Snell's Law Calculator
Snell's Law describes how light bends when crossing the boundary between two materials with different refractive indices. This calculator finds the refraction angle, checks for total internal reflection, and shows the critical angle.
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Formula
n₁ sin(θ₁) = n₂ sin(θ₂)
Snell's Law: n₁ × sin(θ₁) = n₂ × sin(θ₂). Rearranged: θ₂ = arcsin(n₁ × sin(θ₁) / n₂). Total internal reflection occurs when n₁ > n₂ and the incidence angle exceeds the critical angle θ_c = arcsin(n₂/n₁). Refractive index n = c/v, where c is the speed of light in vacuum and v is the speed in the medium.
How to use the Snell's Law Calculator
- 1
Enter your refractive index of medium 1 (n₁)
- 2
Enter your angle of incidence (θ₁)
Value should be in °.
- 3
Enter your refractive index of medium 2 (n₂)
- 4
Read your results instantly
Results update in real time as you type.
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Why light bends at boundaries
Light bends when it crosses from one medium to another because its speed changes. In vacuum, light travels at c = 299,792,458 m/s. In glass (n ≈ 1.5), it slows to c/1.5 ≈ 200,000,000 m/s. The refractive index n = c/v quantifies this slowdown.
The bending occurs because the wavefront must remain continuous at the boundary — the part of the wave that enters the new medium first changes speed, causing the whole wavefront to pivot. This is analogous to a marching band turning: the outside edge marches faster than the inside, and the whole line curves. Higher refractive index means slower light and more bending.
Total internal reflection and fiber optics
When light travels from a denser medium (higher n) to a less dense one (lower n), the refracted angle is larger than the incidence angle. At a critical incidence angle, the refracted ray emerges at 90° (grazes the surface). Beyond this critical angle, no refraction occurs — all light reflects back into the denser medium. This is total internal reflection (TIR).
For glass-air (n_glass = 1.5, n_air = 1): critical angle = arcsin(1/1.5) = 41.8°. Any ray hitting the surface at more than 41.8° reflects completely. Optical fibers exploit this: light entering the fiber core bounces down it via repeated total internal reflections, carrying data across continents with minimal loss.
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Dispersion and rainbows
The refractive index of glass and water varies slightly with wavelength — a phenomenon called dispersion. Red light (700 nm) has a slightly lower n than violet light (400 nm), so it bends less. When white sunlight enters a water droplet, each color refracts at a slightly different angle, exits at a slightly different angle, and the result is a rainbow arc.
Prisms work the same way: a glass prism disperses white light into its spectrum. Newton used this to demonstrate that white light is a mixture of all visible colors, not a pure entity as previously believed. Chromatic aberration in camera lenses occurs for the same reason — different colors focus at slightly different distances.
Tips & Insights
Common refractive indices
Air/vacuum: 1.0. Water: 1.33. Glass (crown): 1.52. Glass (flint): 1.62. Diamond: 2.42. A higher n means slower light and more bending.
Going from dense to sparse increases the angle
Glass to air: angle gets larger. Water to air: angle gets larger. Air to glass: angle gets smaller. Light bends toward the normal when entering a denser medium.
Diamond's critical angle is 24.4°
Diamond's high n (2.42) gives a critical angle of only arcsin(1/2.42) ≈ 24.4°. Most light entering a diamond undergoes TIR, creating its characteristic sparkle (brilliance).
Worked Examples
Air to glass
sin(θ₂) = 1 × sin(45°)/1.5 = 0.7071/1.5 = 0.4714. θ₂ = arcsin(0.4714) ≈ 28.1°. Light bends toward the normal entering glass.
Glass to air (TIR check)
sin(θ₂) = 1.5 × sin(45°)/1 = 1.061 > 1. Total internal reflection occurs — critical angle is 41.8°, and 45° exceeds it.
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Frequently Asked Questions
What is Snell's Law?
n₁ sin(θ₁) = n₂ sin(θ₂). When light crosses a boundary between media, the product of refractive index and sine of angle is conserved.
What is the refractive index?
n = c/v — the ratio of the speed of light in vacuum to its speed in the medium. Air: ≈1.0. Water: 1.33. Glass: ~1.5. Diamond: 2.42.
What is total internal reflection?
When light in a dense medium hits a less dense boundary at an angle greater than the critical angle, it reflects entirely back into the dense medium instead of refracting through.
How do fiber optics use Snell's Law?
Light travels along a glass fiber by repeatedly undergoing total internal reflection at the core-cladding boundary. The cladding has a lower refractive index than the core, trapping the light.
Why does a straw look bent in water?
Light from the underwater part of the straw refracts as it exits the water, changing direction. Your brain traces it back in a straight line, making the straw appear displaced or bent.
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