Refraction & Refractive Index

The absolute refractive index ($n$) of a material is a dimensionless ratio that compares the speed of light in a vacuum ($c$) to the speed of light in that specific material ($v$). It is expressed by the formula $n = \frac{c}{v}$.

Because the speed of light in any physical medium is always less than its speed in a vacuum ($3.00 \times 10^8 \text{ m s}^{-1}$), the value of $n$ for any material is always greater than or equal to $1$. For air, $n$ is approximately $1.00$ and is usually treated as such in calculations.

A material with a higher refractive index is described as being more optically dense. In such materials, light travels slower and undergoes a more significant change in direction when entering from a vacuum or air.

Snell's Law provides the governing equation for the path of light across a boundary: $n_1 \sin \theta_1 = n_2 \sin \theta_2$. This relationship shows that the product of the refractive index and the sine of the angle to the normal is constant across the interface.

When light enters a more dense medium ($n_2 > n_1$), the ray slows down and bends towards the normal, meaning $\theta_2 < \theta_1$. Conversely, when entering a less dense medium ($n_2 < n_1$), the ray speeds up and bends away from the normal, meaning $\theta_2 > \theta_1$.

If a ray is incident exactly along the normal ($\theta_1 = 0^{\circ}$), it will pass through without changing direction ($\theta_2 = 0^{\circ}$), although its speed and wavelength still change.

The Critical Angle ($C$) is a specific angle of incidence that occurs only when light travels from a more dense medium to a less dense medium. It is the angle at which the refracted ray emerges at exactly $90^{\circ}$ to the normal, skimming the boundary.

The formula for the critical angle, derived from Snell's Law where $\theta_2 = 90^{\circ}$, is $\sin C = \frac{n_2}{n_1}$. If the second medium is air ($n_2 = 1$), this simplifies to $\sin C = \frac{1}{n}$.

Total Internal Reflection (TIR) occurs when the angle of incidence exceeds the critical angle ($\theta_1 > C$). In this state, no refraction occurs; instead, $100\%$ of the light is reflected back into the original medium, following the standard law of reflection where the angle of incidence equals the angle of reflection.

Always check the normal: A common mistake is measuring the angle between the ray and the boundary surface. Ensure you subtract from $90^{\circ}$ if the problem provides the surface angle.

Sanity check the index: If your calculated refractive index $n$ is less than $1$, you have likely inverted your ratio. $n$ must always be $\ge 1$.

TIR Conditions: If asked to explain why TIR occurs, you must state both conditions: the light must be in the more dense medium, and the angle of incidence must be greater than the critical angle.

Calculator Mode: Ensure your calculator is in Degrees mode, as most physics problems use degrees rather than radians for refraction.