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Snell's Law of Refraction & Critical Angle Calculator

Snell’s Law governs the refraction and bending of light as it crosses the boundary between two optical media with differing refractive indices (n₁·sin(θ₁) = n₂·sin(θ₂)).

Index of incident medium (Air = 1.0003, Water = 1.333, Glass = 1.52, Diamond = 2.42).

Index of transmitting medium.

Angle of incoming light ray measured relative to the surface normal.

Calculated Result
22.03°

Refraction Angle (θ₂)

Refraction State

22.03°

Critical Angle

N/A

Speed in Medium 1

299,792 km/s

Speed in Medium 2

224,901 km/s

Calculation Breakdown

  1. Phase Velocity of Light in MediaMedium 1: 299,792 km/s, Medium 2: 224,901 km/sv=cn,c=299,792 km/sv = \frac{c}{n}, \quad c = 299{,}792 \text{ km/s}
  2. Snell's Law Governing Relation\sin(\theta_2) = 0.3751n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
  3. Critical Angle for Total Internal ReflectionNone (n1 <= n2, light enters denser medium)θc=arcsin⁡(n2n1)\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)
  4. Angle of Refraction22.03°θ2=arcsin⁡(n1n2sin⁡θ1)\theta_2 = \arcsin\left(\frac{n_1}{n_2} \sin\theta_1\right)

Refraction Angle vs Incident Angle (Air n=1.0 into Water n=1.333)

Interactive visualization based on your current inputs

Refracted Angle (°)
0.0122335470°15°30°45°60°75°Incident Angle (°)Refracted Angle (°)

What Is the Snell's Law of Refraction & Critical Angle Calculator?

Snell’s Law, named after Dutch astronomer Willebrord Snellius, is the fundamental law of geometric optics describing how light changes direction across material interfaces.

It governs lenses, prisms, optical fiber waveguides, atmospheric mirages, and gemstones.

How Does the Snell's Law of Refraction & Critical Angle Calculator Work?

The law states that the ratio of the sines of the angles of incidence and refraction equals the inverse ratio of the indices of refraction.

If the calculated sin(θ₂) exceeds 1.0, the mathematical solution has no real angle, indicating complete 100% Total Internal Reflection.

Snell's Law of Refraction & Critical Angle Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

n1sin⁡θ1=n2sin⁡θ2  ⟹  θ2=arcsin⁡(n1n2sin⁡θ1)n_1 \sin\theta_1 = n_2 \sin\theta_2 \implies \theta_2 = \arcsin\left(\frac{n_1}{n_2} \sin\theta_1\right)

Fermat’s principle of least time dictates that light bends toward the normal in denser media where phase velocity is slower.

How to Use the Snell's Law of Refraction & Critical Angle Calculator

  1. Enter the refractive index of the first medium (where the ray originates).
  2. Enter the refractive index of the second medium.
  3. Supply the incident angle relative to the surface normal.

Step-by-Step Example Calculation

Air into Water Refraction

Input Values:

n1:1.0
n2:1.333
incidentAngle:30
Worked Steps: A light ray entering water (n = 1.333) from air (n = 1.0) at 30° bends toward the normal, refracting at an angle of 22.03° with speed 224,897 km/s.

Understanding Your Result

Refraction Angle (θ₂): Path of the transmitted ray inside medium 2.

Critical Angle (θ_c): Threshold angle beyond which refraction ceases.

TIR Flag: Indicates whether light is 100% reflected back into medium 1.

Phase Velocities: Actual speed of light in both materials.

Factors That Affect the Result

  • Refractive Index Difference: Larger index contrast causes sharper ray bending.
  • Wavelength: Refractive index varies slightly with color (dispersion), causing rainbows through prisms.

When Should You Use This Calculator?

  • Optical engineering: Lens design, prism dispersion, and fiber optic core-cladding sizing.
  • Physics education: Demonstrating wave refraction and boundary conditions.

Assumptions & Limitations

  • Assumes isotropic, homogeneous, non-magnetic optical materials.
  • Assumes monochromatic light (neglects chromatic dispersion).

Frequently Asked Questions

Calculation Accuracy & Reference Note

Deterministic analytical solution conforming to classical Maxwellian electromagnetic boundary equations.

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