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Snell's Law Calculator

Refracted angle theta two equals arcsine of the quantity n one over n two times sine of theta one

Solution

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Refracted Angle (θ₂)

Given the refractive indices of both media and the angle of incidence, find the angle the refracted ray makes with the normal on the far side of the interface. This is the most common Snell's law solve — used to predict where light goes when it crosses from air into glass, water, or any other transparent material.

θ₂ = arcsin((n₁ / n₂) × sin θ₁)

Incident Angle (θ₁)

Solve for the angle of incidence when you already know the refracted angle. Useful for back-calculating the input ray geometry — for example, when you observe a refracted ray angle and want to determine how the original beam was aimed.

θ₁ = arcsin((n₂ / n₁) × sin θ₂)

Refractive Index n₁

Determine the refractive index of the first medium from a known index of the second medium and both measured angles. Used to identify an unknown transparent material by measuring how it refracts light from a calibration medium.

n₁ = n₂ × sin(θ₂) / sin(θ₁)

Refractive Index n₂

The mirror solve for the second medium's refractive index. Often used in introductory physics labs where students measure both angles, know n₁ (typically air at 1.0003 ≈ 1), and compute n₂ for an unknown sample.

n₂ = n₁ × sin(θ₁) / sin(θ₂)

How It Works

Snell's law relates the angles a light ray makes with the normal on either side of a boundary between two transparent media. The product of each medium's refractive index and the sine of its angle from the normal is the same on both sides: n₁ × sin θ₁ = n₂ × sin θ₂. Light bends toward the normal when entering a denser medium (higher index) and away from the normal when entering a less-dense one. When light tries to leave a denser medium at a steep angle, the math demands sin θ₂ > 1 — which has no solution. That's the condition for total internal reflection.

Example Problem

A laser pointer in air shines at the surface of a swimming pool at a 45° angle from the vertical. The water has refractive index n₂ = 1.333. At what angle does the beam travel through the water?

  1. Identify the knowns: incident medium is air with n₁ = 1.000, refracted medium is water with n₂ = 1.333, and the incident angle from the normal is θ₁ = 45°.
  2. Choose what to solve for: the refracted angle θ₂ inside the water.
  3. Write Snell's law in its general form: n₁ × sin(θ₁) = n₂ × sin(θ₂).
  4. Solve for sin(θ₂): sin(θ₂) = (n₁ / n₂) × sin(θ₁) = (1.000 / 1.333) × sin(45°).
  5. Substitute the trig value: sin(45°) ≈ 0.7071, so sin(θ₂) ≈ 0.7505 × 0.7071 ≈ 0.5305.
  6. Take the arcsine: θ₂ = arcsin(0.5305) ≈ 32.04°. The beam bends toward the normal, as expected when entering a denser medium.

If the geometry had been reversed — laser inside the water aimed up at the surface at 45° — Snell's law would give sin(θ₂) ≈ 0.9428, refracting to about 70.5° in air. Push that incident angle past θ_c ≈ 48.6° and the beam wouldn't escape at all — total internal reflection.

Key Concepts

The refractive index n of a medium is defined as the ratio of the speed of light in vacuum c to the phase velocity v in the medium: n = c / v. Vacuum has n = 1 by definition; air is ≈ 1.0003 (often rounded to 1); water is ≈ 1.333; common crown glass is ≈ 1.50; diamond is ≈ 2.42. A higher index means light slows down more in that medium, and Snell's law shows that's exactly when the ray bends more sharply toward the normal as it enters. The reverse — leaving a denser medium for a less-dense one — bends light away from the normal, which is why a fish in a pond appears to be in a different location than where it actually is.

Applications

  • Optical fibers — total internal reflection traps light inside a glass core, enabling long-distance telecom signals with minimal loss
  • Eyeglasses and contact lenses — lens curvature and refractive index together bend light to correct nearsightedness, farsightedness, and astigmatism
  • Camera and microscope optics — multi-element lens stacks chain refractions to focus images and minimize chromatic aberration
  • Swimming pool depth illusion — water bends light from the bottom upward toward the normal, so the pool appears shallower than it really is
  • Rainbow formation — sunlight refracts entering and leaving spherical raindrops, with internal reflection in between, spreading colors by wavelength-dependent index
  • Spear fishing — a submerged fish appears displaced from its true position because light from the fish refracts at the water surface; experienced spearfishers aim below the apparent target
  • Mirage formation — gradients in air refractive index near hot ground bend light, producing the illusion of distant water on a road

Common Mistakes

  • Measuring angles from the surface instead of from the normal — Snell's law always uses the angle between the ray and the perpendicular to the interface, not the angle to the surface itself
  • Mixing degrees and radians — the calculator accepts degrees at the input boundary; if you compute by hand make sure your calculator is in DEG mode for arcsin
  • Forgetting which medium is which — n₁ is the medium the light starts in, n₂ is the medium it ends up in; swapping them gives a refracted angle on the wrong side of the geometry
  • Trying to solve for θ₂ past the critical angle (when n₁ > n₂) — the equation has no real solution; that's total internal reflection, not a calculation error
  • Assuming the refracted ray is at the same angle as the incident ray — only when n₁ = n₂ (same medium) does no bending occur

Frequently Asked Questions

How do you calculate the refracted angle using Snell's law?

Apply θ₂ = arcsin((n₁ / n₂) × sin θ₁). Multiply the index ratio n₁ / n₂ by the sine of the incident angle (measured from the normal), then take the arcsine of the result to get the refracted angle. Both angles are measured from the normal, not from the surface. For air-to-water at 45°, the math gives arcsin((1.0 / 1.333) × sin 45°) = arcsin(0.5305) ≈ 32.04°.

What is the formula for Snell's law of refraction?

Snell's law states n₁ × sin(θ₁) = n₂ × sin(θ₂), where n₁ and n₂ are the refractive indices of the two media and θ₁ and θ₂ are the incident and refracted angles measured from the normal. The same equation can be rearranged to solve for any of the four quantities — refracted angle, incident angle, n₁, or n₂ — when the other three are known.

What is the critical angle and how is it calculated?

The critical angle θ_c is the angle of incidence above which light cannot refract into a less-dense medium and is instead totally reflected back. It is defined only when n₁ > n₂ (light traveling from a denser medium toward a less-dense one) and given by θ_c = arcsin(n₂ / n₁). For glass (n = 1.5) to air (n = 1.0), θ_c ≈ 41.81°. For water (n = 1.333) to air, θ_c ≈ 48.6°.

What is total internal reflection?

Total internal reflection occurs when light traveling from a denser medium strikes the interface with a less-dense medium at an angle greater than the critical angle. Instead of refracting and crossing into the second medium, the light reflects entirely back into the first medium with no loss. This is the operating principle behind optical fibers, prismatic binoculars, and the sparkle of cut diamonds.

What is the refractive index of water?

Pure water has a refractive index of approximately 1.333 for visible light (specifically the sodium D line at 589 nm at 20 °C). Common transparent media: air ≈ 1.0003 (rounded to 1.0 for most calculations), ice ≈ 1.31, ethanol ≈ 1.36, crown glass ≈ 1.50, flint glass ≈ 1.62, and diamond ≈ 2.42. Refractive indices depend slightly on wavelength — this dispersion is what splits white light into colors when it passes through a prism.

Why does light bend when it enters water?

Light slows down when it enters water because the photons interact with the water molecules' electrons, effectively traveling at about 75% of their vacuum speed (c / 1.333 ≈ 2.25 × 10⁸ m/s). The wavefront that hits the surface first slows first, so the rest of the wavefront pivots around it like a marching band changing direction — that pivot is the refraction we see geometrically as the ray bending toward the normal.

Does Snell's law work for any two media?

Yes, as long as both media are transparent (or at least let some light through) and have well-defined refractive indices, Snell's law applies. It also applies to other types of waves — sound, seismic waves, and radio waves all refract according to the same sine-ratio relationship when crossing a boundary between media with different propagation speeds. The relevant 'refractive index' is just the ratio of speeds.

What angle is used in Snell's law — from the surface or from the normal?

Always from the normal — the line perpendicular to the interface at the point where the ray hits the surface. A ray grazing the surface has a 90° incident angle in Snell's law, not 0°. A ray traveling along the normal (straight down into the interface) has a 0° incident angle and passes straight through without bending, regardless of the refractive indices.

Reference: Hecht, Eugene. 2017. Optics. Pearson. 5th ed. — chapter 4 covers Snell's law derivation from Fermat's principle and from electromagnetic boundary conditions.

Worked Examples

Photography & Imaging

How much does a lens bend incoming light at 35°?

A camera lens element made of crown glass (n = 1.52) receives light at a 35° angle of incidence from the surrounding air (n = 1.00). What's the refracted angle inside the glass?

  • Knowns: n₁ = 1.00 (air), n₂ = 1.52 (crown glass), θ₁ = 35°
  • Snell's law: n₁ × sin θ₁ = n₂ × sin θ₂
  • sin θ₂ = (1.00 / 1.52) × sin 35° = 0.6579 × 0.5736 = 0.3774
  • θ₂ = arcsin(0.3774)

θ₂ ≈ 22.17° — the ray bends toward the normal inside the lens

Real camera optics chain many such refractions across multiple element surfaces, each tuned to converge incoming rays onto the image sensor.

Telecommunications

At what angle does light totally reflect inside an optical fiber core?

A fiber's silica core has refractive index n₁ = 1.46 and is surrounded by a cladding of n₂ = 1.44. Find the critical angle above which light is trapped inside the core by total internal reflection.

  • Knowns: n₁ = 1.46 (core), n₂ = 1.44 (cladding)
  • Critical angle: θ_c = arcsin(n₂ / n₁) = arcsin(1.44 / 1.46)
  • θ_c = arcsin(0.9863)

θ_c ≈ 80.57° — light hitting the core–cladding interface at angles greater than this reflects back into the core

This small index difference (only 0.02) is what keeps light bouncing along the fiber over tens of kilometers with minimal loss — the operating principle behind every fiber-optic internet connection.

Marine Biology

What angle of refraction makes a fish appear shallower than it is?

Light from a fish 3 m underwater leaves the water (n₁ = 1.333) at an angle of 30° from the vertical and enters the air (n₂ = 1.00). At what angle does the observer see the light?

  • Knowns: n₁ = 1.333 (water), n₂ = 1.00 (air), θ₁ = 30°
  • Snell's law: n₁ × sin θ₁ = n₂ × sin θ₂
  • sin θ₂ = (1.333 / 1.00) × sin 30° = 1.333 × 0.5 = 0.6665
  • θ₂ = arcsin(0.6665)

θ₂ ≈ 41.78° — the light bends away from the normal when leaving the water, making the fish appear higher and closer to the surface than it really is

Spearfishers learn to aim below the apparent fish position. Below the critical angle θ_c ≈ 48.6° for water-to-air, light escapes; beyond it, the underwater surface looks mirrored when viewed from below.

Snell's Law Formula

Snell's law relates the angles a light ray makes with the surface normal on either side of a boundary between two transparent media:

n₁ × sin(θ₁) = n₂ × sin(θ₂)

Where:

  • n₁ — refractive index of the medium the light is coming from (dimensionless; vacuum = 1)
  • n₂ — refractive index of the medium the light is entering
  • θ₁ — angle of incidence, measured between the incident ray and the surface normal
  • θ₂ — angle of refraction, measured between the refracted ray and the same normal on the other side of the interface

When n₁ > n₂ (light traveling from a denser to a less-dense medium), there is a critical incident angle θ_c above which no refraction can occur. The light is totally reflected back into the first medium — a phenomenon called total internal reflection:

θ_c = arcsin(n₂ / n₁)

Refraction Geometry

The diagram below illustrates the canonical setup: an incident ray in medium n₁ strikes the horizontal interface, refracts, and continues into medium n₂. Both angles are measured from the dashed normal line, not from the surface. When n₂ > n₁ (as drawn), the refracted ray bends toward the normal.

Snell's Law refraction diagramnormalincident rayrefracted rayθ₁θ₂n₁n₂

n₁, n₂ — refractive indices of the upper and lower media · θ₁ — angle of incidence (incident ray to normal) · θ₂ — angle of refraction (refracted ray to normal). The interface is the horizontal solid line; the dashed vertical line is the surface normal at the point of incidence.

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