What exactly is the index of refraction?

It is a unitless number that describes how much light slows down in a specific material compared to its speed in a perfect vacuum. For example, water has an index of about 1.33, meaning light travels 1.33 times faster in a vacuum than it does in water.

What happens if light goes from a higher index to a lower index?

When light moves into a material with a lower index (like from water into air), it speeds up and bends away from the normal line. If the starting angle is wide enough, the light might not escape at all and will perfectly reflect back inside, which is called total internal reflection.

Does Snell's law only apply to light?

No! Snell's law applies to any wave that changes speed when crossing a boundary. This includes sound waves and water waves, although it is most commonly taught and used in optics with light waves.

How do you use Snell's law?

Snell's law is a formula used to calculate exactly how much a light ray bends when it crosses the boundary between two different transparent materials, like air and water. You use it by plugging in the indices of refraction for both materials and the angle at which the light hits the boundary to find the new angle of the light ray.

Think of a shopping cart rolling from a smooth parking lot into thick grass at an angle. The wheels that hit the grass first slow down, causing the whole cart to pivot. Light does something similar when it enters a material where it travels at a different speed, and Snell's law is the mathematical tool we use to predict that pivot.

The Formula

The equation for Snell's law is $n_1 \\sin(\\theta_1) = n_2 \\sin(\\theta_2)$ . In this formula, $n_1$ and $n_2$ are the indices of refraction for the first and second materials. The index of refraction is just a number that tells you how much light slows down in that material. The angles, $\\theta_1$ and $\\theta_2$ , represent the angle of the light ray before and after it crosses the boundary.

How to set up the problem

To use Snell's law correctly, you always need to draw a reference line called the "normal." The normal is an imaginary line that is exactly perpendicular (at a 90-degree angle) to the boundary between the two materials. You must always measure your angles from this normal line to the light ray, never from the boundary surface itself.

Where students slip up

The most common mistake students make is using the wrong angle. If a problem states "a light ray hits the surface of the water at a 30-degree angle to the surface," you cannot plug 30 degrees into Snell's law. Because the normal is 90 degrees to the surface, the correct angle to use is $90 - 30 = 60$ degrees.

Worked through

A ray of light traveling in air ( $n_1 = 1.00$ ) strikes a flat piece of glass ( $n_2 = 1.50$ ) at an angle of 45 degrees to the normal. What is the angle of refraction (the new angle) inside the glass?

First, we set up Snell's law: $n_1 \\sin(\\theta_1) = n_2 \\sin(\\theta_2)$ . Next, we plug in the values we know: $1.00 \\sin(45^\\circ) = 1.50 \\sin(\\theta_2)$ . We know that $\\sin(45^\\circ)$ is approximately 0.707. So, the equation becomes $0.707 = 1.50 \\sin(\\theta_2)$ . To isolate the sine function, divide both sides by 1.50: $\\sin(\\theta_2) = 0.471$. Finally, take the inverse sine (also called arcsin) of both sides to find the angle:$ \theta_2 = \sin^{-1}(0.471) \approx 28.1^\circ$$. The light bends toward the normal, traveling through the glass at an angle of 28.1 degrees.

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Where this comes from: OpenStax College Physics: Geometric Optics · Khan Academy: Refraction and Snell's law