S4.10.5 Chapter 5

Reflection and refraction

A ray of light meets a mirror and makes an angle of 20° with its surface. At what angle was the ray arriving, in the way scientists measure it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A 20°, because that is the angle the ray makes.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 70°, because scientists measure from the line at right angles to the mirror.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 110°, because the ray bends back by the same amount.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw ray diagrams for reflection and refraction at a boundary
Explain total internal reflection and its use in optical fibres

Light travels in straight lines called rays. A ray is drawn as a straight line with an arrow that shows the way the light goes. This chapter follows rays when they meet a mirror, and when they meet the edge of glass or water.

Every angle is measured from the normal. The normal is a dashed line at 90° to the surface, drawn at the point where the ray meets it. The ray that arrives is the incident ray. The angle of incidence is the angle between the incident ray and the normal. The ray that leaves a mirror is the reflected ray. The angle of reflection is the angle between that ray and the normal.

The law of reflection: the angle of incidence equals the angle of reflection. Both are measured from the normal.

The law of reflection says two things. The angle of incidence equals the angle of reflection. And the incident ray, the normal and the reflected ray all lie in one flat plane. On a diagram, that plane is the page.

A very common slip is to measure the angle from the mirror, not from the normal. If a question says a ray makes 30° with the mirror, the angle of incidence is not 30°. It is 90° − 30° = 60°. Always draw the normal first.

The law holds for every ray on every surface. A rough surface, such as paper, is made of tiny patches that face different ways. Each ray obeys the law at its own patch, but the patches point in many directions, so the light scatters. A smooth mirror keeps parallel rays parallel, and you see an image.

Now look at your own image in a flat mirror. Rays from your face reflect from the mirror and reach your eye. Your eye follows each ray back in a straight line, to a point behind the mirror, where the image seems to be.

The image is as far behind the mirror as the object is in front. The rays only seem to come from it.

The image in a flat mirror is the same size as the object. It is as far behind the mirror as the object is in front. It is upright. It is left and right reversed. It is a virtual image: light does not really pass through it, so you cannot catch it on a screen.

Now turn to refraction. Light travels at 3 × 10⁸ m/s in a vacuum and almost as fast in air. In glass and in water it is slower. The refractive index n of a material is the speed of light in a vacuum divided by the speed in the material.

Refractive index and the speed of light in m/s. Ordinary glass is about 1.5. Other kinds of glass differ a little.

When a ray goes into a material where light is slower, it bends towards the normal. When it goes into a material where light is faster, it bends away from the normal. A ray that travels along the normal does not bend, but it still changes speed.

Air to glass: the light slows down and bends towards the normal.

Glass to air: the light speeds up and bends away from the normal.

How much does a ray bend? Snell's law gives the answer. For light going from air into a material of index n, sin i ÷ sin r = n. Here i is the angle of incidence and r is the angle of refraction, both from the normal. The sine of an angle is the ratio of two sides of a right angled triangle.

In general, light goes from a material of index n₁ into one of index n₂. Then n₁ × sin i = n₂ × sin r. In every case the ray bends towards the normal when n goes up, and away when n goes down.

A ray enters water at 45° to the normal.

Worked example

A ray goes from air into water, whose refractive index is 1.33. The angle of incidence is 45°. Find the angle of refraction.

Use sin i ÷ sin r = n, so sin r = sin i ÷ n.
sin 45° = 0.7071, so sin r = 0.7071 ÷ 1.33 = 0.5317.
The angle whose sine is 0.5317 is r = 32.1°.

Answer: 32.1°

A rectangular block of glass shows both bends. The ray bends towards the normal as it goes in, and away from the normal as it comes out. The two bends are equal and opposite, so the ray that leaves is parallel to the ray that went in. It is moved sideways.

Through a glass block, the ray leaves parallel to the way it came in. It is at 40° to the normal again, but shifted sideways.

Now go the other way. A ray inside glass or water meets the edge with air. It bends away from the normal. As the angle of incidence grows, the bent ray swings closer to the edge. At one angle it runs along the edge. That angle is the critical angle, c.

At the critical angle, the angle of refraction is 90°. So sin c = 1 ÷ n for light going from a material of index n into air. For glass of index 1.5, c is about 42°.

Past the critical angle, no light can leave. All of it is reflected back inside, by the law of reflection. This is total internal reflection.

Three rays in glass. Under the critical angle the light goes out. At it, the light runs along the edge. Beyond it, all the light is reflected.

Total internal reflection has two conditions. The light must go from a higher n to a lower n, such as glass to air. And the angle of incidence must be greater than the critical angle. Light going from air into glass can never be totally reflected.

Even when light does cross an edge, a little of it is also reflected. The diagrams leave that weak reflection out. Total internal reflection is different: there, all the light is reflected.

A ray in water at 60° to the normal is past the critical angle. All of it is reflected.

Worked example

Water has a refractive index of 1.33. What is the critical angle for light going from water into air?

At the critical angle, sin c = 1 ÷ n.
sin c = 1 ÷ 1.33 = 0.7519.
The angle whose sine is 0.7519 is c = 48.8°.

Answer: 48.8°

Diamond has a very high index. Its critical angle is small, so light that enters a cut diamond is reflected inside again and again before it comes out. This is part of why diamonds sparkle.

An optical fibre is a thin, flexible thread of glass. Light that enters one end meets the wall at an angle greater than the critical angle. It is totally reflected, again and again, and follows the fibre round bends to the far end.

A ray follows a bent fibre by total internal reflection. Every meeting with the wall is past the critical angle.

A real fibre has a core of glass inside a coating of glass of slightly lower index. The total internal reflection happens where the core meets the coating. A fibre cannot be bent too sharply. The light would then meet the wall at less than the critical angle and leak out.

Fibres carry telephone calls and the internet as pulses of light, which you meet in the last chapter. Doctors also use bundles of fibres in a thin tube, an endoscope, to look inside the body without a large cut.

Try it

Test the law of reflection

You need: a small flat mirror, a torch, two cards held with a narrow gap between them, a sheet of white paper, a pencil and a ruler, something to hold the mirror upright, such as a lump of clay or a book.

Be safe: Never shine the torch into anyone's eyes, and never use the Sun or a mirror in sunlight for this. Tape the edges of a glass mirror, which can cut. Keep the mirror and the clay away from young children.

Draw a straight line across the paper. Stand the mirror upright on the line, with its shiny face towards you. Draw a dashed line at right angles to the mirror, as the normal. Mark where it meets the mirror.
Lay the torch behind the two cards. A thin beam should come through the gap. It should cross the paper at a slant and meet the mirror at the foot of the normal. Keep the beam low and flat on the paper.
Mark two small dots on the beam that goes to the mirror, and two on the beam that leaves it. Take away the mirror and the torch, and draw lines through the dots.
Mark a point on each line exactly 10 cm from the mirror point. Measure how far each of the two points is from the normal.
If the two distances are equal, the two angles are equal.

What you will see: The two distances come out almost the same, within a millimetre or so. A point 10 cm along a ray is 10 cm times the sine of the angle away from the normal. So equal distances mean equal angles.

If you cannot try it, this is what happens.

Try it

The silver surface of water

You need: a clear glass or jar, filled with water, a table.

Be safe: Use a glass or jar that cannot tip over, away from the edge of the table. If you have a clear plastic cup, use that. Wipe up any spills.

Stand the glass of water on a table, away from the edge.
Crouch so that your eye is a little lower than the top of the water. Look up at the underside of the water surface, through the side of the glass.
Look steeply upwards. Then look at a slant, nearly along the underside of the surface.

What you will see: Looking steeply up, you see through the surface to the ceiling. At a shallow slant, the surface looks like a silver mirror. You see the bottom of the glass or the objects in the water. This is total internal reflection.

If you cannot try it, this is what happens.

A wrong idea many people have

A ray that makes 30° with a mirror has an angle of incidence of 30°.

Why it feels right: The angle you can see is the one between the ray and the mirror. It is the first angle we think to measure on a flat surface.

The test that breaks it: Draw the normal first. The ray makes 30° with the mirror, and the mirror makes 90° with the normal. So the ray makes 90° − 30° = 60° with the normal. The law of reflection uses the angle from the normal.

What is true: Angles of incidence, reflection and refraction are always measured from the normal, never from the surface.

A wrong idea many people have

Light always bends towards the normal when it changes material.

Why it feels right: Our first picture of refraction is a ray going into water or glass, and that ray bends towards the normal.

The test that breaks it: Look at the ray going from glass into air. It bends away from the normal, and the angle in the air is bigger than the angle in the glass.

What is true: A ray bends towards the normal when it goes into a material where light is slower. It bends away from the normal when it goes into a material where light is faster.

A wrong idea many people have

Total internal reflection can happen when light goes from air into glass, if the angle is large enough.

Why it feels right: A large angle seems to be all it takes, and light reflects from a glass window at a slant.

The test that breaks it: From air into glass, sin r = sin i ÷ 1.5. This is always less than 1, so a refracted ray always exists. A window at a slant also reflects a little, but some light always goes in.

What is true: Total internal reflection needs two things. The light goes from a higher index to a lower index. The angle of incidence is greater than the critical angle.

Key words

angle of incidence: the angle between the incident ray and the normal.
angle of reflection: the angle between the reflected ray and the normal.
virtual image: an image made where rays seem to come from, so it cannot be caught on a screen.
refractive index: the speed of light in a vacuum divided by its speed in the material. It is written n.
critical angle: the angle of incidence in a denser material at which the refracted ray runs along the edge. sin c = 1 ÷ n.
total internal reflection: all the light reflected inside a material, when it meets the edge beyond the critical angle.
optical fibre: a thin glass thread that carries light along it by total internal reflection.

Back to your guess

The answer is B: 70°, because scientists measure from the line at right angles to the mirror.

Scientists measure the angle from the normal, the line at 90° to the mirror. A ray that makes 20° with the surface makes 90° − 20° = 70° with the normal. That is the angle of incidence.

Keep it

Copy this into your exercise book.

Angles are from the normal. Reflection: i = r. Slower material: bends towards the normal. Total internal reflection: denser to less dense, beyond the critical angle, sin c = 1 ÷ n.

Draw: A glass edge with three rays: one bending out, one along the edge, one reflected, and the normal as a dashed line.