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Home / Physics / Relation Between Critical Angle and Refractive Index

Relation between Critical Angle and Refractive Index: With Definitions, Derivation

Last Updated on Jan 24, 2025
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Refractive index of the material is one of the most important optical parameters for the definition of optical properties of the material. It helps in calculating and tracing the true dimensions and location of submerged objects visible from the top of water. The high speed internet you are enjoying uses of optical fibre for its transmission across the globe which in fact works on the principle of critical angle and total internal reflection. This article discusses the relation between critical angle and refractive index.

What is Critical angle?

When a ray of light travels from an optically denser medium (medium having a larger value of μ) to an optically rarer medium (medium having a smaller value μ), the angle of refraction, r will be greater than the corresponding angle of incidence, i we have,

< 1

If we gradually keep increasing the value of angle of incidence, i, the corresponding angle of refraction, r will also increase and at a certain stage r will become 90°. Let’s say the angle of incidence for this case is, . If the angle of incidence is increased further, there will be no angle of refraction,r, which can satisfy Snell’s law. As a result, the ray of light will not be refracted. The entire light will then be reflected back into the first medium. This phenomenon is termed as total internal reflection, and that particular value of angle,, is called the critical angle for the given pair of media.

Usually, the critical angle of a medium is quoted for light going from some denser medium to the air. In that case, refractive index of medium 2 that is, = 1( since the refractive index of air is, 1) and let’s say, , then the Snell’s law gives,

or,

This is the required expression for the critical angle .

What is the Refractive Index?

We all know that light is an electromagnetic wave having both electric and magnetic fields alternating with each other. In a vacuum, electromagnetic waves propagate freely. However, when the same electromagnetic wave travels through any material medium, the medium now consists of substances that have their own electric as well as magnetic properties.

These electric and magnetic properties of the medium interact with the electric and magnetic properties of waves and cause them to slow down, bend down or get absorbed. These properties define the refractive index of the medium.

The refractive index or sometimes referred to as the index of refraction is the measure of the bending of a ray of light while passing from one medium to another. It is mathematically expressed as the ratio of the speed of light in the vacuum to the speed of light in a specific medium.

It is represented using the symbol,n, and can be calculated using,

Here, n = Refractive index of medium

c = speed of light

v = speed of light in that specific medium

Relation between Critical Angle and Refractive Index

The relation between critical angle and refractive index can be given as Critical angle is inversely proportional to refractive index. 

Therefore, the relationship between critical angle and refractive index is given by 
 

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Relationship between Critical Angle and Refractive Index Derivation

The mathematical expression for the relationship between refractive index and critical angle can be derived as follows:

Let’s consider a ray of light travelling in a rarer medium, ‘a’ incident on some plane surface S and got refracted to a denser medium, ‘b’. Such that,

  • angle of incidence, i = critical angle, C(say)
  • angle of refraction, r = 90°
  • Refractive index of rarer medium =
  • Refractive index of denser medium =

Now, from Snell’s law, we obtained

—(since, I is the angle of incidence equals to critical angle, C and r is the angle of refraction equals to 90°)

—-(1)

This is the required relation between the Refractive index of medium (a & b) and critical angle, C.

We can also, rewrite the equation, (1) as

Where, , represents the refractive index of medium ‘b’ with respect to the refractive index of medium ‘a’.

We can also write Critical angle in terms of refractive index as,

Hope this article was able to give you in detailed explanation about Relation between Critical Angle and Refractive Index. You can also check out other Physics topics too. Boost your exam preparations with the help of the Testbook App. Testbook provides online video lectures, mock test series, and much more. Download the app now to avail exciting offers!

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Relation Between Critical Angle and Refractive Index FAQs

The critical angle is a particular measure of the angle of incidence for which the angle of refraction comes out to be 90°.It can be calculated using,

As the refractive index of the material increases, the measure of critical angle decreases.

Yes, the critical angle is inversely proportional to the refractive index of the medium.

From the relationship between the Critical angle and refractive index, it can be observed that the critical angle is inversely proportional to the refractive index. This means as the refractive index increases, the critical angle decreases and vice versa.

The refractive index of the inner part of optical fibre is 1.68 and the refractive index of the outer part is 1.44 and the measure of the critical angle of optical fibre is 82°

The relation between critical angle and refractive index is given by .
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