Refraction through a glass prism

A piece of glass that bends light the same way twice

In the last chapter, we saw that a ray that goes through a glass slab comes out parallel to the first ray. The two faces of a slab are parallel, so the bending at the second face undoes the bending at the first.

What if the two faces are not parallel? Then the two bends add up. This is what a prism does.

The glass prism

A glass prism is a piece of glass with two triangular ends and three rectangular faces. We study it from the side, so that it looks like a triangle.

• The two faces that the light goes through are the refracting surfaces.
• The angle between the two refracting surfaces is the angle of the prism, $A$. For the common prism, it is $60^\circ$.
• The third face is the base.

The path of a ray

1. At the first surface, the ray goes from air into glass. It bends toward the normal. This is the first refraction, $i$ to $r_1$.
2. Inside, the ray goes straight.
3. At the second surface, the ray goes from glass into air. It bends away from the normal. This is the second refraction, $r_2$ to $e$.

The normals of the two surfaces are not parallel, so the ray does not get back to its first direction. The ray comes out bent toward the base of the prism.

The angle between the incident ray (carried forward) and the emergent ray (traced back) is the angle of deviation, $\delta$.

The rule for the angles

For a prism of angle $A$:

The second rule shows what the deviation depends on: the angle of the prism, the angle of incidence and the refractive index.

You should find that the ray always bends toward the base. As you raise the angle of incidence from $30^\circ$, the deviation first falls, then rises again. It is smallest when the ray inside runs level with the base. Make the glass denser or the prism sharper, and the deviation grows.

How does the deviation change with the angle of incidence?

For a small angle of incidence, the deviation is large. As the angle becomes larger, the deviation becomes smaller, until it reaches a minimum value. Then it grows again. At the minimum deviation, the ray goes through the prism symmetrically: $i = e$ and $r_1 = r_2 = A/2$.

A worked example

A ray falls at $55^\circ$ on a $60^\circ$ prism of glass ($n = 1.5$). Find the angles inside, the angle of emergence and the deviation.

1. First face. $\sin r_1 = \dfrac{\sin 55^\circ}{1.5} = \dfrac{0.819}{1.5} = 0.546$, so $r_1 = 33.1^\circ$.
2. Inside. $r_2 = A - r_1 = 60^\circ - 33.1^\circ = 26.9^\circ$.
3. Second face. $\sin e = 1.5 \times \sin 26.9^\circ = 0.679$, so $e = 42.7^\circ$.
4. Deviation. $\delta = i + e - A = 55^\circ + 42.7^\circ - 60^\circ = 37.7^\circ$.

Practise with new numbers

Try it in Python

The program follows a ray through a $60^\circ$ prism of $n = 1.5$ for seven angles of incidence.

Remember these

• A prism bends a ray toward its base at both surfaces, so the emergent ray is not parallel to the incident ray.
• The angle between the first ray and the last ray is the angle of deviation.
• $r_1 + r_2 = A$ and $i + e = A + \delta$.
• The deviation is the smallest when the ray goes through symmetrically.

Check yourself

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

1. Why does a glass slab not deviate a ray, but a prism does?

2. A ray goes through a prism of angle $60^\circ$. The angle of refraction at the first face is $30^\circ$. What is the angle of incidence at the second face inside the glass?

3. When is the deviation of a prism the smallest?

4. A ray enters a $60^\circ$ prism at $52^\circ$ and leaves at $46^\circ$. Find the deviation.