25.3 The Law of Refraction

Light bends when it moves between different materials. This bending, called refraction, follows Snell's law. Understanding refraction helps explain everyday phenomena like why a straw looks bent in a glass of water or how rainbows form.

The speed of light changes in different materials, and that change in speed is what causes the bending. The relationship between speed and bending is captured by the index of refraction. These ideas are foundational for understanding lenses, fiber optics, and many natural light phenomena.

The Law of Refraction

Angle of Refraction Calculation

Snell's law connects the angle of incidence ($\theta_1$) and the angle of refraction ($\theta_2$) when light crosses a boundary between two media with different indices of refraction ($n_1$ and $n_2$):

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

All angles are measured from the normal line, which is an imaginary line perpendicular to the surface at the point where the light hits.

To calculate the angle of refraction:

1. Identify the indices of refraction for the two media ($n_1$ and $n_2$).
2. Measure the angle of incidence ($\theta_1$) relative to the normal.
3. Rearrange Snell's law to solve for $\theta_2$:

$\theta_2 = \arcsin\left(\frac{n_1}{n_2} \sin \theta_1\right)$

Two rules to remember about which direction light bends:

• Lower to higher index (e.g., air to water): light bends toward the normal.
• Higher to lower index (e.g., water to air): light bends away from the normal.

This makes intuitive sense if you think of it this way: light slows down when entering a denser medium, and that slowing causes it to bend toward the normal. When it speeds up entering a less dense medium, it bends away.

Light Speed in Materials

The speed of light in a vacuum ($c$) is approximately $3 \times 10^8$ m/s. In every other medium, light travels slower. The index of refraction ($n$) quantifies how much slower:

$n = \frac{c}{v}$

where $v$ is the speed of light in that medium. Since $v$ is always less than or equal to $c$, the index of refraction is always $\geq 1$.

• A higher index means light moves slower in that material. Diamond has $n = 2.42$, so light in diamond travels at roughly $\frac{3 \times 10^8}{2.42} \approx 1.24 \times 10^8$ m/s.
• A lower index means light moves faster. Air has $n \approx 1.0003$, which is so close to 1 that we typically round it to $n = 1$.

You'll sometimes see materials described by their optical density. Optically denser media (like glass, $n \approx 1.5$) have higher indices and slower light speeds. Optically less dense media (like air) have lower indices and faster light speeds. "Optical density" here is about how light interacts with the material, not the material's physical mass density.

Applications of Snell's Law

Here's a general approach for solving Snell's law problems:

1. Identify the two media and their indices of refraction ($n_1$ and $n_2$).
2. Determine what's unknown: the angle of refraction ($\theta_2$), or the index of refraction of one medium ($n_2$)?
3. Plug known values into Snell's law and solve.

Finding an unknown angle of refraction:

Suppose light passes from air ($n_1 = 1.00$) into water ($n_2 = 1.33$) at a $30°$ angle of incidence.

$\sin \theta_2 = \frac{n_1}{n_2} \sin \theta_1 = \frac{1.00}{1.33} \sin 30° = \frac{1.00}{1.33} \times 0.5 = 0.376$

$\theta_2 = \arcsin(0.376) \approx 22.1°$

The refracted angle is smaller than the incident angle, confirming that light bent toward the normal (as expected going from a lower to higher index).

Finding an unknown index of refraction:

Suppose light passes from air ($n_1 = 1.00$) into an unknown medium at $45°$ incidence and refracts to $30°$.

$n_2 = \frac{n_1 \sin \theta_1}{\sin \theta_2} = \frac{1.00 \times \sin 45°}{\sin 30°} = \frac{0.707}{0.5} = 1.41$

This index is close to that of glass, so the unknown medium is likely glass.

Special cases:

• Normal incidence ($\theta_1 = 0°$): Light hits the surface straight on. Since $\sin 0° = 0$, Snell's law gives $\theta_2 = 0°$. The light passes through without bending.
• Total internal reflection: When light travels from a higher-index medium to a lower-index medium (e.g., water to air), there's a maximum angle of incidence beyond which no refraction occurs. Instead, all the light reflects back. This threshold is the critical angle:

$\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)$

For water to air: $\theta_c = \arcsin\left(\frac{1.00}{1.33}\right) \approx 48.8°$. At any angle greater than $48.8°$, light in the water reflects entirely and doesn't pass into the air. This is the principle behind fiber optics.

Additional Optical Phenomena

• Dispersion: Different wavelengths (colors) of light refract by slightly different amounts because the index of refraction depends on wavelength. This is why a prism separates white light into a rainbow of colors.
• Frequency stays constant: When light enters a new medium, its frequency doesn't change. Its speed changes, and its wavelength adjusts accordingly ($v = f\lambda$, so if $v$ decreases, $\lambda$ decreases too).
• Polarization: Light waves can vibrate in many directions perpendicular to their travel. Polarization restricts vibration to a single plane. While not directly caused by refraction, light can become partially polarized when it reflects or refracts at a surface.