Critical Angle and Total Internal Reflection: Total Internal Reflection is the phenomenon that makes beautiful rainbows or diamonds dazzle with brilliant colours. It is the phenomenon behind your high-speed internet connectivity. To know what exactly is this phenomenon and what are its practical applications, let us understand it from the basics with examples and applications.

What is Refraction of Light?

Light is one of the most fascinating and mysterious phenomena that we see in our daily life. To most of us, light seems very ordinary and obvious things present in nature and usually, we don’t give it much of a thought but believe us, some of the most important paradigm shifting discoveries have happened in physics by observing the nature of light. One of the most fascinating properties of light is its speed. In a vacuum, the speed of light is \(3 \times {10^8}\;{\rm{m}}\;{{\rm{s}}^{ – 1}}\) with respect to all observers and this in fact is the speed limit of everything in the universe. No matter how much we accelerate an object having mass, it can never reach the speed of light! When light travels in a medium, it slows down and travels at a lesser speed. Based on this, we define the optical density of a medium. 

In very layman’s terms, we can say that a medium is optically denser as compared to another if light slows down much more in it as compared to the other medium. As the optical density of a medium increases, the speed of light in it decreases. Also, the light follows a straight line path in a vacuum or if it is travelling in only one medium. But when light travels from one medium to the other, then at the boundary separating the two media, it may deviate from its path. This phenomenon is known as the refraction of light. Lenses, Prisms, and Glass Slabs, all work according to the principle of refraction of light. Following are the key observations regarding the refraction of light:
1. When a ray of light travels at an angle to the normal from a rarer medium to a denser medium, the ray bends towards the normal.
2. When a ray of light travels at an angle to the normal from a denser medium to a rarer medium, it bends away from the normal.
3. When a ray of light travels perpendicular to the surface, it travels without any deviation, irrespective of whether it is going from rarer to denser or vice versa.

Refractive Index

Refractive Index or Index of Refraction of a medium is defined as the ratio of speed of light in vacuum to its speed in that medium. Mathematically, refractive Index of a medium is given by
\(n = \frac{c}{v}\)
where,
\(c = 3 \times {10^8}{\rm{m}}{{\rm{s}}^{ – 1}}\) is the speed of light in vacuum.
\(v\) is the speed of light in the medium.
We can see that the lower the speed of light in a medium, the higher is the refractive index of the medium.
The refractive indices of some common materials are:

Material	Refractive Index
Air	\(1.0003\)
Ice	\(1.31\)
Water	\(1.3333\)
Human eye lens	\(1.386\)
Glass	\(1.52\)
Diamond	\(2.417\)

Laws of Refraction of Light

1. The incident ray, refracted ray and the normal to the surface at the point of incidence all lie in the same plane.
2. The ratio of the sine of the angle of incidence \((i)\) to the sine of the angle of refraction \((r)\) is a constant. This law is also called Snell’s law.
   \(\frac{{\sin \sin \,(i)}}{{\sin \sin \,(r)}} = \frac{{{n_2}}}{{{n_1}}} = {\rm{ constant }}\)
   When light is passing from medium \(1\) to medium \(2,{n_1}\) is the refractive index of medium \(1,\) and \({n_2}\) is that of the medium \(2.\)
   Incident ray: The ray of light that hits the surface is the incident ray. The angle between this ray and normal is the angle of incidence, \(i.\)
   Refracted ray: It is the ray that bends after exiting the surface. The angle between this ray and normal is the angle of refraction, \(r.\)
   Normal: It is the line or plane that is perpendicular to a surface.
   Point of incidence: It is the point where the incident ray touches the surface.

What is the Critical Angle?

We define critical angle as the angle of incidence in the denser medium that causes the angle of refraction in the rarer medium to be \({\rm{9}}{{\rm{0}}^{\rm{o}}}\).
When light travels from a denser medium to a rarer medium, the angle of refraction is more than the angle of incidence. As the angle of incidence increases, the angle of refraction increases further. At a certain angle of incidence, the refracted light passes along the surface of the rarer medium, that is, the angle of refraction becomes \({\rm{9}}{{\rm{0}}^{\rm{o}}}\). The angle of incidence corresponding to which the angle of refraction is \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) is known as the critical angle.

Derivation of Critical Angle Formula

According to Snell’s law,
\(\frac{{\sin \sin (t)}}{{\sin \sin (r)}} = \frac{n}{{{n_1}}}\)
or,
\({n_1}\sin \sin (i) = {n_2}\sin (r)\)
For critical angle, \({\theta _c},\) angle of refraction is \({90^{\rm{o}}}\).
\({n_1}\sin {\theta _c} = {n_2}\sin {90^{\rm{o}}}\)
or, 
\({n_1}\sin {\theta _c} = {n_2}\)
This means that critical angle is,\({\theta _c} = {\sin ^{ – 1}}\frac{{{n_2}}}{{{n_1}}}\)

What is Total Internal Reflection?

While travelling from a denser to a rarer medium, if the angle of incidence of a ray of light is more than the critical angle, then instead of refracting into the rarer medium, the ray of light is reflected back into the denser medium. This phenomenon is known as total internal reflection.
So, for total internal reflection to happen, the following two necessary and sufficient conditions should be met:
a. The ray of light should be travelling from a denser \(\left( {{n_1}} \right)\) to a rarer \(\left( {{n_2}} \right)\) medium.
\({n_1} > {n_2}\)
b. The angle of incidence should be more than the critical angle.
\(i > {\theta _c}\)

Examples of Total Internal Reflection

Total internal reflection occurs in nature, giving rise to a play of light and colours.

Rainbows

Raindrops are spherical in shape. Sun rays get refracted when they enter the drops from the air. Sunlight is a mixture of colours, each having its own wavelength. So, each colour refracts at a different angle. If the angle is more than the critical angle inside the drop, all these colours get reflected inside the drop. While leaving the drop, the colours get refracted further. Therefore, we see distinct colours of the rainbow.

Mirage

While driving on a hot day, sometimes we see water on the road and reflections of passing vehicles. However, it is not water, but an optical illusion called mirage. During a hot day, the warm air rises. So, there are layers of air formed with different temperatures. Air near the ground becomes the rarer medium, while it is comparatively denser at some height above the ground. Light rays from distant objects undergo total internal reflection at the boundary between dense and rare layers. So, we see the reflections of the objects as if reflected from a sheet of water. Mirage is a common optical phenomenon in hot deserts.

Sparkling of Diamonds

The sparkling of diamonds in bright colours is due to the way they are cut. Diamonds have a high refractive index, so the critical angle is small. A diamond is cut with many faces. Due to low critical angle, a light ray undergoes multiple reflections within the faces before coming out in brilliant colours.

Fisheye

A fish can see the world above water only in a circle. When the angle goes beyond the critical angle, only the objects underwater get reflected, and the fish or a diver cannot see beyond this circle. The objects within the circle are distorted due to refraction at various angles. Therefore, we have a term called “fisheye”. We have fisheye camera lenses to capture an image in a distorted way within a circle.

Applications of Critical Angle and Total Internal Reflection

Total internal reflection causes \(100%\) reflection, unlike a regular mirror that absorbs some amount of light. This phenomenon is made use in many applications.

Optical Fibres

An optical fibre is made of very thin wire of glass or plastic fibre. Due to a higher refractive index, the light entering the fibre never exits outside. However, it keeps getting reflected till it reaches the end of the fibre. It does not get affected even when the fibre bends or twists. Since no light is lost, the entire light that enters the fibre is received at the end.
(i). Optical fibres are used in communication. Having the highest speed, light in optical fibres is a good way of transmission of data.
(ii). In the medical field, it is used in non-intrusive surgery such as endoscopy.

Optical Devices

DSLR (Digital Single Lens Reflex) cameras have a glass prism near the viewfinder. It is called a pentaprism due to five faces. The angles of the faces are such that total internal reflection occurs, and the light from the lens reaches the viewer. It is used to show the viewer the same image that will fall on the sensor so that there
is no difference between what the viewer sees and what is captured.
Two prisms are used in each of the view tubes of binoculars that use total internal reflection to change light paths. Periscopes also use two prisms.

Cat Eye Reflectors

We have seen reflectors at night that glow brightly when headlights of vehicles fall on them. This is because they have tiny prisms that use total internal reflection to bounce back the light that falls on them.

Sample Problems – Critical Angle and Total Internal Reflection

Q.1. The refractive index of diamond is \(2.417.\) Determine its critical angle.
Ans: Critical angle formula is,\({\theta _c} = {\sin ^{ – 1}}\frac{{{n_2}}}{{{n_1}}}\)
The first medium is diamond. The second medium is air.
\({n_1} = 2.417\)
\({n_2} = 1.0003\)
\({\theta _c} = {\sin ^{ – 1}}\frac{{1.0003}}{{2.417}}\)
\({\theta _c} = {24.44^{\rm{o}}}\)
The critical angle of a diamond is almost \({25^{\rm{o}}}\).

Q.2. The Refractive index of glass is \(1.52.\) Find the speed of light in glass.
Ans: Refractive index, \(n\) is,
\(n = \frac{c}{v}\)
where,
\(c\) is the speed of light in vacuum. It is \(3 \times {10^8}\;{\rm{m}}/{\rm{s}}\)
\(v\) is the speed of light in another medium such as glass.
Therefore,
\(v = \frac{c}{n} = \frac{{3 \times {{10}^8}}}{{1.52}} = 1.97 \times {10^8}\;{\rm{m}}/{\rm{s}}\)
The speed of light in glass is almost \(2 \times {10^8}\,{\rm{metres}}\,{\rm{per}}\,{\rm{second}}{\rm{.}}\)

Summary

Total Internal Reflection is the phenomenon that makes beautiful rainbows or diamonds dazzle with brilliant colours. It is the phenomenon behind your high-speed internet connectivity. In this article, we learnt the phenomenon of Total Internal Reflection and what are its practical applications. We also learned the definition of critical angle, that is, as the angle of incidence in the denser medium that causes the angle of refraction in the rarer medium to be \({\rm{9}}{{\rm{0}}^{\rm{o}}}\). In very layman’s terms, we can say that a medium is optically denser as compared to another if light slows down much more in it as compared to the other medium. Two main factors to keep in mind:

• 1. When a ray of light travels at an angle to the normal from a rarer medium to a denser medium, the ray bends towards the normal.
• 2. When a ray of light travels at an angle to the normal from a denser medium to a rarer medium, it bends away from the normal.

While travelling from a denser to a rarer medium, if the angle of incidence of a ray of light is more than the critical angle, then instead of refracting into the rarer medium, the ray of light is reflected back into the denser medium. In addition, we studied the Laws of Refraction of Light along with examples, applications, and solved examples of critical angle and total internal reflection.

FAQs on Critical Angle and Total Internal Reflection

Q.1. Does total internal reflection happen when light passes from a rare medium to a dense medium?
Ans: No. It occurs only when light passes from dense medium to rare medium.

Q.2. Do rainbows form due to total internal reflection?
Ans: Yes. We see rainbows only when the light rays get reflected inside the raindrops.

Q.3. Why do we see colours in a diamond?
Ans: A diamond has a high refractive index. So, its critical angle is small, causing multiple internal reflections inside it. As a result, when light leaves the diamond, it comes out in brilliant colours.

Q.4. Why is the knowledge of critical angles necessary?
Ans: Designing optical fibres, prisms, and other optical instruments need this angle to know when total internal reflection occurs.

Q.5. How can we see the reflection of fish from the water surface in a fish tank?
Ans: The water surface acts like a mirror reflecting the light.

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