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?
Definition: 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).
At values above the critical angle, the total internal reflection occurs, that is, the refracted ray disappears and a strong reflected ray appears.
A critical angle is the angle of incidence for which the refracted ray emerges tangent to the surface of the angle.
How to Find Critical Angle
Critical Angle Formula
The critical angle formula is
θc = sin-1 (1 / n) [Where θc = critical angle, n = refractive index]
We can also use
θc = sin-1 (na / ng)
Therefore, the critical angle formula that can help us to find critical angle based on the question available to us are: θc = sin-1 (1 / n) or θc = sin-1 (na / ng). Where: θc = critical angle, n = refractive index.
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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 θc / sin 900 = gna = 1 / ang = 1 / n = na / ng
i = Angle of incidence
r = Angle of refraction = 900
θc = Critical Angle
gna = Refractive index of the ray of light from glass to air
ang = 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 θc
Thus, sin i / sin r = sin θc / sin 900 = sin θc / 1 = sin θc = gna = 1 / ang = 1 / n = na / ng
As stated earlier, the critical angle formula can be written as: θc = sin-1 (1 / n) or θc = sin-1 (na / ng). Where: θc = critical angle, n = refractive index.
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.
Relationship Between Refractive Index and Critical Angle
If the refractive angle is 900
By applying Snell’s law, the refractive index for glass – air boundary is: gna = (sin θc) / (sin 900)
Now, the refractive index from air to glass is: ang = (sin 900) / (sin θc)
But sin 900 = 1
Therefore, the refractive index from air to glass is: ang = 1 / (sin θc)
Total Internal Reflection
When a ray of light passes from an optically denser medium to an optically less dense medium, there exists a weak internal reflection and a strong refraction. But as the angle of incidence increases, the angle of refraction also increases at the same time.
The intensity of the reflected ray gets stronger and that of the refractive ray becomes weaker. When the angle of incidence exceeds the critical angle, there exists no refraction and a total reflection occurs. We refer to this phenomenon in physics as total internal reflection.
The conditions under which total internal reflection occurs are:
- Light rays must travel through a denser medium to a less dense medium
- The angle of incidence must exceed the critical angle.
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, θc = sin-1 (1 / n)
Solution
We will insert our data into the formula
θc = sin-1 (1 / n) = sin-1 (1 / 1.33)
We will now have
θc = 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, na = 1.67
We also have the refractive index of glass, ng = 1
Additionally, we have a refractive index of the medium is, n = na / ng = 1 / 1.67
Unknown
Critical angle, θc = ?
Formula
The equation for Critical angle, θc = sin-1 (na / ng)
Solution
We will add our data to the formula
Critical angle, θc = sin-1 (na / ng) = sin-1 (1 / 1.67)
The above expression will give us
θc = sin-1 0.599
And our final result for the critical angle will become
θc = 36.80
Problem 3
The refractive index of a medium relative to air is 1.8. Calculate to the nearest degree, the critical angle for the medium.
Data: The information from the question that will help us solve the problem
The refractive index, n = 1.8
Unknown: What we need to find
Critical angle, θc = ?
Formula: The equation that will help us solve the problem
Critical angle, θc = sin-1 (1 / n)
Solution
To solve the problem, we will apply the critical angle formula above:
θc = sin-1 (1 / n) = sin-1 (1 / 1.8) = sin-1 (0.55) = 33.750
Therefore, we can abbreviate the final answer as 340
Problem 4
The refractive index for a given transparent medium is 1.4. What is the minimum angle for total internal reflection to take place in the medium?
Data: The information from the question
The refractive index for the medium, n = 1.4
Unknown: Unavailable data from the question
Critical angle, θc = ?
Formula: The equation that will help us solve the problem
Critical angle, θc = sin-1 (1 / n)
Solution
Critical angle, θc = sin-1 (1 / n) = sin-1 (1 / 1.4) = sin-1 (0.714) = 45.60 = 460
Therefore, the critical angle of the medium is 460
Problem 5
If the refractive index of glass is 1.5, what is the critical angle at the air-glass interface?
Data: The information from the question
Refractive index, n = 1.5 = 3 / 2
Unknown: What we need to find
Critical angle, θc = ?
Formula: The equation to solve the problem
Critical angle, θc = sin-1 (1 / n)
Solution
Critical angle, θc = sin-1 (1 / n) = sin-1 (1 / 1.5) = sin-1 (1 / (3/2))
Therefore, the critical angle, θc= sin-1 (2 / 3) = sin-1 (0.667) = 41.80 = 420
Problem 6
The refractive indices of glass and water are 1.5 and 1.3 respectively. What will be the critical angle when the angle of refraction in the water medium is 900?
Data:
The refractive index of water, nw = 1.3
Refractive index of glass, ng = 1.5
Refractive angle, θ = 900
Unknown:
Critical angle, θc = ?
Formula:
θc = sin-1 (nw sin900)/ ng
Solution
sin θc = (nw sin900)/ ng = (1.3 x 1) / 1.5 = 0.866
θc = sin-1 (0.866) = 600
Table for Substances Their Critical Angles and Refractive Indexes
| substance | Refractive index (n) | Calculation θc = sin-1 (1 / n) | Critical angle (θc) |
| Vacuum | 1.00 | θc = sin-1 (1 / 1) | 900 |
| Pure Water | 1.33 | θc = sin-1 (1 / 1.33) | 48.750 |
| Carbondioxide | 1.0005 | θc = sin-1 (1 / 1.0005) | 88.190 |
| Ice | 1.31 | θc = sin-1 (1 / 1.31) | 49.760 |
| Ethyl Alcohol | 1.36 | θc = sin-1 (1 / 1.36) | 47.330 |
| Quartz | 1.46 | θc = sin-1 (1 / 1.46) | 43.230 |
| Vegetable oil | 1.47 | θc = sin-1 (1 / 1.47) | 42.860 |
| Olive oil | 1.48 | θc = sin-1 (1 / 1.48) | 42.510 |
| Acrylic | 1.49 | θc = sin-1 (1 / 1.49) | 42.160 |
| Table Salt | 1.51 | θc = sin-1 (1 / 1.51) | 41.470 |
| Glass | 1.52 | θc = sin-1 (1 / 1.52) | 41.140 |
| Sapphire | 1.77 | θc = sin-1 (1 / 1.77) | 34.400 |
| Zircon | 1.92 | θc = sin-1 (1 / 1.92) | 31.390 |
| Cubic zirconia | 2.16 | θc = sin-1 (1 / 2.16) | 27.580 |
| Diamond | 2.42 | θc = sin-1 (1 / 2.42) | 24.410 |
| Gallium phosphide | 3.50 | θc = sin-1 (1 / 3.50) | 16.600 |
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