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APEX CLASS ESTD 2018

IITJEE: Lens/Mirror Combinations, Prisms & YDSE Interference Visualizer

By

Ashish Kumar Bahekar

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Apex Class • IITJEE Ray & Wave Optics Lab

Lens/Mirror Combinations, Prisms & YDSE Interference Visualizer

Master Silvered Lenses, Liquid Immersion, Prism Deviation, and YDSE Optical Slab Fringe Shifts!

🔬 Select a High-Frequency JEE Ray Optics Setup:
F₁ F₂ Object (u) Real Image (v) Lens Master Equations 1/v – 1/u = 1/f | m = v / u = f / (f + u) • Lens Maker: 1/f = (μ_g/μ_m – 1)(1/R₁ – 1/R₂) • Min Distance (Real Obj & Real Img): D_min = 4f • Displacement Method: f = (D² – x²) / (4D) • Also in Displacement Method: h_obj = √(I₁ · I₂)! 🎯 Equiconvex Shortcut: For a glass lens (μ = 1.5) with R₁ = +R and R₂ = -R in air, 1/f = (0.5)(2/R) ⇒ f = R! Case A: Glass Lens (μ_g = 1.5) in Water (μ_l = 4/3) f_water = +4 × f_air! Power drops to 1/4th, but still Converging Case B: In Denser Liquid (μ_l = 1.65 > μ_g = 1.5) f_liquid < 0 (Diverging!) Convex Lens acts like a Concave Lens! 💧 Liquid Immersion Ratio: f_liquid / f_air = (μ_g – 1) / (μ_g/μ_l – 1) | Note: A mirror’s focal length f = R/2 NEVER changes in water! 1. Silvered Lens = Equivalent Mirror P_eq = 2 P_lens + P_mirror 1 / f_eq = 1 / f_m – 2 / f_L • Silvered Flat Side (f_m = ∞): f_eq = R / [ 2(μ – 1) ] • Silvered Curved Side: f_eq = R / (2μ) 2. Cutting a Symmetric Lens (Focal Length f) ✂️ Case A: Cut Vertically (⊥ to Principal Axis): One surface becomes flat (R₂ = ∞) → f_new = 2f! ✂️ Case B: Cut Horizontally (∥ to Principal Axis): Both radii R₁, R₂ unchanged → f_new = f (Same!) (Only aperture area & image brightness become half!) 🪞 Why 2 P_lens in a Silvered Lens? Because the light ray passes through the lens TWICE—once going in, and once after reflecting! A r₁ = r₂ = A/2 (Parallel to Base!) At δ_min: i = e = (A + δ_m) / 2 δ Incidence Angle (i) → ★ Min Deviation δ_m (Only when i = e!) i = e 🌈 Prism Formula: μ = sin[(A + δ_m)/2] / sin(A/2) | Thin Prism (A < 10°): δ = (μ – 1)A | No Emergence (TIR): A > 2 i_c!
Mirror vs Lens Formula
1/v ± 1/u = 1/f
+ for Mirror, – for Lens
Lenses in Contact (d Gap)
1/f = 1/f₁ + 1/f₂ – d/(f₁f₂)
Power P = P₁ + P₂ – d·P₁P₂
Critical Angle & TIR
sin(i_c) = μ_rarer / μ_denser
Light Bulb Cone Area = π h²/(μ²-1)
Optical Instruments (M)
M_tele = f_o / f_e
Tube Length L = f_o + f_e (Normal)
🌊 Young’s Double Slit Experiment (Fringe Width β = λD/d). Select YDSE Setup:
S₁ S₂ ↕ d ← Screen Distance D (D ≫ d) → Central Maxima (Δx = 0, I = 4I₀) 1st Bright (y₁ = β = λD/d) I_net = 4 I₀ cos²(φ / 2) Phase φ = (2π / λ) × Path Diff Δx ✨ All YDSE fringes have EQUAL width β = λD/d, and Energy is strictly conserved (I_avg = (I_max + I_min)/2 = 2I₀)! S₁ S₂ Slab (μ, t) Old Center O (y = 0) Shift Δy ↑ Optical Path & Fringe Shift Formulas 1. Extra Optical Path: Δx_slab = (μ – 1) t 2. Fringe Shift: Δy = (μ – 1) t D / d 3. Number of Fringes Shifted: N_shift = Δy / β = (μ – 1) t / λ • Note: Fringe Width β stays UNCHANGED! 🧱 Slab Rule: Pattern ALWAYS shifts TOWARD the slit covered by the slab, while fringe width β = λD/d remains unchanged! 1. In Air (Refractive Index = 1) Fringe Width β_air = λ D / d 2. Immersed in Water (μ = 4/3) New Width: β_liquid = β_air / μ! Angular Width: θ = β/D = λ / (μ d) 💧 Liquid Immersion Rule: Wavelength shrinks to λ/μ, so the entire interference pattern shrinks by factor μ (β’ = β/μ)! Single-Slit Diffraction (Slit Width ‘a’) ← Central Max Width W = 2λD / a (2× β!) → -λ/a +λ/a Polarization Laws (Malus & Brewster) 1. Unpolarized Light (I₀) thru 1st Polarizer: I₁ = I₀ / 2 (Always halves first!) 2. Malus’s Law (Through Analyzer at angle θ): I₂ = I₁ cos²θ = (I₀ / 2) cos²θ 3. Brewster’s Angle: tan(i_B) = μ (i_B + r = 90°!) 🕶️ Diffraction vs YDSE Trap: In Single-Slit Diffraction, a sinθ = nλ gives DARK minima (not bright maxima)!
Intensity & Amplitude
I_max / I_min = (√I₁+√I₂)²/(√I₁-√I₂)²
Also = (a₁ + a₂)² / (a₁ – a₂)²
Fringe Width (β)
β = λ D / d
Angular Fringe Width θ = λ / d
nᵗʰ Dark Fringe Position
y_n = (2n – 1) λD / (2d)
1st Dark is at y = ½ β
Central Diffraction Max
W = 2 λ D / a (θ = 2λ/a)
Twice as wide as side maxima!
🔥 2 High-Yield JEE Wave Optics Shortcuts:
1. Slit Width Ratio in YDSE: Intensity is directly proportional to slit width: I₁ / I₂ = w₁ / w₂ = a₁² / a₂².
2. Coincident Fringes (Two Wavelengths λ₁ & λ₂): Minimum distance from central maxima where bright fringes coincide is y_min = n₁ β₁ = n₂ β₂ where n₁ / n₂ = λ₂ / λ₁ (in simplest integer ratio)!
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