Refraction

Snell's Law provides the governing equation for refraction: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$. In this formula, $\theta_1$ is the angle of incidence and $\theta_2$ is the angle of refraction, both measured from the normal line.

During refraction, the frequency of the light wave remains constant because it is determined by the source. Consequently, as the speed ($v$) changes, the wavelength ($\lambda$) must also change to satisfy the wave equation $v = f \lambda$.

The direction of bending follows specific rules: light bends towards the normal when entering a denser medium ($n_1 < n_2$) and away from the normal when entering a rarer medium ($n_1 > n_2$).

Step 1: Identify the Normal: Always draw a perpendicular line (the normal) at the point where the incident ray hits the boundary. All angles must be measured relative to this line, not the surface.

Step 2: Apply Snell's Law: Substitute the known refractive indices and the incident angle into $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$ to solve for the unknown angle or index.

Step 3: Check for Critical Angle: If light is moving from a denser to a rarer medium, calculate the critical angle $\theta_c = \arcsin(\frac{n_2}{n_1})$. If the incident angle exceeds this value, refraction will not occur.

Total Internal Reflection occurs when a light ray is completely reflected back into the original medium, with no light passing through the boundary. This only happens when light travels from a medium of higher refractive index to one of lower refractive index.

The Critical Angle ($\theta_c$) is the specific angle of incidence that results in an angle of refraction of exactly $90^\circ$. At this angle, the light ray skims along the boundary interface.

For TIR to occur, two conditions must be met: the incident medium must be more optically dense than the second medium ($n_1 > n_2$), and the angle of incidence must be greater than the critical angle ($\theta_1 > \theta_c$).

The Normal Line Rule: Examiners often provide angles relative to the surface to trick students. Always subtract the given surface angle from $90^\circ$ to find the true angle of incidence or refraction.

Sanity Check: If light enters a denser medium (like water or glass from air), the calculated angle of refraction MUST be smaller than the angle of incidence. If it is larger, you likely swapped $n_1$ and $n_2$.

Unitless Index: Remember that the refractive index $n$ is a ratio of speeds and therefore has no units. If your calculation results in a value less than $1.00$ for a physical material, re-evaluate your algebra.

Frequency Change: A common mistake is assuming the frequency of light changes during refraction. Frequency is a property of the source and remains constant; only speed and wavelength change.

TIR Direction: Students often attempt to calculate a critical angle when light moves from air to glass. TIR is physically impossible in this scenario because the light is moving into a denser medium.

Calculator Mode: Ensure your calculator is set to 'Degrees' rather than 'Radians' when performing trigonometric functions for optical problems, as most exam questions use degree measurements.