[Physics Class Notes] on Relation Between Critical Angle And Refractive Index Pdf for Exam

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 a critical value of 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:

 [theta _c] = sin^-1 n_r / n_i

Here, [theta _c] is the critical angle of refraction, and n_r and n_i 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:

sin C= 1/ µ^a_b

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 = µ_a / µ_b

Therefore, µ_b sin C = µ_a sin90^o

Therefore, µ_b / µ_a = 1 / sin C

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

µ^a_b = 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 [n_1sintheta _i=n_2sintheta _r]

In order to get [frac{sintheta _i}{sintheta _r}] , you must note that this ratio is [frac{n_2}{n_1}]

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 [n_1sintheta _i=n_2sintheta _r]

So, for getting [frac{n_2}{n_1}],  this ratio is [sintheta _i=sintheta _r]

On putting values for  θ_i= 30 and θ_r= 45

Therefore, [frac{sin45}{sin30}] = 2

Applications of Critical Angle

The principle of critical angle is used in several practical ways in our day-to-day life. The most widely used is fibre optic cables. The concept is the base for the construction of fibre optic cables and how they work.

Applications of total internal reflection are multi-touch screens, spatial filtering of lights, prismatic binoculars, automotive rain sensors, fluorescence microscopes, and ubiquitous fibre optics communications.

Do It Yourself

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

(a) False 

(b) True

Ans: (b) True

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

(a) zero 

(b) smaller 

(c) higher 

(d) negative

Ans: (c) Higher 

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

Ans: (a) Light’s speed in a medium