Reflection and Refraction: Laws, Snell's Law & Critical Angle Explained

A straw in a glass of water looks bent. A diamond ring sparkles more than a glass one. A mirage shimmers on a hot road. Rainbows arc across the sky after rain. Every one of these is caused by light changing direction — either bouncing off a surface, or bending as it crosses from one medium into another. Reflection and refraction are not separate phenomena with separate explanations: both arise from the same underlying fact that light travels at different speeds in different materials.

The mathematics is clean. The law of reflection (angle in = angle out) needs one line. Snell's law (n₁sinθ₁ = n₂sinθ₂) needs two. But the applications branch out into fibre optic cables, camera lenses, glasses prescriptions, and the total internal reflection that makes diamonds brilliant. Getting the physics right opens all of those.

The Law of Reflection

The angle of incidence (θ_i) equals the angle of reflection (θ_r). Both angles are measured from the normal — the line perpendicular to the surface at the point of incidence. The incident ray, reflected ray, and normal all lie in the same plane.

This law holds for all types of reflection — flat (plane) mirrors, curved mirrors, and even rough surfaces (where the normal varies across the surface, causing diffuse reflection rather than specular reflection).

Refraction and the Refractive Index

When light passes from one medium to another, its speed changes. The refractive index n of a medium is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in that medium (v):

Since light always travels slower in matter than in vacuum, n ≥ 1 for all real materials. Air: n ≈ 1.0003 (effectively 1.0). Water: n = 1.33. Glass: n ≈ 1.5. Diamond: n = 2.42.

Snell's Law

Snell's Law (also called the law of refraction) gives the angle of refraction when light crosses a boundary:

where n₁ is the refractive index of the first medium, θ₁ is the angle of incidence, n₂ is the refractive index of the second medium, and θ₂ is the angle of refraction. Both angles are measured from the normal.

Key consequences:

• Light entering a denser medium (n₂ > n₁) bends toward the normal (θ₂ < θ₁).

• Light entering a less dense medium (n₂ < n₁) bends away from the normal (θ₂ > θ₁).

• Light at normal incidence (θ₁ = 0°) does not bend — it passes straight through.

Worked Example: Snell's Law

Light in air (n = 1.0) hits a glass surface (n = 1.5) at 40° to the normal. Find the refraction angle.

Light bends toward the normal (25.4° < 40°) on entering the denser glass. ✓

Total Internal Reflection

When light travels from a denser medium to a less dense medium (n₁ > n₂), the refracted angle θ₂ > θ₁. As θ₁ increases, at some point θ₂ reaches 90° — the refracted ray travels along the surface. This angle is the critical angle θ_c:

For glass (n = 1.5) to air (n = 1.0): sinθ_c = 1/1.5 = 0.667 → θ_c = 41.8°.

For angles greater than θ_c, total internal reflection occurs — all light is reflected back into the denser medium. No refraction takes place. This is the principle behind:

Optical fibres: light enters a glass fibre and is totally internally reflected along its entire length with almost no loss, enabling high-bandwidth telecommunications. The fibre core has higher n than the surrounding cladding, ensuring TIR at all incidence angles above the critical angle.

Diamond brilliance: diamond's very high n (2.42) gives a critical angle of only 24.4° — much smaller than glass. Most light entering a diamond exceeds the critical angle and undergoes TIR many times before exiting, producing the characteristic sparkle (fire and brilliance) that makes cut diamonds so visually striking.

Prisms in binoculars: use TIR to reflect light and fold the optical path without the energy loss of metallic mirrors.

Why Does Refraction Cause a Straw to Look Bent?

When you look at a straw in a glass of water, the part below the water surface appears displaced from the part above. Your eye traces the light rays reaching it backward in a straight line — but the rays have been bent at the water-air interface. The brain, assuming straight-line travel, places the underwater straw tip at the wrong apparent position. This is why objects seen through water appear shallower than they are: apparent depth = real depth / n.

The Law of Reflection

When a light ray strikes a smooth surface, the angle of incidence θ_i equals the angle of reflection θ_r, both measured from the normal (perpendicular) to the surface:

The incident ray, reflected ray, and normal are all in the same plane. This applies to all types of waves — light, sound, water waves. A flat mirror produces a virtual image the same distance behind the mirror as the object is in front of it (plane mirror image).

Refraction and Snell's Law

When light passes from one medium to another, it changes speed and direction. The relationship between angles is given by Snell's Law:

where n₁, n₂ are the refractive indices of the two media and θ₁, θ₂ are angles from the normal. The refractive index n = c/v (speed of light in vacuum / speed of light in medium). Common values: air ≈ 1.000, water = 1.333, glass (crown) = 1.52, diamond = 2.42.

Worked Example 1: Refraction at Glass Surface

Light strikes air-glass boundary at 30° to the normal. n_glass = 1.52. Find the refraction angle.

Light bends toward the normal when entering a denser medium (n₂ > n₁).

Critical Angle and Total Internal Reflection

When light travels from a denser medium to a less dense one (n₁ > n₂), refraction bends away from the normal. At the critical angle θ_c, the refracted ray travels along the boundary (θ₂ = 90°):

Above θ_c: total internal reflection — all light reflects back into the denser medium. For glass-air: sin θ_c = 1/1.52 → θ_c = 41.1°. For diamond (n=2.42): θ_c = 24.4° — the small critical angle means most light entering a cut diamond undergoes total internal reflection many times, creating the characteristic sparkle.

Applications

Optical fibres: light travels along a glass core (n ≈ 1.5) surrounded by cladding (n ≈ 1.45). Total internal reflection keeps light trapped in the core through bends. Fibre optic cables carry internet and telephone signals as light pulses — one fibre thinner than a hair can carry thousands of simultaneous phone calls. Prisms: use refraction and internal reflection to separate colours (dispersion) or redirect beams. Binoculars use prisms to fold the optical path, making them compact. Mirage: hot air near the ground has lower refractive index than cooler air above. Light from the sky curves upward through total internal reflection, creating the appearance of water on a hot road.

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