Physics · 3 Waves

3.2.4 Dispersion of light

Before you start

Refraction from 3.2.2: light entering glass from air slows down and bends towards the normal; light leaving glass into air speeds up and bends away from the normal.
The terms normal, angle of incidence and angle of refraction.
The wave equation $v = f λ$ from 3.1.
That the frequency of a wave does not change when it crosses from one material into another; its speed and wavelength change.
Extended: refractive index as a ratio of speeds, and $n = \frac{\sin i}{\sin r}$ , from 3.2.2.

Syllabus objectives covered (3)

Describe the dispersion of light as illustrated by the refraction of white light by a glass prism
Know the traditional seven colours of the visible spectrum in order of frequency and in order of wavelength
Recall that visible light of a single frequency is described as monochromatic

Part 1 of 6

White light is a mixture of colours

Light from the Sun or from a filament lamp looks white, but it is not one kind of light. White light is a mixture of all the colours of the visible spectrum, and each colour is a different range of frequencies.
A glass prism does not add colour to the light. It separates colours that were already in it. You can test this: let only one colour from the spectrum through a narrow slit into a second prism. That colour is bent again, but no new colours appear.
Dispersion is the splitting of white light into its separate colours. In a prism it happens by refraction: the change of direction of light when it crosses into a material where it travels at a different speed.
The band of colours that you see on a screen is called a spectrum. Keep the two words apart: dispersion is the process, and the spectrum is the result.

Part 2 of 6

What happens at each face of the prism

Start from what you already know. At the first face, light goes from air into glass, slows down and bends towards the normal. At the second face, it goes from glass into air, speeds up and bends away from the normal.
Now the new idea. In glass, the colours do not all travel at the same speed. Violet light is slowed the most and red light is slowed the least.
Because the colours slow down by different amounts, they bend by different amounts. Violet is refracted most and red is refracted least. The other colours lie between them, in order.
So the colours start to separate at the first face. At the second face, each colour bends away from the normal by a different amount again, and the colours spread out further.
ray diagram — not drawn yet
A triangular glass prism with its apex at the top; a single ray labelled 'white light' strikes the left face, refracts towards the normal and splits inside the glass into a narrow fan; at the right face the fan refracts away from the normal and spreads wider onto a vertical screen, with red at the top (deviated least) and violet at the bottom (deviated most), and the seven colours labelled in order between them; normals drawn as dashed lines at both faces.
The colours separate at the first face and spread further at the second. Violet changes direction the most and red the least.
On a screen beyond the prism you see the spectrum. Red is at one edge and violet is at the other. The deviation (the total change in direction of the light) is greatest for violet.
A full description has three parts, in this order. One: the light is refracted as it enters the glass. Two: each colour is refracted by a different amount, violet most and red least. Three: the light is refracted again as it leaves, and the colours spread further.
The keyword is refraction. Reflection and diffraction are different effects. If you write one of them where refraction belongs, the point is not credited, because keywords must be used in the correct scientific context.

Part 3 of 6

Why a rectangular block gives no spectrum

Shine white light into a rectangular glass block and you do not see a spectrum, although the glass is the same. The difference is the shape.
In a rectangular block, the two faces are parallel. Each colour bends towards the normal at the first face and bends back by the same amount at the second face. So every colour leaves parallel to the ray that went in.
The colours leave parallel to each other, so they do not fan out. They overlap and the light still looks white.
The two faces of a triangular prism are at an angle to each other. The second refraction adds to the spreading instead of undoing it. That is why a triangular prism is used to show dispersion.
Raindrops disperse sunlight too. Sunlight is refracted as it enters a drop, reflected inside it and refracted again as it leaves. The colours separate and form a rainbow.

Part 4 of 6

The seven colours, in order

The spectrum is a smooth band: the colours shade into each other with no sharp edges. The traditional way to name it uses seven colours: red, orange, yellow, green, blue, indigo, violet.
Learn the order as a word or a sentence. ROY G BIV, or 'Richard Of York Gave Battle In Vain'. The first letters give the colours in order.
Red has the lowest frequency and the longest wavelength. Violet has the highest frequency and the shortest wavelength.
So the two orders are the same list, read in opposite directions. In order of increasing frequency: red, orange, yellow, green, blue, indigo, violet. In order of increasing wavelength: violet, indigo, blue, green, yellow, orange, red.
The two orders must be opposite. In a vacuum, all colours travel at the same speed. In $v = f λ$ , if $v$ is fixed, a higher frequency must go with a shorter wavelength.
$λ = \frac{v}{f}$
So memorise one fact only: red is the low-frequency end. Work out everything else from it.
Visible wavelengths go from about $7 \times 10^{- 7}$ m at the red end to about $4 \times 10^{- 7}$ m at the violet end. The syllabus does not ask you to memorise these values. It asks you to know the order.
Before you answer an order question, check two things. Does it ask for frequency or for wavelength? Does it start from the highest or the lowest? Then write all seven colours. Indigo is one of the traditional seven, so do not leave it out.
Notice the link with dispersion. In glass, the higher the frequency, the more the light is refracted. Violet has the highest frequency and is bent the most.

Part 5 of 6

Extended: a different refractive index for each colour

In 3.2.2 you met refractive index as a ratio of speeds:
$n = \frac{speed of light in air}{speed of light in glass}$
Violet travels more slowly in glass than red does. So the glass has a slightly larger refractive index for violet than for red.
Now use $n = \frac{\sin i}{\sin r}$ , rearranged for the angle of refraction:
$\sin r = \frac{\sin i}{n}$
For the same angle of incidence, a larger $n$ gives a smaller $\sin r$ , and so a smaller angle of refraction. A smaller angle of refraction means the light ends up closer to the normal: it is refracted more. That is the reason violet bends most.
The difference is small. In the third worked example below it is less than half a degree. A narrow beam and a screen some distance from the prism let the small difference in direction become a visible separation.
Neither $n = \frac{\sin i}{\sin r}$ nor the speed ratio is given to you on Paper 4. You must recall both.

Part 6 of 6

Monochromatic light

Monochromatic light is light of a single frequency. 'Mono' means one, and 'chromatic' refers to colour.
Put the word frequency in your definition. A colour name such as 'red' covers a range of frequencies, so 'light of one colour' is less precise than the syllabus wording.
Frequency does not change when light crosses a boundary. So monochromatic light still has a single frequency inside the glass. It has one speed in the glass and one refractive index.
So a beam of monochromatic light is refracted by a prism but it is not dispersed. It bends at both faces and leaves as a single beam of the same colour.
ray diagram — not drawn yet
The same triangular glass prism with its apex at the top; a single ray labelled 'monochromatic light (laser)' strikes the left face, refracts towards the normal, crosses the prism as one ray, refracts away from the normal at the right face and reaches the screen as one small spot; normals drawn as dashed lines at both faces.
One frequency gives one angle of refraction at each face, so the beam changes direction but does not spread.
A laser is the standard example of a source whose light is very nearly monochromatic. White light is the opposite case: a mixture of many frequencies, which is why a prism can spread it into a spectrum.

Worked examples

Example 1. A narrow beam of white light is directed at one face of a triangular glass prism. A spectrum appears on a white screen on the far side of the prism. Describe what happens to the light as it passes through the prism.

Show the working

Start at the first face. Name the effect and give the direction: the light slows down as it enters the glass, so it is refracted towards the normal.
Next, say why the colours separate: each colour is refracted by a different amount. Name the two ends, because that shows you know which way the spectrum goes: violet most, red least.
Then the second face: the light is refracted again as it leaves the glass, away from the normal, and the colours spread out further.
Finish with the result and the name of the process: a spectrum from red to violet on the screen, produced by dispersion.
Read your answer once for the keyword. It must say 'refracted', not 'reflected'.

Answer: The white light is refracted (towards the normal) as it enters the glass. Each colour is refracted by a different amount, violet most and red least, so the colours separate. At the second face the light is refracted again (away from the normal) and the colours spread out further. A spectrum from red to violet forms on the screen. This splitting of white light is called dispersion.

Example 2. Two beams of light, A and B, travel in a vacuum at $3.0 \times 10^{8}$ m/s. Beam A has a frequency of $4.6 \times 10^{14}$ Hz. Beam B has a frequency of $6.8 \times 10^{14}$ Hz. Calculate the wavelength of each beam, and deduce which beam is nearer the violet end of the visible spectrum.

Show the working

You know the speed and the frequency, and you want the wavelength. Recall $v = f λ$ and divide both sides by $f$ , because $f$ is what multiplies $λ$ .
$λ = \frac{v}{f}$
Beam A.
$λ_{A} = \frac{3.0 \times 10^{8}}{4.6 \times 10^{14}} = 6.5 \times 10^{- 7} m$
Beam B.
$λ_{B} = \frac{3.0 \times 10^{8}}{6.8 \times 10^{14}} = 4.4 \times 10^{- 7} m$
Violet is the end with the highest frequency and the shortest wavelength. Beam B has the higher frequency and the shorter wavelength, so B is nearer the violet end.
The two tests agree, which is a good check. Higher frequency and shorter wavelength always go together here, because the speed is the same for both beams.
It follows that a glass prism would refract beam B more than beam A.

Answer: $λ_{A} = 6.5 \times 10^{- 7}$ m and $λ_{B} = 4.4 \times 10^{- 7}$ m. Beam B is nearer the violet end, because it has the higher frequency and the shorter wavelength.

Example 3. Extended. For a certain glass, the refractive index is $1.51$ for red light and $1.53$ for violet light. A narrow beam of white light passes from air into the glass at an angle of incidence of $50$ °. Calculate the angle of refraction for the red light and for the violet light, and state which colour is refracted more.

Show the working

Recall $n = \frac{\sin i}{\sin r}$ . It is not given on Paper 4. You want $r$ , so rearrange for $\sin r$ .
$\sin r = \frac{\sin i}{n}$
Red light: substitute $i = 50$ ° and $n = 1.51$ .
$\sin r = \frac{\sin 50^{\circ}}{1.51} = 0.507 \Rightarrow r = {30.5}^{\circ}$
Violet light: the same angle of incidence, but $n = 1.53$ .
$\sin r = \frac{\sin 50^{\circ}}{1.53} = 0.501 \Rightarrow r = {30.0}^{\circ}$
Violet has the smaller angle of refraction, so it ends up closer to the normal. It is refracted more.
Notice how small the difference is: less than half a degree. The colours separate only slightly at the first face. The second face of the prism spreads them further.

Answer: Red: $30.5$ °. Violet: $30.0$ °. Violet light is refracted more.

Example 4. A student shines a beam of white light at one face of a glass prism. She then shines the beam from a green laser at the same face, at the same angle. Compare the beams that leave the prism in the two cases.

Show the working

A comparison needs a statement about both beams. Give a similarity and a difference.
Similarity: both beams are refracted at both faces of the prism, so both leave in a different direction from the one in which they entered.
Difference: the white light is dispersed into a spectrum of colours. The laser beam leaves as a single green beam and is not dispersed.
Give the reason for the difference. Laser light is very nearly monochromatic: it has a single frequency, so it has one refractive index and one angle of refraction. White light is a mixture of frequencies, and each one is refracted by a different amount.

Answer: Both beams are refracted and change direction. The white light is dispersed into a spectrum from red to violet. The laser beam is not dispersed and leaves as a single green beam, because laser light is (very nearly) monochromatic: it has a single frequency.

Where people go wrong

Mistake: Saying that red light is refracted most by a prism because it has the longest wavelength.

Instead: Violet is refracted most and red is refracted least.

Violet light is slowed down most in glass, so the glass has its largest refractive index for violet, and violet changes direction the most.

Mistake: Writing 'reflection' or 'diffraction' as the cause of dispersion in a prism.

Instead: Dispersion in a prism is caused by refraction.

The colours separate because each one changes direction by a different amount when its speed changes at a boundary. That change of direction is refraction.

Mistake: Thinking that the prism adds the colours to white light.

Instead: The prism separates colours that are already in the white light.

If one colour from the spectrum goes through a second prism, it is bent again but no new colours appear. The prism only sorts the light it receives.

Mistake: Giving the colours in order of frequency when the question asks for wavelength, or starting from the wrong end.

Instead: Red has the lowest frequency and the longest wavelength. Reverse the list when you change from frequency to wavelength.

All colours have the same speed in a vacuum, so from $v = f λ$ the higher the frequency, the shorter the wavelength.

Mistake: Defining monochromatic light as 'light of one colour'.

Instead: Monochromatic light is light of a single frequency.

A colour name such as 'green' covers a range of frequencies. The syllabus definition is about one frequency.

Check yourself

What is dispersion, and which effect causes it in a glass prism?
Answer
Dispersion is the splitting of white light into its separate colours (a spectrum). In a prism it is caused by refraction.
Which colour is refracted most by a glass prism, and which colour is refracted least?
Answer
Violet is refracted most. Red is refracted least.
Give the seven traditional colours of the visible spectrum in order of increasing frequency.
Answer
Red, orange, yellow, green, blue, indigo, violet.
Give the same seven colours in order of increasing wavelength.
Answer
Violet, indigo, blue, green, yellow, orange, red.
Light X has a longer wavelength than yellow light. Which of the traditional seven colours could light X be?
Answer
Orange or red. Longer wavelength means towards the red end of the spectrum.
Define monochromatic light.
Answer
Light of a single frequency.
A beam of monochromatic light passes through a glass prism. Is it refracted? Is it dispersed?
Answer
It is refracted at both faces, so it changes direction. It is not dispersed: it has only one frequency, so it leaves as a single beam of one colour.
Why does a rectangular glass block not produce a spectrum from a beam of white light?
Answer
Its faces are parallel. Each colour bends back at the second face by the same amount that it bent at the first face, so all the colours leave parallel to each other. They overlap and the light still looks white.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A student directs a narrow beam of white light at one face of a triangular glass prism. A band of colours appears on a white screen on the other side of the prism.

(a) State the name given to the splitting of white light into its colours. 1 mark No calculator

Mark scheme

B1 dispersion; insufficient: refraction (this is the cause, not the name)

(b) Describe what happens to the light as it passes into, through and out of the prism. 3 marks No calculator

Mark scheme

B1 the light is refracted / bent (towards the normal) as it enters the glass; CON: reflected
B1 different colours are refracted by different amounts / violet refracted most and red least; accept: different colours travel at different speeds in glass / glass has a different refractive index for each colour
B1 the light is refracted again (away from the normal) as it leaves the glass, so the colours spread out further

(c) Identify the colour that is refracted most by the prism and the colour that is refracted least. 2 marks No calculator

Mark scheme

B1 violet refracted most
B1 red refracted least

Question 2. The visible spectrum is traditionally divided into seven colours. A laser produces a narrow beam of monochromatic light of frequency $5.6 \times 10^{14}$ Hz.

(a) State the seven colours of the visible spectrum in order of increasing wavelength, starting with the colour of shortest wavelength. 2 marks No calculator

Mark scheme

B1 all seven colours named, including indigo
B1 in the order violet, indigo, blue, green, yellow, orange, red

(b) State what is meant by monochromatic light. 1 mark No calculator

Mark scheme

B1 light of a single frequency; accept: single wavelength (owtte); insufficient: 'light of one colour' with no reference to frequency

(c) The speed of light in air is $3.0 \times 10^{8}$ m/s. Calculate the wavelength of the laser light in air. 3 marks

Mark scheme

C1 v = fλ recalled, or λ = v / f
C1 λ = 3.0 × 10⁸ / 5.6 × 10¹⁴
A1 5.4 × 10⁻⁷ m (accept 5.36 × 10⁻⁷ m, 540 nm or any correct equivalent); unit required

(d) The laser beam is directed at one face of a glass prism. Predict what happens to the beam as it passes through the prism. 2 marks No calculator

Mark scheme

B1 the beam is refracted / changes direction (at each face)
B1 it is not dispersed / does not split into colours / leaves as a single beam of one colour

Question 3. Extended. For a certain glass, the refractive index for red light is $1.50$ and the refractive index for violet light is $1.53$ . A narrow beam of white light in air strikes the surface of the glass at an angle of incidence of $45$ °.

(a) Calculate the angle of refraction of the red light. 3 marks

Mark scheme

C1 n = sin i / sin r recalled
C1 sin r = sin 45° / 1.50 (= 0.471)
A1 28° (accept 28.1°); unit required

(b) The angle of refraction of the violet light is $27.5$ °. Explain how the two angles of refraction show that the white light is dispersed by the glass. 2 marks No calculator

Mark scheme

B1 the angles of refraction are different, so red and violet light travel in different directions in the glass / separate; ecf from (a)
B1 violet has the smaller angle of refraction, so it is refracted more (closer to the normal) than red; ORA

How this is examined

Describing what a prism does to white light, face by face: a Describe worth two or three marks about a narrow beam of white light passing through a triangular glass prism onto a screen; a version can ask you to complete a ray diagram instead Three separate points: 'the light is refracted (towards the normal) as it enters the glass'; 'each colour is refracted by a different amount, violet most and red least'; 'it is refracted again as it leaves, so the colours spread further'. The word refraction must be there, not reflection.
Extended: using a different refractive index for each colour: a Calculate with two refractive indices (one for red, one for violet) and one angle of incidence, followed by an Explain of what the two answers show Recalling $n = \frac{\sin i}{\sin r}$ (it is not given on Paper 4), rearranging to $\sin r = \frac{\sin i}{n}$ , and giving each angle in degrees. Then: 'the larger refractive index gives the smaller angle of refraction, so violet is refracted more and the colours separate'.
Putting the seven colours in order by frequency or by wavelength: a State of one or two marks that asks for the colours in order, or a question that asks which of two named colours has the higher frequency or the longer wavelength All seven colours, indigo included, in the right direction. Increasing frequency: red, orange, yellow, green, blue, indigo, violet. Increasing wavelength: the same list reversed. Check which quantity and which end the question asks for before you write.
What monochromatic means, and what a prism does to it: a one-mark State asking what is meant by monochromatic; a Predict about a laser beam passing through a prism can follow it The definition 'light of a single frequency'. For the prediction: 'the beam is refracted (changes direction)' and 'it is not dispersed: it leaves as a single beam of one colour'.
Which colour bends most, and why: an Identify or State of one or two marks about the two ends of the spectrum produced by a prism 'Violet is refracted most' and 'red is refracted least': state both ends, because each is a separate mark. If a reason is asked for: 'violet travels more slowly in glass than red'.

Words to know

Dispersion: The splitting of white light into its separate colours. In a glass prism it is caused by refraction, because each colour is refracted by a different amount.
Spectrum: The band of colours produced when white light is dispersed, from red at one end to violet at the other.
White light: A mixture of all the colours of the visible spectrum.
Visible spectrum: The colours of light the eye can see. Traditionally seven: red, orange, yellow, green, blue, indigo, violet, in order of increasing frequency.
Monochromatic light: Light of a single frequency.
Refraction: The change of direction of light when it crosses into a material where it travels at a different speed.
Refractive index (Extended): The ratio of the speed of light in air (or a vacuum) to its speed in the material; also $n = \frac{\sin i}{\sin r}$ . In glass it is slightly larger for violet light than for red light.
Deviation: The total change in direction of a ray of light as it passes through a prism. It is greatest for violet and least for red.

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