Refraction of Light

Lessons

Senior 3 · Physics

Refraction of Light

5 lessons

Lesson 5 of 5

Refraction Through a Prism, Dispersion and Lens Defects

30 min

Colours from a glass of water

On a sunny morning, a glass of water standing on a windowsill can throw a small band of colours onto the floor or wall: red at one edge, violet at the other. Nobody painted those colours. They were already in the sunlight, and the curved water and glass simply sorted them. In 1666 Isaac Newton did the same with a glass prism, and it changed how scientists understood light. This lesson follows a ray through a prism, explains the colours, and shows why the same effect blurs pictures made by simple lenses.

Refraction through a triangular prism

A prism here means a block of glass with two flat faces at an angle to each other. The angle between them is the refracting angle $A$ (the apex angle). A ray refracts twice: towards the normal on entering, and away from the normal on leaving. Because the two faces are not parallel, the ray does not come out parallel to the incident ray, as it did with the rectangular block in lesson 1. It is turned towards the base of the prism.

The angle between the direction of the incident ray and the direction of the emergent ray is the angle of deviation $D$ . Geometry of the triangle formed by the two normals gives two useful results:

r_1 + r_2 = A \qquad\qquad D = i + e - A

Definition

Refracting angle $A$ of a prism: the angle between the two faces through which light passes. Example: 60° for an equilateral prism.

Angle of deviation $D$ : the angle between the incident ray (extended forward) and the emergent ray. Example: 38.5° in the diagram above.

Worked example 1: a ray through a prism

A ray strikes one face of an equilateral glass prism ( $A = 60^\circ$ , $n = 1.50$ ) at an angle of incidence of 40°. Find the angle of emergence and the angle of deviation.

First face: $\sin r_1 = \frac{\sin 40^\circ}{1.50} = \frac{0.643}{1.50} = 0.429$ , so $r_1 = 25.4^\circ$ .

Inside the prism: $r_2 = A - r_1 = 60^\circ - 25.4^\circ = 34.6^\circ$ . This is less than the critical angle (41.8°), so the ray can leave.

Second face: $\sin e = 1.50 \times \sin 34.6^\circ = 1.50 \times 0.568 = 0.852$ , so $e = 58.5^\circ$ .

D = i + e - A = 40^\circ + 58.5^\circ - 60^\circ = 38.5^\circ

Minimum deviation

If you turn the prism slowly while the ray stays fixed, the deviation first falls and then rises again. It is smallest, $D_{\min}$ , when the ray passes through the prism symmetrically, with $i = e$ and $r_1 = r_2 = \frac{A}{2}$ . Measuring $D_{\min}$ gives an accurate way to find the refractive index of the glass:

n = \frac{\sin\left(\frac{A + D_{\min}}{2}\right)}{\sin\left(\frac{A}{2}\right)}

For the prism above ( $n = 1.50$ ), $D_{\min} = 37.2^\circ$ , just less than the 38.5° found at $i = 40^\circ$ .

Worked example 2: refractive index from minimum deviation

An equilateral prism gives a minimum deviation of 40.0°. Find the refractive index of its glass.

n = \frac{\sin\left(\frac{60^\circ + 40^\circ}{2}\right)}{\sin 30^\circ} = \frac{\sin 50^\circ}{\sin 30^\circ} = \frac{0.766}{0.500} = 1.53

Dispersion: white light is a mixture

Send a narrow beam of white light into a prism and a band of colours, a spectrum, appears on a screen beyond it: red, orange, yellow, green, blue, indigo and violet. Red is deviated least and violet most.

The reason is that the refractive index of glass is slightly different for each colour. Violet light travels a little more slowly in glass than red light, so glass has a larger refractive index for violet and bends it more.

Colour	Red	Orange	Yellow	Green	Blue	Violet
$n$ for crown glass	1.512	1.514	1.518	1.519	1.524	1.530

In a vacuum all colours travel at the same speed, so a vacuum causes no dispersion. Newton showed that the prism only separates colours that are already in white light: when he passed one colour from the spectrum through a second prism, it bent again but stayed the same colour, and a second prism turned the other way could recombine the spectrum into white light.

Definition

Dispersion: the splitting of white light into its colours, because the refractive index of a medium is different for different colours. Example: a prism spreading sunlight into a spectrum.

Spectrum (visible): the band of colours, red, orange, yellow, green, blue, indigo and violet, obtained by dispersing white light.

Worked example 3: how far apart are red and violet?

White light strikes a crown glass prism ( $A = 60^\circ$ ) at $i = 40^\circ$ . Using $n_{\text{red}} = 1.512$ and $n_{\text{violet}} = 1.530$ , the method of worked example 1 gives:

	$r_{1}$	$r_{2}$	$e$	$D$
Red	25.2°	34.8°	59.8°	39.8°
Violet	24.8°	35.2°	61.8°	41.8°

Violet is deviated about 2° more than red. That is why the spectrum is narrow close to the prism and needs a distant screen to spread out.

The rainbow

A rainbow is dispersion by millions of raindrops. Sunlight enters each drop and is refracted and dispersed, reflects once off the back of the drop, and is refracted again as it leaves. Each colour leaves at a slightly different angle: red at about 42° to the line from the Sun through your head, violet at about 40°. So you see a rainbow only when the Sun is behind you and rain is falling in front of you, with red on the outside of the bow and violet on the inside.

Lens defects and how they are corrected

A single lens with spherical surfaces does not form a perfect image. Two defects matter most.

Spherical aberration

The left diagram was traced ray by ray with Snell's law for a plano-convex lens of $n = 1.50$ . Rays passing near the edge of the lens are bent too much and cross the axis closer to the lens than rays near the centre. There is no single sharp focus, and the image is blurred. It is reduced by:

a stop (a ring-shaped barrier, like the aperture in a camera) that blocks the edge rays, at the cost of a dimmer image;
lenses whose surfaces are specially shaped, not spherical (aspheric lenses), or combinations of lenses.

Chromatic aberration

A lens is like a ring of small prisms, so it disperses light too. Violet is refracted more than red and comes to a focus closer to the lens. For a crown glass lens with a focal length of 20.0 cm for red light, the violet focus is at about 19.3 cm. The image has coloured fringes. It is corrected with an achromatic doublet: a converging lens of crown glass cemented to a weaker diverging lens of flint glass. Flint glass disperses colours more strongly, so the weak diverging lens cancels the colour spread of the converging lens while the pair as a whole still converges. Curved mirrors reflect all colours alike, so they have no chromatic aberration; this is one reason the largest telescopes use mirrors.

Definition

Spherical aberration: blurring of an image because rays through the edge of a spherical lens focus closer to the lens than rays near the axis.

Chromatic aberration: coloured fringes on an image because a lens refracts different colours by different amounts, so each colour has its own focus.

Applications of refraction: a summary

Application	Effect used	Lesson
Clear water looks shallower; aiming at a fish	Refraction at a flat surface (apparent depth)	1
Optical fibres for internet, telephone and endoscopes	Total internal reflection	2
Prism periscopes, binoculars, bicycle reflectors	Total internal reflection in 45° prisms	2
Sparkle of diamonds; mirages on hot roads	Small critical angle; total internal reflection in hot air	2
Camera, eye, projector, magnifying glass	Images formed by converging lenses	3 and 4
Spectacles	Converging and diverging lenses of chosen power	4
Spectrum of white light; the rainbow	Dispersion	5
Achromatic lenses in cameras and microscopes	Correcting chromatic aberration	5

Warning

Common mistakes

Saying the prism adds colour to white light. It only separates colours already present.
Saying red is bent most. Violet has the larger refractive index and is deviated most; remember "violet = very bent".
Forgetting the critical angle at the second face: if $r_{2}$ is larger than $c$ , no light leaves that face.
Using $D = i + e + A$ . The refracting angle is subtracted: $D = i + e - A$ .

Tip

ROY G BIV. The spectrum, from least to most deviated, spells the name "Roy G. Biv": Red, Orange, Yellow, Green, Blue, Indigo, Violet.

Newton's Prism Experiment (MIT K12 Videos). Watch how a second prism shows that the colours were already in the white light, and that a single colour cannot be split again.

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Practice

List the colours of the visible spectrum in order, starting with the one deviated least by a prism.
A ray strikes an equilateral glass prism ( $A = 60^\circ$ , $n = 1.50$ ) at an angle of incidence of 30°. Find $r_{1}$ , $r_{2}$ , the angle of emergence and the angle of deviation.
For the same prism, a ray strikes the first face at 20°. Show that it cannot leave through the second face.
An equilateral prism gives a minimum deviation of 48.0°. Find the refractive index of its glass.
Find the speed of violet light in crown glass ( $n = 1.530$ ), and explain why it differs from the speed of red light.
Explain why a simple lens forms images with coloured edges, and describe how the defect is corrected.
A camera lens gives sharper pictures when its aperture is made smaller. Which lens defect does this reduce, and what is the disadvantage?
Explain why a rainbow is seen in the afternoon only when you face away from the Sun.

Show answersHide answers

Red, orange, yellow, green, blue, indigo, violet.
$\sin r_1 = \frac{0.500}{1.50}$ , so $r_1 = 19.5^\circ$ ; $r_2 = 60^\circ - 19.5^\circ = 40.5^\circ$ ; $\sin e = 1.50 \times \sin 40.5^\circ = 0.975$ , so $e = 77.1^\circ$ ; $D = 30^\circ + 77.1^\circ - 60^\circ = 47.1^\circ$ .
$r_1 = 13.2^\circ$ , so $r_2 = 60^\circ - 13.2^\circ = 46.8^\circ$ . The critical angle is 41.8°, and 46.8° is larger, so the ray is totally internally reflected at the second face. (Check: $1.50 \times \sin 46.8^\circ = 1.09$ , more than 1.)
$n = \frac{\sin 54^\circ}{\sin 30^\circ} = \frac{0.809}{0.500} = 1.62$ .
$v = \frac{3.00 \times 10^{8}}{1.530} = 1.96 \times 10^{8}\,\mathrm{m\,s^{-1}}$ . Red has a smaller refractive index (1.512), so it travels slightly faster ( $1.98 \times 10^{8}\,\mathrm{m\,s^{-1}}$ ).
Chromatic aberration: the lens refracts violet more than red, so the colours focus at different points. It is corrected with an achromatic doublet: a converging crown glass lens cemented to a weaker diverging flint glass lens.
Spherical aberration: the smaller aperture blocks the edge rays. The disadvantage is that less light gets in, so the image is dimmer (a longer exposure is needed).
Rainbow light leaves the drops at about 40° to 42° from the line from the Sun through your eye, travelling back towards the side the Sun shines from. So the drops that send it to you must be in front of you with the Sun behind you; in the afternoon the Sun is in the west, so the rainbow appears in the east.

Key points

A prism deviates light towards its base: $r_{1} + r_{2} = A$ and $D = i + e - A$ ; at minimum deviation $n = \frac{\sin\left(\frac{A + D_{\min}}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ .
Dispersion: glass has a larger refractive index for violet than for red, so a prism spreads white light into a spectrum, violet deviated most.
A rainbow is dispersion, refraction and reflection in raindrops, with the Sun behind you.
Spherical aberration is reduced with a stop or aspheric lenses; chromatic aberration is corrected with an achromatic doublet.

1Colours from a glass of water

2Refraction through a triangular prism

3Minimum deviation

4Dispersion: white light is a mixture

5The rainbow

6Lens defects and how they are corrected

Spherical aberration

Chromatic aberration

7Applications of refraction: a summary

Colours from a glass of water

Refraction through a triangular prism

Minimum deviation

Dispersion: white light is a mixture

The rainbow

Lens defects and how they are corrected

Applications of refraction: a summary