Snell’s Law Angle of Refraction, Critical Angle & Brewster Angle Calculator

Compute n₁·sin(θ₁) = n₂·sin(θ₂), critical angle, and Brewster’s angle.

Angle of Refraction (θ₂)
22.17 deg
Brewster Polarization Angle 56.66 deg
Normal Incidence Reflectance (R₀) 4.26 %
Optical Speed in Medium 2 197,231.88 km/s
Inputs
n₁
Refractive index of incident medium n₁ (1.0003 air, 1.333 water).
n₂
Refractive index of transmitting medium n₂ (1.52 crown glass, 2.42 diamond).
deg
Angle of incidence θ₁ from surface normal in degrees.
Results & Graph
Angle of Refraction (θ₂)
22.17 deg
Brewster Polarization Angle
56.66 deg
Normal Incidence Reflectance (R₀)
4.26 %
Optical Speed in Medium 2
197,231.88 km/s

Angle of Refraction (θ₂)

Graph 1 • Index N1 Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 deg 18.6 deg 37.2 deg 50%: 22.17 deg 22.2 50% 75%: 22.17 deg 22.2 75% 100%: 22.17 deg 22.2 100% 125%: 28.14 deg 28.1 125% 150%: 34.47 deg 34.5 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Angle of Refraction (θ₂) 22.17 deg Brewster Polarization Angle 56.66 deg Normal Incidence Reflectance (R₀) 4.26 % Optical Speed in Medium 2 197,231.88 km/s
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Angle of Refraction (θ₂)22.17 degBrewster Polarization Angle56.66 degNormal Incidence Reflectance (R₀)4.26 %Optical Speed in Medium 2197,231.88 km/s
Summary: Light entering n₂=1.52 from n₁=1 at 35° refracts to 22.17° (Brewster angle 56.66°).
Summary: Light entering n₂=1.52 from n₁=1 at 35° refracts to 22.17° (Brewster angle 56.66°).
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Formula & Mathematical Methodology: Snell’s Law Refraction Physics & Engineering Equation

θ₂ = arcsin((n₁ / n₂) × sin θ₁)
θBrewster = arctan(n₂ / n₁)
R₀ = (n₁ − n₂n₁ + n₂)²
θ₂ = arcsin((n₁ / n₂) × sin θ₁); θ_Brewster = arctan(n₂ / n₁); R₀ = ((n₁ − n₂)/(n₁ + n₂))²
  • indexN1: Index N1 (n₁)
  • indexN2: Index N2 (n₂)
  • incidentAngleDeg: Incident Angle Deg (deg)

Rounding: Physical constants follow CODATA/NIST values; output quantities display to 2 decimal places.

Worked Example: Physics & Engineering Benchmark: Snell’s Law Refraction

Compute n₁·sin(θ₁) = n₂·sin(θ₂), critical angle, and Brewster’s angle.

  1. Enter the physical dimensions, electrical parameters, or thermodynamic state variables.
  2. Apply the governing equation: θ₂ = arcsin((n₁ / n₂) × sin θ₁); θ_Brewster = arctan(n₂ / n₁); R₀ = ((n₁ − n₂)/(n₁ + n₂))².
  3. Inspect the primary SI output, derived quantities, and 3 visual SVG graphs.
Result: Computes Snell’s Law Refraction via θ₂ = arcsin((n₁ / n₂) × sin θ₁); θ_Brewster = arctan(n₂ / n₁); R₀ = ((n₁ − n₂)/(n₁ + n₂))².

Frequently Asked Questions

What equation powers the Snell’s Law Angle of Refraction, Critical Angle & Brewster Angle Calculator?

It evaluates θ₂ = arcsin((n₁ / n₂) × sin θ₁); θ_Brewster = arctan(n₂ / n₁); R₀ = ((n₁ − n₂)/(n₁ + n₂))² using IEEE-754 double precision and renders three synchronized SVG plots.

Can I test parameter sensitivity across multiple values?

Yes. Graph 1 automatically plots a 50%–150% parameter response curve, and you can pin up to 3 custom scenarios.

Are all units consistent with SI standards?

Yes. Each input field specifies its exact SI or engineering unit and converts internally where needed.

Science & Engineering · Optics, Photonics & Acoustics

Snell’s Law Angle of Refraction, Critical Angle & Brewster Angle Calculator: inputs, method, and how to read the result

Compute n₁·sin(θ₁) = n₂·sin(θ₂), critical angle, and Brewster’s angle.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Index N1: Refractive index of incident medium n₁ (1.0003 air, 1.333 water).
  • Index N2: Refractive index of transmitting medium n₂ (1.52 crown glass, 2.42 diamond).
  • Incident Angle Deg: Angle of incidence θ₁ from surface normal in degrees.

What the calculation includes

With the default values shown, Angle of Refraction (θ₂) is 22.17 deg. The calculation follows Snell’s Law Refraction Physics & Engineering Equation; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Index N1 from 1 n₁ to 2 n₁ changes Angle of Refraction (θ₂) from 22.17 deg to 49 deg. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses standard SI physical constants (g = 9.80665 m/s², c = 299,792,458 m/s, σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴).
  • Engineering reference model; apply appropriate safety factors and empirical calibration for critical hardware designs.

Scope: SI / NIST Physics & Engineering Standard Check the assumptions and limitations against your situation before relying on the estimate.

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