Light bends when it passes between materials with different refractive indices. Enter the two indices and the angle of incidence to get the refraction angle, the critical angle and whether total internal reflection occurs.
Enter both refractive indices.
Enter the angle of incidence from the normal.
Read the refraction and critical angles.
Snell's law: n₁ sin θ₁ = n₂ sin θ₂
Critical angle: sin θc = n₂ ÷ n₁
Light changes direction when it crosses from one transparent material into another. Snell's law, n₁ sin θ₁ = n₂ sin θ₂, predicts by how much. This calculator takes the refractive indices of the two media and the angle of incidence measured from the normal, and returns the angle of refraction. It also finds the critical angle when light goes from a denser to a rarer medium, and reports total internal reflection when no refracted ray can exist.
Refraction is a Class 10 topic and returns in Class 12 optics. It explains why a pool looks shallower than it is, why a straw seems bent in a glass of water, how optical fibres carry internet data under Indian cities and why diamonds sparkle. The calculator lets you check these effects with real numbers.
1. Identify the medium the light starts in (n₁) and the medium it enters (n₂). Air is about 1.00, water 1.33, glass about 1.5.
2. Measure the angle of incidence θ₁ from the normal, the line perpendicular to the surface, not from the surface itself.
3. Compute sin θ₂ = n₁ × sin θ₁ ÷ n₂.
4. If sin θ₂ is 1 or less, find θ₂ = sin⁻¹(sin θ₂).
5. If sin θ₂ is greater than 1, there is no refracted ray: total internal reflection occurs.
6. If n₁ is greater than n₂, find the critical angle θc = sin⁻¹(n₂ ÷ n₁).
The refractive index of a medium is n = c ÷ v, the ratio of the speed of light in vacuum to its speed in that medium. When a wavefront hits a boundary at an angle, one edge slows down first, so the wavefront swings round, like a line of marching students stepping from road into sand. Geometry of the wavefronts in the same time interval gives sin θ₁ ÷ v₁ = sin θ₂ ÷ v₂, which becomes n₁ sin θ₁ = n₂ sin θ₂. Entering a slower medium bends light towards the normal.
Going from a denser to a rarer medium, the refracted ray bends away from the normal. As θ₁ grows, θ₂ reaches 90° first, grazing along the surface. The incidence angle at which that happens is the critical angle, where sin θc = n₂ ÷ n₁. Beyond it, Snell's law would need a sine greater than 1, which is impossible, so all the light reflects back. For water to air θc is about 48.8°, for glass to air about 41.8° and for diamond about 24.4°.
The refractive index depends slightly on wavelength, being larger for violet than for red in glass. This dispersion is why a prism splits white light and why rainbows form. Textbook indices are usually quoted for yellow light. At any real boundary some light is also partly reflected even below the critical angle; Snell's law gives only the direction of the transmitted ray, not how bright it is. The reflected share grows as the incidence angle approaches grazing.
Diya shines a waterproof torch from under the water in a swimming pool so that the beam meets the surface at 55° from the normal, going from water (1.33) into air (1.00).
Snell's law: n₁ sin θ₁ = n₂ sin θ₂ sin θ₂ = 1.33 × sin 55° ÷ 1 = 1.089472
Angle of refraction: sin θ₂ > 1 → total internal reflection, no refracted ray
Critical angle: θc = sin⁻¹(1 ÷ 1.33) = 48.7535°
Answer: Angle of refraction Total internal reflection; Critical angle 48.7535°
Measuring the angle from the surface instead of from the normal.
Swapping n₁ and n₂, which bends the ray the wrong way.
Having a calculator in radians while entering degrees.
Looking for a critical angle when light goes from rarer to denser; none exists then.
Assuming total internal reflection is partial; beyond the critical angle there is no refracted ray at all.
Class 10 and Class 12 refraction and total internal reflection problems.
Understanding how optical fibres guide light for broadband and telecom networks.
Explaining apparent depth of pools, tanks and rivers.
Designing prisms in binoculars and periscopes that use total internal reflection.
Gemmology, where the critical angle and index help identify stones.
Common refractive indices?
Air 1.00, water 1.33, glass about 1.5, diamond 2.42.
Why do diamonds sparkle?
Their high index gives a small critical angle (about 24°), so light reflects internally many times.