Physics 2021 Paper I 50 marks Compulsory Calculate

Q5

(a) Given that the electric potential of a system of charges is V = 12/r² + 1/r³ volt. Calculate the electric field vector at the Cartesian point (4, 2, 3) m. (10 marks) (b) Eight indistinguishable balls are to be arranged in six distinguishable boxes. Calculate the total number of ways in which the above can be done. (10 marks) (c) A rod of length l is perpendicular to a uniform magnetic field B. The rod revolves at an angular speed ω about an axis passing through one end of the rod and parallel to the magnetic field B. Find the voltage induced across the rod's ends. (10 marks) (d) Calculate the critical constants for CO₂ for which the Van der Waals constants are given by a = 0·0072 and b = 0·002. Also calculate the Boyle's temperature of CO₂. The unit of pressure is atmosphere and the unit of volume is that of a gm-mole of the gas at NTP. (10 marks) (e) Consider the two branch parallel circuit shown in the diagram. Determine the resonant frequency of the circuit. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक आवेशों के तंत्र का वैद्युत विभव V = 12/r² + 1/r³ वोल्ट से दिया गया है । कार्तीय निर्देशांक (4, 2, 3) m पर स्थित बिंदु के लिए वैद्युत क्षेत्र सदिश की गणना कीजिए । (10 अंक) (b) आठ अविभेदित गेंदों को छः विभेदित डिब्बों में रखना है । इस बात की गणना कीजिए कि उपर्युक्त कार्य को कुल कितनी विधियों से कर सकते हैं । (10 अंक) (c) l लंबाई की एक छड़ एकसमान चुंबकीय क्षेत्र B के लंबवत है । यह छड़ अपने एक सिरे से गुजरते हुए तथा चुंबकीय क्षेत्र B के समांतर अक्ष के चारों तरफ कोणीय चाल ω से घूर्णन कर रही है । छड़ के सिरों के आर-पार प्रेरित वोल्टता ज्ञात कीजिए । (10 अंक) (d) CO₂ के क्रांतिक नियतांक की गणना कीजिए जिसके लिए वान्डर वाल्स नियतांक a = 0·0072 और b = 0·002 दिया गया है । CO₂ के लिए बॉयल ताप की भी गणना कीजिए । दाब का मात्रक (इकाई) वायुमंडल (atmosphere) और आयतन का मात्रक (इकाई) NTP पर गैस के एक gm-mole के समान है । (10 अंक) (e) चित्र में दिए गए द्वि-शाखी समांतर परिपथ के लिए अनुनाद आवृत्ति की गणना कीजिए । (10 अंक)

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Directive word: Calculate

This question asks you to calculate. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

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How this answer will be evaluated

Approach

Calculate requires precise numerical solutions with full derivations. Structure: begin with stating relevant formulas for each sub-part, show step-by-step calculations with proper unit handling, and conclude with final numerical answers. Allocate ~20% time each to (a), (c), (d), (e) which involve multi-step derivations, and ~20% to (b) which is direct combinatorics. For (e), explicitly construct the impedance expression before solving for resonant frequency.

Key points expected

  • For (a): Apply E = -∇V in spherical coordinates, convert to Cartesian components at point (4,2,3) where r = √29, obtaining Ex, Ey, Ez values
  • For (b): Apply Bose-Einstein statistics for indistinguishable particles: ways = (n+k-1)!/(n!(k-1)!) = 13!/(8!×5!) = 1287
  • For (c): Derive motional EMF using ε = ∫(v×B)·dl with v = ωr, integrating from 0 to l to get ε = ½Bωl²
  • For (d): Calculate critical constants using Vc = 3b, Tc = 8a/(27Rb), Pc = a/(27b²), then Boyle temperature TB = a/(Rb)
  • For (e): Set up parallel LC circuit admittance Y = 1/(R+jωL) + jωC, find ω where Im(Y) = 0 for resonance

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all five physical concepts: gradient of potential for (a), Bose-Einstein counting for (b), motional EMF integration for (c), van der Waals critical point mathematics for (d), and complex admittance cancellation for (e)Identifies most concepts correctly but confuses distinguishable vs indistinguishable statistics in (b) or uses series resonance formula for (e)Fundamental errors like using E = -dV/dr scalar form for (a), Maxwell-Boltzmann statistics for (b), or ignoring phase in (e)
Derivation rigour20%10Complete derivations: full ∇V expansion with unit vector conversion for (a), explicit integration limits for (c), derivation of critical point conditions from (∂P/∂V)T = 0 for (d), admittance rationalization for (e)Skips intermediate steps like Cartesian conversion in (a) or assumes critical formulas without derivation in (d)Jumps to final formulas without derivation, missing key steps like the vector nature of E-field or integration procedure
Diagram / FBD10%5Clear diagrams for (c) showing rod rotation, B-field direction, velocity vectors and dl element; circuit diagram for (e) with labeled components and current directionsBasic sketch for (c) without vector labels or incomplete circuit diagram for (e)No diagrams where essential, or misleading diagrams showing incorrect geometry
Numerical accuracy30%15Precise calculations: r = √29 ≈ 5.385 m for (a), exact combinatorial result 1287 for (b), correct critical constants with proper units (atm, K, L/mol) for (d), simplified resonant frequency expression for (e)Minor arithmetic errors or unit inconsistencies, like using wrong R value in (d) or approximate r value affecting final digitsMajor calculation errors, wrong order of magnitude, or missing units throughout
Physical interpretation20%10Interprets E-field direction toward origin for (a), explains why result is large for (b), relates EMF to flux cutting rate for (c), discusses CO₂ liquefaction relevance for (d), and explains resonance condition meaning for (e)Brief mention of physical meaning without elaboration, or correct interpretation for only some sub-partsPurely mathematical treatment with no physical insight, or incorrect interpretation of results

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