Snell's Law Calculator
Choose two media from a table of refractive indices (or enter your own), give the angle of incidence, and get the refracted ray, the angle it bends through, the critical angle, the share of light reflected and a ray diagram. Other modes solve Snell's law for any unknown index or angle, or just the critical angle. Free, no sign-up.
Results
22.04°
Angle of refraction (θ₂)
7.963°
Bend from the straight path
None going into a denser medium
Critical angle
53.12°
Brewster's angle
2.139%
Light reflected (unpolarised)
97.86%
Light transmitted
2.997 × 10⁸ m/s
Speed of light in medium 1
2.249 × 10⁸ m/s
Speed of light in medium 2
The working
- n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = 1 × sin(30°) ÷ 1.333 = 0.3752 and θ₂ = 22.04°
Refractive indices of common materials
| Material | Index n | Speed of light (10⁸ m/s) |
|---|---|---|
| Vacuum | 1.00 | 2.998 |
| Air (0 °C, 1 atm) | 1.000293 | 2.997 |
| Ice | 1.31 | 2.288 |
| Water (20 °C) | 1.333 | 2.249 |
| Ethanol | 1.361 | 2.203 |
| Fused silica | 1.458 | 2.056 |
| Vegetable oil | 1.47 | 2.039 |
| Glycerol | 1.4729 | 2.035 |
| Acrylic (PMMA) | 1.49 | 2.012 |
| Benzene | 1.501 | 1.997 |
| Crown glass (N-BK7) | 1.5168 | 1.976 |
| Rock salt (NaCl) | 1.544 | 1.942 |
| Carbon disulfide | 1.628 | 1.841 |
| Dense flint glass (N-SF10) | 1.7283 | 1.735 |
| Sapphire | 1.77 | 1.694 |
| Diamond | 2.417 | 1.24 |
Sodium D line (589 nm) values from Hecht, Optics; gases at 0 °C and 1 atm, liquids at 20 °C. The two optical glasses are Schott catalogue values at 587.6 nm. Real indices vary with wavelength and temperature.
How to Use the Snell's Law Calculator
- 1. Choose what to find. Find the refracted ray gives θ₂ from the two media and θ₁. Solve for any term finds an unknown index or angle from the other three. Critical angle needs only the two media.
- 2. Pick the media. Medium 1 is where the light starts; medium 2 is where it goes. Choose from sixteen materials, or enter your own index.
- 3. Enter the angle. Angles are measured from the normal — the line perpendicular to the surface — not from the surface itself.
- 4. Read the result and the drawing. The calculator reports the refraction angle (or total internal reflection), the bend, Brewster's angle, the fraction of light reflected and the speed of light in each medium.
Snell's Law
When light crosses the boundary between two transparent media, it changes direction according to n₁ sin θ₁ = n₂ sin θ₂. Each n is the refractive index of its medium — the ratio of the speed of light in a vacuum to the speed in the medium, n = c ÷ v — and each θ is measured from the normal. Light going into a denser medium (larger n) bends toward the normal; light going into a less dense medium bends away from it. At normal incidence (θ₁ = 0°) it does not bend at all.
The law is named for Willebrord Snellius, who found it in 1621, though the Persian scientist Ibn Sahl described it around 984 and Descartes published the sine form in 1637.
Worked Example: Air Into Water
A ray in air meets a water surface at 30° from the normal (the calculator's starting example, using n = 1.000293 for air and 1.333 for water):
- sin θ₂ = 1.000293 × sin 30° ÷ 1.333 = 0.3752, so θ₂ = 22.04°. With air taken as exactly 1 the answer is 22.03°.
- The ray bends through 30° − 22.04° = 7.963° toward the normal.
- About 2.14% of unpolarised light is reflected and 97.86% enters the water, from the Fresnel equations.
- Light travels at 299,792,458 ÷ 1.333 = 2.249 × 10⁸ m/s in water, about 75% of its vacuum speed.
Critical Angle and Total Internal Reflection
Going from a denser to a less dense medium, the refracted ray bends away from the normal, and for large enough θ₁ it would have to leave at more than 90° — which is impossible. At the critical angle θc = arcsin(n₂ ÷ n₁) the refracted ray skims along the surface, and beyond it no light is transmitted: all of it is reflected, a phenomenon called total internal reflection. From water into air θc is about 48.6°, from crown glass (N-BK7) into air about 41.3°, and from diamond into air only 24.4°, which is why a cut diamond traps light and sparkles.
Total internal reflection is lossless and so keeps light inside optical fibres, where the core's index is slightly higher than the cladding's, and in the prisms of binoculars and periscopes. It needs n₁ to be greater than n₂; light entering a denser medium always refracts.
Refractive index depends on wavelength
The table gives the index at the sodium D line, 589 nm. The index of a transparent material is slightly higher for blue light than for red, which is why a prism or raindrop splits white light. For water the difference is about 1% across the visible spectrum, small enough to ignore in a homework problem but large enough to make a rainbow.
Frequently Asked Questions
- What is Snell's law?
- n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media and θ₁ and θ₂ are the angles of the light with the normal on each side of the boundary. It tells you how much a ray bends when it crosses from one medium into another.
- How do you find the angle of refraction?
- Rearrange to sin θ₂ = n₁ sin θ₁ ÷ n₂, then take the inverse sine. For air into glass (n = 1.5) at 45°, sin θ₂ = 0.7071 ÷ 1.5 = 0.4714, so θ₂ = 28.1°. If the value you get for sin θ₂ is above 1, there is no refracted ray.
- What is the critical angle?
- The angle of incidence, going from a higher to a lower index, at which the refracted ray travels along the boundary: θc = arcsin(n₂ ÷ n₁). Beyond it, total internal reflection occurs. For glass (n = 1.5) into air it is about 41.8°.
- Is the angle measured from the surface or the normal?
- From the normal, the line at right angles to the surface. If a problem gives the angle with the surface, subtract it from 90° first. Using the wrong reference is the most common error in refraction problems.
- What is the refractive index of air, water and glass?
- About 1.0003 for air (often taken as 1), 1.333 for water at 20 °C, and 1.5 to 1.6 for common glass; N-BK7 crown glass is 1.5168. The table above lists sixteen materials, from ice at 1.31 to diamond at 2.417.
- What is Brewster's angle?
- The angle of incidence, arctan(n₂ ÷ n₁), at which reflected light is completely polarised parallel to the surface. For air to water it is 53.12°, and it is the reason polarising sunglasses cut glare from water and roads.