Refraction and Snell's Law

Refraction is the bending of a light ray as it passes from one transparent material into another. It happens because light travels at different speeds in different materials. When a ray crosses the boundary (the interface) between two media at an angle, the change in speed causes the ray to change direction. This is why a straw in a glass of water looks bent, and why a swimming pool always looks shallower than it really is.

The amount of bending depends on how much the speed of light changes between the two materials. To describe this precisely, physicists use a number called the index of refraction.

The index of refraction (or refractive index), \( n \), of a material tells you how much slower light travels in that material compared to a vacuum. It is defined as:

\( n = \dfrac{c}{v} \)

where \( c \) is the speed of light in a vacuum (about \( 3.00 \times 10^8 \) m/s) and \( v \) is the speed of light inside the material. Because light always slows down when entering a medium, \( n \) is always greater than or equal to \( 1 \). Air has an index of refraction very close to \( 1.00 \), water is about \( 1.33 \), and typical glass is around \( 1.5 \). A higher \( n \) means light travels more slowly and bends more sharply when it enters that medium.

Snell's law is the equation that connects the angles of incidence and refraction to the refractive indices of the two media. Angles are always measured from the normal, an imaginary line drawn perpendicular to the interface at the point where the ray strikes it.

Snell's law states:

\( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)

Here \( n_1 \) and \( \theta_1 \) are the refractive index and angle in the first medium, and \( n_2 \) and \( \theta_2 \) are the refractive index and angle in the second medium. Notice the pattern: if \( n_2 \) is larger than \( n_1 \) (light entering a denser medium), then \( \theta_2 \) must be smaller than \( \theta_1 \), so the ray bends toward the normal, exactly as shown in the diagram above.

A ray of light travels through air (\( n_1 = 1.00 \)) and strikes the surface of water (\( n_2 = 1.33 \)) at an angle of incidence of \( 40^\circ \). Find the angle of refraction.

Start with Snell's law and solve for \( \sin\theta_2 \):

\( \sin\theta_2 = \dfrac{n_1 \sin\theta_1}{n_2} = \dfrac{(1.00)\sin(40^\circ)}{1.33} \)

\( \sin\theta_2 = \dfrac{0.643}{1.33} \approx 0.483 \)

\( \theta_2 = \sin^{-1}(0.483) \approx 28.9^\circ \)

The ray bends toward the normal, from \( 40^\circ \) down to about \( 28.9^\circ \), because it entered a medium with a higher refractive index.

When light travels from a denser medium into a less dense one (for example, from glass into air), it bends away from the normal. As the angle of incidence increases, the angle of refraction gets closer and closer to \( 90^\circ \). At a certain angle, called the critical angle \( \theta_c \), the refracted ray travels exactly along the interface. Beyond this angle, no light escapes into the second medium at all, and the ray is completely reflected back into the first medium. This is total internal reflection.

Setting \( \theta_2 = 90^\circ \) in Snell's law gives the formula for the critical angle:

\( \sin\theta_c = \dfrac{n_2}{n_1} \)

This effect only happens when light moves from a medium with a higher index of refraction into one with a lower index, and it is the principle behind fiber-optic cables, which trap light inside a glass or plastic core using total internal reflection.

Refraction explains many familiar effects: a pencil looking bent where it enters water, a magnifying glass bending light rays to focus them at a point, and a prism spreading white light into a rainbow because different colors (wavelengths) refract by slightly different amounts. The way curved glass or plastic surfaces bend light using this same principle is exactly how lenses in eyeglasses, cameras, and telescopes work to focus images.