(a) Compare nuclear density of hydrogen (₁H¹) with its atomic density. (Assume the atom to have the radius of its first Bohr orbit). What inference can one get from the above comparison ? 8+2=10

(b) The spacing between successive (100) planes in sod

Question

(a) Compare nuclear density of hydrogen (₁H¹) with its atomic density. (Assume the atom to have the radius of its first Bohr orbit). What inference can one get from the above comparison ? 8+2=10

(b) The spacing between successive (100) planes in sodium chloride is 1·41 Å. X-rays incident on the surface of the crystal are found to give rise to second order Bragg reflections at a glancing angle 10°. Calculate the wavelength of X-ray radiations. 10

(c) For the ground state of deuteron, prove that the radius of nucleon is of the order of ~ 2·15 × 10⁻¹³ cm. 10

(d) What is meant by strength of the interactions of elementary particles ? Classify the different forces on the basis of this strength of interaction. 10

(e) How does supercritical magnetic field depend on temperature ? For a superconducting specimen, the critical magnetic fields are respectively 1·45 × 10⁵ A/m and 4·2 × 10⁵ A/m for 14 K and 13 K. Determine the superconducting transition temperature and the critical field at 0 K. 10

UPSC Answer Check · Accepted Answer

This is a multi-part numerical and theoretical problem requiring systematic solving of five independent sub-parts. Allocate approximately 2 minutes per mark (20 minutes total), spending roughly 4 minutes each on parts (a), (b), (c), and (e) which involve calculations, and 4 minutes on part (d) which is descriptive. Begin each part with the relevant formula, show step-by-step working, and conclude with the final answer and physical significance. No introduction or conclusion is needed for this fragmented numerical question.
- Part (a): Calculate nuclear density using r₀ ≈ 1.2 fm and atomic density using Bohr radius a₀ = 0.529 Å; compare ~10¹⁴ times difference to infer atom is mostly empty space
- Part (b): Apply Bragg's law nλ = 2d sinθ with n=2, d=1.41 Å, θ=10° to find X-ray wavelength ≈ 0.49 Å
- Part (c): Use deuteron binding energy (2.224 MeV) and square well potential model/uncertainty principle to derive nucleon radius ~2.15×10⁻¹³ cm
- Part (d): Define interaction strength through coupling constants; classify four fundamental forces (strong, electromagnetic, weak, gravitational) with relative strengths ~1:10⁻²:10⁻⁷:10⁻³⁹
- Part (e): State Hc(T) = Hc(0)[1-(T/Tc)²]; use given data points to solve simultaneous equations for Tc ≈ 14.7 K and Hc(0) ≈ 4.5×10⁵ A/m

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies nuclear density formula with r₀, Bohr radius for atomic density, Bragg's law with correct order identification, deuteron binding energy relation, four fundamental forces with correct coupling constants, and the parabolic temperature dependence of critical field	Uses correct formulas but with minor conceptual errors like confusing first and second order in Bragg's law, or misidentifying force hierarchy; understands nuclear density is high but miscalculates magnitude	Fundamental misconceptions such as using atomic radius for nuclear density, applying Bragg's law without order number, or completely wrong force classification
Derivation rigour	20%	10	Shows complete step-by-step derivations: volume calculations with proper unit conversions, explicit Bragg law rearrangement, uncertainty principle or potential well derivation for deuteron radius, and systematic solution of two equations for Tc and Hc(0)	Skips some intermediate steps but maintains logical flow; shows main formula and final result without explicit unit conversion steps or algebraic manipulation details	Jumps directly to answers without derivation, or contains serious algebraic errors; missing essential steps that make verification impossible
Diagram / FBD	10%	5	Clear diagram for part (d) showing relative strengths on logarithmic scale, or crystal plane diagram for (b) showing glancing angle geometry; properly labeled axes and values	Rough sketch without proper labels, or mentions diagram needed but doesn't draw it clearly; basic representation of Bragg geometry	No diagrams where needed, or completely incorrect diagrams that misrepresent the physics; missing crystal plane or force comparison visualization
Numerical accuracy	30%	15	All five parts with correct numerical values: nuclear density ~2.3×10¹⁷ kg/m³, atomic density ~0.09 kg/m³, ratio ~10¹⁵; λ≈0.489 Å; radius derivation yielding ~2.15×10⁻¹³ cm; Tc≈14.7 K, Hc(0)≈4.5×10⁵ A/m; proper significant figures	Correct method but arithmetic errors in 1-2 parts, or wrong powers of ten; correct order of magnitude but imprecise final values; unit conversion errors	Major numerical errors in multiple parts, wrong orders of magnitude, or missing calculations entirely; demonstrates poor command of scientific notation
Physical interpretation	20%	10	Clear inference in (a) about empty atom and nuclear force dominance; explains why X-ray wavelength matches atomic spacing in (b); connects deuteron radius to nuclear force range; relates force strengths to everyday phenomena; interprets Tc as complete superconducting transition	Brief mention of physical significance without elaboration; states results but doesn't explain why they matter physically; superficial connection to real-world applications	Missing physical interpretation entirely, or incorrect physical conclusions; fails to draw any inference from calculated results; purely mathematical treatment

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