Snell's Law Calculator
Calculate the angle of refraction when light passes between two media using Snell's Law (n₁sin θ₁ = n₂sin θ₂). Also determines the critical angle for total internal reflection. See also our Frequency to Wavelength Converter and Speed Calculator.
How to Calculate Refraction Using Snell's Law
Snell's Law describes how light bends when it passes from one transparent medium to another. When a ray of light hits the boundary between two materials at an angle, it changes direction because the speed of light differs in each medium. This bending is called refraction and is the principle behind lenses, prisms, fiber optics, and even the apparent bending of a straw in a glass of water.
To use Snell's Law, you need three pieces of information: the refractive index of the first medium (n₁), the refractive index of the second medium (n₂), and the angle of incidence (θ₁) measured from the normal (perpendicular) to the surface. The law then gives you the angle of refraction (θ₂). If light moves from a less dense to a more dense medium (n₁ < n₂), it bends toward the normal. If it moves from more dense to less dense (n₁ > n₂), it bends away from the normal.
The refractive index of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium: n = c/v. A higher refractive index means light travels more slowly in that material. Air has n ≈ 1.0003, water has n ≈ 1.333, and diamond has n ≈ 2.417. The large refractive index of diamond is what gives it its brilliant sparkle — light undergoes total internal reflection at many facets.
Snell's Law Formula
Snell's Law:
n₁ × sin(θ₁) = n₂ × sin(θ₂)
Solving for θ₂:
θ₂ = arcsin[(n₁/n₂) × sin(θ₁)]
Critical Angle (for n₁ > n₂):
θ_c = arcsin(n₂/n₁)
Refractive Index:
n = c/v = speed of light in vacuum / speed in medium
Wavelength Relationship:
λ_medium = λ_vacuum / n
Brewster's Angle:
θ_B = arctan(n₂/n₁)
Example Calculation
Light travels from air (n₁ = 1.0) into glass (n₂ = 1.5) at an angle of incidence of 30°. Find the angle of refraction:
Given: n₁ = 1.0, n₂ = 1.5, θ₁ = 30°
Apply Snell's Law: n₁×sin(θ₁) = n₂×sin(θ₂)
1.0 × sin(30°) = 1.5 × sin(θ₂)
1.0 × 0.5 = 1.5 × sin(θ₂)
sin(θ₂) = 0.5 / 1.5 = 0.3333
θ₂ = arcsin(0.3333) = 19.47°
The light bends toward the normal (from 30° to 19.47°)
because it enters a denser medium.
Critical angle (glass to air): arcsin(1.0/1.5) = 41.81°
Refractive Index Reference Table
| Medium | Refractive Index (n) |
|---|---|
| Vacuum | 1.0000 |
| Air (STP) | 1.0003 |
| Water | 1.3330 |
| Ethanol | 1.3610 |
| Fused Silica | 1.4585 |
| Crown Glass | 1.5200 |
| Flint Glass | 1.6600 |
| Sapphire | 1.7700 |
| Cubic Zirconia | 2.1500 |
| Diamond | 2.4170 |
| Silicon | 3.4500 |
| Germanium | 4.0000 |
Frequently Asked Questions
What is Snell's Law?
Snell's Law (also called the law of refraction) describes the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices. It states that n₁×sin(θ₁) = n₂×sin(θ₂), where n₁ and n₂ are the refractive indices and θ₁ and θ₂ are the angles measured from the normal to the surface. This law was discovered by Willebrord Snellius in 1621 and independently by René Descartes.
What is total internal reflection?
Total internal reflection occurs when light traveling in a denser medium (higher n) hits the boundary with a less dense medium at an angle greater than the critical angle. At this point, no light passes through — it is all reflected back. The critical angle is θ_c = arcsin(n₂/n₁). This phenomenon is the basis of fiber optic communication, where light bounces along glass fibers with virtually no loss, and is also responsible for the sparkle of diamonds and the mirage effect on hot roads.
What is the critical angle?
The critical angle is the angle of incidence above which total internal reflection occurs. It only exists when light travels from a denser medium to a less dense medium (n₁ > n₂). It is calculated as θ_c = arcsin(n₂/n₁). For glass (n=1.5) to air (n=1.0), the critical angle is about 41.8°. For water to air, it is about 48.6°. For diamond to air, it is only about 24.4°, which is why diamonds trap and reflect so much light internally.
Why does light bend when entering a different medium?
Light bends because its speed changes when entering a different medium. The wavefronts of light arrive at the boundary at an angle, and the part that enters the new medium first slows down (or speeds up), causing the wavefront to pivot. This is analogous to a car driving from pavement onto sand at an angle — the wheel that hits sand first slows down, causing the car to turn. The amount of bending depends on the ratio of speeds (refractive indices) in the two media.
What is Brewster's angle?
Brewster's angle is the angle of incidence at which reflected light becomes completely polarized. At this angle, the reflected and refracted rays are perpendicular to each other. It is calculated as θ_B = arctan(n₂/n₁). For air to glass (n=1.5), Brewster's angle is about 56.3°. Polarizing sunglasses exploit this principle — they block horizontally polarized light reflected from surfaces like water and roads, reducing glare.
Does Snell's Law apply to all types of waves?
Yes, Snell's Law applies to any wave that changes speed when crossing a boundary between two media. This includes sound waves (refraction in the atmosphere causes sound to bend), seismic waves (used in geophysics to map Earth's interior), water waves (refraction causes waves to align with shorelines), and radio waves. The formula is the same: the ratio of sines equals the ratio of wave speeds in the two media.
Applications of Snell's Law
Snell's Law is fundamental to optical engineering and has countless applications. Eyeglasses and contact lenses use refraction to correct vision by bending light to focus properly on the retina. Camera lenses use multiple refracting elements to form sharp images. Fiber optic cables rely on total internal reflection to transmit data over thousands of kilometers with minimal signal loss. Prisms use refraction to separate white light into its component colors (dispersion), since the refractive index varies slightly with wavelength.
In nature, refraction explains why pools appear shallower than they are, why the sun appears above the horizon when it has already set (atmospheric refraction), and why mirages form on hot roads. Geophysicists use seismic wave refraction to map underground rock layers and locate oil deposits. Medical ultrasound imaging accounts for refraction at tissue boundaries to produce accurate images.