The refractive index of a medium is n = sin i ÷ sin r for light entering from air. It is also c ÷ v, and for a transparent object viewed from above it is real depth ÷ apparent depth.
This lesson is part of light and optics. It leads to explaining total internal reflection.
What do you draw first?
Draw the boundary, then the normal, then the two rays. Label the angle in air and the angle in the medium, both measured from the normal.
Light entering a denser medium bends toward the normal, so r is smaller than i. Light leaving bends away from the normal.
Worked example: finding the refracted angle
Light in air meets glass with n = 1.60. The angle of incidence is 40°.
- sin r = sin i ÷ n = sin 40° ÷ 1.60 = 0.6428 ÷ 1.60 = 0.4017.
- r = sin⁻¹(0.4017) = 23.7°.
The angle in the glass is smaller than 40°, which matches bending toward the normal.
Worked example: speed of light in a medium
For the same glass, v = c ÷ n = 3.00 × 10⁸ ÷ 1.60 = 1.88 × 10⁸ m s⁻¹. The light is slower in glass, as a value of n greater than 1 requires.
Worked example: real and apparent depth
A pool is 2.0 m deep, and the refractive index of water is 1.33. Apparent depth = real depth ÷ n = 2.0 ÷ 1.33 = 1.5 m. The pool looks a quarter shallower than it really is.
| Quantity | Formula | Answer |
|---|---|---|
| Refracted angle | sin r = sin i ÷ n | 23.7° |
| Speed in glass | v = c ÷ n | 1.88 × 10⁸ m s⁻¹ |
| Apparent depth | real ÷ n | 1.5 m |
The mistake that costs marks
The common slip is measuring the angle from the surface instead of the normal. Another is putting sin r on top, which gives a value of n below 1.
| Slip | Effect | Fix |
|---|---|---|
| Angle taken from the surface | Wrong angle in the formula | Draw the normal and measure from it |
| n = sin r ÷ sin i | n < 1, impossible for glass | n = sin i ÷ sin r |
| Real depth × n | Apparent depth too large | Divide by n |
If your answer for n is below 1, or the depth seems larger than the real one, something is upside down. The units and significant figure checker can help you check your values.
Check yourself
Light in air meets water (n = 1.33) at an angle of incidence of 45°. Find the angle of refraction.
Answer
sin r = sin 45° ÷ 1.33 = 0.7071 ÷ 1.33 = 0.5317.
r = sin⁻¹(0.5317) = 32.1°.
The angle is smaller than 45°, so the ray bends toward the normal, as expected for light entering water.
What to study next
Go on to explaining total internal reflection, which uses the critical angle. You can also try the light and optics practice set.
To have a teacher check your diagrams and angles, see online one-to-one Physics tuition.