Snell's Law of Refraction Calculator
Calculate the angle of refraction, angle of incidence, refractive index, and critical angle using Snell's Law (n₁ sin θ₁ = n₂ sin θ₂). Supports standard refraction, critical angle detection, and Brewster's angle calculations. Perfect for optics students, engineers, and anyone working with light and lenses.
What Is Snell's Law?
Snell's Law (also known as the law of refraction) describes the relationship between the angles of incidence and refraction when light passes through the boundary between two different isotropic media. It is named after the Dutch astronomer and mathematician Willebrord Snellius, who discovered the relationship in 1621.
The law is expressed by the equation:
n₁ · sin θ₁ = n₂ · sin θ₂
Where:
- n₁ is the refractive index of the first medium (where light originates).
- θ₁ is the angle of incidence (measured from the normal).
- n₂ is the refractive index of the second medium (where light enters).
- θ₂ is the angle of refraction (measured from the normal).
The refractive index 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): n = c / v. Since light travels fastest in a vacuum, the refractive index is always ≥ 1.
How Does the Snell's Law Calculator Work?
This calculator supports three modes of calculation:
1. Refraction Angle: n₁ · sin θ₁ = n₂ · sin θ₂
Solve for θ₂, θ₁, n₁, or n₂ given the other three values.
2. Critical Angle: θc = sin⁻¹(n₂ / n₁)
The maximum angle of incidence at which refraction can occur when light travels from a denser to a less dense medium.
3. Brewster's Angle: θB = tan⁻¹(n₂ / n₁)
The angle at which reflected light is completely polarized perpendicular to the plane of incidence.
When solving for the angle of refraction, the calculator automatically detects if total internal reflection (TIR) occurs. TIR happens when light travels from a denser medium to a less dense medium (n₁ > n₂) and the angle of incidence exceeds the critical angle. In this case, no refracted ray exists — all light is reflected back into the first medium.
Why Use This Snell's Law Calculator?
- Three Modes: Supports refraction angle, critical angle, and Brewster's angle calculations.
- Flexible Solving: In refraction mode, solve for θ₂, θ₁, n₁, or n₂.
- Auto TIR Detection: Automatically detects and reports total internal reflection.
- Quick Presets: Choose from common material pairs like air→water, air→glass, and diamond→air.
- Visual Chart: See the refraction angle plotted against the incidence angle.
- Reference Table: Built-in refractive index reference for common materials.
- Free & Private: No registration, no data storage.
Understanding Refraction
Refraction is the bending of light as it passes from one medium to another. This bending occurs because light travels at different speeds in different materials. When light enters a denser medium (higher refractive index), it slows down and bends toward the normal. When it enters a less dense medium, it speeds up and bends away from the normal.
This principle is fundamental to many optical devices, including:
- Lenses: Glasses, cameras, microscopes, and telescopes rely on refraction to focus light.
- Prisms: Split white light into its component colors through dispersion.
- Fiber Optics: Use total internal reflection to transmit data over long distances.
- Rainbows: Formed by refraction, dispersion, and internal reflection in water droplets.
- The Human Eye: The cornea and lens refract light to focus images on the retina.
Common Applications of Snell's Law
- Optical Engineering: Designing lenses, prisms, and optical fibers.
- Meteorology: Explaining atmospheric refraction and mirages.
- Medicine: Understanding how light penetrates biological tissues.
- Underwater Vision: Explaining why objects appear distorted underwater.
- Gemology: Identifying gemstones by their refractive indices and brilliance.
❓ Snell's Law Calculator FAQ
What is Snell's Law?
Snell's Law describes the relationship between the angles of incidence and refraction when light passes between two media: n₁ · sin θ₁ = n₂ · sin θ₂. It explains how light bends when it enters a material with a different refractive index.
What is the formula for Snell's Law?
The formula is 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).
What is the refractive index?
The refractive index (n) of a medium is the ratio of the speed of light in a vacuum to the speed of light in that medium: n = c / v. It is a dimensionless number ≥ 1. For example, water has n ≈ 1.33, meaning light travels 1.33 times faster in a vacuum than in water.
What is the critical angle?
The critical angle is the angle of incidence beyond which total internal reflection occurs. It is defined as θc = sin⁻¹(n₂ / n₁), where n₁ > n₂. At angles greater than θc, no light refracts into the second medium; all light is reflected back.
What is total internal reflection?
Total internal reflection (TIR) occurs when light traveling from a denser medium to a less dense medium (n₁ > n₂) strikes the boundary at an angle greater than the critical angle. All light is reflected back into the first medium. TIR is used in fiber optics and binoculars.
What is Brewster's angle?
Brewster's angle is the angle of incidence at which light reflected from a surface is completely polarized perpendicular to the plane of incidence. It is given by θB = tan⁻¹(n₂ / n₁). At this angle, the reflected and refracted rays are perpendicular to each other.
What is the refractive index of air?
The refractive index of air at standard temperature and pressure is approximately 1.0003. In most calculations, it is rounded to 1.00. Only in high-precision work is the difference from 1.0 significant.
What is the refractive index of water?
The refractive index of water at 20°C is approximately 1.333. This value varies slightly with temperature and wavelength of light.
What is the refractive index of glass?
The refractive index of common crown glass is approximately 1.52. Flint glass has a higher index of about 1.66. Different types of glass have different refractive indices depending on their composition.
What is the refractive index of diamond?
Diamond has a refractive index of approximately 2.42, which is very high. This high refractive index gives diamond its exceptional brilliance and fire through total internal reflection and dispersion.
Why does light bend when entering water?
Light bends when entering water because it slows down. The speed of light in water (≈ 225,000 km/s) is slower than in air (≈ 300,000 km/s). This change in speed causes the light to change direction according to Snell's Law.
What is the difference between refraction and reflection?
Refraction is the bending of light as it passes through a medium (transmission). Reflection is the bouncing of light off a surface (no transmission). Both occur at boundaries between media, but refraction involves light entering the second medium while reflection involves light bouncing back.
Can I use this calculator for real-world engineering applications?
Yes. This calculator is suitable for physics homework, optics engineering, and real-world applications such as lens design, fiber optics, and gemology. Always verify results with professional tools for critical applications.
What happens when light travels along the normal?
When light travels along the normal (θ₁ = 0°), it enters the second medium without changing direction (θ₂ = 0°). This is because sin(0°) = 0, so Snell's Law gives n₁ · 0 = n₂ · sin θ₂, which means θ₂ = 0°.
What is the relationship between refractive index and the speed of light?
The refractive index is inversely proportional to the speed of light in the medium: n = c / v. A higher refractive index means light travels slower in that medium. For example, light travels at c in a vacuum (n = 1) and at c / 1.33 in water.