📐 Degrees mode
Your calculator must be working in degrees. Look for a small D or DEG on the display. Quick check: type sin 30. If the answer is 0.5, you are in degrees mode. If it is anything else, change the mode before you start.
Light and Sound » Refractive Index
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 3.18
- What the refractive index of a material tells you
- The equation n = sin i / sin r, which you must learn
- How to use the sin and sin−1 keys on your calculator
- How to calculate a refractive index, an angle of refraction or an angle of incidence
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Unlock This CourseWhat is refractive index?
Remember, light slows down and bends towards the normal when it goes from air into glass or water. Some materials slow light more than others, so they bend it more. We need a number that says how strongly a material does this. That number is the refractive index.
Key terms:
- Refractive index (n): a number that shows how much a material slows light down and so how much it bends light. The bigger the refractive index, the more the light is slowed and the more it bends.
- Sine (sin): a calculator function that turns an angle into a number. For example, sin 30° = 0.5.
- Inverse sine (sin−1): the calculator function that goes backwards, from the number to the angle. For example, sin−1 0.5 = 30°.
Refractive index has no unit. It is just a number. For the materials you will meet it is always bigger than 1.
| Material | Refractive index | What it means |
|---|---|---|
| Water | about 1.33 | Slows and bends light a little |
| Glass | about 1.5 | Slows and bends light more than water |
| Diamond | about 2.4 | Slows and bends light a lot |
The exact value for glass depends on the type of glass, so a question will always give you the number or the angles you need.
The equation
In words: refractive index = sine of the angle of incidence ÷ sine of the angle of refraction.
In symbols: n = sin i / sin r
This equation is not given in the exam. You must learn it.
- i is the angle of incidence, the angle in the air, for light going from air into the material.
- r is the angle of refraction, the angle in the material.
- Both angles are measured from the normal, in degrees.
Why use sines and not just the angles? Look at these values for light going from air into water. Dividing the angles gives a different answer each time. Dividing the sines gives the same answer every time, and that answer is the refractive index of water.
| i in ° | r in ° | i ÷ r | sin i ÷ sin r |
|---|---|---|---|
| 20 | 14.9 | 1.34 | 1.33 |
| 40 | 28.9 | 1.38 | 1.33 |
| 60 | 40.6 | 1.48 | 1.33 |
| 80 | 47.8 | 1.67 | 1.33 |
Using your calculator
🔢 sin and sin−1
To turn an angle into a number, press sin, type the angle, then press =. To turn a number back into an angle, press SHIFT (or 2nd), then sin, type the number, then press =. Use sin−1 whenever the question asks for an angle.
You can rearrange the equation in two ways:
| To find | Use | Then |
|---|---|---|
| Refractive index | n = sin i ÷ sin r | Done. No unit |
| Angle of refraction | sin r = sin i ÷ n | Use sin−1 to get r |
| Angle of incidence | sin i = n × sin r | Use sin−1 to get i |
Worked examples
Worked example 1: finding the refractive index
Question: A ray of light goes from air into a clear block. The angle of incidence is 50° and the angle of refraction is 30°. Calculate the refractive index of the block.
n = sin i / sin r
n = sin 50° ÷ sin 30°
n = 0.7660 ÷ 0.5
n = 1.53 (no unit)
Worked example 2: finding the angle of refraction
Question: A ray of light from a torch hits the surface of a pond at an angle of incidence of 45°. The refractive index of water is 1.33. Calculate the angle of refraction.
n = sin i / sin r, so sin r = sin i ÷ n
sin r = sin 45° ÷ 1.33
sin r = 0.7071 ÷ 1.33 = 0.5317
r = sin−1 0.5317
r = 32°
Check: 32° is smaller than 45°, as it should be for light going into water.
Worked example 3: finding the angle of incidence
Question: Light goes from air into a glass window of refractive index 1.5. The angle of refraction in the glass is 20°. Calculate the angle of incidence.
n = sin i / sin r, so sin i = n × sin r
sin i = 1.5 × sin 20°
sin i = 1.5 × 0.3420 = 0.5130
i = sin−1 0.5130
i = 31°
Worked example 4: light coming out into air
Question: A ray of light inside a glass block of refractive index 1.5 meets the surface at 30° to the normal and comes out into the air. Calculate the angle between the ray in the air and the normal.
The sine of the angle in the air always goes on top of the equation, whichever way the light is travelling. For a ray coming out of glass, a question will call the 30° in the glass the angle of incidence and the angle in the air the angle of refraction. The names swap, but the angle in the air still goes on top.
n = sin (angle in air) ÷ sin (angle in glass), so sin (angle in air) = n × sin (angle in glass)
sin (angle in air) = 1.5 × sin 30° = 1.5 × 0.5 = 0.75
angle in air = sin−1 0.75
angle in air = 49°
Check: the angle in the air is bigger than the angle in the glass, because the ray bends away from the normal as it comes out.
Comparing materials
Send the same ray, at an angle of incidence of 45°, into three different materials. The equation gives these angles of refraction.
💧 Water, n = 1.33
r = 32°. The ray is bent the least.
🔍 Glass, n = 1.5
r = 28°. The ray is bent more, so it ends up closer to the normal.
💎 Diamond, n = 2.42
r = 17°. The ray is bent the most.
So for the same angle of incidence, a bigger refractive index gives a smaller angle of refraction.
Common mistakes
- Dividing the angles instead of their sines. Writing n = 50 ÷ 30 is wrong. Take the sine of each angle first.
- Putting the equation upside down. If your refractive index comes out smaller than 1, you have divided the wrong way round. The angle in the air goes on top.
- Stopping at sin r. A value like 0.5317 is not an angle. Use sin−1 to finish.
- Having the calculator in the wrong mode. Check that sin 30 gives 0.5.
- Rounding too early. Keep four figures for the sines and round only the final answer.
- Giving the refractive index a unit, such as degrees. It has none.
Exam-style question
A ray of light travels from air into a block of clear plastic. The angle of incidence is 55° and the angle of refraction is 34°.
(a) State the equation linking refractive index, angle of incidence and angle of refraction. [1 mark]
(b) Calculate the refractive index of the plastic. [2 marks]
(c) A second ray meets the same block at an angle of incidence of 30°. Calculate the angle of refraction for this ray. [3 marks]
(d) The plastic block is replaced by a block of diamond, which has a larger refractive index. The angle of incidence is still 30°. State and explain how the angle of refraction changes. [2 marks]
Model answer
(a) n = sin i / sin r (1)
(b) n = sin 55° ÷ sin 34° = 0.8192 ÷ 0.5592 (1)
n = 1.46 (1) (1.5 also gains the mark; no unit)
(c) sin r = sin i ÷ n = sin 30° ÷ 1.46 (1)
sin r = 0.5 ÷ 1.46 = 0.3425 (1)
r = sin−1 0.3425 = 20° (1)
(d) The angle of refraction is smaller (1), because a material with a larger refractive index slows the light more and so bends it more towards the normal (1).
Exam tip
In part (c), write down the value of sin r before you press sin−1. If you slip on the last step, the examiner can still give you the marks for the substitution and the rearrangement.