Critical Angle and Total Internal Reflection (Leaving Cert Physics): Revision Notes

Critical Angle and Total Internal Reflexion

What is the critical angle?

When light travels from an optically denser medium (like glass or water) to an optically rarer medium (like air), something special happens as we increase the angle of incidence. The critical angle is the specific angle of incidence at which the refracted ray travels along the boundary between the two media - essentially parallel to the surface with an angle of refraction of exactly 90°.

At angles smaller than the critical angle, light continues to refract normally into the second medium. However, when the angle of incidence equals the critical angle, the refracted ray skims along the surface of the boundary.

Total internal reflexion

When the angle of incidence becomes greater than the critical angle, something remarkable occurs - the light ray cannot enter the second medium at all. Instead, it is completely reflected back into the first medium, following the normal laws of reflexion where the angle of incidence equals the angle of reflexion.

This phenomenon is called total internal reflexion because:

• All of the incident light is reflected (none is transmitted)
• The reflexion follows the law of reflexion perfectly
• No light energy is lost in the process

Essential conditions for total internal reflexion

Mathematical relationship between refractive index and critical angle

We can derive a useful mathematical relationship using Snell's law. At the critical angle, the angle of refraction is 90°, so:

$n_1 \sin C = n_2 \sin 90°$

Since $\sin 90° = 1$ and if the second medium is air ($n_2 = 1$):

$n_1 \sin C = 1$

Therefore: $n = \frac{1}{\sin C}$

Or rearranged: $\sin C = \frac{1}{n}$

Where:

• $n$ is the refractive index of the denser medium
• $C$ is the critical angle

Worked examples

Applications: Prisms and optical devices

Total internal reflexion has practical applications in optical devices. A common example is the 45°-90°-45° prism, which can turn a light ray through 90° or 180°.

When light enters the prism at 45° to the hypotenuse face, it undergoes total internal reflexion because this angle exceeds the critical angle for most optical materials. This allows the prism to redirect light efficiently without any loss of intensity.

Snell's window phenomenon

An interesting natural application occurs when observing light underwater. When you look up from beneath the water surface, you can see the entire above-water world compressed into a circular region called Snell's Window.

Key mathematical relationships

Formula	Description	When to use
$\sin C = \frac{1}{n}$	Find critical angle from refractive index	When $n$ is known
$n = \frac{1}{\sin C}$	Find refractive index from critical angle	When $C$ is known
Angle of incidence > $C$	Condition for total internal reflexion	To predict if TIR occurs