Snell's Law: Refraction at an Interface

When a beam of light crosses a boundary between two transparent materials — air to water, glass to air, water to diamond — it almost always changes direction. That bending is called refraction, and the rule that governs it is one of the oldest quantitative laws in physics: Snell's law, discovered independently by Ibn Sahl in 984, Willebrord Snell in 1621, and René Descartes in 1637.

$n_{1} \sin\theta_{1} = n_{2} \sin\theta_{2}$

Here $n_{1}$ and $n_{2}$ are the refractive indices of the two media, $\theta_{1}$ is the angle of incidence (measured from the normal to the surface, not from the surface itself), and $\theta_{2}$ is the angle of refraction.

The refractive index $n$ of a material is the factor by which light slows down in that material compared to vacuum. Vacuum has $n = 1$ exactly, air is $n \approx 1.0003$ (which we round to 1.00), water is $n \approx 1.33$, glass varies from about 1.5 to 1.9, and diamond is $n = 2.42$.

Solving the problem (method 1 — direct)

$n_{1} = 1.00$ (air), $\theta_{1} = 45°$, $n_{2} = 1.50$ (glass).

$\sin\theta_{2} = \dfrac{n_{1}}{n_{2}} \sin\theta_{1} = \dfrac{1.00}{1.50} \sin 45° = \dfrac{0.7071}{1.50} = 0.4714$

$\theta_{2} = \arcsin(0.4714) \approx 28.13°$

The light bends toward the normal (from 45° to 28°) because it's entering a denser medium ($n_{2} > n_{1}$). This makes physical sense: light slows down in glass, and the component of velocity parallel to the surface decreases, pulling the wave front closer to the perpendicular.

Why light bends — the wave explanation (method 2)

Imagine ocean waves hitting a beach at an angle. The part of the wave that reaches shallow water first slows down, while the rest is still in deep water moving fast. The result: the wave front swings around and aligns more nearly parallel to the shore. Light does the same thing.

More precisely: light is a wave with speed $v = c/n$ in a medium with refractive index $n$. When a plane wave hits a boundary at an angle, the side of the wave front that enters the new medium first changes speed. This causes the wave front to rotate. The amount of rotation is exactly what Snell's law predicts.

The key insight is that the wavelength changes when light enters a new medium ($\lambda_{\text{medium}} = \lambda_{\text{vacuum}} / n$), but the frequency stays the same (it's set by the source). Since $v = f\lambda$, slower speed means shorter wavelength and more tightly packed wave fronts.

Worked example with different media

Air → Water: $n_{1}=1.00$, $\theta_{1}=45°$, $n_{2}=1.33$. $\theta_{2} = \arcsin\!\left(\frac{1.00}{1.33}\sin 45°\right) = \arcsin(0.5317) \approx 32.1°$

Water → Glass: $n_{1}=1.33$, $\theta_{1}=30°$, $n_{2}=1.50$. $\theta_{2} = \arcsin\!\left(\frac{1.33}{1.50}\sin 30°\right) = \arcsin(0.4433) \approx 26.3°$

Glass → Diamond: $n_{1}=1.50$, $\theta_{1}=20°$, $n_{2}=2.42$. $\theta_{2} = \arcsin\!\left(\frac{1.50}{2.42}\sin 20°\right) = \arcsin(0.2119) \approx 12.2°$

Notice the pattern: light always bends toward the normal when entering a denser medium, and the denser the second medium, the more it bends.

Glass → Air (reverse): $n_{1}=1.50$, $\theta_{1}=20°$, $n_{2}=1.00$. $\theta_{2} = \arcsin(1.50 \sin 20°) = \arcsin(0.5130) \approx 30.9°$

Going from dense to less-dense, light bends away from the normal. This is the setup for total internal reflection (Problem 152) — when $\theta_{2}$ would exceed 90°, the light can't exit at all.

Refractive indices of common materials

• Air: 1.00
• Ice: 1.31
• Water: 1.33
• Ethanol: 1.36
• Fused silica: 1.46
• Crown glass: 1.52
• Flint glass: 1.62
• Sapphire: 1.77
• Cubic zirconia: 2.17
• Diamond: 2.42

These numbers are approximate (they depend on wavelength — that's dispersion, Problem 153).

Real-world phenomena explained by refraction

• A straw looking "bent" in a glass of water: Light from the submerged part of the straw bends away from the normal as it exits the water surface. Your brain traces the light back in a straight line, placing the submerged straw at the wrong position.
• Swimming pools looking shallower than they are: Light from the bottom bends at the water-air interface, making the bottom appear closer to the surface. The apparent depth is $d_{\text{app}} = d/n \approx d/1.33$.
• Mirages on hot roads: A layer of hot air near the road surface has slightly lower $n$ than cooler air above. Light from the sky grazes this layer and bends upward (away from the surface normal), making the road look like it's reflecting the sky — a "puddle."
• Rainbows: Sunlight enters a raindrop (air→water refraction), reflects off the back (internal reflection), and exits (water→air refraction). The two refraction events, combined with the wavelength dependence of $n$, separate white light into colors.

Common mistakes

• Measuring the angle from the surface instead of the normal. This is the single most common error. Snell's law uses the angle from the normal (the perpendicular to the surface), not the angle from the surface. If a problem says "light hits the surface at 30° to the surface," the angle of incidence is $90° - 30° = 60°$.
• Getting the ratio upside-down. It's $\sin\theta_{2} = (n_{1}/n_{2})\sin\theta_{1}$. When going into a denser medium, $n_{2} > n_{1}$, so the ratio is less than 1, and $\theta_{2} < \theta_{1}$.
• Forgetting that $n_{2}\sin\theta_{2} = n_{1}\sin\theta_{1}$ can have no solution. When $n_{1} > n_{2}$ and $\theta_{1}$ is large enough, $\sin\theta_{2} > 1$, which is impossible. That's total internal reflection — see Problem 152.

Try it in the visualization

Change the angle of incidence from 0° to 89° and watch the refracted beam swing from nearly vertical to nearly horizontal. Switch between different material pairs (air-water, air-glass, air-diamond, water-glass) and compare how much bending each combination produces. Notice that at 0° incidence (perpendicular to the surface), there's no bending at all — the light passes straight through.