16.2 Refraction

Refraction and Snell's Law

Light bends when it crosses from one material into another. This bending, called refraction, happens because light changes speed depending on the medium it's traveling through. Understanding refraction is central to how lenses, fiber optics, and even rainbows work.

more resources to help you study

practice questions

Light direction changes between media

Light travels fastest in a vacuum and slows down when it enters any material. The refractive index ($n$) of a medium quantifies how much it slows light compared to a vacuum. A higher refractive index means light moves more slowly in that material. For reference, air has $n \approx 1.00$, water has $n \approx 1.33$, and diamond has $n \approx 2.42$.

When light passes from one medium into another with a different refractive index, it changes direction at the boundary. The rule for which way it bends:

• Light bends toward the normal when moving into a higher-$n$ medium (e.g., air → water). The light slows down, so the refracted ray angles closer to the perpendicular.
• Light bends away from the normal when moving into a lower-$n$ medium (e.g., water → air). The light speeds up, so the refracted ray angles farther from the perpendicular.

The normal line is an imaginary line drawn perpendicular to the boundary surface at the point where the light hits. All angles in refraction problems are measured from this normal, not from the surface itself.

Snell's Law gives you the exact relationship between the angles and the refractive indices:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

• $n_1$ = refractive index of the medium the light starts in
• $n_2$ = refractive index of the medium the light enters
• $\theta_1$ = angle of incidence (between the incoming ray and the normal)
• $\theta_2$ = angle of refraction (between the refracted ray and the normal)

Example: Light in air ($n_1 = 1.00$) hits water ($n_2 = 1.33$) at an angle of incidence of 45°. To find the angle of refraction:

1. Write Snell's law: $1.00 \times \sin 45° = 1.33 \times \sin \theta_2$
2. Solve for $\sin \theta_2$: $\sin \theta_2 = \frac{1.00 \times 0.707}{1.33} = 0.531$
3. Take the inverse sine: $\theta_2 = \sin^{-1}(0.531) \approx 32.1°$

The light bent toward the normal, which makes sense since it moved into a higher-$n$ medium.

Critical angles and total internal reflection

When light travels from a higher-$n$ medium into a lower-$n$ medium (like water into air), it bends away from the normal. As you increase the angle of incidence, the refracted ray bends farther and farther from the normal. At a specific angle called the critical angle ($\theta_c$), the refracted ray skims right along the boundary surface at 90° from the normal. Beyond that angle, no light passes through at all. Instead, it all bounces back into the original medium. This is total internal reflection.

To find the critical angle, set $\theta_2 = 90°$ in Snell's law (since that's the threshold where refraction just barely fails):

$\sin \theta_c = \frac{n_2}{n_1}$

• $n_1$ must be the higher refractive index (the medium the light is in)
• $n_2$ must be the lower refractive index (the medium the light would enter)

Example: For light going from water ($n_1 = 1.33$) into air ($n_2 = 1.00$):

$\sin \theta_c = \frac{1.00}{1.33} = 0.752 \implies \theta_c \approx 48.8°$

Any light hitting the water-air boundary at an angle greater than 48.8° from the normal will be totally internally reflected.

Two things to keep straight: total internal reflection only happens when light goes from a higher-$n$ medium to a lower-$n$ medium, and only when the angle of incidence exceeds the critical angle.

Applications of total internal reflection:

• Fiber optic cables trap light inside thin glass strands by keeping it bouncing at angles above the critical angle, allowing high-speed data transmission over long distances
• Prisms in binoculars and periscopes use total internal reflection to redirect light with almost no loss, often more efficiently than silvered mirrors
• Medical endoscopes use bundles of optical fibers to carry images from inside the body

Refraction reference points

• The normal line is perpendicular to the boundary at the point where light strikes. All angles are measured from the normal, not the surface.
• The angle of incidence is between the incoming ray and the normal. The angle of refraction is between the refracted ray and the normal on the other side of the boundary.
• Don't confuse refraction with reflection. Reflection is light bouncing off a surface (angle of incidence equals angle of reflection). Refraction is light passing through the boundary and changing direction.
• Optical density is a qualitative way of describing how much a medium slows light. A more optically dense medium has a higher refractive index.

Dispersion and Rainbows

Dispersion in prisms and rainbows

The refractive index of a material isn't exactly the same for every color of light. It varies slightly with wavelength, and this is what causes dispersion: the separation of white light into its component colors.

Shorter wavelengths (violet, blue) have a slightly higher refractive index in most materials than longer wavelengths (red, orange). That means violet light bends more than red light when passing through the same boundary. In a single refraction, this difference is tiny. But a prism refracts light twice (entering and exiting), and the geometry of the prism amplifies the angular spread between colors.

How a prism produces a spectrum:

1. White light enters one face of the prism and refracts. Each wavelength bends by a slightly different amount.
2. The light travels through the prism and hits the second face.
3. Each wavelength refracts again as it exits, increasing the angular separation between colors.
4. The result is a visible spectrum from red (least bent) to violet (most bent).

How rainbows form:

1. Sunlight enters a spherical water droplet and refracts at the front surface, partially separating into colors.
2. The light reflects off the back interior surface of the droplet.
3. The light refracts again as it exits the front of the droplet, further separating the colors.
4. Each color exits at a slightly different angle. Red light exits at about 42° relative to the incoming sunlight, and violet at about 40°.
5. You see red at the top of the rainbow and violet at the bottom because droplets at different heights in the sky send different colors to your eyes at the correct viewing angle.