When light crosses a boundary between two transparent materials it changes direction, unless it hits along the normal. For a ray going from air into a material, refractive index n = sin i ÷ sin r, where i is the angle of incidence and r is the angle of refraction, both measured from the normal.
The depth needed depends on your syllabus year and route, so check the Cambridge IGCSE Physics 0625 page for what is required. This lesson follows constructing a reflected ray because it uses the same normal habit, and it sits inside light and imaging.
Why does light bend at a boundary?
Light travels at different speeds in different materials. It is fastest in air (or a vacuum) and slower in glass or water. When a ray meets a boundary at an angle, one side of the wave slows first, and the ray turns.
Entering a denser material (slower light), the ray bends toward the normal. Leaving a denser material for a less dense one (faster light), the ray bends away from the normal.
Along the normal, the ray slows down but does not change direction. The frequency stays the same, while the speed and wavelength change.
How do I use n = sin i ÷ sin r, step by step?
- Draw or picture the normal at the point where the ray crosses the boundary.
- Identify i and r, both from the normal. i is in the material where the ray starts (air), and r is in the second material.
- Check the direction: into glass or water, r should be smaller than i. If your answer says otherwise, recheck.
- Write the equation and substitute. Use the sine key with your calculator in degree mode.
- Round sensibly, state the unit (n has none), and check the result is greater than 1 for glass or water.
To find an angle instead, rearrange: sin r = sin i ÷ n. Then use the inverse sine key to turn sin r back into an angle.
Worked example
A ray of light in air meets a flat glass block with i = 50° and r = 31°. (Invented example data.) Then a second ray in air hits glass of refractive index 1.5 with an angle of incidence of 30°. Find the refractive index in the first case and the angle of refraction in the second.
Step 1, first case, write the equation: n = sin i ÷ sin r.
Step 2, substitute: sin 50° = 0.7660 and sin 31° = 0.5150. n = 0.7660 ÷ 0.5150 = 1.487, which is 1.5 to two significant figures.
Step 3, second case, rearrange: sin r = sin i ÷ n = sin 30° ÷ 1.5 = 0.5 ÷ 1.5 = 0.3333.
Step 4, use the inverse sine: r = sin⁻¹(0.3333) = 19.47°, so r ≈ 19°.
Step 5, check: in the second case, entering glass, r must be smaller than i. 19° is smaller than 30°, so the direction is right. Checking the first case, 31° is smaller than 50°, which also fits.
The mistake to watch for
The most common slip is to divide the angles instead of the sines.
Mistaken answer: n = 50 ÷ 31 = 1.61
The student used the angles directly. That gives a different number that would change with the angle, so it cannot be a property of the material.
The correction is to take the sine of each angle first: n = sin 50° ÷ sin 31° = 1.49. A second slip to check is swapping the two sines, which gives n < 1 for glass. Since light slows down in glass, n must be greater than 1, and that alone tells you the fraction is upside down.
Check yourself
Try these with a calculator in degree mode, then open each answer.
1. A ray in air enters a plastic block with i = 45° and r = 28°. Calculate n.
Show answer
sin 45° = 0.7071 and sin 28° = 0.4695. n = 0.7071 ÷ 0.4695 = 1.506, so n ≈ 1.5.
2. Light in air enters water (n = 1.33) with an angle of incidence of 40°. Find the angle of refraction.
Show answer
sin r = sin 40° ÷ 1.33 = 0.6428 ÷ 1.33 = 0.4833. r = sin⁻¹(0.4833) = 28.9°, so r ≈ 29°. This is smaller than 40°, as expected for light entering water.
3. A ray in glass (n = 1.5) meets the glass-to-air boundary with an angle of incidence of 20°. Find the angle of refraction in the air. Which way does the ray bend?
Show answer
For light leaving glass, sin r = n × sin i = 1.5 × sin 20° = 1.5 × 0.3420 = 0.5130. r = sin⁻¹(0.5130) = 30.9°, so r ≈ 31°. The ray bends away from the normal, because the angle in air (31°) is larger than the angle in glass (20°).
Where this leads next
Next, see what happens when the angle in glass gets large enough that no light leaves: total internal reflection conditions. When you reach lenses, refraction at the curved glass surfaces is what bends the rays in drawing a simple lens image.
Students who know n = sin i ÷ sin r but still get the wrong answer usually slip at the calculator step or the direction check. Our teachers watch for exactly that in online one-to-one Physics tuition.