Snell’s law calculator
n₁ · sin θ₁ = n₂ · sin θ₂
Three refraction calculations: the refracted angle from n₁·sin θ₁ = n₂·sin θ₂ (with the bending direction explained, and total internal reflection served as a verdict when the critical angle is exceeded), the critical angle itself. Which only exists going from the denser medium to the less dense one, and the tool says so, and the Brewster angle where the reflection is fully polarised.
28.13°
- Calculation
- sin θ₂ = n₁·sin θ₁ ÷ n₂ = 1 × sin(45°) ÷ 1.5 → θ₂ = 28.13°
The ray is refracted at 28.13°: it bends toward the normal (denser medium).
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasn₁ · sin θ₁ = n₂ · sin θ₂
n₁ · sin θ₁ = n₂ · sin θ₂
critical angle: θc = arcsin(n₂/n₁), if n₁ > n₂
Brewster: θB = arctan(n₂/n₁)
Angles are measured from the normal to the surface, the convention that makes the formula work, and the most frequent measurement mistake. Toward a denser medium (n₂ > n₁) the ray bends toward the normal; toward a less dense one it bends away, up to the critical angle beyond which sin θ₂ would exceed 1: there is then no refracted ray at all, total internal reflection. At the Brewster angle, reflected and refracted rays are perpendicular and the reflection is fully polarised.
- Refractive index (n)
- · the ratio of the speed of light in vacuum to its speed in the medium. It depends on wavelength: that is dispersion, which makes the prism rainbow.
- Critical angle (θc)
- · the incidence beyond which refraction becomes impossible, going from the denser medium to the less dense one. The guiding principle of optical fibre.
- Brewster angle (θB)
- · the incidence where the component polarised in the plane of incidence is no longer reflected at all: the reflected light is fully polarised, the principle of polarising filters in photography.
Validity domainThe entered indices are assumed constant: the tool does not model dispersion (n depends on wavelength, the usual values are given for sodium yellow).
The entered indices are assumed constant: the tool does not model dispersion (n depends on wavelength, the usual values are given for sodium yellow). It handles plane interfaces between homogeneous, transparent. Isotropic media: no index gradients (mirages), no absorption, no birefringence. Reflected and transmitted intensities (Fresnel coefficients) are not computed, only the angles are, plus the polarisation verdict at Brewster.
Common pitfall: measuring the angle from the surfaceSnell’s law uses the angle to the normal.
Snell’s law uses the angle to the normal. A ray “at 30° to the water surface” actually hits at 60° from the normal, applying the formula to the wrong angle gives a plausible, wrong result. Another directional trap: total internal reflection only exists from denser to less dense; from air into glass there is always a refracted ray, whatever the incidence.