Snell's Law Calculator

Frequently Asked Questions

What is Snell's law?

n₁·sinθ₁ = n₂·sinθ₂, describing how light bends when passing between media of different refractive index. Angles are measured from the surface normal.

What is the critical angle?

When light goes to a less dense medium, the angle of incidence at which the refracted ray grazes the surface: θ_c = arcsin(n₂/n₁). Beyond it, total internal reflection occurs.

Why does light bend toward the normal entering water?

It slows in the denser medium (higher n), so the wavefront pivots toward the normal. Going the other way it bends away.

Where is total internal reflection used?

Fibre-optic cables, prisms in binoculars, and the sparkle of diamonds (n = 2.42 gives a small critical angle, trapping light).

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How This Calculator Works

Snell's Law governs how light bends when it crosses a boundary between two transparent media. The relationship is n₁ × sinθ₁ = n₂ × sinθ₂, where n is the refractive index of each medium and θ is the angle measured from the normal (the line perpendicular to the surface), not from the surface itself. This tool rearranges that single equation four ways: solving for the refraction angle θ₂, the incidence angle θ₁, or either refractive index. When solving for θ₂ it computes sinθ₂ = (n₁ × sinθ₁) ÷ n₂, then takes the inverse sine. If that intermediate value exceeds 1 the inverse sine has no real solution - physically this is total internal reflection (TIR), and the calculator instead reports the critical angle. Built-in presets fill common indices: air 1.0003, water 1.333, glass 1.5, and diamond 2.417.

Worked Example: Air → Water

A light ray strikes a calm pond at 30° from the normal. With n₁ = 1.0003 (air) and n₂ = 1.333 (water): sinθ₂ = (1.0003 × sin 30°) ÷ 1.333 = (1.0003 × 0.5) ÷ 1.333 ≈ 0.3752. Then θ₂ = arcsin(0.3752) ≈ 22.03°. The ray bends toward the normal because it slows entering the denser medium. Reverse it - light leaving water into air bends away from the normal, and beyond the critical angle ≈ 48.6° it cannot escape at all, reflecting entirely back into the water. That trapped-light effect is exactly why a swimming pool's surface looks mirror-like from below at a shallow angle.

Refractive Index & the Speed of Light

The refractive index n = c ÷ v is the ratio of light's speed in vacuum (c ≈ 3 × 10⁸ m/s) to its speed v in the material. A higher n means light travels slower and bends more sharply. Diamond's very high index of 2.417 produces a tiny critical angle (≈ 24.4°), so light entering a cut diamond bounces internally many times before exiting - the optical origin of its "fire" and brilliance. Index also varies slightly with wavelength (dispersion): violet light has a higher n than red, which is why a prism fans white light into a spectrum and why lenses show chromatic aberration.

Total Internal Reflection & Critical Angle

Condition: TIR occurs only when light travels from a denser medium to a less dense one (n₁ > n₂) and the incidence angle exceeds the critical angle θc.

Critical angle: θc = arcsin(n₂ ÷ n₁) - the incidence angle at which the refracted ray would graze the surface (θ₂ = 90°).

Fiber optics: signals travel kilometres down a glass core because every bounce exceeds θc, losing almost no light.

Endoscopes & binoculars: use TIR prisms instead of mirrors for near-perfect, tarnish-free reflection.

Common Mistakes & Limits

The single biggest error is measuring angles from the surface instead of the normal - a ray 20° above a surface is at a 70° incidence angle. Snell's Law assumes flat boundaries, monochromatic light, and isotropic, non-absorbing media; it does not by itself give how much light reflects versus refracts (that requires the Fresnel equations) and breaks down for birefringent crystals like calcite, where two indices apply. It also says nothing about intensity loss, scattering, or diffraction. Indices here are typical values at roughly 589 nm (sodium yellow) and around 20°C; real materials shift with temperature, pressure, and wavelength.

Frequently Asked Questions

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