# Critical Angle Formula

Critical Angle can be described as the angle of incidence that offers an angle of refraction of 90 degrees. Remember that the critical angle is defined as an angle of incidence value. The critical angle will be 48.6 degrees for water-air boundaries and 61.0 degrees for crown glass-water boundary.

The real value of the critical angle depends completely on the combination of materials that are available on each side of the boundary. Let us consider two different medium

1. Incident medium
2. Refractive medium

Critical angle = Θi that gives a Θr value of 90-degrees

According to Snell’s Law equation, a generic equation for finding the critical angle can be derived easily.

$n_{i}Sin\theta_{i}=n_{r}Sin\theta _{r}$
$n_{i}Sin\theta_{crit}=n_{r}Sin90$
$n_{i}Sin\theta_{crit}=n_{r}$
$Sin\theta_{crit}=n_{r}/n_{i}$
Θcrit= sine-1 (nr/ni)

## Solved Examples

Example 1

Find the critical angle for the crown glass-air boundary

The solution to the problem requires the use of the above equation for the critical angle.

Θcrit = sin-1 (nr/ni)

nr=refractive index of glass=1.52

ni=refractive index of air=1.000

Θcrit = sin-1 (1.000/1.52) = 41.1 degrees

Example 2

Find the critical angle for the diamond-air boundary.

The solution to the problem requires the use of the above equation for the critical angle.

nr=refractive index of diamond=2.42

ni=refractive index of air=1.000

Θcrit = sin-1 (nr/ni)

Θcrit = sin-1 (1.000/2.42) = 24.4 degrees