Recall and use the equation n = 1 / sin c

3.2.2 Refraction of Light

Learning Objective

Recall and use the equation \$n = \frac{1}{\sin c}\$ where n is the refractive index of a medium and c is the critical angle for total internal reflection.

Key Concepts

• Refraction: The change in direction of a light ray when it passes from one transparent medium to another with a different optical density.
• Refractive Index (n): A dimensionless number that describes how fast light travels in a medium compared to vacuum. \$n = \frac{c0}{v}\$ where \$c0\$ is the speed of light in vacuum and \$v\$ is the speed in the medium.
• Snell’s Law: Relates the angles of incidence (\$i\$) and refraction (\$r\$) to the refractive indices of the two media. \$n1 \sin i = n2 \sin r\$
• Critical Angle (c): The angle of incidence in the denser medium for which the angle of refraction is \$90^{\circ}\$. Beyond this angle total internal reflection occurs.
• Equation for Critical Angle: Rearranging Snell’s law for \$r = 90^{\circ}\$ gives \$\sin c = \frac{n2}{n1}\$ for \$n1 > n2\$. Solving for \$n1\$ yields the target equation \$n1 = \frac{1}{\sin c}\$ when the second medium is air (\$n_2 \approx 1\$).

Derivation of \$n = \frac{1}{\sin c}\$

1. Start with Snell’s law: \$n1 \sin i = n2 \sin r\$
2. For the critical condition, the refracted ray grazes the interface, so \$r = 90^{\circ}\$ and \$\sin r = 1\$.
3. Replace \$i\$ with the critical angle \$c\$: \$n1 \sin c = n2\$
4. If the second medium is air, \$n2 \approx 1\$, giving \$n1 = \frac{1}{\sin c}\$

Typical Refractive Indices

Worked Example

Question: Light travelling in glass (n = 1.52) strikes the glass‑air interface. Calculate the critical angle \$c\$ for total internal reflection.

1. Use the critical‑angle form of Snell’s law: \$\sin c = \frac{n{\text{air}}}{n{\text{glass}}} = \frac{1.00}{1.52}\$
2. Calculate: \$\sin c = 0.658\$
3. Find \$c\$ using the inverse sine: \$c = \sin^{-1}(0.658) \approx 41.1^{\circ}\$

Therefore, any incident angle greater than \$41.1^{\circ}\$ will result in total internal reflection.

Practice Questions

1. Given a critical angle of \$48^{\circ}\$ for a liquid, determine its refractive index using \$n = \frac{1}{\sin c}\$.
2. A ray of light passes from water into glass. If the angle of incidence in water is \$30^{\circ}\$, calculate the angle of refraction in glass (use \$n{\text{water}} = 1.33\$, \$n{\text{glass}} = 1.52\$).
3. Explain why a diamond sparkles more than glass, referring to the values of \$n\$ and the resulting critical angles.

Common Mistakes to Avoid

• Confusing the critical angle with the angle of incidence that produces the maximum refraction; the critical angle is specific to total internal reflection.
• Using the equation \$n = \frac{1}{\sin c}\$ for a situation where the second medium is not air; in such cases the full Snell’s law must be applied.
• Forgetting to convert degrees to radians only when using a calculator set to radian mode.