Refraction of light and its laws

The bent pencil

Put a pencil in a glass of water and look from the side. The pencil seems to be broken at the surface of the water. Put a coin in an empty bowl and move back until you cannot see it. Then ask a friend to pour water into the bowl. The coin comes into view again.

Nothing has happened to the pencil or to the coin. What changed is the path of the light. When light goes from one transparent material to another, its path bends at the boundary. This bending is called refraction.

Why does light bend?

Light travels at different speeds in different materials. It is fastest in a vacuum and a little slower in air. It is slower in water and slower again in glass.

Think of a bicycle that rides from a smooth road on to soft sand, and the front wheel enters the sand at an angle. The wheel that touches the sand first slows down, and the other wheel is still on the road. So the bicycle turns. Light does the same when one side of a beam enters a slower material before the other side does.

The words we use

• The incident ray falls on the boundary, and the refracted ray goes into the second material.
• The normal is a line at right angles to the boundary at the point of incidence.
• The angle of incidence $i$ is the angle between the incident ray and the normal.
• The angle of refraction $r$ is the angle between the refracted ray and the normal.

Which way does the ray bend?

A material in which light travels slower is called optically denser. A material in which light travels faster is optically rarer.

• When light goes from a rarer material to a denser one (air to glass), it bends toward the normal. The angle of refraction is smaller than the angle of incidence: $r < i$.
• When light goes from a denser material to a rarer one (glass to air), it bends away from the normal. The angle of refraction is larger than the angle of incidence: $r > i$.
• A ray that falls along the normal ($i = 0^\circ$) does not bend at all. It goes straight on, but it still changes its speed.

The laws of refraction

1. The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane.
2. For light of one colour and a given pair of materials, the ratio $\dfrac{\sin i}{\sin r}$ is a constant.

The second law is called Snell's law, after the scientist who found it. The constant is the refractive index of the second material with respect to the first. We study it in the next lesson. For light going from air into glass, the constant is about $1.5$.

You should find that $\dfrac{\sin i}{\sin r}$ stays the same for every angle, as long as the materials stay the same. At $0^\circ$ nothing bends. Swap the materials, with the denser one on top, and the ray bends away from the normal.

A worked example

A ray goes from air into glass with an angle of incidence of $40^\circ$. The constant for air to glass is $1.5$. Find the angle of refraction.

1. The law. $\dfrac{\sin i}{\sin r} = 1.5$. We want $r$, so $\sin r = \dfrac{\sin i}{1.5}$.
2. Put in the numbers. $\sin 40^\circ = 0.643$, so $\sin r = \dfrac{0.643}{1.5} = 0.429$.
3. The angle. $r = 25.4^\circ$.

The ray bends toward the normal, as it must when it goes into glass. The answer is smaller than $40^\circ$, so it makes sense.

Practise with new numbers

Try it in Python

The program finds the angle of refraction for several angles of incidence when light goes from air into glass.

Remember these

• Refraction is the bending of light when it goes from one transparent material to another, because its speed changes.
• Into a denser material, the ray bends toward the normal. Into a rarer material, it bends away from the normal.
• A ray along the normal does not bend.
• Snell's law: $\dfrac{\sin i}{\sin r}$ is a constant for a pair of materials.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ray goes from glass into water. Does it bend toward or away from the normal?

2. Why does a pencil in water look broken?

3. Light enters a glass block along the normal. Does it bend? Does its speed change?

4. A ray goes from air into water ($n = 1.33$) at $40^\circ$. Find the angle of refraction.