Snell's Law
Snell's Law (also called the law of refraction) states that the product of the refractive index and the sine of the angle of incidence is constant across the interface between two media: n₁ sin θ₁ = n₂ sin θ₂. It quantitatively describes how a light ray changes direction when it transitions between media of different optical densities. Snell's Law is the cornerstone of lens design, fibre optic engineering, and the correction of refractive vision errors.
Key Formula
n1 × sin(θ1) = n2 × sin(θ2)
LaTeX: n_1 \sin\theta_1 = n_2 \sin\theta_2
| Symbol | Meaning | Unit |
|---|---|---|
| n₁ | Refractive index of incident medium | dimensionless |
| θ₁ | Angle of incidence measured from normal | degrees |
| n₂ | Refractive index of refracting medium | dimensionless |
| θ₂ | Angle of refraction measured from normal | degrees |
Worked Example
Problem
A light ray in water (n = 1.33) hits the water-glass interface (n = 1.60) at 45°. Calculate the refracted angle in the glass.
Solution
Step 1: Apply Snell's Law: n₁ sin θ₁ = n₂ sin θ₂ Step 2: 1.33 × sin 45° = 1.60 × sin θ₂ Step 3: sin 45° = 0.7071, so 1.33 × 0.7071 = 0.9404 Step 4: sin θ₂ = 0.9404 / 1.60 = 0.5878 Step 5: θ₂ = arcsin(0.5878) ≈ 36.0°
Answer
Angle of refraction in glass ≈ 36.0° (ray bends toward the normal as it enters denser glass)
Snell's Law: Refraction Angles for Air-to-Medium Interface (n_air = 1.00)
| Medium | n | Incident Angle (°) | Refracted Angle (°) | Bending Direction |
|---|---|---|---|---|
| Water | 1.33 | 30 | 22.1 | Toward normal |
| Water | 1.33 | 60 | 40.6 | Toward normal |
| Glass (crown) | 1.52 | 30 | 19.2 | Toward normal |
| Glass (crown) | 1.52 | 60 | 35.2 | Toward normal |
| Diamond | 2.42 | 30 | 12.0 | Toward normal |
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Related Terms
Light Refraction
Light refraction is the bending of a light ray as it passes from one transparent medium into another of different optical density, caused by a change in the wave's speed. The greater the difference in refractive indices between the two media, the more the ray bends toward or away from the normal. Refraction is responsible for phenomena such as the apparent bending of a straw in water, the formation of rainbows, and the focusing action of lenses.
Total Internal Reflection
Total Internal Reflection (TIR) occurs when a light ray travelling through a denser medium strikes the boundary with a less dense medium at an angle greater than the critical angle, causing the ray to be completely reflected back into the denser medium rather than refracted out. The critical angle θ_c is defined as sin θ_c = n₂/n₁, where n₁ > n₂. TIR is the operating principle behind optical fibres, diamonds' brilliance, and binocular prisms.
Lens (Optics)
In optics, a lens is a transmissive optical element, typically made of glass or transparent plastic, that refracts light to converge or diverge rays, thereby forming images. Lenses work by exploiting the refraction of light at curved surfaces, and their shape (convex or concave) determines whether rays are brought together (converging) or spread apart (diverging). Lenses are fundamental components of eyeglasses, cameras, microscopes, telescopes, and the human eye itself.
Named after Dutch mathematician Willebrord Snellius (1580–1626), who derived the relationship empirically around 1621. René Descartes independently rediscovered it and published it in "La Dioptrique" (1637). The law was also known as the Snell–Descartes law and was derived from Fermat's principle of least time by Pierre de Fermat in 1662.