Refraction & Refractive Index

• The Refractive Index ($n$) is a dimensionless constant that quantifies how much a medium reduces the speed of light compared to a vacuum. It is defined by the ratio of the speed of light in a vacuum ($c$) to the speed of light in the substance ($v$).

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

• Because light travels fastest in a vacuum, the value of $n$ for any material is always greater than 1. For practical calculations, the refractive index of air is treated as $1.00$, as light slows down negligibly in air compared to a vacuum.

• A higher refractive index indicates a more optically dense material, which causes a greater reduction in the speed of light and more significant bending of the ray.

• Snell's Law provides the mathematical relationship between the refractive indices of two media and the angles at which light crosses the boundary. It states that the product of the refractive index and the sine of the angle to the normal is constant across the interface.

Key Equation: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

• In this equation, $n_1$ and $\theta_1$ refer to the incident medium and angle, while $n_2$ and $\theta_2$ refer to the refractive medium and angle. This law allows for the calculation of an unknown refractive index or the path of a light ray through multiple layers of material.

• If a ray strikes the boundary at an angle of $0^{\circ}$ to the normal (perpendicular to the surface), it does not change direction, although its speed still changes according to the refractive index.

• Total Internal Reflection (TIR) is a phenomenon where light is completely reflected back into the original medium instead of refracting into the second medium. This only occurs when light travels from a more dense medium to a less dense medium (e.g., glass to air).

• The Critical Angle ($C$) is the specific angle of incidence that results in an angle of refraction of exactly $90^{\circ}$, meaning the light ray travels along the boundary interface.

Critical Angle Formula: $\sin(C) = \frac{n_2}{n_1}$ (If $n_2$ is air, $\sin(C) = \frac{1}{n}$)

• For TIR to occur, two conditions must be met: the light must be moving toward a medium with a lower refractive index, and the angle of incidence must be greater than the critical angle.

• Always check the Normal: Examiners often provide angles measured from the surface of the material to trick students. You must subtract this from $90^{\circ}$ to find the correct angle for Snell's Law.

• Sanity Check for n: If your calculation results in a refractive index $n < 1$, you have likely swapped the values of $v$ and $c$ or $n_1$ and $n_2$. A refractive index must always be $\ge 1$.

• Calculator Mode: Ensure your calculator is in Degrees mode rather than Radians when performing trigonometric functions for refraction problems, as most physics problems use degrees for ray optics.

• Ray Diagrams: When drawing, use a ruler and ensure the bending direction is qualitatively correct. A ray entering a denser medium MUST be drawn closer to the normal than the incident ray.