Physics 2025 Paper I 50 marks Compulsory Solve

Q5

(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 प्रश्न 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)

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Approach

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.

Key points expected

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

Evaluation rubric

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies image charge placement in (a), distinguishes angle with plane vs angle with normal in (b), states KCL/KVL precisely with sign conventions in (c), recognizes displacement current as source of magnetic field in (d), and applies entropy equality (not inequality) for reversible engine in (e)	Minor errors in image charge position or angle interpretation; KCL/KVL statements present but sign conventions unclear; recognizes displacement current exists but confuses D and E; applies first law correctly but entropy condition partially wrong	Fundamental errors: treats charge above conductor as isolated, uses θ=60° directly in torque formula, states laws without physical meaning, ignores displacement current completely, or applies Carnot efficiency formula incorrectly for three reservoirs
Derivation rigour	20%	10	Complete derivations: field superposition with vector components in (a), torque from force on current elements in (b), continuity equation to KCL and Faraday's law to KVL in (c), Maxwell-Ampère integral form to H in (d), entropy balance with algebraic solution in (e); all steps logically connected	Derivations present but skips key steps: assumes image method without justification, quotes torque formula without derivation, states conservation laws without showing connection to circuit laws, uses differential vs integral form inconsistently, algebraic solution present but entropy derivation abbreviated	Missing derivations: jumps to final formulas, no justification for image method, no derivation of KCL/KVL from Maxwell's equations, confuses conduction and displacement current, or solves numerically without showing entropy conservation equation
Diagram / FBD	15%	7.5	Clear diagrams: (a) shows charge, image charge, field vectors at observation point with coordinate axes; (b) depicts coil orientation, normal vector, B-field direction with 30° angle labeled; (c) illustrates junction and loop with current/voltage conventions; (d) shows capacitor cross-section with E and H directions; (e) energy flow diagram with three reservoirs	Diagrams present but incomplete: missing image charge in (a), angle mislabeled in (b), generic circuit without specific KCL/KVL labeling, capacitor shown without field directions, engine schematic without heat flow arrows	No diagrams or seriously flawed: wrong geometry, missing essential elements, or diagrams that contradict written solution; pure algebraic treatment without visual representation
Numerical accuracy	25%	12.5	All calculations correct: (a) E = 9√2 N/C at 45° in xy-plane, normal component verified zero; (b) τ = 0.75 Nm; (c) no numerical values required; (d) H = 2.21×10⁻⁴ A/m at r=10 cm; (e) Q₁ = -300 kJ, Q₂ = -500 kJ with algebraic check; proper units and significant figures throughout	One or two numerical errors: arithmetic mistakes in vector addition, incorrect angle substitution (using 60° gives wrong answer), unit conversion errors (cm to m), sign errors in heat interactions, or algebraic solution with calculation slip	Multiple errors: wrong constants (k, ε₀, μ₀), order of magnitude mistakes, completely wrong answers (negative torque, heat flowing from cold to hot reservoir without work justification), or missing numerical values in parts requiring them
Physical interpretation	20%	10	Interprets results physically: explains why field must be normal to conductor surface (boundary condition), discusses torque direction and equilibrium orientation, relates KCL/KVL to fundamental conservation laws in practical circuits, identifies H as azimuthal and explains displacement current equivalence, verifies entropy generation is zero and discusses engine reversibility	Brief interpretation after calculations: states boundary condition without explanation, mentions torque tends to align dipole, notes laws are conservation principles, recognizes H exists due to changing E, states reversible means no entropy production without elaboration	No physical interpretation: pure calculation with no discussion of results, fails to verify answers make physical sense (e.g., heat flows into all reservoirs), or misinterprets direction of torque, field orientation, or entropy change signs

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