Snell's Law Calculator

Quick Presets

Total Internal Reflection — sin θ₂ > 1 for this combination. Light does not pass through.

Solve For

Medium 1

Material

n₁

Angle θ₁ (°)

Medium 2

Material

n₂

Angle θ₂ (°)

Select material or enter refractive index manually. Angles in degrees.

What Is Snell's Law?

Snell's Law (the Law of Refraction), named after Dutch mathematician Willebrord Snellius (1621), governs how light changes direction when passing from one transparent medium to another:

n₁ · sin θ₁ = n₂ · sin θ₂

n₁, n₂ = refractive indices; θ₁, θ₂ = angles from the normal

Light bends toward the normal when entering a denser medium (higher n) and away from the normal when entering a rarer medium (lower n).

Refractive Index Explained

The refractive index n = c/v, where c = 2.998 × 10⁸ m/s is the speed of light in vacuum and v is the speed in the medium. A higher n means light slows down more and bends more at the interface.

v = c / n | λ_medium = λ_vacuum / n

Speed of light in water: v = 2.998×10⁸ / 1.333 ≈ 2.25×10⁸ m/s. In diamond: v ≈ 1.24×10⁸ m/s — less than half the vacuum speed.

Critical Angle and Total Internal Reflection

When light travels from a denser medium (n₁) to a rarer medium (n₂ < n₁), the refracted ray bends away from the normal. At the critical angle θ_c, the refracted ray travels along the interface (θ₂ = 90°):

θ_c = arcsin(n₂ / n₁)

For incidence angles greater than θ_c, all light is reflected back — Total Internal Reflection (TIR). This powers optical fiber, cat-eye road reflectors, prism binoculars, and the dazzling appearance of cut diamonds.

Refractive Index Table (20+ Materials)

Material	n	Critical angle from air
Vacuum	1.0000	—
Air	1.0003	—
Ice	1.31	49.8°
Water	1.333	48.6°
Sugar Solution 25%	1.372	46.8°
Human Cornea	1.376	46.6°
Ethanol	1.36	47.3°
Olive Oil	1.47	42.9°
Acrylic / Plexiglas	1.49	42.2°
Glycerin	1.473	42.8°
Optical Fiber Core	1.475	42.7°
Crown Glass	1.52	41.1°
Quartz / Salt NaCl	1.544	40.4°
Polycarbonate	1.58	39.3°
Flint Glass	1.62	38.2°
Sapphire	1.77	34.4°
Cubic Zirconia	2.16	27.6°
Diamond	2.417	24.4°

Applications: Fiber Optics, Cameras, Mirages

Optical Fiber

Core (n=1.475) surrounded by cladding (n≈1.46). Light is totally internally reflected and guided over thousands of km.

Camera Lenses

Multi-element lens designs use different glass types (different n) to correct chromatic aberration via opposing dispersions.

Mirages

Hot desert air near the ground has lower n; light from the sky refracts upward and undergoes TIR, creating the illusion of water.

Diamond Brilliance

Diamond's n=2.417 means θ_c=24.4°. Nearly all light entering a well-cut diamond undergoes multiple TIR, exiting only from the top face.

Worked Examples

Example 1 — Air → Water 30°

n₁=1.0003, θ₁=30°, n₂=1.333

sin θ₂ = n₁ sin θ₁ / n₂ = 1×0.5/1.333

sin θ₂ = 0.3750 → θ₂ = 22.08°

Example 2 — Crown Glass → Air Critical

n₁=1.52, n₂=1.0003

θ_c = arcsin(1.0003 / 1.52)

θ_c = arcsin(0.6581) = 41.1°

Example 3 — Air → Diamond 45°

n₁=1.0003, θ₁=45°, n₂=2.417

sin θ₂ = 1×0.7071/2.417 = 0.2926

θ₂ = 17.0°

Example 4 — Fiber Optic Critical Angle

n_core=1.475, n_cladding=1.46

θ_c = arcsin(1.46/1.475)

θ_c = arcsin(0.9898) = 81.8°

Frequently Asked Questions

Snell's Law (also called the law of refraction) states that when light passes from one medium to another, n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal. It governs how light bends at the interface between two transparent materials.

The refractive index n of a material is the ratio of the speed of light in vacuum (c = 2.998 × 10⁸ m/s) to the speed of light in the material: n = c/v. A higher n means light travels more slowly in that material. Air ≈ 1.0003, water ≈ 1.333, glass ≈ 1.52, diamond ≈ 2.417.

The critical angle θ_c is the angle of incidence (in a denser medium) at which the refracted ray travels exactly along the interface (θ₂ = 90°). For angles greater than θ_c, total internal reflection occurs instead of refraction. θ_c = arcsin(n₂/n₁), valid only when n₁ > n₂.

Total internal reflection (TIR) occurs when light traveling from a denser medium hits the interface at an angle greater than the critical angle. All light is reflected back into the denser medium — none passes through. TIR is the fundamental principle behind optical fiber communication, diamonds' brilliance, and mirages.

When light exits water (n≈1.333) into air (n≈1.0003), it bends away from the normal according to Snell's Law. Your brain traces the light ray back in a straight line, placing the apparent position of a submerged object at a shallower depth than it actually is. A straw in a glass of water appears bent at the water surface for the same reason.

Optical fibers use total internal reflection to transmit light signals over long distances. The fiber core (n ≈ 1.475) is surrounded by cladding (n ≈ 1.46). Light entering at an angle greater than the critical angle (~80°) is totally reflected and guided along the fiber without loss. A single fiber can carry terabits of data per second.

At the sodium D line (589 nm): water ≈ 1.333, crown glass ≈ 1.52, flint glass ≈ 1.62, diamond ≈ 2.417. Diamond's very high refractive index gives it a critical angle of only 24.4° from air, meaning light bounces internally many times before exiting — creating the characteristic sparkle. Refractive index varies slightly with wavelength (dispersion).

Snell's Law applies to all electromagnetic wavelengths, but the refractive index n varies with wavelength — a phenomenon called dispersion. Short wavelengths (violet, blue) refract more than long wavelengths (red) in most materials. This dispersion is what causes a prism to split white light into a rainbow spectrum and why chromatic aberration occurs in lenses.

Results

Step-by-Step Solution