Snell's Law Calculator

What is Snell's Law?

Snell's law describes how light bends when passing from one transparent material to another. When light crosses the boundary between air and water, glass and diamond, or any two media with different optical densities, it changes direction. This refraction explains why a straw looks bent in a glass of water, why pools appear shallower than they are, and how lenses focus light to form images.

Picture sunlight striking a lake surface at a 30° angle from the vertical (angle of incidence). Air has refractive index n₁ = 1.00, water has n₂ = 1.33. Snell's law states n₁ sin(θ₁) = n₂ sin(θ₂). Plugging in: 1.00 × sin(30°) = 1.33 × sin(θ₂). So 0.5 = 1.33 × sin(θ₂), giving sin(θ₂) = 0.376. The refracted angle is θ₂ = arcsin(0.376) = 22.1°. The light bends toward the vertical, making underwater objects appear shifted from their true position.

Willebrord Snellius discovered this relationship in 1621, though Ibn Sahl described it in Baghdad 600 years earlier. The law emerges from light's wave nature — light slows in denser media, and the change in speed causes the wavefront to pivot. The refractive index n = c/v measures how much light slows: n = 1.33 means light travels 1.33 times slower in water than in vacuum. This fundamental principle enables eyeglasses, cameras, microscopes, fiber optics, and rainbows.

How it Works: Formulas Explained

Snell's law is n₁ sin(θ₁) = n₂ sin(θ₂), where n₁ and n₂ are the refractive indices of the first and second media, and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal (perpendicular) to the surface. All angles in Snell's law measure from the normal, not from the surface — a common source of errors. If a ray strikes a surface at 40° from the surface, the angle from normal is 90° - 40° = 50°.

Refractive index quantifies how much a material slows light. Vacuum has n = 1 exactly. Air is nearly 1 (1.0003, usually rounded to 1.00). Water is 1.33, crown glass 1.52, flint glass 1.62, and diamond 2.42. Higher index means more bending. When light enters a higher-index material (air to water), it bends toward the normal (θ₂ < θ₁). When entering lower-index (water to air), it bends away from normal (θ₂ > θ₁).

The law can be rearranged to solve for any variable. For the refracted angle: θ₂ = arcsin[(n₁/n₂) × sin(θ₁)]. For the incident angle: θ₁ = arcsin[(n₂/n₁) × sin(θ₂)]. For the second index: n₂ = n₁ × sin(θ₁)/sin(θ₂). When n₁ > n₂ and the incident angle exceeds the critical angle, total internal reflection occurs — no light escapes. The critical angle is θ_c = arcsin(n₂/n₁).

Working through a complete example: A laser beam in air (n₁ = 1.00) strikes a glass prism (n₂ = 1.52) at 45° from normal. The refracted angle is θ₂ = arcsin[(1.00/1.52) × sin(45°)] = arcsin[0.658 × 0.707] = arcsin(0.465) = 27.7°. Inside the glass, the beam travels at 27.7° from normal — bent 17.3° toward the perpendicular. When it exits the opposite face back into air, it bends away from normal by the same amount, returning to 45°.

Step-by-Step Guide

1. Identify the two media and their refractive indices. Look up n values: air = 1.00, water = 1.33, typical glass = 1.50-1.52, diamond = 2.42, ethanol = 1.36, sapphire = 1.77. For unknown materials, you may need to solve for n using measured angles. Note which medium the light starts in (n₁) and which it enters (n₂).
2. Measure the angle of incidence from the normal. The normal is an imaginary line perpendicular to the surface at the point where light strikes. If the problem gives the angle from the surface, subtract from 90°. A ray at 35° to the surface has θ₁ = 90° - 35° = 55° from normal. Always use angles from normal in Snell's law.
3. Write Snell's law with your known values. n₁ sin(θ₁) = n₂ sin(θ₂). Example: Light goes from water (n₁ = 1.33) to air (n₂ = 1.00) at θ₁ = 40°. Equation: 1.33 × sin(40°) = 1.00 × sin(θ₂). Calculate left side: 1.33 × 0.643 = 0.855. So sin(θ₂) = 0.855.
4. Solve for the unknown using inverse sine. θ₂ = arcsin(0.855) = 58.7°. The light bends away from normal (58.7° > 40°) as expected when going from higher to lower index. If sin(θ₂) > 1, you've exceeded the critical angle — total internal reflection occurs. No light escapes; it all reflects internally.
5. Check for total internal reflection. When light travels from higher to lower index (water to air), check if θ₁ exceeds the critical angle θ_c = arcsin(n₂/n₁). For water to air: θ_c = arcsin(1.00/1.33) = arcsin(0.752) = 48.8°. Any incident angle greater than 48.8° produces total internal reflection. This is how fiber optics trap light inside glass fibers.
6. Calculate the deviation angle if needed. The angle between the original ray direction and the refracted ray is the deviation: δ = |θ₂ - θ₁|. In the water-to-air example above, δ = |58.7° - 40°| = 18.7°. For a prism, total deviation depends on both entry and exit refractions plus the prism apex angle.

Real-World Examples

Example 1: Apparent depth of a swimming pool. A pool is actually 2.0 m deep. Looking straight down (θ₁ ≈ 0°), light from the bottom travels water (n₁ = 1.33) to air (n₂ = 1.00). For small angles, apparent depth = actual depth × (n₂/n₁) = 2.0 × (1.00/1.33) = 1.50 m. The pool looks only 1.5 meters deep — 75% of actual depth. This is why diving into shallow water is dangerous; the bottom appears farther away than it really is.

Example 2: Fiber optic total internal reflection. Optical fiber has a core (n₁ = 1.48) surrounded by cladding (n₂ = 1.46). Critical angle: θ_c = arcsin(1.46/1.48) = arcsin(0.986) = 80.6°. Light must strike the core-cladding boundary at more than 80.6° from normal (less than 9.4° from the fiber axis) to be trapped. This narrow acceptance cone is why fibers must be aligned precisely for efficient light coupling.

Example 3: Prism dispersion and rainbows. White light enters a glass prism (n = 1.52 for green) at 50°. Green light refracts to θ₂ = arcsin[sin(50°)/1.52] = arcsin(0.766/1.52) = 30.3°. But blue light has n = 1.53, so θ₂ = arcsin(0.766/1.53) = 30.0°. Red light has n = 1.51, so θ₂ = arcsin(0.766/1.51) = 30.6°. The 0.6° spread between red and blue creates the rainbow spectrum when light exits the prism.

Example 4: Eyeglass lens correction. A myopic (nearsighted) person needs a diverging lens. The lens maker's formula uses Snell's law: 1/f = (n-1)(1/R₁ - 1/R₂). For a lens with n = 1.50, R₁ = -20 cm (concave), R₂ = +20 cm (convex): 1/f = 0.50 × (-1/20 - 1/20) = 0.50 × (-0.1) = -0.05 cm⁻¹. So f = -20 cm = -0.20 m. Lens power = 1/f(m) = -5.0 diopters — a typical moderate myopia correction.

Example 5: Mirages on hot roads. Hot air near asphalt has lower density and lower refractive index (n₁ ≈ 1.0002) than cooler air above (n₂ ≈ 1.0003). Light from the sky traveling downward bends away from normal as it enters hotter air. At a shallow enough angle, total internal reflection occurs — light from the sky reflects upward, appearing as a shimmering "puddle" on the road. The angle difference is tiny (fractions of a degree), but over tens of meters, it creates a visible effect.

Common Mistakes to Avoid

Measuring angles from the surface instead of the normal. Snell's law requires angles from the perpendicular (normal) to the surface. If a problem states "light strikes at 30° to the surface," the correct angle for Snell's law is 90° - 30° = 60° from normal. Using 30° directly gives wrong answers. Always draw the normal line and measure angles from it.

Confusing which index is n₁ and which is n₂. The subscript 1 refers to the medium where light originates; subscript 2 is where it's going. Light going from air to water: n₁ = 1.00, n₂ = 1.33. Reversed (water to air): n₁ = 1.33, n₂ = 1.00. Swapping them inverts the bending direction. Label your diagram clearly and double-check before calculating.

Forgetting that sin(θ) cannot exceed 1. When solving for an angle, if you get sin(θ) > 1, you haven't made a calculation error — you've discovered total internal reflection. This happens when light tries to go from higher to lower index at too steep an angle. The critical angle marks the boundary: beyond it, no refraction occurs, only reflection. Recognize this physical limit.

Using degrees vs. radians incorrectly. Calculators must be in the correct mode. If your calculator is in radian mode but you enter 45 (thinking degrees), it computes sin(45 radians) = sin(2578°) ≈ 0.85, not sin(45°) = 0.707. Always verify calculator mode. Most physics problems use degrees for angles. In programming, math libraries often expect radians — convert with radians = degrees × π/180.

Pro Tips

Use the small-angle approximation for quick estimates. For angles below about 10°, sin(θ) ≈ θ (in radians). Snell's law becomes n₁θ₁ ≈ n₂θ₂. A ray entering water at 5° (0.087 rad): θ₂ ≈ (1.00/1.33) × 0.087 = 0.065 rad = 3.7°. Exact calculation gives 3.7° — identical to this approximation. Useful for paraxial optics (rays near the axis) in lens design.

Remember the reversibility principle. Light paths are reversible. If light goes from air to glass at 40° and refracts to 25°, then light going from glass to air at 25° will refract to 40°. This symmetry simplifies many problems. Trace light backward from your eye to understand what you see through optical systems.

Apply Snell's law twice for parallel-sided slabs. When light passes through a glass window (air → glass → air), it refracts at entry and again at exit. The exit angle equals the entry angle (if faces are parallel), but the beam is laterally displaced. Displacement d = t × sin(θ₁ - θ₂)/cos(θ₂), where t is thickness. A 5 mm glass pane at 45° incidence displaces the beam by about 1.5 mm.

Understand wavelength dependence (dispion). Refractive index varies with wavelength — blue light bends more than red. This is why prisms create spectra and lenses suffer chromatic aberration. Crown glass: n = 1.53 at 400 nm (blue), n = 1.51 at 700 nm (red). Achromatic lenses combine crown and flint glass to cancel this effect, bringing two wavelengths to the same focus.

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