(a) What is the minimum energy required to break a ₂He⁴ nucleus into free protons and neutrons?

[ Given, m_H = 1·007825 amu, m_n = 1·008665 amu, m_e = 0·00055 amu and m_He = 4·002603 amu ]
15

(b) (i) Consider a uranium nucleus (₉₂U²³⁶) breaking up

Question

(a) What is the minimum energy required to break a ₂He⁴ nucleus into free protons and neutrons?

[ Given, m_H = 1·007825 amu, m_n = 1·008665 amu, m_e = 0·00055 amu and m_He = 4·002603 amu ]
15

(b) (i) Consider a uranium nucleus (₉₂U²³⁶) breaking up spontaneously into two equal parts. Estimate the reduction of electrostatic energy of the nucleus considering uniform charge distribution.

[ Assume that nuclear radius is 1·2×10⁻¹³ A¹/³ cm ]
15

(ii) Is it possible for a photon to transfer all its energy to a free electron? Give reasons.
5

(c) Explain the cause of hysteresis phenomenon in ferromagnetic materials. What does the area of the hysteresis loop signify?
10+5=15

UPSC Answer Check · Accepted Answer

This is a multi-part numerical-cum-descriptive question requiring precise calculations for (a) and (b)(i), conceptual reasoning for (b)(ii), and explanatory analysis for (c). Allocate approximately 35% time to part (a) for careful mass-energy conversion, 35% to part (b) including both calculation and reasoning, and 30% to part (c) with a clear hysteresis diagram. Begin with the binding energy calculation using atomic masses correctly, proceed through electrostatic energy estimation with proper radius scaling, address the photon-electron collision physics with reference to Compton scattering constraints, and conclude with domain theory explanation and energy dissipation interpretation.
- Part (a): Correct identification that atomic mass of hydrogen includes electron, so use m_p = m_H - m_e or appropriate atomic mass accounting; calculation of mass defect Δm = [2m_H + 2m_n - m_He] or equivalent; conversion to energy using 1 amu = 931.5 MeV/c² yielding ~28.3 MeV
- Part (b)(i): Application of electrostatic energy formula U = (3/5)(Z²e²)/(4πε₀R) for uniform sphere; correct radius scaling R ∝ A^(1/3) with R_He = R₀(4)^(1/3) and R_U = R₀(236)^(1/3); calculation of energy ratio and reduction factor considering two fragments each with Z/2 and A/2
- Part (b)(ii): Recognition that photon-electron energy transfer requires momentum conservation; explanation that free electron cannot absorb photon completely due to simultaneous energy-momentum conservation violation; reference to Compton scattering or need for bound electron/third body
- Part (c): Explanation of hysteresis via domain wall movement, irreversible domain rotation, and pinning by impurities/crystal defects; clear distinction between reversible and irreversible magnetization processes; area interpretation as energy dissipated per unit volume per cycle (hysteresis loss)
- Part (c): Qualitative or quantitative sketch of B-H loop showing saturation, remanence, coercivity; labeling of key points and proper loop orientation

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	22%	11	Demonstrates flawless understanding across all parts: correct distinction between atomic and nuclear masses in (a), proper electrostatic energy model for nuclear fission in (b)(i), accurate application of relativistic kinematics for photon-electron interaction in (b)(ii), and precise domain theory with pinning mechanisms for hysteresis in (c)	Shows adequate conceptual grasp with minor errors: may confuse atomic/nuclear masses in (a) or use approximate formulas in (b)(i); gives partially correct reasoning for (b)(ii) without full conservation law treatment; explains hysteresis qualitatively without domain-level detail	Fundamental misconceptions: treats hydrogen mass as proton mass without electron correction, uses wrong electrostatic formula or radius scaling, claims photon can transfer all energy to free electron without qualification, or describes hysteresis as merely 'lag' without physical mechanism
Derivation rigour	18%	9	Presents complete, logically structured derivations: explicit mass defect calculation with clear algebraic steps in (a); systematic derivation of electrostatic energy ratio with proper handling of fragment parameters in (b)(i); rigorous momentum-energy conservation proof for (b)(ii); step-by-step domain energy analysis for (c)	Shows main derivation steps with gaps or shortcuts: states key formulas without full derivation, skips intermediate algebraic manipulations, or presents correct final expressions with incomplete justification	Missing derivations or incorrect logic: jumps to answers without showing work, uses dimensional guesses, or presents circular reasoning especially in (b)(ii) and energy interpretation in (c)
Diagram / FBD	12%	6	Clear, properly labeled B-H hysteresis loop for part (c) showing: saturation magnetization (M_s), remanent magnetization (M_r), coercive field (H_c), initial magnetization curve, and proper axes with units; may include schematic domain structure diagrams illustrating pinned walls	Acceptable sketch of hysteresis loop with basic labels (saturation, remanence, coercivity) but missing some details or with imperfect proportions; or adequate verbal description substituting for minor diagram deficiencies	No diagram provided for (c), or seriously flawed diagram with wrong shape (e.g., straight lines, no loop), mislabeled axes, or confusion between B-H and M-H representations; irrelevant diagrams for numerical parts
Numerical accuracy	28%	14	Precise calculations with correct significant figures: binding energy ~28.3 MeV (accepting 26.7-28.8 MeV range depending on mass convention), proper handling of 1 amu = 931.5 MeV/c²; fission energy reduction with correct numerical factor (~35-40% estimate acceptable with clear method); all unit conversions accurate	Correct method with minor arithmetic errors or slight unit conversion mistakes; correct order of magnitude but imprecise final values; acceptable approximations in (b)(i) with stated assumptions	Order-of-magnitude errors, wrong formulas leading to nonsensical results, missing unit conversions, or failure to complete numerical parts; confusion between mass defect and binding energy per nucleon
Physical interpretation	20%	10	Insightful physical reasoning throughout: connects binding energy to nuclear stability and magic numbers for He-4; relates electrostatic energy release to fission energetics and chain reactions; explains why Compton scattering requires energy sharing; directly links hysteresis area to power loss in Indian transformer cores and magnetic storage applications	Adequate physical interpretation with standard explanations: states binding energy significance, notes energy release in fission, mentions momentum conservation for photons, and identifies hysteresis loss without application context	Missing physical interpretation: treats calculations as purely mathematical exercises, fails to explain why results matter, or gives incorrect physical significance (e.g., hysteresis area as 'stored energy' rather than dissipated energy)

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