Snell's Law and Refraction, Explained
After reading this you can predict how much light bends when it crosses between two materials, compute the critical angle for total internal reflection, and know when the simple index model stops being trustworthy.
What refraction is and why light bends
Light travels at speed c in vacuum, about 3.00 \times 10^8 m/s. Inside a transparent material it moves slower. The refractive index n is just the ratio of those two speeds:
Here v is the speed of light in the material. Water has n \approx 1.33, so light moves at 3.00 \times 10^8 / 1.33 \approx 2.26 \times 10^8 m/s in water. Crown glass sits near 1.52, diamond near 2.42.
When a beam meets the boundary at an angle, one side of the wavefront enters the slower medium before the other side. That side slows first, so the wavefront pivots. The result: the beam changes direction at the surface. Enter a slower medium and the beam bends toward the normal (the line perpendicular to the surface). Enter a faster medium and it bends away.
The hook you have seen: a straw in a glass of water looks broken at the waterline. Light from the underwater part bends as it leaves the water, and your eye traces it back along the wrong line.
The formula and where it comes from
Snell's law relates the two angles, both measured from the normal, through the two indices:
The symbols: n_1 and \theta_1 are the index and angle in the first medium (where the light starts), and n_2, \theta_2 are the index and angle in the second. The product n \sin\theta is conserved across the boundary.
To find the refracted angle, solve for \theta_2:
The physics behind it is Fermat's principle: light takes the path that minimizes travel time. Because the two media have different speeds, the fastest route is not a straight line but a bent one, and minimizing the time reproduces exactly the sine relation above. You can derive it by writing the total travel time as a function of the crossing point and setting the derivative to zero.
Always measure angles from the normal, not from the surface. A beam skimming almost parallel to the surface has \theta_1 near 90°, not near 0°. Mixing the two conventions is the single most common numerical error.
Critical angle and total internal reflection
When light goes from a slower medium into a faster one (n_1 \gt n_2), it bends away from the normal. Push \theta_1 higher and at some point the refracted beam would need \theta_2 = 90°, lying flat along the surface. That incidence angle is the critical angle \theta_c. Set \theta_2 = 90° so \sin\theta_2 = 1:
Beyond \theta_c there is no valid refracted angle, and the argument of the arcsine exceeds 1. All the light reflects back into the first medium. This is total internal reflection, and it is why optical fibres carry light for kilometres with tiny loss: rays strike the core wall above the critical angle and bounce along instead of leaking out.
The critical angle only exists when n_1 \gt n_2. Going from air into glass there is no critical angle, because the light is entering a slower medium and can always find a refracted path.
A worked example: air into water at 30°
Reproducing the demo values
Take the tool defaults: from vacuum or air (n_1 = 1.00), into water (n_2 = 1.33), incidence angle \theta_1 = 30°.
- Compute the conserved quantity on the incident side: n_1 \sin\theta_1 = 1.00 \times \sin 30° = 1.00 \times 0.5000 = 0.5000.
- Divide by n_2 to get \sin\theta_2 = 0.5000 / 1.33 = 0.3759.
- Take the arcsine: \theta_2 = \arcsin(0.3759) = 22.08°.
The beam bends toward the normal, from 30° down to about 22.1°, exactly as expected when entering a slower medium. Since the light is entering a denser medium, no critical angle applies here, and no total internal reflection can occur.
Reverse the direction (water to air) and the critical angle appears: \theta_c = \arcsin(1.00/1.33) = 48.75°. A ray inside water hitting the surface steeper than about 48.8° from the normal is trapped.
Seeing the whole curve
The refracted angle is not proportional to the incidence angle. For air into water, small incidence angles bend by a modest ratio, but the curve flattens as \theta_1 nears 90° because \sin\theta_1 saturates at 1. The largest possible refracted angle for air into water is \arcsin(1/1.33) = 48.75°, reached only at grazing incidence.
Reading and interpreting the result
Three outcomes are possible, and each means something physical.
- A refracted angle less than the incidence angle
- Light entered a denser medium (n_2 \gt n_1) and bent toward the normal. Example: air to glass, 40° becomes about 25°.
- A refracted angle greater than the incidence angle
- Light entered a less dense medium and bent away from the normal. This is only valid up to the critical angle.
- No refracted angle (total internal reflection)
- You asked for n_1 \gt n_2 with \theta_1 \gt \theta_c. The arcsine has no real solution because its argument exceeds 1. The energy reflects back.
| From medium | n1 | Critical angle |
|---|---|---|
| Water | 1.33 | 48.75° |
| Acrylic (PMMA) | 1.49 | 42.16° |
| Crown glass | 1.52 | 41.14° |
| Diamond | 2.42 | 24.41° |
Diamond's low critical angle of 24.4° is why cut diamonds sparkle: light entering the top hits many internal faces above that angle and bounces around before escaping toward your eye.
Common mistakes
Do not feed the calculator an angle beyond the critical angle and expect a refracted ray. When (n_1/n_2)\sin\theta_1 \gt 1, there is no real arcsine. That is not an error in the math, it is total internal reflection, and the correct answer is that no light transmits.
Other traps to watch:
- Wrong index order. Swapping n_1 and n_2 reverses the bending. The from and into media are not interchangeable.
- Angle from the surface. If a problem gives 20° from the surface, the incidence angle is 70° from the normal.
- Assuming one index fits all colours. Index depends on wavelength (dispersion). Crown glass is about 1.514 for red and 1.523 for violet. That 0.6% difference is what splits white light into a spectrum in a prism.
- Treating air as exactly 1. Air is 1.000293. For most work rounding to 1.00 changes angles by well under 0.02°, but it matters in precision surveying and astronomy.
Related tools
Refraction is one piece of geometric optics. Once you know how rays bend at a surface, the Lens and mirror equation calculator tells you where an image forms. If you are working with the wave nature of light, the Photon energy calculator connects wavelength to energy, and the de Broglie wavelength calculator extends wave behaviour to matter. For the frequency shifts of moving sources see the Doppler effect calculator. When you measure an index in the lab and want to report its uncertainty, the Error propagation calculator and the Percent error calculator handle the statistics.
Frequently asked questions
Why does light bend when it enters water?
Because it slows down. The part of the wavefront that enters the water first is delayed relative to the rest, so the whole front pivots. Snell's law quantifies the pivot: n_1 \sin\theta_1 = n_2 \sin\theta_2.
What is the critical angle in plain terms?
It is the steepest angle from the normal at which light can still escape into a faster medium. Steeper than that and it reflects entirely. For water to air it is 48.75°, for crown glass to air about 41.1°.
Can total internal reflection happen going into glass from air?
No. It only happens when light travels toward a lower-index medium, meaning n_1 \gt n_2. Air into glass is the opposite direction, so there is always a refracted ray.
Does Snell's law work for sound or water waves?
The same sine relation applies to any wave that changes speed at a boundary, with the index replaced by the ratio of wave speeds. Seismic waves and sonar both refract this way. The optics-specific value n = c/v only applies to light.
Why do different colours refract by different amounts?
The refractive index depends slightly on wavelength, an effect called dispersion. Violet light sees a higher index than red in glass, so it bends more. A prism spreads white light into a spectrum precisely because of this small difference, often less than one percent in index.
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