष के अनुदिश अपने एक किनारे के परितः कोणीय वेग $\omega$ से घूर्णन कर रहा है। उसके कोणीय संवेग और उसकी गतिज ऊर्जा के लिए व्यंजकों को प्राप्त कीजिए।
(दिया गया है, $I_{XX} = \frac{2}{3} Ma^2$, $I_{YX} = -\frac{1}{4} Ma^2$ और $I_{ZX} = -\frac{1}{4} Ma^2$) (20 अंक)
Answer approach & key points
Derive the threshold population inversion for laser oscillations in part (a) by balancing gain and cavity losses, showing how mirror reflectivity and spontaneous emission enter the condition. For part (b), calculate the Doppler-broadened linewidth using the Maxwell-Boltzmann distribution at 300 K. In part (c), construct the inertia tensor for the cube and use it to find angular momentum components and rotational kinetic energy, noting that ω is along x-axis but L has non-zero y,z components due to non-diagonal elements. Allocate roughly 40% time to (a), 20% to (b), and 40% to (c) based on marks distribution.
Part (a): Rate equation analysis showing gain coefficient γ(ν) = (N₂-N₁)B₂₁hνg(ν)/c and threshold condition γₜₕ = α + (1/2L)ln(1/R₁R₂)
Part (a): Final threshold population inversion expression (N₂-N₁)ₜₕ = 8πν²τₛₚ/c³g(ν₀) × loss terms, or equivalent with cavity lifetime τc
Part (b): Doppler broadening formula ΔνD = (2ν₀/c)√(2kTln2/m) or ΔωD = (2ω₀/c)√(2kTln2/m) for He-Ne with m = 20 amu (Neon)
Part (b): Numerical substitution yielding ΔωD ≈ 2π × 1.5 GHz or ~9.4 × 10⁹ rad/s (order of magnitude check essential)
Part (c): Recognition that ω⃗ = (ω, 0, 0) and use of Lᵢ = Σⱼ Iᵢⱼωⱼ giving Lx = (2/3)Ma²ω, Ly = -(1/4)Ma²ω, Lz = -(1/4)Ma²ω
Part (c): Kinetic energy calculation K = ½ω⃗·L⃗ = ½Iₓₓω² = (1/3)Ma²ω², or equivalently ½ΣᵢⱼIᵢⱼωᵢωⱼ
Part (c): Physical explanation that angular momentum is NOT parallel to angular velocity due to non-diagonal inertia tensor (principal axes ≠ coordinate axes)
(a) Consider a thick lens of thickness t made of a material of relative refractive index n. Let $R_1$ and $R_2$ be the radii of curvature of its two surfaces. Obtain the system matrix of the lens. (15 marks)
(b) Consider multiple reflections from a plane parallel film of thickness h and refractive index $n_2$ and derive an expression for the total reflectivity from the surface of the film. (20 marks)
(c) A solid shaft of mass M, length $l$ and radius r is to be replaced by a lighter hollow shaft of the same length $l$ and having the same ratings of $\tau/\theta$, where $\tau$ is the couple and $\theta$ is the angle of twist. Estimate the percentage reduction in mass of the hollow shaft if the outer radius of the shaft is twice the inner radius. Assume the material of the new shaft is same as that of the replaced shaft. (15 marks)
हिंदी में पढ़ें
(a) सापेक्ष अपवर्तनांक n के एक पदार्थ से निर्मित मोटाई t के एक मोटे लेंस को लीजिए। मान लीजिए कि उसके दो पृष्ठों की वक्रता के अर्ध्व्यास $R_1$ और $R_2$ हैं। लेंस की निकाय मैट्रिक्स (आव्यूह) प्राप्त कीजिए। (15 अंक)
(b) अपवर्तनांक $n_2$ और मोटाई h की एक समतल समांतर फिल्म से होने वाले बहुल परावर्तनों को लीजिए और फिल्म के पृष्ठ से होने वाली कुल परावर्तकता के लिए व्यंजक की व्युत्पत्ति कीजिए। (20 अंक)
(c) एक द्रव्यमान M, लंबाई $l$ और अर्ध्व्यास r के ठोस कूपक (शाफ्ट) को समान लंबाई $l$ और समान $\tau/\theta$ की रेटिंग, जहाँ $\tau$ बल-युग्म और $\theta$ व्यावर्तन कोण है, के एक हल्के खोखले कूपक द्वारा प्रतिस्थापित किया जाना है। यदि खोखले कूपक का बाह्य अर्ध्व्यास उसके आंतरिक अर्ध्व्यास का दो गुना है, तो उसके द्रव्यमान में प्रतिशत कमी का आकलन कीजिए। मान लीजिए कि नए कूपक और प्रतिस्थापित कूपक का पदार्थ समान है। (15 अंक)
Answer approach & key points
Begin with a brief introduction distinguishing matrix optics from Gaussian optics. For part (a), derive the system matrix by multiplying refraction and translation matrices in correct order. For part (b), use the method of summing infinite geometric series of reflected amplitudes with proper phase considerations. For part (c), equate torsional rigidity C = τ/θ for both shafts and solve for mass ratio. Allocate approximately 30% time to (a), 40% to (b) as it carries highest marks, and 30% to (c). Conclude with brief remarks on practical applications in optical instruments and mechanical engineering.
Part (a): Correct identification of individual matrices — refraction at first surface (R1), translation through thickness t, and refraction at second surface (R2) with proper sign convention
Part (a): Proper matrix multiplication order R2 × T × R1 yielding final system matrix with elements A, B, C, D satisfying AD-BC=1 for unimodular property
Part (b): Application of Fresnel coefficients at each interface with correct amplitude reflection/transmission coefficients r12, t12, r23, t23
Part (b): Inclusion of phase factor δ = (4πn2h cosθ2)/λ and summation of infinite series leading to Airy formula for reflectivity
Part (c): Expression for torsional rigidity C = πGr⁴/(2l) for solid shaft and C = πG(r₂⁴-r₁⁴)/(2l) for hollow shaft
Part (c): Setting equal rigidity ratings, substituting r₂ = 2r₁, solving for r₁ in terms of r, then calculating mass ratio and percentage reduction
Clear statement of assumptions: paraxial approximation for (a) and (b), same material (same G, ρ) for (c), thin film interference conditions
50MCompulsorysolveElectromagnetism and thermodynamics
(a) Consider a point charge of 5 nC placed at a distance of 1 m from a perfect conducting plane (z = 0) of infinite extent. Find the electric field at a point (2, 2, 0) m and show that it is normal to the plane. (10 marks)
(b) A rectangular coil consists of 50 closely wrapped turns and has dimensions of 0·5 m × 0·4 m. It carries a current of 1·5 A. If a uniform magnetic field B = 0·1 T is applied such that the direction of the magnetic field makes an angle of 60° with respect to the plane of the coil, what is the torque exerted on the coil by the magnetic field? (10 marks)
(c) State and explain Kirchhoff's current law and Kirchhoff's voltage law. Derive these laws from the principles of charge conservation and energy conservation. (10 marks)
(d) A parallel plate capacitor having circular plates of radius 10 cm is being charged. If the electric field at any instant within the capacitor changes at the rate 5·0 V m⁻¹ s⁻¹, calculate the magnetic intensity |H⃗| inside the capacitor. (10 marks)
(e) A reversible heat engine operates with three reservoirs at 300 K, 400 K and 1200 K. It absorbs 1200 kJ energy as heat from the reservoir at 1200 K and delivers 400 kJ work. Determine the heat interactions with the other two reservoirs. (10 marks)
हिंदी में पढ़ें
(a) अनंत विस्तार के एक आदर्श चालकीय समतल (z = 0) से 1 m की दूरी पर स्थित 5 nC के एक बिंदु आवेश को लीजिए। एक बिंदु (2, 2, 0) m पर विद्युत-क्षेत्र ज्ञात कीजिए और दर्शाइए कि यह समतल के लम्बवत है। (10 अंक)
(b) सुसंकुलित रूप से लपेटे गए 50 फेरों की एक आयताकार कुंडली की विमाएँ 0·5 m × 0·4 m हैं। इसमें 1·5 A की विद्युत धारा प्रवाहित होती है। यदि एक एकसमान चुंबकीय क्षेत्र B = 0·1 T इस प्रकार प्रयुक्त किया जाता है कि चुंबकीय क्षेत्र की दिशा कुंडली के समतल के सापेक्ष 60° का कोण बनाती है, तो चुंबकीय क्षेत्र द्वारा कुंडली पर प्रयुक्त बल-आघूर्ण क्या है? (10 अंक)
(c) किरखॉफ के धारा नियम और किरखॉफ के वोल्टता नियम का उल्लेख और व्याख्या कीजिए। आवेश संरक्षण और ऊर्जा संरक्षण के सिद्धांतों से इन नियमों की व्युपत्ति कीजिए। (10 अंक)
(d) अर्ध्व्यास 10 cm की वृत्ताकार प्लेटों से बने एक समांतर प्लेट संधारित्र को आवेशित किया जा रहा है। यदि संधारित्र के अंदर किसी क्षण पर विद्युत-क्षेत्र 5·0 V m⁻¹ s⁻¹ की दर से परिवर्तित होता है, तो संधारित्र के अंदर चुंबकीय तीव्रता |H⃗| की गणना कीजिए। (10 अंक)
(e) एक उत्क्रमणीय ऊष्मा इंजन 300 K, 400 K और 1200 K पर तीन भंडारों के साथ संक्रियत है। यह 1200 K पर भंडार से ऊष्मा के रूप में 1200 kJ ऊर्जा अवशोषित करता है और 400 kJ का कार्य प्रदान करता है। अन्य दो भंडारों के साथ ऊष्मा अन्योन्यक्रियाओं को निर्धारित कीजिए। (10 अंक)
Answer approach & key points
Solve each sub-part systematically with equal time allocation (~20% each) since all carry equal marks. Begin with method of images for (a), torque formula for (b), clear statement-derivation pairs for (c), displacement current for (d), and entropy balance for (e). Present derivations before substituting numerical values, and conclude each part with physical verification of results.
Part (a): Apply method of images with image charge -5 nC at (0,0,-1); calculate field at (2,2,0) from both charges and prove tangential component vanishes on z=0 plane
Part (b): Use torque formula τ = NIAB sinθ with θ = 30° (angle between normal and B), not 60°; calculate magnitude correctly as 0.75 Nm
Part (c): State KCL (ΣI = 0 at junction) and KVL (ΣV = 0 in loop); derive KCL from ∮J·dA = -dQ/dt and KVL from ∮E·dl = -dΦB/dt = 0 for electrostatics
Part (d): Apply Maxwell-Ampère law with displacement current; use ∮H·dl = ε₀(dΦE/dt) to find H = (r/2)(dD/dt) = (rε₀/2)(dE/dt) at radius r
Part (e): Apply entropy conservation for reversible engine: Q₁/T₁ + Q₂/T₂ + Q₃/T₃ = 0 with Q₃ = +1200 kJ, W = 400 kJ; solve simultaneous equations for Q₁ and Q₂
50MderiveElectromagnetism, AC circuits and statistical mechanics
(a) Consider a long straight wire of length L carrying a current I. Determine the magnetic vector potential $\vec{A}$ at a point P located at distance x from the wire. (20 marks)
(b) As shown in the figure, a series circuit connected across a 200 V, 60 Hz line consists of a capacitor of capacitive reactance of 30 Ω, a non-inductive resistor of 44 Ω and a coil of inductive reactance 90 Ω and resistance 36 Ω.
Determine:
(i) Power factor of the circuit
(ii) Power absorbed by the circuit
(iii) Power dissipated in the coil
(c) Consider a mixture of N_A molecules of a monatomic gas A and N_B molecules of a monatomic gas B. For this mixture, obtain the Helmholtz free energy and pressure. (The particle partition function for a monatomic gas is q = (2π mkT/h²)^(3/2) V).
हिंदी में पढ़ें
(a) वि�िद्युत धारा I प्रवाहित लम्बाई L के एक सीधे लम्बे तार को लीजिए। तार से दूरी x पर अवस्थित बिन्दु P पर चुंबकीय सदिश विभव $\vec{A}$ ज्ञात कीजिए। (20 अंक)
(b) जैसा कि चित्र में दर्शाया गया है, एक 200 V, 60 Hz लाइन से संबद्ध एक श्रेणी परिपथ में 30 Ω की धारिता प्रतिघात का एक संधारित्र, 44 Ω का एक अप्रेरणिक प्रतिरोधक और 36 Ω प्रतिरोध तथा 90 Ω की प्रेरणिक प्रतिघात की एक कुंडली है।
ज्ञात कीजिए:
(i) परिपथ का शक्ति गुणांक
(ii) परिपथ द्वारा अवशोषित शक्ति
(iii) कुंडली में क्षतिग्रस्त शक्ति
(c) एक एकपरमाणुक गैस A के N_A अणुओं और एक एकपरमाणुक गैस B के N_B अणुओं के एक मिश्रण को लीजिए। इस मिश्रण के लिए, हेल्महोल्ट्ज़ मुक्त ऊर्जा और दाब ज्ञात कीजिए। (एक एकपरमाणुक गैस के लिए कण संवितरण फलन है q = (2π mkT/h²)^(3/2) V).
Answer approach & key points
Derive the magnetic vector potential for the finite wire in part (a) using proper integration limits and Coulomb gauge, then solve the AC circuit problem in part (b) by calculating impedance, phase angle, and power quantities stepwise, and finally derive the Helmholtz free energy for the gas mixture in part (c) using Maxwell-Boltzmann statistics. Allocate approximately 40% effort to part (a) given its 20-mark weight in the original scheme, 35% to part (b) for its three numerical sub-parts, and 25% to part (c) for the statistical mechanics derivation.
Part (a): Setup of integral for vector potential using A = (μ₀I/4π)∫(dl'/|r-r'|) with proper coordinate system and limits from -L/2 to +L/2
Part (a): Final expression A = (μ₀I/4π)ln[(L/2+√(x²+L²/4))/(-L/2+√(x²+L²/4))]ẑ or equivalent, with discussion of infinite wire limit
Part (b)(i): Calculation of total impedance Z = √[(R₁+R₂)²+(X_L-X_C)²] = √[80²+60²] = 100Ω, leading to power factor cosφ = 80/100 = 0.8 lagging
Part (b)(ii)-(iii): Power absorbed P = VIcosφ = 200×2×0.8 = 320W (or I²R_total = 4×80 = 320W), and power in coil = I²R_coil = 4×36 = 144W
Part (c): Derivation of Helmholtz free energy F = -kT[N_A ln(q_A/N_A) + N_B ln(q_B/N_B) + N_A + N_B] using Stirling's approximation
Part (c): Pressure derivation P = -(∂F/∂V)_T = (N_A+N_B)kT/V = nRT/V, showing ideal gas mixture law with Dalton's law implicit
50MderiveGibbs phase rule, Van der Waals equation, conducting sphere in electric field
(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 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)
Answer approach & key points
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.
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
50MderiveElectromagnetic waves, chemical potential equilibrium, Planck's radiation law
(a) (i) In free space, an electric field ($\vec{E}$) is given by the following expression :
$$\vec{E} = 10 \cos (\omega t - 100 x)\hat{j} \text{ V/m}$$
Find the angular frequency $\omega$ and the displacement current. (10 marks)
(ii) An electromagnetic wave has its magnetic field $|\vec{B}| = 55 \times 10^{-8}$ T. Determine the magnitude of the Poynting vector. (5 marks)
(b) Explain why, at equilibrium, the chemical potential of a component must be the same in all coexisting phases. Derive the equilibrium condition for a binary liquid-vapour system in terms of chemical potential. (15 marks)
(c) Derive the Planck's radiation law for blackbody radiation using the Bose-Einstein distribution function. Explain how results from quantum statistics differ from classical results derived from the Rayleigh-Jeans law. (20 marks)
हिंदी में पढ़ें
(a) (i) मुक्त आकाश में एक विद्युत-क्षेत्र ($\vec{E}$) निम्नलिखित व्यंजक द्वारा व्यक्त किया गया है :
$$\vec{E} = 10 \cos (\omega t - 100 x)\hat{j} \text{ V/m}$$
कोणीय आवृत्ति $\omega$ और विस्थापन धारा ज्ञात कीजिए। (10 अंक)
(ii) एक विद्युत-चुंबकीय तरंग का चुंबकीय क्षेत्र $|\vec{B}| = 55 \times 10^{-8}$ T है। पॉइंटिंग सदिश का परिमाण ज्ञात कीजिए। (5 अंक)
(b) व्याख्या कीजिए कि क्यों, साम्यावस्था पर एक घटक का रासायनिक विभव सभी सहविद्यमान प्रावस्थाओं में एकसमान होना चाहिए। एक द्वयी द्रव-वाष्प निकाय के लिए रासायनिक विभव के पदों में साम्यावस्था प्रतिबंध की व्युत्पत्ति कीजिए। (15 अंक)
(c) बोस-आइंस्टाइन बंटन फलन का प्रयोग करके कृष्णिका विकिरण के लिए प्लांक विकिरण नियम की व्युत्पत्ति कीजिए। रेले-जीन्स नियम से व्युत्पन्न क्लासिकी परिणामों से किस प्रकार क्वांटम सांख्यिकी परिणाम भिन्न हैं, इसकी व्याख्या कीजिए। (20 अंक)
Answer approach & key points
Begin with a concise introduction stating the electromagnetic wave parameters in (a), then systematically solve for angular frequency using c = ω/k, displacement current via Maxwell's equations, and Poynting vector magnitude. For (b), explain chemical potential equality using entropy maximization/Gibbs free energy minimization, then derive the binary liquid-vapor equilibrium condition μₗ = μᵥ for each component. For (c), derive Planck's law by applying Bose-Einstein statistics to photon gas, obtaining energy density u(ν,T), then explicitly contrast with Rayleigh-Jeans divergence at high frequencies (ultraviolet catastrophe). Allocate approximately 25% time to (a), 35% to (b), and 40% to (c) based on mark distribution.
For (a)(i): Calculate ω = ck = 3×10⁸ × 100 = 3×10¹⁰ rad/s, and displacement current density J_d = ε₀(∂E/∂t) = 10ε₀ω sin(ωt-100x) A/m² with correct magnitude
For (a)(ii): Apply Poynting vector magnitude S = B²c/μ₀ = (55×10⁻⁸)² × 3×10⁸/(4π×10⁻⁷) ≈ 72.4 W/m² with proper unit conversion
For (b): Explain that at equilibrium, dS = 0 requires equal chemical potentials to prevent particle flow; derive μᵢˡ = μᵢᵛ for binary system using Gibbs-Duhem or equality of fugacities
For (c): Derive Planck's law starting from Bose-Einstein distribution ⟨n⟩ = 1/(e^(hν/kT)-1), obtaining u(ν,T) = (8πhν³/c³)/(e^(hν/kT)-1) and energy density integration
For (c) contrast: Explicitly show Rayleigh-Jeans u(ν,T) = 8πν²kT/c³ diverges as ν→∞ (ultraviolet catastrophe) while Planck's law converges, introducing quantum h