Refraction through a glass slab

A ray through a window glass

Look at a car outside through a thick window glass, and then move your head from side to side. The car seems to jump a little at the edge of the glass. This is because the light that goes through the glass is shifted sideways. Let us follow one ray.

The ray through a glass slab

A glass slab has two flat faces that are parallel to each other.

1. At the first face, the ray goes from air into glass. It bends toward the normal. The angle of refraction is $r$.
2. Inside the glass, the ray goes straight.
3. At the second face, the ray goes from glass into air. It bends away from the normal. The angle at which it leaves, the angle of emergence $e$, is equal to the angle of incidence $i$.

The two faces are parallel, so the angle of refraction inside the glass is the same $r$ at both faces. The bending at the first face is exactly undone at the second face. So the emergent ray is parallel to the incident ray, and $e = i$.

Lateral displacement

The emergent ray is parallel to the incident ray, but it is not on the same line. The perpendicular distance between the two lines is the lateral displacement $d$.

The lateral displacement is larger when

• the slab is thicker,
• the angle of incidence is larger,
• the refractive index is larger.

When the ray falls along the normal, $d = 0$. A thin sheet of window glass shifts the ray by a very small amount, and we do not notice it.

For a slab of thickness $t$, the lateral displacement is

You do not need to remember this formula for the examination, but the simulation and the program below use it to show the pattern.

You should find that the angle of emergence always equals the angle of incidence. The red bar grows with a thicker slab, a larger angle and a larger index. At $0^\circ$ it vanishes.

A worked example

A ray falls at $60^\circ$ on a glass slab ($n = 1.5$). Find the angle inside the glass and the angle of emergence.

1. First face. $\sin r = \dfrac{\sin 60^\circ}{1.5} = \dfrac{0.866}{1.5} = 0.577$, so $r = 35.3^\circ$.
2. Inside. The ray meets the second face at the same $35.3^\circ$, because the faces are parallel.
3. Second face. Going out, the bend is undone: $e = 60^\circ$, parallel to the incident ray.

Try it in Python

The program finds the lateral displacement in a $6\ \text{cm}$ slab of glass for different angles of incidence.

Why does a pool look shallower?

Look at a coin at the bottom of a tank of water.

A ray from the coin goes up to the surface and leaves the water. It goes from a denser to a rarer material, so it bends away from the normal. When this ray enters your eye, you see the coin at the place where the bent ray seems to come from. This place is higher than the real place of the coin.

So a tank, a pool or a pond looks shallower than it really is. For a person who looks straight down,

For water, $n = 1.33$. A pool that is $3\ \text{m}$ deep seems to be $\dfrac{3\ \text{m}}{1.33} = 2.26\ \text{m}$ deep. This is why you must take care when you step into clear water that looks shallow.

Other examples in daily life

• A straw or a pencil in a glass of water seems to be bent or broken at the surface.
• A fish in a pond seems to be nearer to the surface than it really is. A person who spears fish must aim lower than the place where the fish seems to be.
• A thick glass slab placed on a printed page makes the letters under it seem to be raised.

Practise with new numbers

Remember these

• A ray that goes through a glass slab comes out parallel to the incident ray: $e = i$.
• The emergent ray is shifted sideways by the lateral displacement $d$. It is larger for a thicker slab, a larger angle and a larger refractive index.
• Because of refraction, an object under water seems to be nearer to the surface: $\dfrac{\text{real depth}}{\text{apparent depth}} = n$.

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 falls on a glass slab at $40^\circ$. At what angle does it leave the slab?

2. When is the lateral displacement zero?

3. A coin lies $1.2\ \text{m}$ below the surface of water ($n = 1.33$). How deep does it seem to be, when you look straight down?

4. The same ray goes through a slab twice as thick. What happens to the lateral displacement?