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Ray Optics

Part of the JEE Main study roadmap. Physics topic phy-023 of Physics.

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Ray Optics

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Ray Optics — Key Facts for JEE Main • Core formula: Snell’s Law — n₁ sin i = n₂ sin r. The product of refractive index and sine of angle is conserved across a boundary. • Core concept: Light travels in straight lines and changes direction when it crosses a boundary between media of different optical densities. • Most common application: Finding image position by spherical mirrors and thin lenses using the mirror/lens formula. • Key constant to memorise: Speed of light in vacuum, c = 3 × 10⁸ m/s. Refractive index of air ≈ 1.00. • Most tested concept in JEE Main: Lens maker formula 1/f = (n−1)(1/R₁ − 1/R₂) and mirror equation 1/v + 1/u = 1/f. • Common mistake students make: Using incorrect sign convention — always adopt the Cartesian (new) sign convention: distances measured from pole, positive to the right. ⚡ Exam tip: When in doubt, draw a clear ray diagram with angles measured from the normal. For mirrors, focal length f is negative for concave mirrors; for lenses, f is positive for convex lenses.

🟡 Standard — Regular Study (2d–2mo)

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Ray Optics — JEE Main / Advanced Study Guide

Ray optics, also called geometric optics, explains optical phenomena by treating light as straight-line rays. The central principle is Snell’s law, which governs refraction at interfaces: n₁ sin i = n₂ sin r. When light is incident on a boundary concave to the incident medium, it deviates toward the normal if the second medium has a higher refractive index.

Key equations to master:

  • Mirror equation: 1/v + 1/u = 1/f (sign convention: distances positive to the right of the pole)
  • Lens equation: 1/v − 1/u = 1/f for thin lenses
  • Lens maker formula: 1/f = (n − 1)(1/R₁ − 1/R₂)
  • Magnification: m = −v/u (mirror); m = v/u (lens)
  • Power of a lens: P = 1/f (in metres); unit: dioptre (D)

Important concepts:

  • Focal length of a spherical mirror: f = R/2, where R is the radius of curvature.
  • Angle of deviation: δ = i + r − A for a prism; minimum deviation occurs when i = r.
  • Total internal reflection (TIR): Occurs when light tries to go from a denser to a rarer medium at angles greater than the critical angle θc = sin⁻¹(n₂/n₁).
  • Refraction through a slab: Emergent ray is parallel to incident ray; net lateral displacement d = t sin i / cos r.

Common misconceptions: Forgetting that the refractive index depends on wavelength (dispersion); confusing the sign of radius for convex vs concave surfaces.

Practice Numerical 1: A convex lens has a focal length of 20 cm. An object is placed 60 cm from the lens. Find the image distance.

  • Using lens formula: 1/v − 1/u = 1/f → 1/v − (−1/60) = 1/20 → 1/v + 1/60 = 1/20 → 1/v = (3−1)/60 = 2/60 → v = 30 cm (real, inverted, magnified).

Practice Numerical 2: A ray of light incidents on a glass slab (n = 1.5) at 45°. Find the angle of refraction in glass.

  • Snell’s law: n_air sin i = n_glass sin r → 1 × sin 45° = 1.5 sin r → sin r = 0.7071/1.5 = 0.4714 → r ≈ 28.1°.

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Ray Optics — Comprehensive JEE Notes

Deeper theory and derivations:

Derivation of Snell’s law from Fermat’s principle: Fermat’s principle states that light travels along the path that minimises optical path length. For two media with refractive indices n₁ and n₂ separated by a straight interface, minimising the total travel time leads directly to n₁ sin θ₁ = n₂ sin θ₂. This is the foundational result from which all geometric optics flows. Fermat’s principle also explains the law of reflection (angle of incidence equals angle of reflection) as the path with stationary time.

Prism deviation: For a thin prism with apex angle A and refractive index n, the angle of minimum deviation δ_min is given by δ_min = A(n − 1) for small A. This approximation is valid when A < 10° and is frequently used in JEE problems. The more exact formula δ = (i − r₁) + (e − r₂) = i + e − A, with i = angle of incidence, e = angle of emergence. Minimum deviation occurs when i = e, giving r₁ = r₂ = A/2.

Chromatic dispersion: The refractive index of a medium varies with wavelength: n(λ) = A + B/λ² (Cauchy’s equation). For visible light, blue wavelengths refract more than red (higher n for shorter λ). This causes angular dispersion δ_B − δ_R = (n_B − n_R)A. In achromatic doublets, two lenses made of different materials (crown and flint glass) are combined to cancel chromatic aberration: their powers satisfy P₁/P₂ = −(n_B − n_R)/(n’_B − n’_R).

Critical angle and TIR: Derivation: sin θc = n₂/n₁ (n₁ > n₂). Applications: optical fibres (core n₁, cladding n₂), diamond sparkle (high n → large critical angle → most rays undergo TIR). When a ray enters a fibre at angle θ such that θ < θc, it undergoes multiple total internal reflections and travels with negligible loss.

Spherical aberration: For mirrors and lenses with spherical surfaces, marginal rays (rays far from the principal axis) focus at a different point than paraxial rays. This limits image quality. The cause is that spherical surfaces are not ideal parabolic. Mitigations: use aperture stops, aspheric surfaces, or combine multiple elements (Schmidt corrector plate).

Historical notes: Snell’s law was first stated in complete form by René Descartes in 1637, though ibn Sahl had discovered the law of refraction in the 10th century. The wave theory explanation was provided by Augustin-Jean Fresnel in the early 19th century via the Huygens–Fresnel principle. Fermat’s principle, articulated in 1657, provided the unifying variational principle underlying all geometric optics.

Comparative analysis with Wave Optics: Ray optics is the short-wavelength limit (λ → 0) of wave optics. When obstacles or apertures are comparable to the wavelength of light, wave effects (diffraction, interference) become significant and ray optics fails. For typical visible light (λ ≈ 500 nm) and everyday optical elements (cm scale), ray optics is extremely accurate. The transition to wave optics is governed by the Fresnel number F = a²/(Lλ), where a is aperture size and L is distance. When F >> 1, ray optics holds; when F ≈ 1, wave effects dominate.

Advanced solved example: A right-angle prism of refractive index √2 is used to deviate a ray by 90°. Find the incident angle and verify whether TIR occurs at the hypotenuse face. For 90° deviation, by geometry: r = 45° at the second face. Using Snell’s law at the first face: n sin i = n sin r → i = r (for equal refractive indices). At the hypotenuse, a ray inside the prism strikes at angle of incidence = 45°. Critical angle: θc = sin⁻¹(1/√2) = 45°. Since the angle of incidence equals the critical angle, the ray grazes the boundary — at exactly 45°, it is the threshold for TIR. Any slightly larger incidence angle results in total internal reflection.