What Is Critical Angle and Total Internal Reflection?

SL and HL (C.3 Wave phenomena): The critical angle is the angle of incidence in the more optically dense medium for which the refracted ray travels along the boundary. Total internal reflection occurs when the incidence angle exceeds this critical angle, so no wave is transmitted into the second medium.

The Reasoning and Mechanism

When a wave passes from a medium with higher refractive index, such as glass, into one with lower refractive index, such as air, it refracts away from the normal. Snell's law gives

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

As the angle of incidence increases, the angle of refraction also increases. At the critical angle $c$, the angle of refraction is $90^\circ$, so

$$n_1\sin c=n_2\sin 90^\circ$$

and therefore

$$\sin c=\frac{n_2}{n_1},$$

where $n_1>n_2$.

For glass with $n_1=1.50$ and air with $n_2=1.00$:

$$c=\sin^{-1}\left(\frac{1.00}{1.50}\right)=41.8^\circ.$$

Two conditions are required: the wave must travel from a higher to a lower refractive index, and the incidence angle must exceed the critical angle.

Physically, refraction occurs because the wave speed changes at the boundary, while frequency remains constant. Using $n=c_0/v$, a lower refractive index means a higher wave speed. As the refracted angle reaches $90^\circ$, no larger refracted angle is possible. Beyond this point, the wave is reflected back into the original medium, as used in optical fibres and prisms.

A common misconception is that total internal reflection occurs at the critical angle. It does not: at $c$, the refracted ray is still present along the boundary; total internal reflection begins only when $\theta_1>c$.

Exam Technique

In an IB response, measure every angle from the normal, not from the surface. State both conditions for total internal reflection and show the substitution into Snell's law when asked to calculate the critical angle.