Bohr radius r_n ∝ MCQ with Answer and Explanation

Explanation:
Fringe width β = λD/d, where λ wavelength, D screen distance, d slit separation. New d' = 2d, new D' = D/2. Thus β' = λ(D/2)/(2d) = λD/(4d) = β/4. Fringe width reduces to one-fourth. Memory tip: 'β ∝ D/d; changes multiply'. This proportional reasoning problem tests wave optics understanding, frequently appearing in competitive exams. Always track how each parameter change affects the result; combine multiplicative factors for net effect. Verify with dimensional analysis: β has length dimension, consistent with λD/d.

Explanation:
sinθ_c = n₂/n₁ = 1/1.5 = 2/3 ⇒ θ_c = sin⁻¹(0.667) ≈ 42°. Total internal reflection condition. Memory tip: 'θ_c = sin⁻¹(1/n) for denser to air'. Optics calculation frequently tested in competitive exams with standard refractive indices.

Explanation:
U ∝ V². If V' = 2V, U' = 4U. Energy scales with square of voltage. Memory aid: 'Capacitor energy: U ∝ V²; double voltage ⇒ quadruple energy'. Electrostatics proportionality frequently tested in competitive exams.