What if the angle of incidence is zero?

If $ heta_1 = 0$, then $\sin(0) = 0$, requiring $ heta_2 = 0$. The wave travels straight through the boundary without changing direction.

What does it mean if my calculation gives $\sin( heta_2) > 1$?

This indicates total internal reflection. The equation yields no real solution because the incident angle exceeds the critical angle, and all wave energy is reflected back into the first medium.

Can Snell's law be used if the index of refraction is not given?

Yes, provided the wave speeds in both media ($v_1, v_2$) are known. Snell's law can be written equivalently as $$\frac{\sin( heta_1)}{v_1} = \frac{\sin( heta_2)}{v_2}$$.

How to solve a Snell's law refraction problem

Snell's law dictates the relationship between the angles of incidence and refraction when a wave passes through a boundary between two different isotropic media. It applies whenever a light, sound, or other wave changes speed upon entering a new medium, provided the wave does not strike the boundary at exactly perpendicular incidence.

The setup

Identify the two media and their respective indices of refraction, $n_1$ and $n_2$ . Identify the angle of incidence, $heta_1$ , which must be measured from the normal (the line perpendicular to the boundary), not from the surface of the boundary itself.

The steps

Draw a schematic showing the boundary, the normal line, and the incident ray. 2. Write the Snell's law equation: $n_1 \sin( heta_1) = n_2 \sin( heta_2)$ 3. Algebraically isolate the unknown variable (typically $heta_2$ ). 4. Substitute the known values into the equation and compute the result using the inverse sine function.

Checking the result

Evaluate the physical behavior of the calculated angle. If the wave enters a denser medium ( $n_2 > n_1$ ), the ray must bend toward the normal ( $heta_2 < heta_1$ ). If it enters a less dense medium ( $n_1 > n_2$ ), the ray must bend away from the normal ( $heta_2 > heta_1$ ).

Common errors

The most frequent error is measuring the incident or refracted angle relative to the boundary surface instead of the normal. A second common error is performing calculations with a calculator set to radians when the input angles are provided in degrees.

Worked example

A light ray traveling in air ( $n_1 = 1.00$ ) strikes a flat glass surface ( $n_2 = 1.50$ ) at an angle of 45.0 degrees relative to the normal. Determine the angle of refraction inside the glass.

Identify knowns: $n_1 = 1.00$ , $n_2 = 1.50$ , $heta_1 = 45.0^\circ$ . Write the equation: $n_1 \sin( heta_1) = n_2 \sin( heta_2)$ Rearrange for the unknown: $\sin( heta_2) = \frac{n_1}{n_2} \sin( heta_1)$ Substitute values: $\sin( heta_2) = \frac{1.00}{1.50} \sin(45.0^\circ)$ Compute the right side: $\sin( heta_2) = \frac{1}{1.5} (0.7071) = 0.4714$ Compute the inverse sine: $heta_2 = \arcsin(0.4714) = 28.1^\circ$ The ray bends toward the normal, as expected for $n_2 > n_1$ .

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References: University Physics Volume 3 (OpenStax) · Fundamentals of Physics (Halliday & Resnick) · Sears and Zemansky's University Physics (Young & Freedman)