KAHIBARO
Table of Contents
Refraction at a Boundary
Snell's law describes what happens when light passes from one transparent medium into another, such as from air into water or from glass into air. In this situation, the light usually changes direction. This bending is called refraction.
The amount of bending depends on the two media involved and on the angle at which the light strikes the boundary. To describe this clearly, we measure angles from the normal, which is an imaginary line perpendicular to the surface at the point where the light meets it.
If light enters a medium where it travels more slowly, it bends toward the normal. If it enters a medium where it travels more quickly, it bends away from the normal.
The Statement of Snell's Law
Snell's law relates the angle of incidence and the angle of refraction. It is written as
$$
n_1 \sin \theta_1 = n_2 \sin \theta_2
$$
Here, $n_1$ is the refractive index of the first medium, $n_2$ is the refractive index of the second medium, $\theta_1$ is the angle between the incident ray and the normal, and $\theta_2$ is the angle between the refracted ray and the normal.
The refractive index tells us how strongly a medium slows light compared with vacuum. A larger refractive index means light travels more slowly in that medium.
Important rule:
$$
n_1 \sin \theta_1 = n_2 \sin \theta_2
$$
All angles in Snell's law must be measured from the normal, not from the surface.
Meaning of the Refractive Index
The refractive index $n$ is defined by
$$
n = \frac{c}{v}
$$
where $c$ is the speed of light in vacuum and $v$ is the speed of light in the medium.
Since $c$ is larger than $v$ in ordinary materials, the refractive index is usually greater than 1. Because light moves at different speeds in different media, its direction changes when it crosses a boundary.
Combining this idea with Snell's law helps explain why light bends. If light goes from a smaller refractive index to a larger one, then the refracted angle becomes smaller. If it goes from a larger refractive index to a smaller one, then the refracted angle becomes larger.
Common Bending Cases
The most important patterns are easy to summarize.
| Situation | Refractive indices | What happens to the ray |
|---|---|---|
| Air to glass | $n_1 < n_2$ | Bends toward the normal |
| Air to water | $n_1 < n_2$ | Bends toward the normal |
| Glass to air | $n_1 > n_2$ | Bends away from the normal |
| Water to air | $n_1 > n_2$ | Bends away from the normal |
| Same medium to same medium | $n_1 = n_2$ | No bending |
If $n_2 > n_1$, then usually $\theta_2 < \theta_1$, so the ray bends toward the normal.
If $n_2 < n_1$, then usually $\theta_2 > \theta_1$, so the ray bends away from the normal.
Using the Formula
Suppose light goes from air into glass. Take $n_1 = 1.00$, $n_2 = 1.50$, and let the incident angle be $\theta_1 = 30^\circ$.
Using Snell's law,
$$
n_1 \sin \theta_1 = n_2 \sin \theta_2
$$
so
$$
1.00 \sin 30^\circ = 1.50 \sin \theta_2
$$
Since $\sin 30^\circ = 0.5$,
$$
0.5 = 1.50 \sin \theta_2
$$
$$
\sin \theta_2 = \frac{0.5}{1.50} = 0.333
$$
Therefore,
$$
\theta_2 \approx 19.5^\circ
$$
The refracted angle is smaller than the incident angle, so the ray bends toward the normal, as expected.
Special Case of Normal Incidence
If light strikes the boundary straight on, then the incident angle is zero:
$$
\theta_1 = 0^\circ
$$
Since $\sin 0^\circ = 0$, Snell's law gives
$$
n_1 \sin 0^\circ = n_2 \sin \theta_2
$$
so
$$
0 = n_2 \sin \theta_2
$$
which means
$$
\theta_2 = 0^\circ
$$
So a ray hitting the surface along the normal does not bend, even though its speed changes.
Typical Refractive Indices
Some common approximate values are listed below.
| Medium | Refractive index $n$ |
|---|---|
| Vacuum | 1.000 |
| Air | 1.00 |
| Water | 1.33 |
| Glass | 1.5 |
| Diamond | 2.42 |
These values help predict how much light bends at a boundary.
A Geometric View
Snell's law is often easier to understand visually. When light enters a medium with larger refractive index, the refracted ray moves closer to the normal. When it enters a medium with smaller refractive index, it moves farther from the normal.
Practical Importance
Snell's law is one of the basic tools of geometrical optics. It is used to predict how light travels through water, glass, lenses, prisms, and many optical devices. Whenever a light ray crosses from one medium to another, this law gives the direction of the refracted ray.
Key ideas to remember:
Angles are measured from the normal.
$$
n = \frac{c}{v}
$$
and
$$
n_1 \sin \theta_1 = n_2 \sin \theta_2
$$
A larger refractive index means slower light and bending toward the normal when entering that medium.