Physics · Ch 6 — Optics
Critical Angle and Total Internal Reflection
Critical Angle and Total Internal Reflection
When light travels from a denser medium (index ) toward a rarer medium (index ), the refraction angle exceeds the incidence angle ; as is increased, grows even faster and reaches exactly (grazing the boundary) at one particular incidence angle called the critical angle , for which (or when the rarer medium is air, ). If the incidence angle in the denser medium is increased beyond , refraction into the rarer medium becomes impossible altogether: the entire beam is reflected back into the denser medium, a phenomenon called total internal reflection (TIR). The two necessary conditions for TIR are: (a) light must be travelling from a denser medium towards a rarer one, and (b) the angle of incidence in the denser medium must exceed the critical angle, . A lower refractive index (e.g. ice, 1.310) gives a comparatively larger critical angle (49.8°); a much higher refractive index (e.g. rutile, 2.621) gives a much smaller critical angle (22.4°) -- a smaller crit …
| Material | Refractive index | Critical Angle |
|---|---|---|
| Ice | 1.310 | 49.8° |
| Water | 1.333 | 48.6° |
| Fused Quartz (SiO2) | 1.458 | 43.3° |
| Crown Glass | 1.541 | 40.5° |
| Flint Glass | 1.890 | 31.9° |
| Calcite (CaCO2) | 1.658 | 37.0° |
| Diamond | 2.417 | 24.4° |
What this figure shows. A light ray inside a denser medium (n1, e.g. water) approaches a boundary with a rarer medium (n2, e.g. air) at an angle of incidence i. At the special angle where the refracted ray grazes the boundary exactly along the surface (refraction angle r = 90 degrees), the incidence angle shown equals the critical angle. A second ray, drawn at an incidence angle i greater than the critical angle (i > ic), is shown undergoing total internal reflection instead -- reflecting entirely back into the denser medium at r = i, with absolutely n …
Worked out. A ray strikes the first face of an equilateral prism (apex angle A = 60 degrees) at normal incidence (i1 = 0), and is arranged to just graze the second face on emerging (i2 = 90 degrees). Using the deviation formula d = i1 + i2 - A gives d = 0 + 90 - 60 = 30 degrees. Because the emergent ray only just grazes the second face, the light inside the prism must be striking that second face at exactly the critical angle for the prism material; combining this with sin(ic) = 1/n and, from the prism's own 60-degree apex-angle geometry, sin(ic) = sin(30 degrees) = 1/2, gives n = 2 for the material of this particular prism -- a neat example of the critical-angle idea appearing inside a prism problem rathe …