Snell'ss Law Calculator
Calculate snell's law calculator calculate angle of refraction theta2 and critical angle for total internal reflection using Snell's Law (n1 * sin(theta1) = n2 * sin(theta2)). Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Optical Refraction Angle (n_1 · sin θ_1 = n_2 · sin θ_2)
Enter your physical parameters below to compute verified refraction metrics.
Calculation Results
Calculated using verified physical methodology: Snell's Law: n_1 \cdot \sin\thetA1 = n_2 \cdot \sin\thetA2
Critical Angle for TIR: \theta_c = \arcsin\left(\n_2 / n_1\right) \text{ (if } n_1 > n_2)
Quick Summary
The Snell's Law Calculator evaluates light ray optical refraction ($\thetA2 = \arcsin(\n_1 / n_2\sin\thetA1)$) across material boundaries and critical angles for Total Internal Reflection ($\theta_c$).
Formula Explanation
Snell's Law: n_1 \cdot \sin\thetA1 = n_2 \cdot \sin\thetA2
Critical Angle for TIR: \theta_c = \arcsin\left(\n_2 / n_1\right) \text{ (if } n_1 > n_2)
How It Works
The Snell's Law Calculator multiplies incident index ($n_1$) by sine of incident angle ($\sin\thetA1$) and divides by target index ($n_2$). It outputs refraction angle ($\thetA2$), ray deviation angle ($\Delta \theta = |\thetA1 - \thetA2|$), light speed ($v_2 = c / n_2$), and Total Internal Reflection status.
Step-by-Step Worked Example
Practical Problem: A light ray traveling through air ($n_1 = 1.0003$) strikes a calm water surface ($n_2 = 1.333$) at an incident angle $\thetA1 = 30.0^\circ$. Calculate the refraction angle $\thetA2$.
- Step 1: Identify Input Parameters: $n_1 = 1.0003$, $\thetA1 = 30.0^\circ$, $n_2 = 1.333$.
- Step 2: Calculate Sine of Incident Angle ($\sin 30^\circ$): $\sin(30.0^\circ) = 0.500000$.
- Step 3: Apply Snell's Law Formula: $\sin\thetA2 = \n_1 \cdot \sin\thetA1 / n_2 = \1.0003 \times 0.500000 / 1.333 = \0.50015 / 1.333 = 0.375206$.
- Step 4: Execute Inverse Sine ($\arcsin$): $\thetA2 = \arcsin(0.375206) = 22.0416^\circ$.
- Step 5: Format Final Refraction Metrics: Refraction Angle $\thetA2 = 22.04^\circ$ ($0.3847\text{ rad}$). Angular Deviation $\Delta\theta = |30.0 - 22.04| = 7.96^\circ$ (bent toward normal). Light speed in water $v_2 = 224,899,800\text{ m/s}$ (75.0% of $c$).
Real-World Calculation Examples
Scenario 1: Sunlight Entering Water Pool Surface
Parameters: $n_1 = 1.0003$ (Air), $\thetA1 = 30.0^\circ$, $n_2 = 1.333$ (Water)
Result: $\thetA2 = 22.04^\circ$ ($7.96^\circ$ ray deviation toward normal). Water refraction.
Scenario 2: Glass Prism Refraction
Parameters: $n_1 = 1.0003$ (Air), $\thetA1 = 45.0^\circ$, $n_2 = 1.520$ (Crown Glass)
Result: $\thetA2 = 27.73^\circ$ ($17.27^\circ$ ray deviation). Glass prism refraction.
Scenario 3: Fiber Optic Cable Total Internal Reflection
Parameters: $n_1 = 1.48$ (Glass Core), $\thetA1 = 60.0^\circ$, $n_2 = 1.44$ (Cladding)
Result: Total Internal Reflection (TIR Active!). Critical angle $\theta_c = 76.45^\circ$. Fiber optic trap.
Scenario 4: Diamond Gemstone Brilliance Refraction
Parameters: $n_1 = 1.0003$ (Air), $\thetA1 = 45.0^\circ$, $n_2 = 2.417$ (Diamond)
Result: $\thetA2 = 17.02^\circ$ ($27.98^\circ$ severe ray bending). Diamond refraction.
Key Benefits of Using This Calculator
Total Internal Reflection (TIR) Detection
Automatically detects when incident angle exceeds critical angle ($\thetA1 > \theta_c$) for total internal reflection.
Fiber Optics & Lens Engineering
Essential tool for designing optical camera lenses, prisms, and fiber optic telecommunications.
Multi-Unit Readouts
Outputs refraction angle in degrees (°), radians, ray deviation angle, and light speed ($v_2$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is Snell's Law?
Snell's Law (Law of Refraction) states that the ratio of the sines of the angles of incidence and refraction is equal to the inverse ratio of the indices of refraction (n1 * sin(theta1) = n2 * sin(theta2)).
What is the formula for Snell's Law?
n1 * sin(theta1) = n2 * sin(theta2), or sin(theta2) = (n1 / n2) * sin(theta1).
What is Total Internal Reflection (TIR)?
TIR occurs when light traveling in a denser medium (n1 > n2) strikes a boundary at an angle greater than the critical angle theta_c, causing 100% of light to reflect back inside without refracting into medium 2.
What is the critical angle formula?
theta_c = arcsin(n2 / n1) (valid only when n1 > n2).
Who discovered Snell's Law?
Dutch astronomer Willebrord Snellius (Snell) formulated it in 1621; Persian scientist Ibn Sahl discovered it earlier in 984 AD.
Why does a straw look bent in a glass of water?
Because light rays exiting water (n = 1.333) into air (n = 1.00) refract away from the normal, shifting the apparent position of the submerged straw.
What is Fermat's Principle of Least Time?
Fermat's Principle states that light follows the path that takes the minimum time between two points, from which Snell's Law can be derived mathematically.
What is the critical angle for water-to-air boundary?
theta_c = arcsin(1.0003 / 1.333) = ~48.6 degrees.
What is the critical angle for glass-to-air boundary?
theta_c = arcsin(1.0003 / 1.520) = ~41.1 degrees.
How does fiber optic cable use Total Internal Reflection?
Optical fibers feature a high-index glass core (n1 = ~1.48) surrounded by a lower-index cladding (n2 = ~1.44), trapping light signals inside via continuous TIR over long distances.