Recall and use the equation n = sin i / sin r

3.2.2 Refraction of Light

Learning Objective

Recall and use the equation

\$n = \frac{\sin i}{\sin r}\$

where n is the refractive index, i is the angle of incidence and r is the angle of refraction.

Key Concepts

• Refraction: the change in direction of a light ray when it passes from one transparent medium to another.
• Angle of incidence (i): the angle between the incident ray and the normal to the surface.
• Angle of refraction (r): the angle between the refracted ray and the normal.
• Refractive index (n): a measure of how much a medium slows down light compared with vacuum.

Snell’s Law

Snell’s Law relates the angles of incidence and refraction to the refractive indices of the two media:

\$\frac{\sin i}{\sin r} = \frac{v1}{v2} = \frac{n2}{n1}\$

For the IGCSE syllabus we usually work with the simplified form when one medium is air (≈ vacuum, n≈1):

\$n = \frac{\sin i}{\sin r}\$

How to Use the Equation

1. Measure the angle of incidence i (in degrees) with a protractor.
2. Measure the angle of refraction r inside the second medium.
3. Calculate \$\sin i\$ and \$\sin r\$ (use a scientific calculator).
4. Divide \$\sin i\$ by \$\sin r\$ to obtain the refractive index n.
5. If required, rearrange the equation to find an unknown angle:

   • To find \$i\$: \$i = \arcsin (n \sin r)\$
   • To find \$r\$: \$r = \arcsin \left(\dfrac{\sin i}{n}\right)\$

Typical Refractive Indices

Worked Example

Problem: A ray of light strikes a glass slab from air at an angle of incidence \$i = 30^\circ\$. The angle of refraction measured inside the glass is \$r = 19^\circ\$. Calculate the refractive index of the glass.

Solution:

1. Calculate \$\sin i = \sin 30^\circ = 0.500\$.
2. Calculate \$\sin r = \sin 19^\circ = 0.326\$ (to three decimal places).
3. Apply the formula \$n = \dfrac{\sin i}{\sin r} = \dfrac{0.500}{0.326} \approx 1.53\$.

The calculated refractive index (1.53) is consistent with typical crown glass.

Common Mistakes to Avoid

• Mixing up the angles: always measure both angles from the normal, not from the surface.
• Using degrees directly in the sine function without converting to radians (most calculators handle degrees if set correctly).
• For media other than air, remember the full Snell’s Law form \$\dfrac{n1}{n2} = \dfrac{\sin r}{\sin i}\$.

Practice Questions

1. A light ray passes from water (\$n=1.33\$) into air. If the angle of incidence in water is \$45^\circ\$, find the angle of refraction in air.
2. Light enters a plastic block (\$n=1.59\$) from air at an incidence angle of \$20^\circ\$. Determine the angle of refraction inside the plastic.
3. In an experiment, the measured angles are \$i = 25^\circ\$ and \$r = 15^\circ\$. Calculate the refractive index of the unknown material.