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We all are familiar with the concept of the angle of incidence and reflection and refraction. When light strikes the mirror, it reciprocates; however, when the same light is incident at a medium, it gets deviated from its original path; this is actually refraction.

So, the critical angle in Physics is inclusive of incidence, reflection, and refraction. The critical angle is the angle of incidence that produces a 90 degrees angle of refraction.Â

This page describes the critical angle definition, critical angle formula, and the critical angle and total internal reflection by considering real-life situations.

The critical angle definition explains a lot about the laws of refraction and how the angle of incidence can be adjusted to bring a 90-degree angle of refraction. Here, to understand the formula of critical angle, we will introduce the concept of snellâ€™s law as well. So letâ€™s get started.

You already know that Snellâ€™s law is the law of refraction. We also call it the Snell-Descartes law. It is defined in the following manner:

The formula of critical angle describes the relationship between the angles of incidence and refraction, when considering light or other wave crossing a boundary between two different isotropic media, viz: water, glass, or air.

If the light coming from the air medium passes through the jar filled with water, then we observe that there is a difference in the refractive indices of each.

If n1 is the refractive index of air and Sin Î¸_{1} is the angle of incidence when light passes through the air, it suffers refraction in the form of the angle of deviation, i.e., Sin Î¸_{2} with water having a refractive index of n2, then we have the formula for Snellâ€™s law as;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â n1 Sin Î¸_{1} = n2 Sin Î¸_{2} â€¦.(1)

Here,

Sin Î¸_{1} = angle of incidence for the incident medium

&

Sin Î¸_{2 }= angle of refraction or the angle of deviation at the refractive medium

So, now, we will derive the critical angle formula from equation (1):

According to the Critical Angle definition, the incident angle is the critical angle that adjusts the refractive angle to 90-degrees, so in equation (1), we will do the same; letâ€™s do it:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â n1 Sin Î¸critical = n2 Sin 90^{o}

So, we know that Sin 90^{o} is equal to 1 and we get our new equation as;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â n1 Sin Î¸critical = n2

Now, we get the following equation;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â n2/n1 = Sin Î¸critical

According to the inverse trigonometric, we rewrite the above equation in the following manner:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Î¸critical =Â Sin^{-1} (n_{2}/n_{1}) â€¦..(2)

So, equation (2) is the required Critical Angle Formula.

In this heading, we will discuss the following:

Critical angle of glass

Critical angle of diamond

Critical angle of water

Critical Angle of Glass

We know that the refractive index glass,Â n1= 1.52

And, the refractive index of air,Â n2 = 1.00

Using equation (2), and applying these two values as;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Î¸critical = Sin^{-1} (1.00/1.52)

On solving, we get the critical angle of glass as 41.1^{o}

Critical Angle of Diamond

We know that the refractive index diamond,Â n1= 2.42

And, the refractive index of air,Â n2 = 1.00

Using equation (2), and applying these two values as;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Î¸critical = Sin^{-1} (1.00/1.52)

On solving, we get the critical angle of water as 41.1^{o} .

Critical Angle of Water

We know that the refractive index of water,Â n1 = 1.00

And, the refractive index of air,Â n2= 1.33

Using equation (2), and applying these two values as;

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Î¸critical = Sin^{-1} (1.00/1.33)

On solving, we get the critical angle of water to be 48.75^{o} â‰ˆ 49^{o}

Now, the critical angle has a close relationship with the total internal reflection; letâ€™s understand how.

We all must have experienced a mirage (a false reflection of the river) on a hot sunny day especially if we are in a desert; this mirage is actually a total internal reflection.Â

As we know that when the light incidences on the boundary or any medium, it suffers refraction. Now, as this angle of incidence approaches a required limit, which is the critical angle, the angle of refraction becomes 90Â°, at which the refracted ray of light of an electromagnetic wave becomes parallel to the surface.Â

Letâ€™s suppose that the angle of incidence increases beyond the critical angle, at this point, the condition of refraction no longer satisfies; so there is no refracted ray, and the partial reflection becomes total. This is how the critical angle and total internal reflection works in real-life.

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For visible light, the critical angle is approximately 49Â° for when it incidences at the water-to-air boundary and about 42Â° when it incidences the common glass-to-air boundary.

FAQ (Frequently Asked Questions)

1. Illustrate the Real-Life Examples of Refraction.

Ans- We all know that light hits everything where some substances let it reflect at a constant speed, but others slow it down or deviate it from its original path. When the light slows, it bends; and therefore, objects seem to bend, closer, or larger than they really are. Like when we see that the fish is in our sight, however, it is just refraction, in reality, the fish is before or away from the location, we point from our eyes.

Another example is a pencil. When a pencil is dipped in a half-filled glass of water. We notice that the pencil above the water looks normal, but under the water, it looks bent and slightly larger than the normal size. All this happens because of the phenomena of refraction. Now, letâ€™s look at a few more examples on refraction:

Prism

Twinkling stars

Ice crystals

2. Describe Snellâ€™s Window.

Ans - While standing beside an aquarium with your eyes below the water level, you are likely to see fish or submerged objects reflected in the water-air surface. Here. The brightness of the reflected image of the fish is as bright as the "direct" view, which is actually startling.

We can experience a similar effect by opening our eyes while swimming just below the surface of the water. If the water is calm, the surface outside the critical angle (measured from the vertical) appears reflecting/mirror-like, objects seem below. We cannot see the region above the water except overhead, where the hemispherical field of view remains compressed into a conical field known as Snell's window.