Physics 2024 Paper I 50 marks Compulsory Solve

Q5

(a) In spherical coordinates, V = –25 V on a conductor at r = 2 cm and V = 150 V on another conductor at r = 35 cm. The space between the conductors is a dielectric for which ε_r = 3·12. Find the surface charge densities on the conductors. (10 marks) (b) Find the magnetic field strength (H) at the centre of a square current loop of side L. (10 marks) (c) The magnitude of the average electric field normally present in the Earth's atmosphere just above the surface of the Earth is about 150 N/C, directed radially inward, toward the centre of the Earth. What is the total net surface charge carried by the Earth? Assume the Earth to be a conductor. (The radius of the Earth is 6·37×10^6 m) (10 marks) (d) Prove that the work done by a perfect gas during a quasi-static adiabatic expansion is given by $$W = \frac{P_i V_i}{\gamma - 1}\left[1 - \left(\frac{P_f}{P_i}\right)^{\left(\frac{\gamma-1}{\gamma}\right)}\right]$$ where γ is the ratio of specific heats. (10 marks) (e) Calculate the Fermi energy in electron-volt for sodium assuming that it has one free electron per atom. The density of sodium = 0.97 gm/cc and the atomic weight of sodium is 23. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) गोलीय निर्देशांक प्रणाली में r = 2 cm पर एक चालक पर V = –25 V और r = 35 cm पर दूसरे चालक पर V = 150 V है। चालकों के बीच ε_r = 3·12 का एक परावैद्युत है। चालकों पर पृष्ठ आवेश घनत्वों को ज्ञात कीजिए। (10 अंक) (b) भुजा L के एक वर्गाकार धारा लूप के केंद्र पर चुंबकीय क्षेत्र की तीव्रता (H) ज्ञात कीजिए। (10 अंक) (c) पृथ्वी की सतह से ठीक ऊपर पृथ्वी के वायुमंडल में सामान्यतः विद्यमान औसत विद्युत क्षेत्र का परिमाण लगभग 150 N/C है, जो पृथ्वी के केंद्र की ओर त्रिज्यतः निर्देशित है। पृथ्वी द्वारा अधोनत कुल नेट पृष्ठ आवेश क्या है? पृथ्वी को एक चालक मान लीजिए। (पृथ्वी की त्रिज्या 6·37×10^6 m है) (10 अंक) (d) सिद्ध कीजिये कि एक स्थैतिकल्प रूद्धोष्म प्रसार के दौरान एक आदर्श गैस द्वारा किया गया कार्य $$W = \frac{P_i V_i}{\gamma - 1}\left[1 - \left(\frac{P_f}{P_i}\right)^{\left(\frac{\gamma-1}{\gamma}\right)}\right]$$ है, जहाँ γ विशिष्ट ऊष्माओं का अनुपात है। (10 अंक) (e) सोडियम के लिए फर्मी ऊर्जा (इलेक्ट्रॉन-वोल्ट में) की गणना कीजिये, यह मानकर कि इसमें प्रति परमाणु एक मुक्त इलेक्ट्रॉन है। सोडियम का घनत्व = 0.97 gm/cc है और सोडियम का परमाणु भार 23 है। (10 अंक)

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Approach

This is a multi-part numerical problem requiring systematic solution of five independent physics problems. Begin with a brief statement of the governing equations for each part, then solve sequentially: (a) spherical capacitor with Laplace's equation, (b) Biot-Savart law for square loop, (c) Gauss's law application, (d) thermodynamic derivation from first law, and (e) Fermi-Dirac statistics. Allocate time proportionally: ~4 minutes per mark, with careful unit checking throughout. Conclude each part with physical interpretation of results.

Key points expected

  • Part (a): Apply Laplace's equation in spherical coordinates, find potential V(r) = A + B/r, determine constants from boundary conditions, then use D = εE and σ = D·n̂ to find surface charge densities on both conductors
  • Part (b): Use Biot-Savart law dH = Idl×r̂/(4πr²), integrate over four sides of square, exploit symmetry; each side contributes equal H perpendicular to plane, with geometric factor involving L/2 and angle integration
  • Part (c): Apply Gauss's law ∮E·dA = Q_enclosed/ε₀ with spherical Gaussian surface just above Earth; E is radial inward so Q = -4πε₀R²E, yielding net negative charge
  • Part (d): Start from first law dU = δQ - δW with δQ = 0 for adiabatic, use dU = nC_vdT and PV^γ = constant; integrate PdV from V_i to V_f, substitute using adiabatic relations to obtain required form
  • Part (e): Calculate electron number density n = ρN_A/M, then apply Fermi energy formula E_F = (ℏ²/2m)(3π²n)^(2/3), convert to eV; note sodium's BCC structure with one conduction electron per atom

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all governing principles: Laplace's equation for (a), Biot-Savart law for (b), Gauss's law for (c), first law of thermodynamics with adiabatic condition for (d), and Fermi-Dirac statistics with free electron model for (e); no conceptual errors in any partCorrect principles for most parts but minor errors such as using Coulomb's law instead of Laplace's equation for (a), or incorrect assumption about electron density in (e); some confusion between B and H in (b)Fundamental misconceptions: treats (a) as parallel plate capacitor, uses Ampere's circuital law for (b), forgets adiabatic condition in (d), or uses classical statistics for (e)
Derivation rigour20%10Complete mathematical rigor: explicit integration steps in (b) showing angle subtended, clear boundary condition application in (a), systematic elimination of variables in (d) with proper justification of each step, dimensional consistency throughoutCorrect final expressions but skips key steps: omits integration limits in (b), assumes rather than derives potential form in (a), or presents final formula in (d) without showing intermediate PV^γ = constant substitutionMissing derivations entirely or logically flawed: states results without proof, algebraic errors in manipulating exponents in (d), incorrect integration of Biot-Savart components in (b)
Diagram / FBD10%5Clear diagrams for (b) showing square loop with current direction, field point at center, distance geometry (L/2, r = L/2√2); spherical Gaussian surface for (c) with labeled radius and field direction; PV diagram for (d) showing adiabat with shaded work areaBasic sketches present but incomplete: loop diagram without geometric labels, or missing Gaussian surface for (c); diagrams support but don't fully illuminate the solutionNo diagrams where essential (especially part b), or misleading diagrams with wrong geometry; failure to indicate directions of fields and currents
Numerical accuracy35%17.5All numerical values correct with proper significant figures: (a) σ_inner ≈ -1.35 μC/m², σ_outer ≈ 77.4 nC/m²; (b) H = 2√2 I/(πL); (c) Q ≈ -6.8×10⁵ C; (e) E_F ≈ 3.15 eV; unit conversions (cm→m, gm/cc→kg/m³) flawlessCorrect methodology but arithmetic errors: factor of 2 or π mistakes, incorrect powers of 10, wrong unit conversions (uses ε₀ instead of ε = ε_rε₀ in a), or final answers without unitsOrder-of-magnitude errors, wrong formulas substituted, complete failure to calculate numerical values for parts (a), (c), (e); leaves answers in terms of symbols when numbers requested
Physical interpretation15%7.5Insightful interpretation: explains sign of charges in (a) and (c) indicating Earth's negative charge balances atmospheric ionization; discusses why H at center of square differs from circular loop; notes sodium's E_F is typical of metals with free electron behavior; comments on magnitude of Earth's chargeBrief mention of physical meaning without depth: states signs correctly but doesn't explain origin, or notes numerical result without comparing to typical valuesPurely mathematical treatment with no physical insight; fails to recognize that negative Q in (c) implies electron excess, or that E_F >> kT validates degenerate fermion treatment

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