Ray Optics

At 30 cm from the lens, on the same side as the object

Step 1: For the convex lens, u = -30 cm, f = +20 cm. 1/v - 1/(-30) = 1/20 → 1/v = 1/20 - 1/30 = (3-2)/60 = 1/60. So v = 60 cm. Image I₁ forms 60 cm to the right of lens. Step 2: The mirror is 50 cm to the right of the lens. So the image I₁ is 60 - 50 = 10 cm behind the mirror (virtual object for mirror). For the concave mirror, u = +10 cm (virtual object, behind mirror), f = -15 cm. Step 3: Mirror equation: 1/v + 1/u = 1/f → 1/v = 1/(-15) - 1/(10) = -1/15 - 1/10 = (-2-3)/30 = -5/30 = -1/6. v = -6 cm. Image I₂ forms 6 cm in front of mirror (real image). Step 4: This image I₂ is at 50 - 6 = 44 c

45°

Step 1: If the ray grazes the second surface, angle of refraction at second surface r₂ = 90° — no, this means angle of incidence at second surface equals critical angle iₒ. Step 2: Critical angle: sin iₒ = 1/n = 1/√2 → iₒ = 45°. So r₂ = 45°. Step 3: Since r₁ + r₂ = A = 45°, we get r₁ = 45° - 45° = 0°. Step 4: Applying Snell's law at first surface: n₁ sin i₁ = n₂ sin r₁ → 1 × sin i₁ = √2 × sin 0° = 0 → i₁ = 0°. Wait: r₁ = 0° gives i₁ = 0°. But the answer options say 45°. Let me redo: r₁ + r₂ = A = 45°. If grazing at second face, r₂ = iₒ = 45°. Then r₁ = 45° - 45° = 0°. So Snell's law: sin i₁ =

√0.60 ≈ 0.775

Step 1: Numerical aperture formula for optical fibre in air (n₃ = 1): NA = √(n₁² - n₂²). Step 2: n₁ = 1.6 (core), n₂ = 1.4 (cladding). Step 3: NA = √(1.6² - 1.4²) = √(2.56 - 1.96) = √0.60 = 0.7746 ≈ 0.775. Step 4: The acceptance angle iₐ = sin⁻¹(NA) = sin⁻¹(0.775) ≈ 50.8°. Why wrong options: 0.2 = n₁ - n₂ (wrong, that's not the formula); 1.0 has no basis; √0.40 would result from n₁=1.4, n₂=1.0 which mixes up the values.

-10 cm

Step 1: For the silvered lens system, the effective power is P = 2P_lens + P_mirror. Step 2: Lens (plano-convex with flat surface facing object, convex silvered): R₁ = ∞ (flat), R₂ = -20 cm (convex surface, silvered). Power of lens: P_l = (n-1)(1/R₁ - 1/R₂) = (1.5-1)(1/∞ - 1/(-20)) = 0.5 × (0 + 1/20) = 0.5/20 = 1/40 D (in cm units). Step 3: Power of mirror (silvered convex surface): P_m = 1/(-f_m). For the convex surface, R = 20 cm, so f_m = R/2 = 10 cm. But since light hits the inner side (concave from light's perspective), f_m = -10 cm. P_m = 1/(-(-10)) = 1/10. Step 4: Total power P = 2(1/40