3.2.2c Refractive index and critical angle

Refractive index compares the speed of light in two regions

Definition

Refractive index

The ratio of the speed of a wave in one region to its speed in another. Symbol $n$ , and it has no unit because it is a ratio of two speeds.

\frac{v_1}{v_2}

where $v_1$ is the speed in the first region and $v_2$ is the speed in the second.

For a material quoted on its own, the comparison is always with air, so the faster speed goes on top and $n$ comes out greater than $1$ .
Light travels at about $3.0 \times 10^8\ \text{m/s}$ in air and at roughly $2.0 \times 10^8\ \text{m/s}$ in glass, which is where the familiar value $\approx 1.5$ comes from.
A larger refractive index means light is slowed more, so it is bent more sharply at the boundary.
Typical values are about $1.33$ for water, $1.5$ for glass and $2.4$ for diamond, and $n$ is exactly $1$ for a vacuum.
Because it is a ratio of two quantities with the same unit, $n$ never carries a unit of its own.

The same number can be found from the angles instead of the speeds

Measuring the speed of light inside a glass block is far beyond a school laboratory, but the angles are easy to measure with a protractor.
For light passing from air into a material, the refractive index is found from the sines of the two angles.
$\frac{\sin i}{\sin r}$
Here $i$ is the angle in the air and $r$ is the angle in the material, and both are measured from the normal.
Since $r$ is smaller than $i$ on the way in, $sin⁡i\sin i$ is bigger than $sin⁡r\sin r$ and $n$ automatically comes out greater than $1$ .
The equation rearranges to $sin⁡i=nsin⁡r\sin i = n \sin r$ and to $sin⁡r=sin⁡in\sin r = \dfrac{\sin i}{n}$ , so any one of the three quantities can be found from the other two.
If the light is going the other way, out of the material into air, turn the fraction upside down and use the angle in the material on the bottom.

$A ray crossing from air into glass, with the angle of incidence and the angle of refraction both marked from the normal, used to calculate n from sin i over sin r$

Refractive index and critical angle are the same fact in two forms

At the critical angle the light inside the material is at angle $c$ to the normal and the refracted ray grazes the surface at $90∘90^\circ$ .
Putting those angles into the sine relationship, with $sin⁡90∘=1\sin 90^\circ = 1$ , collapses it to a single tidy equation.
$\frac{1}{\sin c}$
Rearranged to find the critical angle itself, it becomes $sin⁡c=1n\sin c = \dfrac{1}{n}$ , so $\sin^{-1}\left(\dfrac{1}{n}\right)$ .
The two quantities move in opposite directions: a bigger refractive index gives a smaller critical angle, so light is trapped more easily.
This is why diamond, with $n = 2.4$ , has a critical angle of only about $24∘24^\circ$ , while water, with $n = 1.33$ , needs about $49∘49^\circ$ .

Common Mistake

Do not calculate $sin⁡(ir)\sin\left(\dfrac{i}{r}\right)$ ; take the sine of each angle first and then divide.
Do not put the angle inside the glass on top when light is going from air into glass, or your $n$ comes out less than $1$ .
Do not give the refractive index a unit, because it is a ratio.
Do not stop at $sin⁡c\sin c$ when the question asks for the critical angle; take the inverse sine to finish the job.
Do not put the slower speed on top in $\dfrac{v_1}{v_2}$ when comparing a material with air, since that inverts the answer.

Exam technique

Choosing which of the three equations to use:
- Two speeds in the question means $\dfrac{v_1}{v_2}$ .
- Two angles at a boundary means $\dfrac{\sin i}{\sin r}$ .
- The words "critical angle", or one angle with a refracted ray along the surface, means $\dfrac{1}{\sin c}$ .
Set your calculator to degrees before you start, since a calculator left in radians turns every one of these into a wrong answer.
You have to recall all three equations, so write the one you are using down before substituting; it often carries a mark of its own.
Show the values of $sin⁡i\sin i$ and $sin⁡r\sin r$ as a middle line, because that is where an error-carried-forward mark can rescue a slip.
Sanity-check every refractive index against $1$ : anything below it means you have the fraction upside down.
Give refractive indices to two significant figures unless the data justifies more, and give angles to the nearest degree.
When a question mixes the two ideas, calculate the critical angle first and then compare it with the angle you were given.

Self review

Define refractive index in terms of speeds, and say why it has no unit.
Write the equation linking $n$ to the angles of incidence and refraction, saying which angle is in which material.
Light enters a liquid at $45∘45^\circ$ and refracts at $32∘32^\circ$ . Calculate the refractive index.
A material has a critical angle of $42∘42^\circ$ . Calculate its refractive index.
Explain why a material with a higher refractive index has a smaller critical angle.

Previous Next

How was this guide?

Teach Genie

Review 3.2.2c Refractive index and critical angle by teaching Genie

Teach it back in your own words, spot gaps, and remember it better.

Start teaching

3.2.2c Refractive index and critical angle

Refractive index compares the speed of light in two regions

Definition

Refractive index

The ratio of the speed of a wave in one region to its speed in another. Symbol $n$ , and it has no unit because it is a ratio of two speeds.

\frac{v_1}{v_2}

where $v_1$ is the speed in the first region and $v_2$ is the speed in the second.

For a material quoted on its own, the comparison is always with air, so the faster speed goes on top and $n$ comes out greater than $1$ .
Light travels at about $3.0 \times 10^8\ \text{m/s}$ in air and at roughly $2.0 \times 10^8\ \text{m/s}$ in glass, which is where the familiar value $\approx 1.5$ comes from.
A larger refractive index means light is slowed more, so it is bent more sharply at the boundary.
Typical values are about $1.33$ for water, $1.5$ for glass and $2.4$ for diamond, and $n$ is exactly $1$ for a vacuum.
Because it is a ratio of two quantities with the same unit, $n$ never carries a unit of its own.

The same number can be found from the angles instead of the speeds

Measuring the speed of light inside a glass block is far beyond a school laboratory, but the angles are easy to measure with a protractor.
For light passing from air into a material, the refractive index is found from the sines of the two angles.
$\frac{\sin i}{\sin r}$
Here $i$ is the angle in the air and $r$ is the angle in the material, and both are measured from the normal.
Since $r$ is smaller than $i$ on the way in, $sin⁡i\sin i$ is bigger than $sin⁡r\sin r$ and $n$ automatically comes out greater than $1$ .
The equation rearranges to $sin⁡i=nsin⁡r\sin i = n \sin r$ and to $sin⁡r=sin⁡in\sin r = \dfrac{\sin i}{n}$ , so any one of the three quantities can be found from the other two.
If the light is going the other way, out of the material into air, turn the fraction upside down and use the angle in the material on the bottom.

$A ray crossing from air into glass, with the angle of incidence and the angle of refraction both marked from the normal, used to calculate n from sin i over sin r$

Refractive index and critical angle are the same fact in two forms

At the critical angle the light inside the material is at angle $c$ to the normal and the refracted ray grazes the surface at $90∘90^\circ$ .
Putting those angles into the sine relationship, with $sin⁡90∘=1\sin 90^\circ = 1$ , collapses it to a single tidy equation.
$\frac{1}{\sin c}$
Rearranged to find the critical angle itself, it becomes $sin⁡c=1n\sin c = \dfrac{1}{n}$ , so $\sin^{-1}\left(\dfrac{1}{n}\right)$ .
The two quantities move in opposite directions: a bigger refractive index gives a smaller critical angle, so light is trapped more easily.
This is why diamond, with $n = 2.4$ , has a critical angle of only about $24∘24^\circ$ , while water, with $n = 1.33$ , needs about $49∘49^\circ$ .

Common Mistake

Do not calculate $sin⁡(ir)\sin\left(\dfrac{i}{r}\right)$ ; take the sine of each angle first and then divide.
Do not put the angle inside the glass on top when light is going from air into glass, or your $n$ comes out less than $1$ .
Do not give the refractive index a unit, because it is a ratio.
Do not stop at $sin⁡c\sin c$ when the question asks for the critical angle; take the inverse sine to finish the job.
Do not put the slower speed on top in $\dfrac{v_1}{v_2}$ when comparing a material with air, since that inverts the answer.

Exam technique

Choosing which of the three equations to use:
- Two speeds in the question means $\dfrac{v_1}{v_2}$ .
- Two angles at a boundary means $\dfrac{\sin i}{\sin r}$ .
- The words "critical angle", or one angle with a refracted ray along the surface, means $\dfrac{1}{\sin c}$ .
Set your calculator to degrees before you start, since a calculator left in radians turns every one of these into a wrong answer.
You have to recall all three equations, so write the one you are using down before substituting; it often carries a mark of its own.
Show the values of $sin⁡i\sin i$ and $sin⁡r\sin r$ as a middle line, because that is where an error-carried-forward mark can rescue a slip.
Sanity-check every refractive index against $1$ : anything below it means you have the fraction upside down.
Give refractive indices to two significant figures unless the data justifies more, and give angles to the nearest degree.
When a question mixes the two ideas, calculate the critical angle first and then compare it with the angle you were given.

Self review

Define refractive index in terms of speeds, and say why it has no unit.
Write the equation linking $n$ to the angles of incidence and refraction, saying which angle is in which material.
Light enters a liquid at $45∘45^\circ$ and refracts at $32∘32^\circ$ . Calculate the refractive index.
A material has a critical angle of $42∘42^\circ$ . Calculate its refractive index.
Explain why a material with a higher refractive index has a smaller critical angle.

Previous Next

How was this guide?

Teach Genie

Review 3.2.2c Refractive index and critical angle by teaching Genie

Teach it back in your own words, spot gaps, and remember it better.

Start teaching

3.2.2c Refractive index and critical angle

Refractive index compares the speed of light in two regions

The same number can be found from the angles instead of the speeds

Refractive index and critical angle are the same fact in two forms

Review 3.2.2c Refractive index and critical angle by teaching Genie

3.2.2c Refractive index and critical angle Revision Guide

3.2.2c Refractive index and critical angle

Refractive index compares the speed of light in two regions

The same number can be found from the angles instead of the speeds

Refractive index and critical angle are the same fact in two forms

Review 3.2.2c Refractive index and critical angle by teaching Genie

3.2.2c Refractive index and critical angle Revision Guide