Snell’s Law
Snell’s law describes how light bends when it crosses a boundary between two transparent materials. The bending — called refraction — happens because light travels at different speeds in different media. Snell’s law gives the exact relationship between the angle of incidence, the angle of refraction, and the refractive indices of the two media. It explains why a straw in a glass of water looks bent, why lenses focus light, why rainbows form, and how optical fibers carry the internet across oceans.
The Law
For a light ray crossing from medium 1 (refractive index \( n_1 \)) into medium 2 (refractive index \( n_2 \)):
$$ n_1 \sin\theta_1 = n_2 \sin\theta_2 $$
where \( \theta_1 \) is the angle of incidence and \( \theta_2 \) is the angle of refraction, both measured from the normal (the line perpendicular to the interface). The law applies at the instant of crossing; the incident ray, refracted ray, and normal all lie in the same plane.
Refractive Index
The refractive index of a medium is defined as \( n = c/v \), where \( c \) is the speed of light in vacuum and \( v \) is the speed of light in the medium. Some common values:
| Medium | Refractive index n |
|---|---|
| Vacuum | 1.000 |
| Air (at STP) | 1.0003 |
| Water | 1.33 |
| Crown glass | 1.52 |
| Diamond | 2.42 |
| Silicon (visible light) | ~3.5 |
Higher \( n \) means slower light in that material. When light enters a denser medium (higher \( n \)), it bends toward the normal; when it enters a less dense medium, it bends away.
Worked Example
A ray of light strikes a flat air-water interface at 30° from the normal. What’s the refraction angle?
Use \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) with \( n_1 = 1.00 \), \( n_2 = 1.33 \), \( \theta_1 = 30° \):
$$ \sin\theta_2 = \frac{1.00 \cdot \sin 30°}{1.33} = \frac{0.500}{1.33} = 0.376 $$
So \( \theta_2 = \arcsin(0.376) \approx 22.1° \). The ray bends from 30° to 22.1° on entering water — toward the normal, as expected for a denser medium.
Total Internal Reflection
When light travels from a denser to a less dense medium (e.g., glass to air), the refraction angle grows faster than the incidence angle. At the critical angle \( \theta_c \), the refraction angle reaches 90° and the ray skims the surface:
$$ \sin\theta_c = \frac{n_2}{n_1} $$
Beyond \( \theta_c \), no refraction is possible — all the light is reflected back into the denser medium. This is total internal reflection. For glass-to-air, \( \theta_c = \arcsin(1/1.52) \approx 41° \); for water-to-air, \( \theta_c \approx 49° \).
This is how optical fibers work: light is trapped inside the fiber by total internal reflection at every glass-to-cladding boundary, even when the fiber bends — carrying signals for kilometers without leaking.
Why Refraction Happens
In a wave picture, when one side of a wavefront crosses into a slower medium before the other side does, the wavefront pivots — the way marching soldiers change direction if one side enters mud first. In Fermat’s principle, light takes the path that minimizes travel time between two points; if the path crosses a slower medium, the time-minimizing path bends. Both pictures predict Snell’s law exactly.
Applications
- Lenses. Every optical lens — eyeglasses, camera, microscope, telescope — works by refracting incoming light to converge at (or appear to come from) a chosen focal point. Lens design is essentially the geometric application of Snell’s law to curved glass surfaces.
- Optical fibers. Long-distance internet, telephone, and cable TV signals travel as light pulses trapped in glass fibers by total internal reflection — Snell’s law applied at every bend.
- Rainbows. Sunlight enters a raindrop, refracts, reflects off the back, and refracts again on exit. Different wavelengths bend by slightly different amounts (dispersion) and emerge in a rainbow spectrum centered around 42° from the antisolar point.
- Prisms and spectroscopy. Glass prisms split white light into its spectrum because n varies slightly with wavelength. Spectroscopy uses this to identify elements by their emission and absorption lines.
- Mirages. Hot air near the ground has a lower refractive index than cooler air above. Light from the sky bends upward into your eye, creating the illusion of water on a hot road.
- Diamond cutting. Diamond’s very high index (2.42) and critical angle (~24°) mean cut facets trap light by total internal reflection — the source of diamond’s brilliant sparkle.
Related study notes: Refraction, Total Internal Reflection, Lenses, Dispersion of Light.
Frequently Asked Questions
What is Snell’s law?
Snell’s law gives the angle a light ray bends to when it crosses a boundary between two transparent materials. It states that n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media and θ₁, θ₂ are the angles measured from the normal to the surface.
What is refractive index?
The refractive index n of a medium is the ratio of the speed of light in vacuum to the speed of light in the medium: n = c/v. Vacuum has n = 1 by definition; air is essentially 1; water is 1.33; common glass is 1.5; diamond is 2.42. Higher n means slower light.
Why does light bend when it enters water?
Because light travels slower in water than in air. When the wavefront crosses the boundary at an angle, the side that enters the water first slows down before the other side, pivoting the wavefront. In Fermat’s principle: light follows the path that minimizes total travel time, which is a bent path when speeds differ.
What is total internal reflection?
When light tries to leave a denser medium (like glass) into a less dense one (like air) at a steep angle, no refraction is possible — all the light reflects back into the denser medium. This happens beyond the critical angle, defined by sin θ_c = n₂/n₁. Optical fibers exploit this to keep light trapped inside the fiber.
How do optical fibers work?
A thin glass core is surrounded by a cladding with a slightly lower refractive index. Light entering one end at a shallow enough angle keeps hitting the core-cladding boundary above the critical angle and undergoes total internal reflection — bouncing along the fiber for kilometers without significant loss. This is how the internet’s backbone carries data across continents.
Who discovered Snell’s law?
The Dutch astronomer Willebrord Snellius derived it in 1621, though Persian scientist Ibn Sahl had written a manuscript with the same content in 984 CE. Descartes published a version in 1637. Today the law is universally called Snell’s law (Snellius’s law on the continent), regardless of the messy historical priority.