Critical Angle Calculator

What is the Critical Angle?

The critical angle is the angle of incidence in the denser medium for which the angle of refraction is 90 degrees. Beyond this angle, all of the light is totally internally reflected back into the denser medium (total internal reflection).

How to Use This Calculator

1. Enter the refractive index of the first medium (n₁). This is typically the medium where the light is initially traveling.
2. Enter the refractive index of the second medium (n₂). This is typically the surrounding medium (e.g., air).
3. Click the "Calculate" button to determine the critical angle.
4. If the critical angle is not applicable (when n₁ ≤ n₂), the calculator will indicate that total internal reflection cannot occur for these indices.

Principle Behind the Calculation

When light passes from a medium with refractive index n₁ to a medium with refractive index n₂, it bends according to Snell's Law:

n₁ sin(θ₁) = n₂ sin(θ₂)

The critical angle (θ_c) occurs when θ₂ = 90 degrees, meaning sin(θ₂) = 1. At this angle:

n₁ sin(θ_c) = n₂

Therefore, sin(θ_c) = n₂ / n₁

Practical Applications

• Optical fibers - where total internal reflection keeps light signals contained within the fiber
• Prism-based optical instruments
• Periscope designs
• Endoscopes in medical imaging

Example

If light passes from water (n ≈ 1.33) to air (n ≈ 1.0), the critical angle would be:

sin(θ_c) = 1.0 / 1.33 ≈ 0.7519

θ_c = arcsin(0.7519) ≈ 48.75°

This means that if light enters water and hits the water-air boundary at an angle greater than 48.75°, it will be totally internally reflected.