Relation Between Critical Angle And Refractive Index

Relation between critical angle and refractive index can be formed because both of them are inversely proportional. But, before going into this detail, you must understand these topics separately.

What Is Critical Angle?

In optics as a topic of Physics, the critical angle makes reference to a particular angle of incidence, which gives an angle of refraction of 90 degrees. Additionally, a water-air boundary has critical value 48.6 degrees. While the critical angle for crown water and glass boundary is 61.8 degrees.

However, the value of critical angle is always dependent on the mediums situated on both sides of a boundary.

Additionally, the equation of critical angle is:

θcric = sin-1 nr / ni

Here, θcric is the critical angle of refraction, and nr and ni are refraction index and incident index, respectively.

What is Refractive Index?

The extent at which rays of light bend when it enters from one medium source to another is known as its refractive index. The refractive index is represented using the alphabet 'n.' Moreover, it can be written as n = c/v, where ‘c’ is light speed or velocity of a specific wavelength in the medium air. On the other hand, v is light’s velocity or speed in other media.

Furthermore, three factors determine this refractive index - medium nature, light colour and physical conditions.

Fact: An optically rarer medium is one where light travels faster through it. Whereas, an optically denser medium is one where light travels slower through it.

Refractive Index and the Critical Angle Relationship

The mathematical representation of their relationship is:

sinC= 1/ µab

Here, C = critical angle, µ = refractive index and a and b are two mediums within which light passes.

Furthermore, take a look at this derivation below!

Snell's Law (also known as the Second Law of refraction) is applied to derive the relation between critical angle and refractive index.

Hence, take a light ray having an incident angle i, refractive angle r = 90 degrees, critical angle = C and refractive index of rarer and denser medium be µa and µb respectively.

So, by applying the second Law of refraction or Snell's Law:

sin i / sin r = µa / µb

Therefore, µb sinC = µa sin90o

Therefore, µb / µa = 1 / sinC

Thus, with the help of this equation, critical angle and refractive index relation can be stated as:

µab = 1/ sinC

Solved Numericals

(i) Find out the ratio between sine of incident angle and the sine of reflected angle where their refractive indices are provided. In medium 1 it is 2.33, while in medium 2 it is 1.66.

Solution. Snell's Law gives N1 sin θi = N2 sin θt

In order to get sin θi/sin θt, you must note that this ratio is N2/N1

After substituting for N1 = 2.33 along with N2 = 1.66

=> 1.66/2.33 = 0.71

(ii) Find the ratio between refractive index of two mediums, 1 and 2. Here, the reflected angle of medium 1 is 300, while that of medium 2 is 450. 

Solution. Snell's Law gives N1 sin θi = N2 sin θt

So, for getting N1/N2, this ratio is sin θt/sin θi

On putting values for θi = 30 and θt = 45

Therefore, sin 45/sin 30 = 2

Do It Yourself

1. Angle of incidence is equal to angle of reflection for perfect reflection. Answer true or false.

(a) False (b) True

2. The higher the value of refractive index of a given medium, the bending of light will be

(a) zero (b) smaller (c) higher (d) negative

3. The refractive index of a medium is the relation between light’s speed in vacuum or air, and

(a) Light’s speed in a medium (b) Can be a or c (c) Speed of sound in a medium (d) none

Along with this discussion above, there are quite a few additional topics which are related to the relation between critical angle and refractive index. You can download our Vedantu app to get access to study material on all these related topics, along with online classes.

FAQ (Frequently Asked Questions)

1. What is the Numerical Formula of Critical Angle?

Ans. Formula for calculating critical angle is θcric = sin-1 nr/ni.

2. Define Critical Angle and Write Its Conditions.

Ans. Critical angle happens when an incident ray causes refraction to make an angle of 90 degrees. Moreover, critical angle value depends upon the two mediums.

3. How do you Calculate the Critical Angle of Refraction?

Ans. With the help of Snell's Law and setting angle of refraction to 90 degrees, critical angle is calculated.