Snell's law gives the mathematical connection between a light ray's direction and the optical properties of two materials. It is the main equation used to predict refraction at a boundary.

What Snell's Law Describes

When light reaches a boundary between two media, the direction of the ray in the first medium and the direction of the ray in the second medium are connected by a precise relationship. Snell's law does not describe the ray with words alone; it gives an algebraic rule that links the two angles to the two refractive indices.

The law applies at the point where the ray meets the surface.

This refraction diagram shows an incident ray crossing a boundary into a second medium, with the normal drawn perpendicular to the interface. The labeled angles help emphasize the correct geometric definition of $\theta_1$ and $\theta_2$ as angles measured from the normal, which is essential for using $n_1\sin\theta_1=n_2\sin\theta_2$ correctly. Source

A line called the normal is drawn perpendicular to the boundary at that point, and both relevant angles are measured from that normal.

Key angles

This is the angle the ray has before it enters the second medium.

Both angles must be measured from the normal, not from the surface itself, or the equation will be used incorrectly.

The material property that appears in Snell's law is the index of refraction.

A larger value of $n$ means the medium has a stronger effect on the direction of the ray in Snell's law.

The equation

This equation shows that the product of refractive index and the sine of angle must match on both sides of the boundary. For a fixed pair of media, you can also rearrange it as $\dfrac{\sin\theta_1}{\sin\theta_2}=\dfrac{n_2}{n_1}$. This form emphasizes that the ratio of the sines depends only on the two media.

Because the law uses sines, the change in angle is not linear. Doubling one angle does not generally double the other angle.

What the subscripts mean

The subscripts do not mean “incident” and “refracted” automatically. They label the two media. Medium 1 is the side where the ray begins, and medium 2 is the side the ray enters. The angle and index with the same subscript must always belong to the same medium.

Interpreting Snell's Law

Snell's law is useful because it connects geometry and material properties in one step.

This OpenStax figure compares two refraction cases: slowing down in a higher-index medium bends the ray toward the normal, while speeding up bends it away. The paired panels reinforce how the geometry (angles to the normal) and material properties (indices of refraction, tied to speed) work together in Snell’s law. Source

• If the two media have different refractive indices, then the two angles will usually be different.

• If the two media have the same refractive index, then $n_1=n_2$, so the equation requires $\theta_1=\theta_2$ for ordinary refraction.

• For the same incident angle, a larger difference between the two refractive indices produces a larger change in ray direction.

• The medium with the larger refractive index corresponds to the smaller angle in the equation, because the sine term must adjust to keep both sides equal.

These statements come directly from the algebraic structure of the law. On AP Physics 2 problems, the main task is usually to identify which quantities are known, substitute them correctly, and solve for the unknown angle or unknown refractive index.

Using Snell's Law in problems

A reliable method helps avoid setup mistakes.

• Identify the two media and assign them as 1 and 2.

• Write the given angle as the angle to the normal.

• Match each angle with the refractive index of the same medium.

• Substitute into $n_1\sin\theta_1=n_2\sin\theta_2$.

• Isolate the unknown quantity algebraically.

• If solving for an angle, use the inverse sine only after the sine is isolated.

When solving numerically, keep enough digits through the intermediate steps so rounding does not shift the final angle too much. Also make sure your calculator is in the intended angle mode.

Snell's law can be used in two common ways:

• to find an unknown refracted angle when both refractive indices and the incident angle are known

• to find an unknown refractive index when both angles and one refractive index are known

If light passes through more than one boundary, Snell's law is applied separately at each boundary rather than all at once.

Common errors to avoid

Many mistakes on refraction questions are not physics mistakes; they are setup mistakes.

• Measuring angles from the surface instead of from the normal

• Pairing $n_1$ with $\theta_2$ or $n_2$ with $\theta_1$

• Forgetting that the equation uses $\sin\theta$, not just $\theta$

• Applying the law without clearly identifying which medium is first and which is second

• Rounding too early in multistep algebra

If your computed value for a sine is greater than 1 or less than $-1$, the setup should be checked carefully because no real angle can come from that result.

This diagram illustrates total internal reflection and the critical angle, where the refracted ray would emerge at $\theta_2=90^\circ$ along the boundary. It provides a geometric reason that Snell’s law can produce an impossible value (a sine outside $[-1,1]$): beyond the critical angle, refraction ceases and the ray reflects instead. Source

In AP-level work, that usually signals that a quantity was entered incorrectly or that the simple refraction situation assumed by the equation is not being met.