Refraction of Light

Introduction

Refraction is the bending of light as it passes from one transparent medium to another. This bending occurs because the speed of light changes when it enters a different medium. The amount of bending depends on the refractive indices of the two media.

Snell's Law

Snell's Law describes the relationship between the angles of incidence and refraction and the refractive indices of the two media. The law is mathematically expressed as:

$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$

Where:

• $n_1$ is the refractive index of the first medium.
• $\theta_1$ is the angle of incidence (the angle between the incident ray and the normal to the surface).
• $n_2$ is the refractive index of the second medium.
• $\theta_2$ is the angle of refraction (the angle between the refracted ray and the normal to the surface).

Refractive Index

The refractive index ($n$) of a medium is a dimensionless number that indicates how much slower light travels in that medium compared to a vacuum. A vacuum has a refractive index of 1. The refractive index of a medium is always greater than or equal to 1.

$$n = \frac{c}{v}$$

Where:

• $c$ is the speed of light in a vacuum (approximately $3.0 \times 10^8 \, m/s$).
• $v$ is the speed of light in the medium.

The Equation: $n = \frac{1}{\sin c}$

This equation is a simplified form of Snell's Law, used when the angle of incidence is very small. When $c$ is very small (close to 0 degrees), $\sin c \approx c$ (in radians). Therefore, $n \approx \frac{1}{c}$, which is the equation we are focusing on.

This equation is useful for calculating the refractive index of a medium when the angle of incidence is small.

Example Calculation

Consider light traveling from air ($n_1 \approx 1$) into water ($n_2 \approx 1.33$) at a very small angle of incidence. We can use the equation $n = \frac{1}{\sin c}$ to find the refractive index of water.

$$n_2 = \frac{1}{\sin c}$$

$$1.33 = \frac{1}{\sin c}$$

$$\sin c = \frac{1}{1.33} \approx 0.7518$$

$$c = \arcsin(0.7518) \approx 48.6^\circ$$

Therefore, the angle of refraction is approximately $48.6^\circ$

Table of Refractive Indices (Approximate Values)

Medium	Refractive Index ($n$)
Vacuum	1.00
Air	1.00
Water	1.33
Glass	1.50 - 1.90 (depending on the type of glass)
Diamond	2.42

Note: These are approximate values. The exact refractive index can vary depending on the specific composition and temperature of the material.