Govt. Exams
θ = 1.22 × λ/D = 1.22 × (500 × 10⁻⁹)/(1 × 10⁻³) = 1.22 × 500 × 10⁻⁶ = 610 × 10⁻⁶ = 6.1 × 10⁻⁴ rad.
Using lensmaker's formula: focal length changes with relative refractive index. f_m/f_a = [(n_lens/n_medium - 1)/(n_lens/n_air - 1)] = [(1.5/1.33 - 1)/(1.5/1 - 1)] = [0.128/0.5] = 0.256. So f_m = f_a/0.256 = 20/0.256 ≈ 78 cm. Recalculating: relative index in medium = 1.5/1.33 ≈ 1.128. f_m = f_a × (1.5 - 1)/(1.128 - 1) ≈ 50.4 cm.
For Fraunhofer diffraction by circular aperture (Airy disk), the radius of first dark ring = 1.22λf/D, which is proportional to λ/D.
Grating equation: d·sin(θ) = m·λ. Here d = 1/(500 × 10³) = 2 × 10⁻⁶ m. For m = 2: (2 × 10⁻⁶)·sin(30°) = 2·λ. (2 × 10⁻⁶)·(0.5) = 2·λ. λ = 500 nm.
Optical path through film = nt. Geometric path = t. Extra optical path = nt - t = (n-1)t. This causes a phase shift equivalent to a path difference of (n-1)t.
Using lens maker's formula: 1/f = (n-1)[1/R₁ + 1/R₂] = (0.5)[1/20 + 1/20] = (0.5)(2/20) = 1/20. Therefore f = 20 cm.
Intensity is proportional to (slit width)². If I₁:I₂ = 4:1, then w₁:w₂ = √4:√1 = 2:1.
Angular width of central maximum = 2λ/a. If slit width a doubles, angular width becomes 2λ/(2a) = λ/a, which is half the original.
tan(θ_B) = n = 1.5. θ_B = arctan(1.5) = 56.31°. At this angle, reflected light is completely polarized.
All three conditions are equivalent and occur at minimum deviation: symmetric path, equal angles, and ray parallel to base.
