What is Critical Angle and its Formula?

Introduction

In this post, i will walk you through what is critical angle and it’s formula. I will also solve a few problems to help you understand the concept of the critical angle.

What is a Critical Angle?

The critical angle is the angle of incidence in the denser medium when the angle of refraction in the less dense medium is ninety degrees (90⁰).

A critical angle is the angle of incidence for which the refracted ray emerges tangent to the surface of the angle.

The formula for finding a critical angle is

θ_c = sin^-1 (1 / n) [Where θ_c = critical angle, n = refractive index]

We can also use

θ_c = sin^-1 (n_a / n_g)

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Explanation of Critical Angle

The critical angle is a term in physics and optics to describe the angle of incidence of a ray of light at which it refracts at an angle of 90 degrees to the normal (the line perpendicular to the surface of the medium it is entering).

Let’s assume we allow a ray of light to move from one medium to another. For example, say air to water or from water to glass. This ray of light will change direction because of the change in the speed of light in any of the mediums.

Therefore, we refer to the change in the direction of the ray of light as refraction. The amount of refraction depends on the angle of incidence of the light ray and the refractive index of the two media.

Therefore, to explain the critical angle in the simplest term. We can say that it’s the angle of incidence at which the refracted ray of light is parallel to the boundary between the two media at 90 degrees. In other words, the angle of incidence results in the refracted ray being bent at a 90-degree angle from the normal.

You may also like to read:

what is the critical angle for light traveling from crown glass (n = 1.52) into water (n = 1.33)?

Snell’s Law

We can calculate critical angle by applying Snell’s law. The law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two media.

sin i / sin r = sin θ_c / sin 90⁰ = _gn_a = 1 / _an_g = 1 / n = n_a / n_g

i = Angle of incidence

r = Angle of refraction = 90⁰

θ_c = Critical Angle

_gn_a = Refractive index of the ray of light from glass to air

_an_g = Refractive index of the ray of light from air to glass

n = Refractive index of glass by definition

Critical Angle Formula

When the angle of refraction is 90 degrees, the sine of the angle of refraction is equal to one (1). Therefore, the sine of the critical angle is similar to the reciprocal of the refractive index of the two media.

sin r = sin 90⁰ = 1

and sin i = sin

Thus, sin i / sin r = sin θ_c / sin 90⁰ = sin θ_c / 1 = sin θ_c = _gn_a = 1 / _an_g = 1 / n = n_a / n_g

The critical angle is an important concept in optics. This is because it determines whether a light ray will be refracted or reflected when it encounters a boundary between two media.

If the angle of incidence is greater than the critical angle. The ray of light will be reflected back into the medium from which it came. This phenomenon is called total internal reflection. We use it in various optical devices, such as fiber optic cables, and prism-based cameras.

Critical Angle Practice Problems

Here are practice problems to help you understand how to calculate critical angle problems:

Problem 1

What is the critical angle for light traveling from water to air? The refractive index of water = 4/3

Data

The refractive index of water, n = 4/3 = 1.33

Unknown

Critical angle = ?

Formula

Critical angle, θ_c = sin^-1 (1 / n)

Solution

We will insert our data into the formula

θ_c = sin^-1 (1 / n) = sin^-1 (1 / 1.33)

We will now have

θ_c = sin^-1 (0.75) = 48.6⁰

Therefore, the critical angle for light traveling from water to air is 48.6 degrees.

Problem 2

A ray of light strikes from a medium with n = 1.67 on a surface of separation with the air with n = 1. Find the value of the critical angle.

Data

The refractive index of air, n_a = 1.67

Refractive index of glass, n_g = 1

The refractive index of the medium is, n = n_a / n_g = 1 / 1.67

Unknown

Critical angle, θ_c = ?

Formula

Critical angle, θ_c = sin^-1 (n_a / n_g)

Solution

We will add our data to the formula

Critical angle, θ_c = sin^-1 (n_a / n_g) = sin^-1 (1 / 1.67)

The above expression will give us

θ_c = sin^-1 0.599

And our final result for the critical angle will become

θ_c = 36.8⁰

Sources:

Google Books