The critical angle
When light travels from a denser medium (e.g. glass) into a less dense medium (e.g. air), it bends away from the normal. As the angle of incidence increases, the refracted ray bends more and more until, at one special angle, the refracted ray travels exactly along the boundary (angle of refraction = 90°). This angle of incidence is called the critical angle, c.
Total internal reflection
If the angle of incidence is greater than the critical angle, no light can escape into the less dense medium. Instead all the light is reflected back into the denser medium, obeying the law of reflection. This is called total internal reflection (TIR).
Remember / Remember
- Two conditions for TIR: light travels from a denser to a less dense medium, AND the angle of incidence exceeds the critical angle.
- Applications: optical fibres, prism periscopes and binoculars, the sparkle of diamonds, mirages.
Key formula / Key formula
At the critical angle, r = 90°, so sin c = 1 / n
where n is the refractive index of the denser medium. A larger n gives a smaller critical angle.
Worked example / Worked example
Find the critical angle of glass (n = 1.50).
sin c = 1 / n = 1 / 1.50 = 0.667.
c = sin⁻¹(0.667) = 41.8°. Any ray inside the glass meeting the surface at more than 41.8° is totally internally reflected.
Worked example / Worked example
The critical angle of a liquid is 49°. Find its refractive index.
n = 1 / sin c = 1 / sin 49° = 1 / 0.755 = 1.33. The liquid is water.
In an optical fibre, light entering one end hits the core-cladding boundary at an angle greater than the critical angle, so it is totally internally reflected again and again, carrying signals over long distances with very little loss.