The refractive index

How much does a material bend light?

Diamond sparkles more than glass, and glass bends light more than water. We need a number for this. The number is the refractive index.

The absolute refractive index

The absolute refractive index $n$ of a material is the speed of light in a vacuum divided by the speed of light in the material:

Here $c = 3.00 \times 10^{8}\ \text{m/s}$ is the speed of light in a vacuum, and $v$ is the speed of light in the material. Light is never faster in a material than in a vacuum, so $n$ is never less than $1$. It has no unit, because it is a ratio of two speeds.

When we want the speed, $v = \dfrac{c}{n}$.

A larger $n$ means that light moves slower, and it bends more when it enters the material from air. Diamond has the highest value in the table, and this is one reason for its sparkle.

You should find that air wins by a hair, and the material with the largest index comes last. Set your material to $1$: it ties with air, the fastest light can go.

Optically denser is not the same as denser

We call a material with a larger refractive index optically denser. This is a statement about light, and not about mass. Kerosene has a smaller mass density than water, because it floats on water, but its refractive index ($1.44$) is larger than that of water ($1.33$). So kerosene is optically denser than water, but it is not denser in mass.

The refractive index of one material with respect to another

When light goes from material 1 to material 2, the constant in Snell's law is the refractive index of material 2 with respect to material 1:

For air to glass, $n_1 = 1$ and $n_2 = 1.5$, so $n_{21} = 1.5$. This is the constant that we used in the last lesson. For water to glass, $n_{21} = \dfrac{1.5}{1.33} = 1.13$. The bending is less than for air to glass, because the two speeds are closer.

See it in a laboratory

The simulation is by PhET Interactive Simulations, University of Colorado Boulder (CC BY 4.0).

Worked example 1

Find the speed of light in kerosene, of refractive index $1.44$.

1. The rule. $v = \dfrac{c}{n}$.
2. Put in the numbers. $v = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{1.44} = 2.08 \times 10^{8}\ \text{m/s}$.
3. Check. It agrees with the table, and it is less than $c$.

Worked example 2: the index from a measurement

A student measures $i = 60^\circ$ and $r = 35^\circ$ for a ray that goes from air into a block of plastic. Find the refractive index of the plastic.

1. The rule. $n = \dfrac{\sin i}{\sin r}$.
2. Put in the numbers. $n = \dfrac{\sin 60^\circ}{\sin 35^\circ} = \dfrac{0.866}{0.574} = 1.51$.
3. So the plastic bends light about as much as glass.

Practise with new numbers

Try it in Python

The program finds the speed of light in four materials.

Remember these

• The absolute refractive index is $n = c/v$. It has no unit and is never less than $1$.
• A larger refractive index means slower light and more bending toward the normal.
• Optically denser means a larger refractive index, and it does not mean a larger mass density.
• $n_{21} = \dfrac{\sin i}{\sin r} = \dfrac{n_2}{n_1}$.

Check yourself

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

1. The speed of light in a dense flint glass is $1.8 \times 10^{8}\ \text{m/s}$. Find its refractive index.

2. Which is optically denser, water ($1.33$) or glass ($1.5$)?

3. Can the refractive index of a material be $0.8$?

4. Light goes from glass ($1.5$) into water ($1.33$). Find the refractive index of water with respect to glass.