Snell's Law Calculator

Free Snell's Law Calculator

This Snell's Law Calculator finds a refracted angle, incident angle or unknown refractive index at the boundary between two transparent media. Enter the known refractive indices and angles in degrees, select the missing quantity, and receive a formula rearrangement, direct substitution and final result. The live diagram labels the two media, normal, incident ray, refracted ray and all values in one clear visual.

It is built for physics homework, CBSE, GCSE, A-level and introductory university optics. Snell's law explains the bending of a straw in water, the focusing action of lenses, prism colours, optical fibres and many camera effects. The calculator turns the equation into both a numerical answer and an easy-to-read picture.

Snell's Law Formula

Snell's law is n₁ sin θ₁ = n₂ sin θ₂. n₁ is the refractive index of the first medium, n₂ is the refractive index of the second medium, θ₁ is the angle of incidence and θ₂ is the angle of refraction. Angles must always be measured from the normal, which is perpendicular to the surface at the point where the ray arrives.

Rearrange to θ₂ = sin⁻¹(n₁ sin θ₁/n₂), θ₁ = sin⁻¹(n₂ sin θ₂/n₁), or n₂ = n₁ sin θ₁/sin θ₂. Refractive index has no unit. Use degree-mode trigonometry when working with the angles shown in a school ray diagram.

How to Use the Snell's Law Calculator

Choose refracted angle when light enters a second medium and you know n₁, n₂ and θ₁. Choose incident angle when θ₂ is known instead. Choose second-medium index when both measured angles and n₁ are known. The calculator displays the correct rearrangement automatically, avoiding an unnecessary algebra step.

First identify which side the incoming ray is travelling through; that is medium 1. Then mark the normal, not the interface itself, before reading the angle. Insert values with their labels, calculate the sine ratio and take inverse sine for an angle. The visual diagram updates alongside the result, providing a quick physical check.

Snell's Law Worked Example

A ray travels from air, n₁ = 1.00, into glass, n₂ = 1.50, at an incident angle of 45°. θ₂ = sin⁻¹(1.00 × sin 45°/1.50) = sin⁻¹(0.4714) = 28.13°. The refracted ray is closer to the normal than the incident ray because glass has a higher refractive index.

For the inverse question, suppose a ray goes from air at 30° to an unknown liquid at 22°. n₂ = 1.00 × sin 30°/sin 22° ≈ 1.34. That is close to water. The calculation provides a quantitative way to identify a material from measured refraction data.

Bending Toward and Away from the Normal

When light enters a higher-index medium, it slows and bends toward the normal. Air to water and air to glass are common examples. The refracted angle is smaller than the incident angle. When light enters a lower-index medium, it speeds up and bends away from the normal, making the refracted angle larger.

A ray that travels exactly along the normal has θ₁ = 0°. It does not change direction because sin 0 is zero, although its speed and wavelength still change. This is an important special case: refraction changes light speed at every boundary, but it only changes direction when the ray arrives at an angle.

Why Refraction Happens

Light behaves as a wave, and a wavefront changes speed when it crosses from one material into another. One side reaches the new material first and changes speed before the other side. This makes the wavefront rotate, changing the ray direction. Snell's law is the precise relationship that predicts the rotation.

The effect makes objects under water look closer to the surface than they are, creates the apparent bend of a pencil in a glass, and lets lenses form images. Frequency does not change at the boundary. Instead, speed and wavelength change together, which preserves the frequency of the source.

Refractive Index Values

Vacuum has n = 1.0000 and air is close to 1.0003. Water is approximately 1.33, typical glass about 1.5, and diamond roughly 2.42 for visible light. Exact values vary with wavelength, temperature, material composition and pressure. A stated value in a problem should be used rather than an approximate table value.

The index ratio, not simply the material name, determines bending. If two media have equal index, light does not bend at their interface. A greater difference between indices generally produces a larger change in angle, provided the incident angle is not zero.

Total Internal Reflection

When light attempts to move from a higher-index medium to a lower-index medium, θ₂ grows as θ₁ grows. At the critical angle, the refracted ray travels along the surface. Beyond it, no real refracted angle can satisfy Snell's law, and the ray reflects entirely back into the first medium. This is total internal reflection.

Optical fibres use total internal reflection to keep light inside a high-index core surrounded by lower-index cladding. It also contributes to the brightness of diamonds. The calculator reports total internal reflection whenever the inverse-sine input is outside the physically possible range. Use the Critical Angle Calculator for the exact threshold angle.

Applications of Snell's Law

Snell's law is used to design eyeglasses, contact lenses, cameras, microscopes, telescopes, projectors, prisms and fibre-optic systems. Lenses focus by refracting light at two curved surfaces. Camera lenses use multiple elements with selected indices to reduce distortion and colour fringes while forming sharp images on a sensor.

Scientists use refractometry and Snell's law to analyse liquids, measure concentration and identify materials. Engineers model underwater viewing, laser paths, medical endoscopes and communication fibres with the same equation. This makes the calculator useful far beyond a classroom ray diagram.

Common Errors and Accuracy

The most common mistake is measuring θ from the surface rather than the normal. If a diagram gives an angle to the surface, subtract it from 90° before using Snell's law. Other errors include swapping n₁ and n₂, using radians while inputs are in degrees, and taking sine instead of inverse sine when solving for an angle.

For high-precision work, use refractive index at the specified wavelength and temperature. Real media may absorb or scatter light, and index can vary with colour. The simple formula assumes isotropic transparent media and a clean boundary, which is ideal for most educational problems and first engineering estimates.

Step-by-Step Snell's Law Method

Start by drawing the boundary and a normal through the point of incidence. Label the medium containing the incoming ray as medium 1 and the other as medium 2. Copy the two indices and angles with their subscripts before selecting a rearrangement. This takes a few seconds but prevents the most frequent error: putting the second index with the first angle.

Next calculate the sine of the known angle, multiply or divide by the index ratio, and only then use inverse sine when the unknown is an angle. The inverse-sine input must lie between −1 and +1. If it is greater than one for light travelling from high index to low index, do not force a decimal answer; state that total internal reflection occurs.

Reading the Live Refraction Diagram

The diagram makes it clear that θ₁ and θ₂ are measured from the dashed normal. It fills the lower medium in blue, draws the incoming ray in amber and the transmitted ray in green, and writes n₁, n₂ and both angles beside the rays. This keeps the equation and geometry aligned while you change values.

Use the diagram to check direction. For air to glass, the green refracted ray should sit nearer the normal than the amber incoming ray. For glass to air, it should move farther away. If you enter an angle beyond the critical angle, the graph replaces the transmitted ray with a reflected ray and a total-internal-reflection label.

Practice Problem Patterns

A common question provides air, glass and an incident angle. Set n₁ = 1.00, n₂ = 1.50 and solve for θ₂. Another gives two measured angles and asks you to identify a liquid; rearrange for n₂. In both cases, a full answer includes the equation, substituted numbers, angle unit and a short statement such as bends toward the normal.

Some questions ask for the apparent depth of water or a prism deviation. Snell's law still controls the ray directions at every boundary, but those applications need additional geometry. Solve one interface at a time, preserve the angle to each local normal and use the calculated outgoing ray as the next incoming ray.

Snell's Law FAQ Summary

Use n₁ sin θ₁ = n₂ sin θ₂, measure both angles from the normal, and label the incident medium first. Higher refractive index bends a ray toward the normal; lower index bends it away. The calculator gives a formula solution and a value-labelled diagram so the result is easy to understand and check.