Snell's Law — Refraction Formula & Critical Angle

A straw in a glass of water looks snapped in two at the surface, and a coin at the bottom of a pool sits shallower than it really is. Both illusions come from one rule: light changes direction when it crosses from one medium into another. Snell's law is the procedure that turns that bending into numbers, letting you find an unknown angle from the two refractive indices and the known angle.

When to use Snell's law

Apply it in ray optics whenever a light ray reaches a boundary and enters a second medium, because light travels at different speeds in different media and so changes direction at the interface. The same setup also tells you when refraction stops altogether and total internal reflection takes over, so reach for this method on water surfaces, glass blocks, prisms, lenses, and optical fibers.

The procedure, step by step

1. Identify the media. Write down the starting medium and the second medium so you can assign $n_1$ and $n_2$ correctly. The order matters: $n_1, \theta_1$ belong to the starting medium and $n_2, \theta_2$ to the second. Swapping them changes the physical situation.

2. Measure from the normal. Use the angle between the ray and the normal, not the surface.

3. Apply the right formula. For refraction,

For the critical angle, when light leaves a higher-index medium ($n_1 > n_2$), set $\theta_2 = 90^\circ$ so the refracted ray grazes the boundary:

This second formula only exists when $n_1 > n_2$; if $n_1 \le n_2$ there is no critical angle in that direction.

4. Check the direction. Light entering a higher-index medium bends toward the normal; entering a lower-index medium it bends away, unless the incident angle exceeds the critical angle and total internal reflection occurs instead. As a quick pre-algebra check: if $n_2 > n_1$ then $\theta_2 < \theta_1$, and if $n_2 < n_1$ then $\theta_2 > \theta_1$.

A full example, start to finish

Light travels from water into air with $n_1 = 1.33$, $n_2 = 1.00$, and $\theta_1 = 40^\circ$. Identify the media (step 1): start in water, end in air. Apply the formula (step 3):

Using $\sin 40^\circ \approx 0.643$,

Checking direction (step 4), $58.8^\circ > 40^\circ$, which is right because the light moves from higher to lower index and bends away from the normal. Now the critical angle for the same pair:

Since $40^\circ < 48.8^\circ$, the ray refracts into air. Had the incident angle exceeded about $48.8^\circ$, this water-to-air setup would have produced total internal reflection.

Where each step tends to break, and how to self-check

• Step 2: measuring from the surface instead of the normal. Self-check: is your angle the gap between the ray and the perpendicular?
• Step 1: reversing $n_1$ and $n_2$ after the diagram is set. Re-read which medium the light starts in.
• Step 3: using the critical-angle formula when light is entering the higher-index medium rather than leaving it. There is no critical angle in that direction.
• Step 4: assuming light always bends toward the normal, or accepting a $\sin \theta_2 > 1$ as a normal refraction result. A sine above $1$ is the signal that total internal reflection should occur.

Where Snell's law is used

It is the first tool for any introductory ray crossing a boundary: water surfaces, glass blocks, prisms, lenses, and optical fibers, and it explains both the bent-straw illusion and why fiber optics can trap light. To practice, let light start in glass with $n_1 = 1.50$ and enter air with $n_2 = 1.00$: find the critical angle first ($\theta_c \approx 41.8^\circ$), then decide whether an incident angle of $35^\circ$ refracts or totally internally reflects. Since $35^\circ < 41.8^\circ$, it refracts.