Result

Critical angle

(O1.4.1) O1.4 · Total Internal Reflection →

sin θ_{c} = \frac{n _{2}}{n _{1}} (n_{1} > n_{2})

Statement

The angle of incidence beyond which light travelling from a denser to a less dense medium reflects entirely rather than refracting.

Symbols and units

$θ_{c}$	critical angle	degrees
$n_{1}, n_{2}$	refractive indices, denser to less dense	—

When it applies

Light travelling from a medium of higher refractive index toward one of lower index.

Don't use when

Only occurs going from a higher-index to a lower-index medium (n₁ > n₂) — it cannot happen traveling the other way, regardless of angle.

Derivation

Set the refraction angle to 90° (a grazing exit ray) in Snell's law and solve for the incidence angle:

n_{1} sin θ_{c} = n_{2} sin 9 0^{\circ} \Rightarrow sin θ_{c} = \frac{n _{2}}{n _{1}}

Watch out for

Occurs only travelling from a MORE dense to a LESS dense medium (n₁ > n₂) — it cannot happen the other way, however steep the angle. Beyond θ_c, no refracted ray exists at all; every bit of light reflects.

Connects to

Snell's law