How to Find Critical Angle of a Light Ray

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.

What is a Critical Angle?

The critical angle is defined as the angle of incidence where light, instead of passing into the second medium, refracts along the boundary between two different media.

• As the angle of incidence increases, the angle of refraction also increases until it reaches 90°.
• When light is refracted along the boundary, the angle of refraction is 90°.
• The angle at which the incident ray refracts at an angle of 90 degrees is known as the critical angle, denoted as C.

Calculation of Critical Angle

The critical angle (C) can be calculated using the formula:

Sin C = 1/n

or

C = sin^-1(1/n)

where,

• C is critical angle
• 1 is refractive index of air or vacuum
• n is refractive index of medium

Critical Angle a Pair of Media

Snell’s law defines the correlation between the angles of incidence (θ₁) and refraction (θ₂​) and the refractive indices ( n1 and n₂ ) of two media:

[ n₁sin θ₁ = n₂sin θ₂]

Now, we know that the concept of critical angle is valid when ray moves from denser medium to rarer medium. Since, at critical angle, the refracted ray travels parallel to the media, thus it makes 90° with the normal. Therefore,

θ₁ = Critical Angle denoted by C

θ₂ = 90°

n₁ sin C = n₂ sin 90°

Sin C = n2/n1

Hence,

C = sin^-1(n₂/n₁)

By setting ​θ₂ to 90 degrees (since refraction is along the boundary at the critical angle) and substituting n₂ = 1 (for air) and putting it in the Snell’s law we get

θ1 = Critical Angle denoted by C

θ2 = 90°

n1 = n

n2 = 1 (air)

n₁sin θ₁ = n₂sin θ₂

n × Sin C = 1 × Sin 90

n × Sin C = 1

⇒ Sin C = 1/n

⇒ C = Sin^-1(1/n)

How to Find Critical Angle

The steps to find critical angle are mentioned below:

Step 1: Identify the refractive indices (n₁ and n₂) of the two media.

Step 2: Recall Snell’s Law and rearrange to make critical angle the subject [ n₁ sin θ₁ = n₂ sin θ₂ ]

Step 3: At the critical angle, the angle of refraction (θr) becomes 90 degrees.

Step 4: Use [ Sin θ_c = n₂/ n₁ ] to find the critical angle.

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

FAQs on How to Calculate Critical Angle

What is the critical angle of a light ray?

The critical angle is defined as the angle of incidence that causes light to refract along the boundary when transitioning between two media. When this angle is exceeded, total internal reflection takes place.

Does the critical angle depend on the colors (wavelengths) of light?

The critical angle is mainly dictated by the refractive indices of the two media in question and is generally independent of the light’s wavelength. Nonetheless, in certain instances, minor changes in the critical angle can be observed for different wavelengths, which can result in phenomena such as chromatic aberration.

What happens if the angle of incidence is less than the critical angle?

If the angle of incidence is less than the critical angle, refraction occurs, and the light ray bends as it enters the second medium.

What are some applications of total internal reflection?

Total internal reflection is used in fiber optics, endoscopes, and critical angle refractometers for measuring liquid refractive indices.

Can the critical angle be greater than 90 degree?

No, because it represents the angle at which the refracted ray travels along the interface between the two media.