Examples on How to Find the Critical Angle

Example 1: A glass cube is held in contact with a liquid and a light ray is directed at the vertical face of the cube. The angle of incidence at the vertical face is 39° and the angle of refraction is 25°. The refractive index of the glass cube is 1.45 and the refractive index of the liquid is 1.32. Calculate the critical angle for the ray at the glass-liquid boundary

Step 1: Recall Snell’s Law and rearrange to make critical angle the subject

n₁ sin θ₁ = n₂ sin θ₂

Step 2: Substitute in the known quantities

n₁ = refractive index of glass cube = 1.45

n₂ = refractive index of liquid = 1.32

θ₁ = C (critical angle)

θ₂ = 90° (The angle of refraction is 90° when at the critical angle)

sin C = n₂ sin(θ₂)/n₁

C = sin^-1( 1.32/1.45)

Step 3: Calculate the critical angle: C = 65.55 °

Example 2: What must be the angle of incidence for there to be a total internal reflection of a ray going from the glass with n_g= 1.7 to liquid with n_l = 1.52?

Solution

Given the indices for the means by which the ray passes,

we use the formula

θ_c = sin^-1 (n_r/n_i)

θ_c = sin^-1 (1.52/1.7) = sin^-1 (0.89)

So,

θc = 62.87

Critical angle is 1.064 rad.

Example 3: A ray of light strikes from a medium with n = 1.52 on a surface of separation with the air with n = 1. Calculate the value of a critical angle.

solution:

Given the indices for both the means.

We know the formula,

θ_c = sin^-1 (n_r/n_i )

θ_c = sin^-1 (1/1.52)

Therefore,

θ_c = 40.54

Example 4: Find the critical angle for the diamond-air boundary.

Solution:

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

n_r = refractive index of diamond = 2.42

n_i = refractive index of air = 1.000

θ_c = sin^-1(n_r/n_i)

θ_c= sin^-1 (1.000/2.42)

θ_c= 24.4 degrees

Also, Check

• Total Internal Reflection
• Brewster’s Law
• Apparent Depth

The critical angle is a key concept in optics, especially when light interacts with boundaries between two dissimilar materials. It determines the exact angle of incidence at which light will refract along the interface between the materials, instead of entering the second medium.

In this article, how to find the critical angle of a light ray, the formula for critical angle, Snell’s law, finding a critical angle, and solve the problem.